JP2003222664A - Device for estimating current density vector and device for estimating electric conductivity - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To measure a magnetic field vector without contacting which is generated in the cases that an electric current field exists inside a sample previously, that electric current flows directly and that eddy current is induced by applying a magnetic field, to determine a distribution of current density vectors in an interested space or an interested region, and thereby to determine a distribution of the conductivity in the interested space or the interested region. <P>SOLUTION: The distribution of current density vectors is determined by solving an integral equation (i.e., Biot-savart law) in the interested space or the interested region being described by data of the magnetic field around the sample which is measured by a non-contact detector such as a device for detecting the magnetic field vector or the like employing detecting elements and array-type elements for detecting magnetism, static magnetic field or electromagnetic wave. Then the relative distribution of conductivity in relation to a referential conductivity being set in the interested space or the interested region is estimated by solving a first spatial partial differential equation being described by current data. <P>COPYRIGHT: (C)2003,JPO

Description

Detailed Description of the Invention

[0001]

TECHNICAL FIELD The present invention relates to a non-destructive physical property / characteristic evaluation / inspection technique for an object, a substance / material, a non-invasive diagnosis / inspection technique for an organism, and an apparatus.

[0002]

2. Description of the Related Art As a conventional technique, a current density vector space is obtained by performing a numerical analysis (for example, a minimum norm minimum method) corresponding to a vertical component of a magnetic field vector component which is measured on the surface of an object to be measured. Many estimates the distribution (Prior Art 1, FIG. 11 (a): See the schematic diagram of the coil part of the magnetic field detector), and other two estimates are made from the tangential component on the surface of the measurement target. There is a method based on the Fourier analysis method and the minimum norm minimum method (Prior Art 2, FIG. 11B: See a schematic view of the sensor part of the magnetic field detector).

As a technique included in the conventional technique 1, there is a technique for realizing a one-dimensional or two-dimensional array type SQUID meter and then performing aperture plane synthesis in order to improve the measurement accuracy of the magnetic field.

[0004]

In the prior arts 1 and 2, it is mathematically impossible to uniquely determine the three-dimensional current density vector distribution in the three-dimensional space of interest, and it is stabilized numerically. Even if it did, only the pseudo current density vector data was obtained, and the accuracy enough to estimate the conductivity spatial distribution was not obtained.

However, if the object to be measured can be approximately treated as a two-dimensional object such as a flat plate or a thin film, it is possible to determine the two-dimensional current density vector space distribution in that plane by the conventional technique 2. However, in the case of the Fourier analysis method, it is necessary to assume the spatial periodicity of the current distribution in the finite two-dimensional region of interest, which causes a problem in estimation near the boundary of the region of interest.

In particular, when the two-dimensional object is surrounded by an insulator (for example, in the air) and the entire two-dimensional object is set as the region of interest, it is impossible to estimate the vicinity of the boundary.

In the present invention, basically, when a sample exists in a space in an arbitrary state, a one-dimensional, two-dimensional or three-dimensional aperture plane synthesis is sometimes performed, and then a three-dimensional magnetic field vector or two-dimensional The 3D or 2D distribution of the magnetic field vector is measured, and the 3D space of interest or the 2D region of interest of a finite size that has been set arbitrarily (in order to handle 3D objects clearly, The three-dimensional current density vector or the two-dimensional current density vector or the two components of the three-dimensional current density vector in the area where the two-dimensional planes intersect and the area of interest set in the two-dimensional object are collectively used. It is an object of the present invention to obtain a current density vector estimation device and an electrical conductivity estimation device that can estimate the electrical conductivity spatial distribution in a three-dimensional space or a two-dimensional region of interest.

[0008]

In order to achieve the above object, a current density vector estimating device according to a first aspect of the present invention comprises:
A current density vector estimation device for estimating a current density vector distribution in a three-dimensional region of interest or a two-dimensional region of interest of a measurement target, a current density distribution in a one-dimensional region of interest, or a time series thereof, A sample setting table to be installed, a magnetic field vector detecting means for detecting a magnetic field vector,
A housing in which the magnetic field vector detecting means is installed so as to measure the measurement target, and a connection to at least one of the sample installation base and the housing for measuring the distribution of the magnetic field vector of the measurement target. Scanning means capable of changing the vertical and horizontal directions, and adjusting means for driving the magnetic field vector detecting means and adjusting the output of the magnetic field vector detecting means for measuring the measurement target,
Data recording means for recording the measurement results output from the adjusting means, and the current density vector distribution or its time series from the magnetic field vector data measured through the application of discrete approximation or finite element approximation to the current density vector distribution or its time series. Data processing means for estimating and distance adjusting means for adjusting the distance between the measurement object and the magnetic field vector detecting means are provided.

According to a second aspect of the present invention, there is provided a current density vector estimation device, wherein a current density vector distribution in the three-dimensional space of interest or a two-dimensional region of interest of the measurement object, or a current density distribution in a one-dimensional region of interest, or The output of the magnetic field vector detecting unit or the output of the adjusting unit, which is measured to obtain those time series, is associated with position / time data indicating the position / time of the measurement and recorded in the data recording unit. Characterize.

Further, the current density vector estimation device according to a third aspect of the present invention further includes, from the outside of the three-dimensional space of interest of the measurement object, within the space or within a two-dimensional region of interest therein, as necessary. In order to generate at least one or more electric field in the one-dimensional region of interest,
It is characterized by having a current field generating means capable of inducing an eddy current by allowing a current to flow directly or by applying a magnetic field.

According to a fourth aspect of the current density vector estimating apparatus, the magnetic vector detecting means is a superconducting quantum interference device (SQUID), a Hall device, a Josephson junction device, a magnetic impedance device, GMR, CMR, S.
It is characterized by using ET and the like.

According to a fifth aspect of the current density vector estimating device, the magnetic field vector detecting means is magnetic,
The present invention is characterized in that a plurality of static magnetic field or electromagnetic wave detection elements are used, the detection directions are set to two directions or three directions orthogonal to each other, and sometimes aperture plane synthesis is performed to collect magnetic field vector data.

According to a sixth aspect of the current density vector estimating apparatus, a plurality of electromagnetic wave detecting elements are used as the magnetic field vector detecting means, and the detecting direction is one direction, two directions orthogonal to each other, or three directions, and One-dimensional, two-dimensional,
Or, constructing a three-dimensional array and sometimes synthesizing aperture planes,
It is characterized by collecting magnetic field data.

According to a seventh aspect of the present invention, there is provided a current density vector estimating apparatus which measures the two-dimensional spatial distribution of a two-dimensional magnetic field vector or its time series using the magnetic field vector detecting means, and from the measured magnetic field vector data, By performing a predetermined numerical analysis based on the application of the discrete approximation or the finite element approximation to the current density vector or its time series, the 2
It is characterized in that the three-dimensional current density vector distribution or its time series is estimated.

Further, the current density vector estimating device according to the present invention measures the three-dimensional or three-dimensional spatial distribution of the two-dimensional magnetic field vector or its time series using the magnetic field vector detecting means, and measures the measured magnetic field. The spatial distribution of the three-dimensional or two-dimensional current density vector in the three-dimensional space of interest by performing predetermined numerical analysis based on the application of discrete approximation or finite element approximation to the current density vector or its time series from the vector data. Alternatively, it is characterized by estimating its time series.

According to a ninth aspect of the present invention, there is provided a current density vector estimating apparatus which measures a three-dimensional or three-dimensional spatial distribution of a two-dimensional magnetic field vector or its time series using the magnetic field vector detecting means, and measures the measured magnetic field. By performing predetermined numerical analysis based on applying discrete approximation or finite element approximation to current density vector or its time series from vector data, three-dimensional or two-dimensional in three-dimensional space of interest or two-dimensional region of interest It is characterized by estimating the spatial distribution of the current density vector or its time series.

According to a tenth aspect of the present invention, there is provided a current density vector estimating device for performing a predetermined numerical analysis based on applying a discrete approximation or a finite element approximation to a current density or a time series thereof to obtain the magnetic field vector detecting means. Is characterized by estimating the spatial distribution of the current density vector component in the direction orthogonal thereto or its time series from the spatial distribution of the one direction component of the magnetic field vector measured using From the three-dimensional spatial distribution data, the three-dimensional spatial distribution or the two-dimensional spatial distribution or the one-dimensional spatial distribution of the current density vector component, or from the two-dimensional spatial distribution data of the magnetic field vector component, the two-dimensional space of the current density vector component Distribution or the one-dimensional spatial distribution, or the current density vector from the one-dimensional spatial distribution data of the magnetic field vector component. It is characterized by estimating the one-dimensional spatial distribution of the cuttle component.

According to the eleventh aspect of the present invention, there is provided an electrical conductivity estimating apparatus, which is an electrical conductivity distribution or a dielectric constant distribution in a three-dimensional space of interest, a two-dimensional region of interest, or a one-dimensional region of interest of a measurement object, or a distribution thereof. In the electric conductivity estimation device for estimating the time series, in the current density vector estimation device,
A sample installation table on which the measurement target is installed, a magnetic field vector detection unit that detects a magnetic field vector, a housing installed so that the magnetic field vector detection unit measures the measurement target, and a magnetic field of the measurement target. In order to measure the vector distribution or its time series, the sample installation table and the housing, a scanning means connected to at least one of the above and below and capable of changing the vertical and horizontal directions, and driving the magnetic field vector detection means, Adjusting means for adjusting the output of the magnetic field vector detecting means for measuring an object to be measured, data recording means for recording the measurement results output from the adjusting means, and current density vector distribution or its time series (or spectrum) discrete The magnetic field vector data measured through the application of the approximation or the finite element approximation or its time series data Estimate the density vector distribution or its time series (or spectrum), normalize each current density vector distribution using its norm (magnitude), and then perform finite difference approximation or finite element approximation to obtain the electrical conductivity distribution. And a dielectric constant distribution, a data processing means for estimating a time series of the electric conductivity distribution and the dielectric constant distribution and frequency dispersion thereof, and a distance adjusting means for adjusting a distance between the measurement object and the magnetic field vector detecting means. And are provided.

According to a twelfth aspect of the present invention, there is provided an electric conductivity estimating apparatus in the three-dimensional space of interest of the measuring object or in
The distribution of the current density vector in the three-dimensional region of interest, the distribution of the current density in the one-dimensional region of interest, or the output of the magnetic field vector detection means measured to obtain the time series thereof, or the output of the adjusting means is used as the position of the measurement. It is characterized in that it is recorded in the data recording means in association with position / time data indicating time.

Further, the electric conductivity estimating apparatus according to the thirteenth aspect of the present invention further includes, as necessary, from outside the three-dimensional space of interest of the measurement object, within the space, or within a two-dimensional region of interest within the space. In order to generate at least one current field in the one-dimensional region of interest, there is provided a current field generation means capable of directly flowing a current or inducing an eddy current by applying a magnetic field. Is characterized by.

Further, in the electric conductivity estimating apparatus according to claim 14, the magnetic field vector detecting means is a superconducting quantum interference device (SQUID), a Hall device, a Josephson junction device, a magnetic impedance device, GMR, CMR, SET.
It is characterized by using such as.

According to a fifteenth aspect of the present invention, in the electric conductivity estimating apparatus, a plurality of magnetic, static magnetic field or electromagnetic wave detecting elements are used as the magnetic field vector detecting means, and the detecting directions are orthogonal to two directions or three directions. Sometimes aperture plane synthesis is performed,
It is characterized by collecting magnetic field vector data.

According to a sixteenth aspect of the present invention, there is provided an electric conductivity estimating device, wherein a plurality of magnetic, static magnetic field or electromagnetic wave detecting elements are used as the magnetic field vector detecting means, and the detecting direction is one direction, two directions orthogonal to each other, or The present invention is characterized in that three directions are formed and a one-dimensional, two-dimensional, or three-dimensional array is formed, and sometimes aperture plane synthesis is performed to collect magnetic field data.

According to a seventeenth aspect of the present invention, there is provided an electrical conductivity estimating apparatus which calculates an electric conductivity distribution in a three-dimensional space of interest, a two-dimensional region of interest, or a one-dimensional region of interest or a time series thereof from a current density vector or its time series. Or its frequency dispersion, the finite difference approximation or the finite element approximation (variation method or Galerkin method) is applied to the electric conductivity distribution or the current density vector distribution expressed in the predetermined first-order spatial partial differential equation. Using a predetermined numerical solution method based on applying

The electrical conductivity estimating apparatus according to the eighteenth aspect of the invention is based on the time series of the current density vector, and the electrical conductivity distribution and the dielectric constant in the three-dimensional space of interest, the two-dimensional region of interest, or the one-dimensional region of interest. When estimating the distribution, time series of electrical conductivity distribution or permittivity distribution and their frequency dispersion,
To apply finite difference approximation or finite element approximation (using variational method or Galerkin method) to the distribution of the conductivity distribution or the spectrum of the current density vector expressed in a given first-order spatial partial differential equation It is characterized by using a predetermined numerical solution based on.

The electrical conductivity estimating apparatus according to a nineteenth aspect of the present invention provides, as a measurement result, a measured magnetic vector distribution, a measured current density distribution, a current density divergence / gradient distribution, a conductivity and a dielectric constant distribution. , Conductivity or permittivity gradient distribution, conductivity or permittivity Laplacian distribution, permittivity to conductivity ratio distribution, permittivity to conductivity ratio gradient distribution, permittivity to conductivity ratio Laplacian distribution Further, it is characterized in that display means is provided for displaying an image of their frequency dispersion and their change over time, their absolute change over time (difference value), and their relative change over time (value of ratio).

[0027]

BEST MODE FOR CARRYING OUT THE INVENTION Embodiments of a current density vector estimating device and an electric conductivity estimating device according to the present invention will be described below in detail with reference to the drawings.

In this embodiment, magnetism, static magnetic field, or
Electromagnetic wave detection element and its one-dimensional, two-dimensional or three-dimensional
Three-dimensional space of interest or two-dimensional region of interest or one-dimensional region of interest that can be set by applying a reference object to an unknown sample as needed using a non-contact detector such as a magnetic field vector detector that uses a three-dimensional array type element group The magnetic field vector or magnetic field vector component is appropriately set so that the relative conductivity distribution with respect to the conductivity reference value (eg, true value, unit value, etc.) given in the reference area in the area can be stably reconstructed. It is possible to accurately measure the required current density vector distribution using the devised numerical analysis method, and to reconstruct the absolute conductivity distribution or the relative conductivity distribution. To do.

The orthogonal Cartesian coordinate system (x, y, z) will be described below.

The simultaneous first-order partial differential equations used when estimating the conductivity distribution described in claim 17 are as follows.

[Equation 1]

However, J ₁ (x, y, z) [= (J _1x (x, y, z), J
_1y (x, y, z), J _1z (x, y, z)) ^T ] and J ₂ (x, y, z) [= (J _2x (x, y, z), J
_2y (x, y, z), J _2z (x, y, z)) ^T ] is any two independent current fields generated by a current source outside the 3D space of interest or outside the 2D region of interest. Is a current density vector distribution of.

Each of these equations is

[Equation 2] Can be expressed as

When there is one current field, each equation (1a),
Formula (1b), Formula (1c), [or Formula (2a), Formula (2b), (2c)
Only one PDE in [Equation] holds. Although it should be avoided, when there are multiple current fields, the partial differential equations are coupled in that number. still,
Formulas (1a) to (1c) and (2a) to (2c) are respectively
It may be handled after multiplying both sides by 1 / σ (x, y, z) and σ (x, y, z).

The following are expressions (1a), (1b) and (1c) below.
[Or (2a), (2b), (2c)] in the current data set, for example, for the current vector J ₁ (x, y, z), (J
_1x (x, y, z), J _1y (x, y, z)) ^T , (J _1y (x, y, z), J _1z (x, y,
z)) ^T , (J _1z (x, y, z), J _1x (x, y, z)) ^T, and each of these three vectors is represented by a current vector J ₁ (x,
It is called a pseudo two-dimensional current density vector of y, z).

For example, an arbitrary z coordinate z in the three-dimensional space of interest
= To estimate the conductivity in the two-dimensional plane (x, y, Z) of Z
As shown in (1a) [or (2a)], the plane
Three-dimensional current density vector J in (x, y, Z)₁pseudo of (x, y, z)
Similar two-dimensional current density vector (J _1x(x, y, Z), J_1y(x, y, Z))
^TNeeds to be measured. Therefore, the measured current density
Depending on the dimensionality of the degree vector and the direction of the
Equation (1b) [or (2b)] relating to the plane (X, y, z) of the mark X, any
Equation (1c) [or (2c) equation] for the plane (x, Y, z) of y coordinate Y is also
Can be used. In addition, the measurement object is a two-dimensional object
When treated as a body, the z coordinate is specified and z = Z
The position of the original object is represented by two-dimensional coordinates (x, y, Z).
In this case, only the formula (1a) [or the formula (2a)] is satisfied.

When a superconductor can be contained in the three-dimensional space of interest or the two-dimensional region of interest, the electric conductivity σ (x, y, z) becomes zero according to the equations (1a) to (1c). If it may contain an insulator (2
Equations (a) to (2c) are used.

On the other hand, the reference conductivity required for the conductivity estimation is generally lnσ (x, y, Z) = lnσ _l (x, y, Z) εw _l (l = 1 ~ N) (3) When determining the conductivity distribution in each z = Z plane (x, y, Z), when two or more current fields are measured, It suffices if the reference value is given at at least one point w _l in that plane. N is the number of reference points. Further, when one field is measured, it is necessary to realize a reference region w _l that extends in a direction in which a current mainly flows in the plane. In this case, N is the number of reference areas.

Therefore, the three-dimensional current density vector, or
If the pseudo two-dimensional current density vector is measured in the three-dimensional space of interest or the two-dimensional region of interest,
Appropriately over the region, the above formula (1a), formula (1b), (1c)
Equation, [or (2a) equation, (2b) equation, (2c) equation] Partial differential equation is established, reference conductivity also within the range given appropriately, sometimes also estimated conductivity Since it can be used as a value, it becomes redundant, but the conductivity distribution can be uniquely expressed. However, in reality, the measurement data always includes an error (noise), and in particular, when only one of the three pseudo two-dimensional current density vector distributions is measured, the spreading direction, size, and position of the reference region are The state becomes improper and the estimation becomes unstable. Therefore, we devised a numerical analysis method (pending)
Is applied to directly estimate the conductivity space distribution in the three-dimensional space, or by estimating for each two-dimensional region of interest, the conductivity space in the three-dimensional region of interest or the two-dimensional region of interest is calculated. Estimate the distribution. However, when a plurality of current density vector distributions are measured, each pseudo two-dimensional current density vector distribution data uses the norm (size) of each two-dimensional distribution in the interest space of the pseudo two-dimensional current density vector. Is normalized. Also, in particular, 1 / σ (x, y, z) on both sides of equation (1a)
When the conductivity distribution 1 / σ (x, y, z) or the current density vector distribution is approximated by a finite difference, σ (x, y, z) is also applied to both sides of equation (2a). When a finite difference approximation is made for the conductivity distribution σ (x, y, z) or the current density vector distribution, 1 / σ (x, y, z) is applied to both sides of equation (1a) to obtain 1 / σ
When the finite element approximation (variational method or Galerkin method) is applied to (x, y, z), σ (x, y, z) is applied to both sides of the equation (2a).
When the finite element approximation (variation method or Galerkin method) is applied to σ (x, y, z) by multiplying by, each pseudo two-dimensional current density vector distribution data and its pseudo two-dimensional current density vector The rotation distribution data is obtained by calculating the power of each two-dimensional distribution in the ROI of the inner product of the pseudo two-dimensional current density vector and the gradient operator for the conductivity and the power of the two-dimensional distribution of the rotation of the pseudo two-dimensional current density vector. It may be normalized using the square root of the sum.

Assuming that a two-dimensional current density vector distribution is measured, a two-dimensional gradient operator D in the x and y directions may be used in regularization (pending application, claim 7, formula 5). It is also possible to obtain the three-dimensional conductivity space distribution by simultaneous equations established in the three-dimensional space of interest, and in that case, the three-dimensional gradient operator D'is used. The procedure for this case is shown below.

For the first-order spatial partial differential equation and initial conditions, the discrete Cartesian coordinate system (I, J, K) to (x / Δx, y /
Δy, z / Δz) (where Z = K′Δz),
Three-dimensional unknown conductivity spatial distribution ln (1 / σ (x, y, z)) or ln
σ (x, y, z) or current density vector distribution J (x, y, z
Finite difference approximation or finite element approximation with respect to z) [Use variational principle or Galerkin method. Conductivity distribution ln (1 / σ (x, y,
z)) or lnσ (x, y, z) and current density vector distribution J
(x, y, z) (= (J _x (x, y, z), J _y (x, y, z), J _z
(x, y, z)) ^T ) three-dimensional basis function used to approximate
(The number of nodes in the finite element is omitted. The same applies below.) Φ _σ (I, J, K, x,
y, z) (ln (1 / σ (x, y, z)) ~ Σ _{I, J, K} φ _σ (I, J, K, x, y, z) ln
(1 / σ (I, J, K)), lnσ (x, y, z) ~ Σ _{I, J, K} φ _σ (I, J, K,
x, y, z) lnσ (I, J, K)) and φ _J (I, J, K, x, y, z)
(J (x, y, z) ~ (Σ _{I, J, K} φ _J (I, J, K, x, y, z) J _x (I, J, K), Σ
_{I, J, K} φ _J (I, J, K, x, y, z) J _y (I, J, K), Σ _{I, J, K} φ _J (I, J, K,
x, y, z) J _z (I, J, K)) ^T ) are both required to be partially differentiable more than once. However, when the variational principle (so-called energy minimization principle) is used for the current density vector distribution (the energy function which is a functional is 1 /
σ (x, y, z), which cannot be used when an insulator is included in the region of interest), and the Galerkin method is used as the basis function to approximate the current density vector distribution. In the case where the gradient of the conductivity distribution is partially integrated, the basis function φ _σ of the conductivity distribution may be DC. ], And substituting the initial conditions, the following simultaneous equations regarding the three-dimensional unknown conductivity spatial distribution JDs = j (4) where s: unknown conductivity spatial distribution ln (1 / σ (I, J, K)) or a vector D representing lnσ (I, J, K): ln (1 / σ (x, y, z)) or x for lnσ (x, y, z) , a matrix J consisting of finite difference approximation (forward difference approximation, backward difference approximation, etc.) constants of a two-dimensional gradient operator in the y direction: a pseudo two-dimensional current density vector distribution (J _x (I, J, K), J _y (I,
J, K)) ^T , (J _y (x, y, z), J _z (x, y, z)) ^T , (J _z (x, y, z),
Matrix determined from J _x (x, y, z)) ^T [However, if multiple current density vector distributions are measured, each pseudo two-dimensional current density vector distribution data is It is normalized using the norm (size) of each two-dimensional distribution in space. ] j: A low-pass filtered pseudo two-dimensional current density vector [normalized] first-order partial differential value vector is obtained.

Finite element method (variational principle or Galerkin method)
Even when used with reference conductivity
3D unknown conductivity spatial distribution ln (1 / σ (I, J, K)) or ln
The simultaneous equations for σ (I, J, K) are derived. Therefore,
Three-dimensional basis function φ used _σ(I, J, K, x, y, z) is the above
Reference area in the 3D space of interest given as the initial value in
Area w_lOr reference point w_lOf the reference conductivity in
It must be possible.

When the finite element method is used for this three-dimensional space of interest, 2 of each z coordinate z = Z (K = K ').
Pseudo two-dimensional current density vector distribution J (x, y, Z) (= (J _x (x, y, Z
Z), J _y (x, y, Z)) ^T ) is measured and only this pseudo two-dimensional current density vector distribution data is used, the two-dimensional conductivity distribution ln in each two-dimensional region of interest is (1 / σ (x, y, Z))
And lnσ (x, y, Z) are two-dimensional basis functions φ _σ (I, J, K ', x, y,
Ln (1 / σ (x, y, Z)) ~ Σ _{I, J} φ _σ (I, J, K ', x, y,
Z) ln (1 / σ (I, J, K ')), lnσ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J,
K ', x, y, Z) lnσ (I, J, K'), and the pseudo two-dimensional current density vector distribution J (x, y, Z) (
= (J _x (x, y, Z), J _y (x, y, Z)) ^T ) is a two-dimensional basis function φ _J (I, J, K ', x, y, Z) and J ( x, y, Z) ~ (Σ _{I, J} φ _J (I, J,
K ', x, y, Z) J _x (I, J, K'), Σ _{I, J} φ _J (I, J, K ', x, y, Z) J _y (I, J,
It may be approximated as K ')) ^T. These two used
The differentiability of the dimensional basis function is the same as the three-dimensional basis function, and in addition, the two-dimensional basis function φ _σ (I, J, K ', x, y,
Z) needs to be able to consistently represent the reference region w _l in each two-dimensional region of interest or the reference conductivity in the reference point w _l . From this, simultaneous equations regarding the two-dimensional unknown conductivity distribution ln (1 / σ (I, J, K ′)) or lnσ (I, J, K ′) in each two-dimensional region of interest are derived.

Also, in the case of targeting a two-dimensional region of interest in a two-dimensional object having az coordinate z = Z (K = K '), the finite difference approximation or the finite element method is used.
[Two-dimensional conductivity distribution ln (1 / σ (x, y, Z)) and lnσ (x,
y, Z) is the two-dimensional basis function φ _σ (I, J, K ', x, y, Z)
(1 / σ (x, y, Z)) ~ Σ _{I, J} φ _σ (I, J, K ', x, y, Z) ln (1 / σ
(I, J, K ')), lnσ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J, K', x, y, Z) ln
It is approximated by σ (I, J, K ') and the two-dimensional current density vector distribution J
(x, y, Z) (= (J _x (x, y, Z), J _y (x, y, Z)) ^T ) is a two-dimensional basis function φ _J (I, J, K ', x, y , Z) using J (x, y, Z) ~
(Σ _{I, J} φ _J (I, J, K ', x, y, Z) J _x (I, J, K'), Σ _{I, J} φ _J (I, J,
K ', x, y, Z) J _y (I, J, K')) ^T. The differentiability of these two-dimensional basis functions used is similar to the three-dimensional basis function, in addition to the two-dimensional basis function φ _σ (I, J,
K ', x, y, Z) is the reference area w _l or the reference point w in the two-dimensional region of interest.
_It must be able to represent the reference conductivity in _l consistently. ], The two-dimensional unknown conductivity distribution ln (1 / σ (I, J, K ')) or lnσ (I, J,
The simultaneous equations for K ') are derived.

In the three-dimensional space of interest, the equation (1a)
~ (1c) and (2a) ~ (2c) on each side
1 / σ (x, y, z) and σ (x, y, z) are multiplied and treated
In the same manner, finite difference approximation is performed or
Uses the finite element method [conductivity distribution 1 / σ (x, y, z)
And σ (x, y, z) are three-dimensional basis functions φ_σ(I, J, K, x, y, z)
1 / σ (x, y, z) ~ Σ_{I, J, K}φ_σ(I, J, K, x, y, z) (1 / σ
(I, J, K)), σ (x, y, z) ~ Σ_{I, J, K}φ_σ(I, J, K, x, y, z) σ
Current density vector distribution J (x, y, z) approximated as (I, J, K)
 (= (J_x(x, y, z), J_y(x, y, z), J_z(x, y, z))^T)
Is a three-dimensional basis function φ_J(I, J, K, x, y, z) J (x, y, z) ~
(Σ_{I, J, K}φ_J(I, J, K, x, y, z) J_x(I, J, K), Σ _{I, J, K}φ_J(I,
J, K, x, y, z) J_y(I, J, K), Σ_{I, J, K}φ_J(I, J, K, x, y, z) J_z(I,
J, K))^TIs approximated by These three-dimensional basis functions used
The differentiability of numbers is as described above, and in addition to the cubic
Element basis function φ_σ(I, J, K, x, y, z) is consistent with reference conductivity
It can be represented. ] By each
3D unknown conductivity distribution 1 / σ (I, J, K) and σ (I, J,
The simultaneous equations for K) are derived.

When the finite element method is used for this three-dimensional space of interest, the two-dimensional plane (x, y, Z) of each z coordinate z = Z (K = K ') is used. Pseudo two-dimensional current density vector distribution J (x, y, Z) (= (J _x (x, y, Z),
If J _y (x, y, Z)) ^T ) is measured and only this pseudo two-dimensional current density vector distribution data is used, similarly, the two-dimensional conductivity distribution 1 in each two-dimensional region of interest 1 / σ (x, y, Z)
And σ (x, y, Z) are two-dimensional basis functions φ _σ (I, J, K ', x, y, Z)
1 / σ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J, K ', x, y, Z) (1
/ σ (I, J, K ')), σ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J, K', x, y, Z)
It may be approximated by σ (I, J, K '), and the pseudo two-dimensional current density vector distribution J (x, y, Z) (= (J _x (x, y,
Z), J _y (x, y, Z)) ^T ) is a two-dimensional basis function φ _J (I, J, K ', x,
y (Z,) J (x, y, Z) ~ (Σ _{I, J} φ _J (I, J, K ', x, y, Z) J
_x (I, J, K '), Σ _{I, J} φ _J (I, J, K', x, y, Z) J _y (I, J, K ')) ^T.

The differentiability of these two-dimensional basis functions used is as described above, and in addition the two-dimensional basis function φ _σ (I, J, K ', x, y, Z) is the reference conductivity. Can be expressed without contradiction. From this, the two-dimensional unknown conductivity distribution in each two-dimensional region of interest 1 / σ (I, J, K ') or σ (I, J, K')
A system of equations for is derived. Also, z coordinate z = Z
In the case of targeting a two-dimensional region of interest within a two-dimensional object of (K = K '), similarly, finite difference approximation is performed or the finite element method is used [two-dimensional conductivity distribution 1 / σ
(x, y, Z) and σ (x, y, Z) are two-dimensional basis functions φ _σ (I, J, K ', x,
y / Z) 1 / σ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J, K ', x, y,
Z) (1 / σ (I, J, K ')), σ (x, y, Z) ~ Σ _{I, J} φ _σ (I, J, K', x,
y, Z) σ (I, J, K '). The differentiability of these two-dimensional basis functions used is as described above, and in addition the two-dimensional basis function φ _σ (I, J, K ', x, y, Z) represents the reference conductivity consistently. It is possible. By this, simultaneous equations regarding the two-dimensional unknown conductivity distribution 1 / σ (I, J, K ′) or σ (I, J, K ′) in the two-dimensional region of interest are derived.

Next, in this case, the finite difference approximation is performed in the three-dimensional space of interest. In this case, the simultaneous equations are solved by using the least squares method. At that time, the regular operator is applied to the JD inverse operator. Amplification of high frequency noise included in j may be suppressed. Even when the finite element method is used, when the derived simultaneous equations are solved by the method of least squares, a regularization may be similarly applied, but the conductivity spatial distribution ln (1 / σ ( x, y, z)), 1 / σ
(x, y, z), lnσ (x, y, z), or when the variational principle is applied to σ (x, y, z)

[Equation 3] In some cases, as will be described later, regularization may be performed when the minimization (variation) is performed. Specifically, using the regularization parameters α1 and α2 (positive value), R _s (s) = || j−JDs || ² + α ₁ || D's || ² + α ₂ || D ' ^T D's || ² (7) where D's: spatial distribution of unknown conductivity ln (1 / σ (x, y, z)) or approximation of three-dimensional gradient of lnσ (x, y, z) D' ^T D's: Minimize the three-dimensional Laplacian approximation of ln (1 / σ (x, y, z)) or lnσ (x, y, z) with respect to s.

Even when the reference area is inappropriate,
Since D's and D' ^T D's are positive definite values, R _s (s) always has one minimum value. By minimizing R _s (s), the regularized regular equation (D ^T J ^T JD + α ₁ D' ^T D '+ α ₂ D' ^T D'D' ^T D ') s = D ^T J ^T j (8 ) Is obtained, and the solution is therefore s = (D ^T J ^T JD + α ₁ D' ^T D '+ α ₂ D' ^T D'D' ^T D ') ^-1 D ^T J ^T j (9).

Here, each of D'and D' ^T D'is a finite difference approximation of a three-dimensional gradient operator and a three-dimensional Laplacian operator when the finite difference approximation is performed, and when the finite element method is used. Finite difference approximation or finite element approximation of 3D gradient operator and 3D Laplacian operator [however,
In this case, the three-dimensional basis function φ _σ (I, J, K, x, y,
z) needs to be partially differentiable more than twice, and can consistently represent the reference conductivity in the reference region w _l or the reference point w _l in the three-dimensional space of interest given as the initial value above. Must be one. ] And its squared norm is used. When the finite element method is used, a regularization parameter α ₀ is newly introduced, and the squared norm of the conductivity distribution s is multiplied and then the functional R _s (s) [(7) Applicable], and minimization may be performed. Regularized normal equation for s derived at that time [(8)
Corresponding to the expression], α ₀ I is derived approximately for s
(I: identity matrix) or exactly derived α ₀ I '(I': 3
The dimensional basis function φ _σ (matrix represented using I, J, K, x, y, z) will also be applied.

Further, the formulas (1a) to (1c) and the formula (2a) to
Equation (2c) has 1 / σ (x, y, z) and σ (x,
y, z) and multiplied by finite difference, or
Even when the finite element method is used, regularization may be similarly performed. Also, z coordinate z = Z (K = K ')
In any of the above cases that target the two-dimensional region of interest itself in the two-dimensional plane (x, y, Z) of, there are cases where it is also regularized, but at that time, finite difference approximation is performed. In this case, the two-dimensional gradient operator and the two-dimensional Laplacian operator finite difference approximation D [used in equation (4)] and D ^T D are used. And the finite difference approximation D of the two-dimensional Laplacian operator
[Used in (4)] and D ^T D is used, or finite element approximation of 2D gradient operator and 2D Laplacian operator
[However, in this case, the two-dimensional basis function φ _σ (I,
(J, K ', x, y, Z) must be partially differentiable more than once,
It is necessary to be able to consistently represent the reference conductivity in the reference region w _l or the reference point w _l in the two-dimensional space of interest given as the initial value above. ] Is used. In addition, when the finite element method is used, similarly, the regularization parameter α ₀ is introduced, and the squared norm of the conductivity distribution s is multiplied, and then the functional R _s (s) [(7) formula is obtained. Applicable to] and may be minimized. In the regularized normal equation [corresponding to Eq. (8)] with respect to s, which is derived at that time, α ₀ I (I: identity matrix) that is derived approximately for s, or is derived exactly α ₀ I '(I': matrix expressed using two-dimensional basis function φ _σ (I, J, K ', x, y, Z)) is applied.

The conductivity spatial distribution ln (1 / σ (x, y,
z)), 1 / σ (x, y, z), lnσ (x, y, z), or σ (x,
When the variation principle is applied to y, z) [Eq. (5-1), (5-2)
[6, (6-1), (6-2)] functional is [J ₁ (x, y, z) (= (J
_1x (x, y, z), J _1y (x, y, z), J _1z (x, y, z)) ^T )] is

[Equation 4] When the measured current density vector distribution is two [J
₁ (x, y, z) and J ₂ (x, y, z) (= (J _2x (x, y, z), J _2y (x, y, z), J
_2z (x, y, z)) ^T )] is

[Equation 5] Becomes Multiple measured current density vector distributions [M
(> 1), i = 1 to M],

[Equation 6] Is. Conductivity distribution s and current density expressed in each functional
Finite element approximation with respect to the degree vector [
Nodal conductivity ln (1 / σ (I, J, K)) or 1 / σ (I, J, K) and
Basis function φ_σ(I, J, K, x, y, z) (partially differentiable more than once)
Using ln (1 / σ (x, y, z)) ~ Σ_{I, J, K}φ_σ(I, J, K, x, y, z)
ln (1 / σ (I, J, K)) or 1 / σ (x, y, z) ~ Σ_{I, J, K} φ_σ(I, J, K,
x, y, z) (1 / σ (I, J, K)), and also corresponds to equation (2a)
Node conductivity lnσ (I, J, K) or σ (I, J, K) and
Basis function φ_σ(I, J, K, x, y, z) (partially differentiable more than once)
Using lnσ (x, y, z) ~ Σ_{I, J, K}φ_σ(I, J, K, x, y, z) lnσ
(I, J, K) or σ (x, y, z) ~ Σ_{I, J, K}φ_σ(I, J, K, x, y, z) σ
(I, J, K), the nodal current (J_1x(I, J, K), J_1y(I, J,
K), J_1z(I, J, K))^TAnd basis function φ_J(I, J, K, x, y, z) (once
(Partially differentiable)₁(x, y, z) ~ (Σ_{I, J, K}φ
_J(I, J, K, x, y, z) J_1x(I, J, K), Σ_{I, J, K}φ_J(I, J, K, x, y, z) J
_1y(I, J, K), Σ_{I, J, K}φ_J(I, J, K, x, y, z) J_1z(I, J, K))^TAnd within
Is inserted. ] Is applied (the number of nodes is omitted). However,
The basis function φ used _σ(I, J, K, x, y, z) are initial values above
Reference region w in 3D space of interest given as a value_lOr
Reference point w_lIt is possible to express the reference conductivity in
And the spatial distribution of conductivity ln (1
/ σ (x, y, z)), 1 / σ (x, y, z), lnσ (x, y, z),
Also, after σ (x, y, z) is approximated by a finite element, the initial value is
It is assigned to the functional as a constant value.

In addition, the two-dimensional flatness of each z coordinate z = Z (K = K ')
Pseudo two-dimensional current density in the two-dimensional region of interest in the plane (x, y, Z)
Degree vector distribution J (x, y, Z) (= (J_x(x, y, Z), J
_y(x, y, Z))^T) Is measured and the pseudo two-dimensional current density
If only Toll distribution data is used, and z-coordinate z
A 2D region of interest in a 2D object of = Z (K = K ')
I in these functionals if_i(s) [Expression (5-1), (5-
2), (6-1), (6-2)]] is the two-dimensional region of interest
Conductivity distribution expressed in each functional by the double integral of
s and the current density vector in the finite element approximation [(1a)
Nodal conductivity ln (1 / σ (I, J, K ')) or 1 / σ of the corresponding finite element
(I, J, K ') and two-dimensional basis function φ_σ(I, J, K ', x, y, Z) (once
Ln (1 / σ (x, y, Z)) ~ Σ_{I, J}
φ_σ(I, J, K ', x, y, Z) ln (1 / σ (I, J, K')) or 1 / σ (x, y, Z) ~
 Σ_{I, J}φ_σ(I, J, K ', x, y, Z) (1 / σ (I, J, K'))
Also, the nodal conductivity lnσ of the finite element corresponding to equation (2a)
(I, J, K ') or σ (I, J, K') and two-dimensional basis function φ_σ(I, J, K ',
x, y, Z) (possible partial differentiation more than once) lnσ (x, y, Z) ~
 Σ_{I, J}φ_σ(I, J, K ', x, y, Z) lnσ (I, J, K') or σ (x, y, Z)
~ Σ _{I, J}φ_σ(I, J, K ', x, y, Z) σ (I, J, K')
Nodal current (J_1x(I, J, K '), J_1y(I, J, K '))^TAnd two-dimensional basis
Base function φ_JUse (I, J, K ', x, y, Z) (partially differentiable more than once)
Wait J₁(x, y, Z) ~ (Σ_{I, J}φ_J(I, J, K ', x, y, Z) J_1x(I, J,
K '), Σ_{I, J}φ_J(I, J, K ', x, y, Z) J_1y(I, J, K '))^TInterpolated with
It ] Is applied. However, the two-dimensional basis function used
φ_σ(I, J, K ', x, y, Z) is the reference area w in the two-dimensional region of interest._l
Or reference point w_lThe reference conductivity in
It should be something that can be used for conductivity spatial distribution
ln (1 / σ (x, y, Z)), 1 / σ (x, y, Z), lnσ (x, y, Z
Z), or after σ (x, y, Z) is finite element approximated, the initial value
Is assigned to the functional as a specified value.

Further, each I _i (s) [Equation (5-1), Equation (5-2), (6-
These functionals for the three-dimensional space of interest expressed by the equations (1) and (6-2)] are minimized with respect to the three-dimensional unknown conductivity spatial distribution, but in the case of regularization, Is
Like all other methods for estimating the three-dimensional conductivity spatial distribution using the finite element method described above, the three-dimensional conductivity spatial distribution ln
(1 / σ (x, y, z)), 1 / σ (x, y, z), lnσ (x, y, z), or σ (x,
y, z) to be used by adding the distribution itself, the three-dimensional gradient distribution, and the square norm of the three-dimensional Laplacian distribution,
Identity matrix I, difference approximation D ′ of three-dimensional gradient operator, and 3
Use the difference approximation D' ^T D'of the three-dimensional Laplacian operator, or
Approximated using the dimensional basis function φ _σ (I, J, K, x, y, z),
The matrix I ′, the finite element approximation of the three-dimensional gradient operator, and the finite element approximation of the three-dimensional Laplacian operator are evaluated. In this case, the three-dimensional basis function φ _σ (I, J, K, x, y, z) used
Must be partially differentiable more than twice, and the initial values given above are also substituted for the conductivity distribution itself, the gradient approximation of the conductivity distribution, and the square norm of the Laplacian approximation. Finally, this regularized functional is the three-dimensional unknown conductivity distribution ln (1 / σ (I, J, K)), 1 / σ (I, J, K),
lnσ (I, J, K) or σ (I, J, K) is minimized, and the estimation result is ln (1 / σ (x, y, z)) = Σ _{I, J, K} φ _σ (I, J,
K, x, y, z) ln (1 / σ (I, J, K)), 1 / σ (x, y, z) = Σ _{I, J, K} φ
_σ (I, J, K, x, y, z) (1 / σ (I, J, K)), lnσ (x, y, z) = Σ _{I, J, K}
φ _σ (I, J, K, x, y, z) lnσ (I, J, K), or σ (x, y, z) = Σ
_{I, J, K} φ _σ (I, J, K, x, y, z) σ (I, J, K) is obtained.

For a two-dimensional region of interest (z = Z),
Functionals are minimized with respect to the two-dimensional unknown conductivity distribution
In the case of
Rate distribution ln (1 / σ (x, y, Z)), 1 / σ (x, y, Z), lnσ (x, y, Z),
Also, the distribution of σ (x, y, Z) itself, the two-dimensional gradient distribution, and
And the square norm of the two-dimensional Laplacian distribution should be used.
Unit matrix I, two-dimensional gradient operator D [used in equation (4)],
And 2D Laplacian D^TUse D or 2D area
Two-dimensional basis function φ of the conductivity distribution adopted in the region_σ
Matrix I ', two-dimensional gradient approximated using (I, J, K', x, y, Z)
Finite element approximation of the operator and two-dimensional Laplacian work
The finite element approximation of the element is evaluated. In this case used
Two-dimensional basis function φ_σ(I, J, K ', x, y, Z) is a partial differential more than twice
It must be possible and the default value given above is
The conductivity distribution itself, the gradient approximation of the conductivity distribution,
It is also assigned to the square norm of the Laplacian approximation.
Finally, this regularized functional has two-dimensional unknown conductivity.
Distribution ln (1 / σ (I, J, K ')), 1 / σ (I, J, K'), lnσ (I, J, K '),
Also, σ (I, J, K ') is minimized and
, Ln (1 / σ (x, y, Z)) = Σ_{I, J}φ_σ(I, J, K ', x, y, Z) ln (1
/ Σ (I, J, K ')), 1 / σ (x, y, Z) = Σ_{I, J}φ_σ(I, J, K ', x,
y, Z) (1 / σ (I, J, K ')), lnσ (x, y, Z) = Σ_{I, J}φ _σ(I, J,
K ', x, y, Z) lnσ (I, J, K'), or σ (x, y, Z) = Σ_{I, J}φ
_σ(I, J, K ', x, y, Z) σ (I, J, K') is obtained.

As a special case, Ii (s) [i = 1 to 1
M] is regularized and minimized in each of i as described above, and then the unknown conductivity distribution s [ln (1 / σ (I, J, J,
K)), 1 / σ (I, J, K), lnσ (I, J, K), σ (I, J, K), ln (1 / σ
(I, J, K ')), 1 / σ (I, J, K'), lnσ (I, J, K '), or σ (I,
J, K ')] may be least squared.

As described above, according to the present method, when a current field already exists in the three-dimensional space of interest or the two-dimensional space of interest, the current is not forced to flow from the outside, and therefore the field is disturbed. Without, it is possible to evaluate the conductivity spatial distribution only by measuring the current density vector distribution.

Regarding the regularization parameters α ₀ , α _1, and α ₂ , in the regularized normal equation [corresponding to the equation (8) when the finite difference approximation is performed], a vector representing the spatial distribution of conductivity is given. The matrix of s is adjusted to a large value so that it is sufficiently positive definite numerically, or adjusted according to the accuracy (SN ratio) of the pseudo two-dimensional current density vector distribution data used (SN ratio). Is small when it is high and large when the SN ratio is low). Also, in particular, 1 / σ on both sides of equation (1a)
(x, y, Z) multiplied by a conductivity distribution 1 / σ (x, y, Z) or a finite difference approximation with respect to the current density vector distribution, or
Both sides of Eq. (2a) are multiplied by σ (x, y, Z) to obtain the conductivity distribution σ (x,
y, Z) or a finite difference approximation with respect to the current density vector distribution, 1 / σ (x, y, Z) is also applied to both sides of equation (1a) and 1 / σ (x, y, Z) If a finite element approximation (variational method or Galerkin method) is applied to, then both sides of equation (2a) have σ
When (x, y, Z) is applied and the finite element approximation (variational method or Galerkin method) is applied to σ (x, y, Z), the pseudo two-dimensional current density vector data and It is adjusted by the accuracy (SN ratio) of the square root of the sum of the power of the two-dimensional distribution data of the inner product with the gradient operator concerning the conductivity and the power of the two-dimensional distribution data of the rotation of the pseudo two-dimensional current density vector (SN ratio). It is small when the ratio is high, and large when the SN ratio is low). According to this, for example,
It may be inversely proportional to the SN power ratio. These power-dependent regularization parameters α ₀ · α ₁ · α ₂ are SN of each pseudo two-dimensional current density vector distribution data when a plurality of pseudo two-dimensional current density vector distribution data are used.
The value is proportional to the sum of the regularization parameter values evaluated from the power ratio. In accordance with this, for example, it may be the sum of values inversely proportional to the SN power ratio of each pseudo two-dimensional vector distribution data.

Regarding the regularization parameters α ₁ and α ₂ , those that differ depending on the direction of the partial differential appearing in the gradient operator and the Laplacian operator (the one that depends on the direction).
As a result, the matrix of the vector s representing the spatial distribution of the conductivity is adjusted to a large value so that it is sufficiently positive definite numerically, or the pseudo 2 used. It may be adjusted by the accuracy (SN ratio) of the two-way component distribution data of the dimensional current density vector (the direction with a high SN ratio is large, and the direction with a low SN ratio is small). Also, in particular, when both sides of equation (1a) are multiplied by 1 / σ (x, y, Z) and a finite difference approximation is made regarding the conductivity distribution 1 / σ (x, y, Z) or the current density vector distribution. , Or both sides of the equation (2a) are multiplied by σ (x, y, Z) to approximate the conductivity distribution σ (x, y, Z) or the current density vector distribution by finite difference approximation, or (1a ) Is multiplied by 1 / σ (x, y, Z) on both sides and the finite element approximation (variation method or Galerkin method) is applied to 1 / σ (x, y, Z), and 2a) is multiplied on both sides by σ (x, y, Z) to obtain σ (x, y,
When the finite element approximation (variation method or Galerkin method) is applied to Z), the two-dimensional component data of the pseudo two-dimensional current density vector used and the first-order bias applied to the conductivity in the same direction as that component are used. Accuracy of the square root of the sum of the power of each two-dimensional distribution data of the product with the differential operator and the power of the two-dimensional distribution data of each first-order partial differential component in the same two directions that represents the rotation of the pseudo two-dimensional current density vector ( SN ratio) (small when the SN ratio is high and large when the SN ratio is low), or the inner product of the pseudo two-dimensional current density vector data used and the gradient operator on the conductivity. Power of each two-dimensional distribution data of the amount of change between Δx and Δy in each direction and each two-dimensional amount of change between Δx and Δy of the first-order partial differential component in the same direction that represents the rotation of the pseudo two-dimensional current density vector Sum of power of distribution data It may be adjusted by the precision of the square root (SN ratio) (small when the SN ratio is high, and large when the SN ratio is low). According to this, for example, it may be proportional to the SN power ratio of each direction distribution data.

When a plurality of pseudo two-dimensional current density vector distribution data are used, the regularization parameters α ₁ and α ₂ depending on these directions are accurate (SN Ratio) is considered (small when the SN ratio is high, large when the SN ratio is low). In particular, when using pseudo two-dimensional current density vector distribution data having different two-direction components, the number of each direction component distribution data to be used may be taken into consideration (a large number of directions indicates a large number, May be proportional to the number of directional component distribution data used, for example. According to this, when only the pseudo two-dimensional current density vector distribution data having the same two-direction component is used, the direction-dependent regularization parameter evaluated in each pseudo two-dimensional current density vector distribution data as described above. The value and the regularization parameter value evaluated from the SN power ratio of each pseudo two-dimensional current density vector distribution data itself are values proportional to the sum of products calculated by weighting the importance, and particularly, When the pseudo two-dimensional current density vector distribution data having different two-direction components is used, the regularization parameter value evaluated from the number of the direction component distribution data is added to the sum of products of both regularization parameter values evaluated in the same manner. It may be a value proportional to the value multiplied by.

Further, the regularization parameters α ₀ · α ₁ · α ₂ are
It may be realized as spatially varying, and as a result, the local matrix for the conductivity of each interest point is adjusted to a large value such that it is sufficiently positive definite numerically, or used. It may be adjusted according to the accuracy (SN ratio) of the pseudo two-dimensional current density vector data of each interest point (small when the SN ratio is high, and large when the SN ratio is low).
Also, in particular, when both sides of equation (1a) are multiplied by 1 / σ (x, y, Z) and a finite difference approximation is made regarding the conductivity distribution 1 / σ (x, y, Z) or the current density vector distribution. , And σ (x, on both sides of equation (2a)
y, Z) and then the conductivity distribution σ (x, y, Z) or the current density vector distribution is approximated by a finite difference, or (1a)
If both sides of the equation are multiplied by 1 / σ (x, y, Z) and the finite element approximation (variation method or Galerkin method) is applied with respect to 1 / σ (x, y, Z), and (2a ) Is multiplied by σ (x, y, Z) to obtain σ
When the finite element approximation (variation method or Galerkin method) is applied to (x, y, Z), the pseudo two-dimensional current density vector data used at each interest point and the gradient operator for the conductivity are Power of inner product data and its pseudo 2
It may be adjusted by the precision of the square root of the sum of the dimensional current density vector and the power of rotation (SN ratio) (small when the SN ratio is high, and large when the SN ratio is low). According to this, for example, it may be inversely proportional to the SN power ratio.

These position-dependent regularization parameters α ₀ · α ₁ · α ₂ are a plurality of pseudo 2 at the same point of interest.
When the dimensional current density vector data is used, the value is proportional to the sum of the regularization parameter values evaluated from the SN power ratio of the pseudo two-dimensional current density vector data used at each interest point. According to this, for example, the sum is a value that is inversely proportional to the SN power ratio of the pseudo two-dimensional current density vector data of each interest point. Also, especially when the number of pseudo two-dimensional current density vector data used at each interest point is different, this number of data may be taken into consideration (large at a position with a large number, small at a position with a small number),
According to this, for example, it may be proportional to the number of pseudo two-dimensional current density vector data to be used. In accordance with this, the regularization parameter value evaluated from the SN power ratio of the pseudo two-dimensional current density vector data used at each interest point and the number of the pseudo two-dimensional current density vector data used at each interest point are evaluated. The value of the regularization parameter to be set may be a value proportional to the sum of products calculated by weighting the importance.

Regarding the regularization parameters α ₁ and α ₂ , those that spatially change as described above and that differ depending on the direction of the partial differential appearing in the gradient operator and the Laplacian operator (depending on the direction) As a result, the local matrix for the conductivity of each point of interest is adjusted to a large value so that it is sufficiently positive definite numerically, or each local matrix used is used. Accuracy of two-way component data of pseudo two-dimensional current density vector of interest point (SN
Ratio) (the direction where the SN ratio is high is large and the direction where the SN ratio is low is small). Also, in particular, conductivity distribution 1 / σ (x, y, Z) is obtained by multiplying both sides of equation (1a) by 1 / σ (x, y, Z).
Or, in case of finite difference approximation with respect to current density vector distribution, also, both sides of equation (2a) are multiplied by σ (x, y, Z) to obtain conductivity distribution σ (x, y, Z) or current density vector distribution. In case of finite difference approximation with respect to, 1 / σ (x, y,
Z) is applied and the finite element approximation (variation method or Galerkin method) is applied to 1 / σ (x, y, Z), and σ (x, y, Z is applied to both sides of equation (2a). ) Is applied and the finite element approximation (variational method or Galerkin method) is applied to σ (x, y, Z), the two-way component of the pseudo two-dimensional current density vector used at each interest point. Each power of the data of the product of the data and its component and the first-order partial differential operator concerning the conductivity in the same direction, and the power of each first-order partial differential component in the same two directions representing the rotation of the pseudo two-dimensional current density vector It is adjusted by the precision of the square root of the sum (SN ratio) (small when the SN ratio is high, large when the SN ratio is low), or the pseudo two-dimensional current density vector data used at each interest point. Δx in two directions of the inner product of the gradient operator and the conductivity
Accuracy of the square root of the sum of each power of the variation data between Δy and each power of the variation data between Δx and Δy of the first-order partial differential component in the same two directions that represents the rotation of the pseudo two-dimensional current density vector ( S / N ratio) (small when the S / N ratio is high, and large when the S / N ratio is low). According to this, for example, it may be proportional to the SN power ratio of each direction data of each interest point.

These position-dependent and direction-dependent regularization parameters α ₁ and α ₂ are used for each pseudo-two-dimensional current density vector data when a plurality of pseudo-two-dimensional current density vector data is used. Accuracy of dimensional current density vector (SN ratio)
Is considered (large when the SN ratio is high, small when the SN ratio is low). In addition, especially at the position where the pseudo two-dimensional current density vector data having different two-direction components is used, the number of each direction component data used may be taken into consideration (a large number of directions is large and a small number of directions is large). The direction may be small), and in accordance with this, for example, it may be proportional to the number of direction component data used. According to this, when only the pseudo two-dimensional current density vectors having the same two-direction components are used, the direction-dependent regularization parameter values and the respective direction-dependent regularization parameter values evaluated in each pseudo two-dimensional current density vector data as described above are used. The value of the regularization parameter evaluated from the SN power ratio of the pseudo two-dimensional current density vector itself is set to a value proportional to the sum of the products calculated by weighting the importance, and particularly, the different two-direction components are set. When using the pseudo-two-dimensional current density vector data that it has, it is proportional to the value obtained by multiplying the sum of products of both regularization parameter values evaluated in the same manner by the regularization parameter value evaluated from the number of direction component data. It may be set to a value.

Here, when Ii (s) (i = 1 to M) itself is subjected to regularization, the regularization parameter can have a negative value, and accordingly, the absolute value of the regularization parameter is It should be noted that when the SN power ratio is adjusted, it may be adjusted so as to be inversely proportional to its square root.

When the SN ratio of the pseudo two-dimensional current density vector distribution data is used, the SN ratio of the pseudo two-dimensional current density vector data measured at the node (I, J, K) of each element,
Or, the SN ratio of the pseudo two-dimensional current density vector data given in continuous coordinates within the element estimated from this may be necessary. From this, the SN ratio of the pseudo two-dimensional current density vector distribution data can be evaluated. Sometimes. Alternatively, the SN ratio of the pseudo two-dimensional current density vector distribution data is obtained, and the pseudo two-dimensional current density vector data of each element is calculated.
The SN ratio and the SN ratio of the pseudo two-dimensional current density vector data of the node of each element may be evaluated. As a result, the spatially uniform regularization parameter α ₀ · α ₁ · α ₂ or the spatially varying regularization parameter α ₀ · α ₁ · α ₂ may be adjusted.

When the one-dimensional conductivity distribution in the one-dimensional region of interest is to be measured, the conductivity distribution may be estimated by using only the current density data that dominantly flows in the orthogonal direction. For example, the conductivity distribution ln (1 / σ in the x-axis direction
(x)), 1 / σ (x), lnσ (x), or σ (x) is estimated,

[Equation 7] In the above, the finite difference approximation is applied as described above, or the finite element method (variational principle or Galerkin method) is used to derive an algebraic equation regarding the unknown conductivity distribution and apply regularization. The conductivity distribution can be estimated stably. Therefore, similar to the case where a plurality of two-dimensional regions of interest are provided for a three-dimensional region of interest, not only two-dimensional regions of interest but also a plurality of one-dimensional regions of interest are provided in the three-dimensional region of interest. In the case of targeting a two-dimensional region of interest, a plurality of one-dimensional regions of interest are provided, and an algebraic equation regarding unknown conductivity distribution is derived in each low-dimensional region of interest,
Regularization may be performed in the high dimensional region of interest or in each low dimensional region of interest.

It is impossible to apply to the three-dimensional conductivity distribution and the two-dimensional conductivity distribution, but when applying the variational principle to these one-dimensional conductivity distributions, the one-dimensional conductivity distribution itself is not used. Without, the one-dimensional distribution s of the first derivative of the one-dimensional conductivity
It is also possible to apply the variational principle to, for example (1
In the equation a-1), the functional concerning s of the one-dimensional distribution ln (1 / σ (x)) when the variational principle is applied to the one-dimensional conductivity distribution ln (1 / σ (x)) is Similar to equation (5-1),

[Equation 8] And the 1-dimensional distribution when the variational principle is applied to the 1-dimensional conductivity distribution 1 / σ (x) after multiplying 1 / σ (x) on both sides of the equation (1a-1). The functional of 1 / σ (x) with respect to s is the same as equation (5-2).

[Equation 9] However, in equation (1a-1), the one-dimensional conductivity ln (1 / σ
distribution of first derivative d (ln (1 / σ (x))) / dx when the variational principle is applied to dx d (ln (1 / σ (x))) / The functional of dx with respect to s is

[Equation 10] Also, the distribution of the first derivative of the one-dimensional conductivity 1 / σ (x) d (1 / σ (x)) / dx after multiplying 1 / σ (x) on both sides of the equation (1a-1) When the variational principle is applied to

[Equation 11] Is. One-dimensional differential equation (2a-1), (1c-1), (2
The same applies to equation (c-1). When estimating the one-dimensional conductivity distribution itself, a one-dimensional gradient operator or a one-dimensional Laplacian operator may be used.

When the time series of the current density vector space distribution is measured, it is assumed that a plurality of current density vector space distributions are measured, and the regularization using the gradient operator or the Laplacian operator is performed as described above. By applying it, the unknown conductivity spatial distribution can be estimated.

The unknown conductivity spatial distribution may change with time, and in this case, the reference conductivity (distribution) value and the reference region where the position / size / state / number can change with time. At each measured time of the current density vector spatial distribution data using Time series of spatial distribution can be obtained. When multiple time series of independent current density vector space distributions are measured, after all simultaneous equations that hold in each of the same state (time) of different time series of the measurement target are simultaneous, as described above, The time series of the unknown conductivity spatial distribution can be obtained by estimating the unknown conductivity spatial distribution by performing regularization using a gradient operator or a Laplacian operator.

Further, when estimating the time series of the unknown conductivity spatial distribution, all equations that hold at each time are simultaneous, and the first derivative in the time direction of the time series of the unknown conductivity spatial distribution is calculated. Regularization may be performed using the squared norm of the second-order partial differential [regularization parameters for each are α ₃ and α ₄ ]. When multiple time series of independent current density vector space distributions are measured, all the equations that are satisfied in each of the same states (time) of different time series of the measurement object are simultaneous, and then the time of the time series is measured. Regularization may be performed using the first-order partial derivative of the direction or the square-norm of the second-order partial derivative [regularization parameters for each are α ₃ and α ₄ ]. The first-order partial differential and the second-order partial differential in the time direction of the time series of the conductivity spatial distribution are obtained when a three-dimensional region of interest is targeted, as in the case of using the above gradient operator or Laplacian operator. Provides a plurality of two-dimensional regions of interest and one-dimensional regions within that region, and when targeting a two-dimensional region of interest, provides a plurality of one-dimensional regions of interest, and unknown conductivity in each low-dimensional region of interest at each time. An algebraic equation for the rate distribution may be derived and regularization may be performed in the high dimensional region of interest or each low dimensional region of interest, as appropriate. In this case, the regularization using the gradient of the unknown conductivity spatial distribution at each time and the Laplacian squared norm may be performed at the same time.

When the time series of the conductivity spatial distribution is estimated, the spatial distribution of the frequency dispersion of the conductivity is approximated by performing spectrum analysis at each position of the space distribution of the time series data of the conductivity. You can ask. When evaluating the spatial distribution of time-series frequency dispersion of conductivity, positively changing the frequency (single) of the current source,
The current density vector space distribution may be measured, or a broadband current source may be used.

The regularization parameters α ₃ and α ₄ are adjusted to a large value so that the matrix of the vector s representing the spatial distribution of conductivity is sufficiently positive definite numerically at each time. Alternatively, it may be adjusted according to the accuracy (SN ratio) of the variation data in the time direction of the pseudo two-dimensional current density vector distribution used (small when the SN ratio is high, and large when the SN ratio is low). Also, especially on both sides of equation (1a)
When 1 / σ (x, y, Z) is multiplied to obtain a finite difference approximation with respect to the conductivity distribution 1 / σ (x, y, Z) or the current density vector distribution,
Also, if both sides of the equation (2a) are multiplied by σ (x, y, Z) to approximate the conductivity distribution σ (x, y, Z) or the current density vector distribution by finite difference approximation, or (1a) If both sides of the equation are multiplied by 1 / σ (x, y, Z) and the finite element approximation (variation method or Galerkin method) is applied with respect to 1 / σ (x, y, Z), and (2a ) Is applied to both sides of σ (x, y, Z) and the finite element approximation (variation method or Galerkin method) is applied to σ (x, y, Z), the pseudo 2 Power of the two-dimensional distribution data of the inner product of the two-dimensional current density vector data and the gradient operator for the conductivity and the power of the two-dimensional distribution of rotation of the pseudo two-dimensional current density vector Adjusted by the precision of the square root of the sum of power (SN ratio) (small when the SN ratio is high, large when the SN ratio is low)
Sometimes. According to this, for example, it may be inversely proportional to the SN power ratio.

These power-dependent regularization parameters α ₃ and α ₄ are used for each pseudo two-dimensional current density vector distribution when a plurality of pseudo two-dimensional current density vector distribution data are used at each time. The value is proportional to the sum of the regularization parameter values evaluated from the SN power ratio of the variation data in the time direction. According to this, for example, it may be the sum of values inversely proportional to the SN power ratio of the variation data in the time direction of each pseudo two-dimensional vector distribution.

The regularization parameters α ₃ and α ₄ may be realized as spatially varying at each time, and as a result, the local matrix relating to the conductivity of each interest point is numerically analyzed. Is adjusted to a sufficiently large positive definite value, or adjusted according to the accuracy (SN ratio) of the variation data in the time direction of the pseudo two-dimensional current density vector of each interest point used (SN ratio). It is small when the ratio is high and large when the SN ratio is low). Also, especially on both sides of equation (1a)
When 1 / σ (x, y, Z) is multiplied to obtain a finite difference approximation with respect to the conductivity distribution 1 / σ (x, y, Z) or the current density vector distribution,
Also, if both sides of the equation (2a) are multiplied by σ (x, y, Z) to approximate the conductivity distribution σ (x, y, Z) or the current density vector distribution by finite difference approximation, or (1a) If both sides of the equation are multiplied by 1 / σ (x, y, Z) and the finite element approximation (variation method or Galerkin method) is applied with respect to 1 / σ (x, y, Z), and (2a ) Is used at each point of interest when σ (x, y, Z) is applied to both sides of the equation and the finite element approximation (variation method or Galerkin method) is applied to σ (x, y, Z). Of the power of the variation data in the time direction of the inner product of the pseudo two-dimensional current density vector data and the gradient operator for the conductivity and the power of the variation data in the time direction of the rotation of the pseudo two-dimensional current density vector It may be adjusted by the precision of the square root of (SN ratio) (small when the SN ratio is high and large when the SN ratio is low). According to this, for example, it may be inversely proportional to the SN power ratio.

These position-dependent regularization parameters α ₃ · α ₄ , if a plurality of pseudo two-dimensional current density vector data are used at the same interest point, the pseudo two-dimensional data used at each interest point The value is proportional to the sum of the regularization parameter values evaluated from the SN power ratio of the change amount data of the current density vector in the time direction. According to this, for example, the sum is a value that is inversely proportional to the SN power ratio of the variation data in the time direction of the pseudo two-dimensional current density vector of each interest point. Also, especially when the number of pseudo two-dimensional current density vector data used at each interest point is different, this number of data may be taken into consideration (large at a position with a large number,
(Small in small numbers), according to this, for example,
It may be proportional to the number of pseudo two-dimensional current density vector data used. According to this, the regularization parameter value evaluated from the SN power ratio of the variation data in the time direction of the pseudo two-dimensional current density vector used at each interest point and the pseudo two-dimensional current density used at each interest point The regularization parameter value evaluated from the number of vector data may be a value proportional to the sum of products calculated by weighting the importance.

Here, when Ii (s) (i = 1 to M) itself is subjected to regularization, the regularization parameter can take a negative value, and accordingly, the absolute value of the regularization parameter is It should be noted that when the above SN power ratio is used as an index, it may be adjusted so as to be inversely proportional to its square root.

Next, a conductivity estimating method for simultaneously estimating the permittivity will be described. According to the above conductivity estimation method,
Current density vector distribution data measured in the region of interest
J _i [i (= 1 to M) refers to the measured independent current density vector distribution J _i , and M is the number (1 or more) of the measured independent current density vector distributions. ] And the reference conductivity (distribution) value given at the reference point or at the reference point w _l (l = 1 to N) as the initial condition, that is, using the equation (3), Equations (1a) and (1b), which are first-order spatial partial differential equations expressing the conductivity distribution σ (x, y, z) and σ (x, y),
Formula (1c) [Equation (2a), (2b), (2c)] is given a numerical solution using the finite element method (variation principle or Galerkin method) or finite difference method and regularization method. By applying, the unknown conductivity distribution σ (x, y, z), σ (x, y), and σ (x) are obtained.

On the other hand, according to the present conductivity estimation method, even if the measured current density vector distribution changes with time, if it changes sinusoidally at the single frequency fi, The dielectric constant reference value can be used together with the conductivity reference value to measure the distribution of the conductivity, the dielectric constant, the distribution of the dielectric constant and the conductivity, or the like. However, i (= 1 to M)
Is the time series J _{i of the} independent current density vector distributions treated
(t), M represents the number of independent time series of the current density vector distribution (1 or more), and t represents the time from the start of acquisition of the current density vector component distribution data. To do.

Specifically, the current density vector in the region of interest is
Time-series data of the spatial distribution of each direction component
In contrast, at each point of interest, Fast Fourier's Transform (FFT)
Spectrum analysis by discrete Fourier transform such as
For example, when targeting a 3D region of interest,
Time series data of flow density vector Ji (x, y, z, t) [= (Jix (x,
y, z, t), Jiy (x, y, z, t), Jiz (x, y, z, t))^T] Spectra
Or single frequency f _i[Constant without depending on i
There is. ] Ji (x, y, z, fi) [= (Jix (x, y, z, fi), Jiy
(x, y, z, fi), Jiz (x, y, z, fi))^T= (Re [Jix (x, y, z, fi)]
+ jIm [Jix (x, y, z, fi)], Re [J_iy(x, y, z, fi)] + jIm [Jiy
(x, y, z, fi)], Re [Jiz (x, y, z, fi)] ＋ jIm [Jiz (x, y, z, f
i)])^T] (However, Re [• (fi)] and Im [• (fi)] are
Spectrum of frequency fi ・ Real and imaginary components of (fi)
And j is an imaginary unit. ) Measured direction component of
According to the distribution data, 3D current density vector time series data
Frequency Ji (x, y, z, t) frequency fi [measured and given
There is a case. ] And conductivity distribution σi (x, y, z) and permittivity εi
Spatial distributions expressed by (x, y, z) Ai (x, y, z) and Bi (x,
y, z), that is,

[Equation 12] The following simultaneous spatial partial differential equations for

[Equation 13] That is,

[Equation 14] Or

[Equation 15] When

[Equation 16] Or

[Equation 17] When

[Equation 18] Or

[Formula 19] , Or a system of spatial partial differential equations consisting of multiple spatial partial differential equations among them, the conductivity spatial distribution σi (x, y, z) and the permittivity spatial distribution εi (x, y , z) for the following initial conditions

[Equation 20] and,

[Equation 21] When

[Equation 22] and,

[Equation 23] (However, w _m σ _{i (t)} and w _m ε _{i (t)} are the same or may intersect with each other.), And are expressed by equations (13a) and (13b), respectively. Ai (x, y, z) and Bi (x, y, z)
At each point of interest, if both the conductivity distribution σi (x, y, z) and the permittivity distribution εi (x, y, z) are unknown, they are also unknown, and the permittivity εi (x, y, z) ) Is a given point, its value is used to calculate the unknown conductivity σi (x, y, z), and the conductivity σi
At the point where (x, y, z) is given, its value is used to calculate the unknown permittivity εi (x, y, z).

[Equation 24] Can be asked.

It should be noted that even in another independent time series i, 3
According to the measured direction component distribution data of the dimensional current density vector, (14a) equation, (14b) equation, (14c) equation [or, (14a ') equation,
(14b '), (14c')] any of the spatial partial differential equations or simultaneous spatial partial differential equations consisting of a plurality of spatial partial differential equations of these, and the conductivity spatial distribution σi (x, y, z) and the permittivity spatial distribution εi (x, y, z) initial condition (15a) and (15
Equations (a ') and (15b) and (15b') are given, and thus by simultaneous equations of all equations obtained from the time series i, the spatial distributions Ai (x, y, z) and Bi ( x, y, z), and the conductivity distribution σi (x, y, z) and the permittivity distribution εi (x, y,
When both z) are unknown, they are also known, and the permittivity εi (x,
y, z) is used to give the unknown conductivity σi (x, y, z), and at the point where the conductivity σi (x, y, z) is given. Value of unknown dielectric constant εi
(x, y, z) can be obtained from the equations (16a) and (16b).

Further, the conductivity distribution σi (x, y, z) and the permittivity εi
Both (x, y, z) or one of them may be unchanged with respect to the time series i, and in this case also, according to the measured directional component distribution data, equation (14a), ( 14b), (14c) [or (14a '), (14b'), (14c ')] spatial partial differential equation or any of these spatial partial differential equations Simultaneous spatial partial differential equations and conductivity σi (x, y,
z) or σ (x, y, z) and the permittivity εi (x, y, z) or ε (x, y, z) initial conditions (15a) and (15a ') or (15b) and (15b ')
Formula [It may or may not depend on the time series i. ] Therefore, by simultaneous equations of all equations, the spatial distribution Ai (x, y, z) or A (x, y, z) and Bi (x, y,
z) or B (x, y, z), and the conductivity σi at each point of interest
(x, y, z) or σ (x, y, z) and permittivity εi (x, y, z) or εi (x, y,
When both z) are unknown, they are also known, and the permittivity εi (x,
y, z) or ε (x, y, z) is given, and its value is used to calculate the unknown conductivity σi (x, y, z) or σ (x, y, z),
In addition, when the conductivity σi (x, y, z) or σ (x, y, z) is given, the unknown dielectric constant εi (x, y, z) is used by using that value.
Alternatively, ε (x, y, z) can be obtained from the equations (16a) and (16b).

In addition, the time series σi (x, y, z, t) of the conductivity distribution which changes with time and the time series εi (x, y, z, t) of the permittivity may be measured. , In that case, the following initial conditions in which the size, position, and number of the reference value itself and the reference region may change depending on the time t and the time series i

[Equation 25] and,

[Equation 26] When

[Equation 27] and,

[Equation 28] (15a) and (15a ') and (15b) and (15b') are used in place of the maximum entropy method.
By obtaining the spatial distribution of each short-time spectrum of the time series of the three-dimensional current density vector by (MEM), etc., in each short time, according to the measured direction component distribution data,
(14a) formula, (14b) formula, (14c) formula [or (14a ') formula, (14b')
Equation, (14c ')] any of the spatial partial differential equations or simultaneous spatial partial differential equations consisting of multiple spatial partial differential equations among them are all simultaneous, and the spatial distribution Ai (x, y, z, t ) And Bi
(x, y, z, t) is obtained, and at each interest point, the conductivity distribution σi (x, y, z, t) and the permittivity distribution εi (x, y, z, t) at that time are both If they are unknown, they also have a dielectric constant εi (x, y, z, t).
Is given, the unknown conductivity σi (x, y, z, t) is given, and the conductivity σi (x, y, z, t) is given. In, the unknown permittivity εi (x, y, z, t) can be obtained from Eqs. (16a) and (16b) using that value. That is, the time series σi (x, y, z, t) of the conductivity distribution and the time series εi (x, y, z, t) of the dielectric constant distribution can be obtained.

Even in the other independent time series i, according to the measured direction component distribution data of the three-dimensional current density vector in each short time, equations (14a), (14b), and (14c)
Equation [or (14a '), (14b'), (14c ')] any one of the spatial partial differential equations or simultaneous spatial partial differential equations consisting of a plurality of these spatial partial differential equations, Time series of conductivity spatial distribution σi (x, y, z, t) and time series of permittivity spatial distribution εi
Initial condition (17a) and (17a ') and (17a) for (x, y, z, t)
b) and (17b ′) are given, and therefore, by simultaneous equations of all equations obtained from the time series i, the spatial distribution time series Ai (x, y, z, t) and Bi (x, y, z, t), and the conductivity distribution σi (x, y, z, t) and the permittivity distribution εi (x,
When y, z, t) are both unknown, both
When i (x, y, z, t) is given, its value is used to calculate unknown conductivity σi (x, y, z, t), and conductivity σi (x, y, z , t)
At the point where is given, the unknown permittivity εi (x, y, z, t) can be obtained from the equations (16a) and (16b) by using that value. That is, the time series of conductivity distribution σi (x, y, z, t)
And the time series εi (x, y, z, t) of the permittivity distribution can be obtained.

Further, both the time series σi (x, y, z, t) of the conductivity distribution and the time series εi (x, y, z, t) of the dielectric constant, or one of them is the time series. In some cases, it is invariable with respect to i, and in each short time also according to the measured direction component distribution data, equation (14a), equation (14b), equation (14c) [or (14
a '), (14b'), (14c ')] spatial partial differential equations or simultaneous spatial partial differential equations consisting of multiple spatial partial differential equations, and time series of conductivity σi (x, y, z,
t) or σ (x, y, z, t) and the time series of permittivity εi (x, y, z, t) or ε (x, y, z, t) Initial condition (17a) or (17a) ') Expression and (1
Expression 7b) or Expression (17b ') [It may or may not depend on the time series i. ], Therefore, by simultaneous equations of all equations, the spatial distribution time series Ai (x, y, z, t) or A
(x, y, z, t) and Bi (x, y, z, t) or B (x, y, z, t) are obtained, and the conductivity σi (x, y, z, t) is obtained at each interest point. ) Or σ (x, y, z, t) and permittivity εi (x, y, z, t) or ε (x, y, z, t) are both unknown, both Rate εi (x, y, z, t) or ε (x, y,
z, t) is used to give unknown conductivity σi (x, y, z, t) or σ (x, y, z, t), and conductivity σi
If (x, y, z, t) or σ (x, y, z, t) is given, the unknown dielectric constant εi (x, y, z, t) or ε is used.
(x, y, z, t) can be obtained from equations (16a) and (16b). Further, by performing Fourier transform on the obtained time series data of conductivity and time series data of permittivity, the respective frequency dispersions can be approximately obtained.

Thus, the time series σi (x, y, z, t) of the conductivity distribution in the region of interest in the equation (14) of the spatial partial differential equation,
Σi (x, y, z) or σ (x, y, z) of conductivity distribution, εi (x, y, z, t) of time series of permittivity distribution, εi (x, of permittivity distribution y, z)
Or ε (x, y, z) is given as a measured value, or a typical value, or is measured, and sometimes a broadband current source is used outside the region of interest, or ,
The spatial distribution of frequency dispersion of these conductivity σi (x, y, z, t) and permittivity εi (x, y, z, t) is approximated by changing the frequency fi of the variable single frequency current source. May be required.

The numerical solution itself follows the above-mentioned conductivity estimation method. That is, as much as possible, a current source and an appropriate measurement system with respect to the reference region of the conductivity distribution or the dielectric constant distribution [that is, for each of the conductivity distribution (time series) and the dielectric constant distribution (time series),
In each two-dimensional plane, a system in which the reference region extends long in the direction in which the current predominantly flows] is configured, and the time series of the measured current density vector distribution is low-pass filtered in the spatial direction. Also, if necessary, a low-pass filter is applied also in the time direction, and after performing the above short-time spectrum analysis, the unknown spatial distribution Ai expressed in the spatial partial differential equation is calculated.
(x, y, z, t) or Ai (x, y, z) or A (x, y, z) and Bi (x, y, z, t)
Or Bi (x, y, z) or B (x, y, z), t) and Bi (x, y, z, t), or a finite difference approximation or a finite element approximation (3D) for the current density vector distribution ( Variational method or Galerkin method) is applied to these unknown spatial distributions Ai (x, y, z, t) [time series] or A (x, y, z, t) [time series] or Ai ( x, y, z) or A (x, y, z) and B
i (x, y, z, t) [time series] or B (x, y, z, t) [time series] or Bi (x,
y, z) or B (x, y, z) is derived, and for each of these unknown distributions, regularization using the spatial distribution itself, the gradient distribution, or the square norm of the Laplacian distribution,
In addition, when these time series are to be measured, in addition to these square norms, the two-dimensional partial differential of the first or second floor in the time direction at each time when each short-time spectrum analysis is performed is performed.
Regularization using the power norm will be applied. The time-series filtering of the current density vector distribution in the time direction may be realized in the frequency domain after performing spectrum analysis. Finally, equation (16a) and (16
From equation (b), the unknown conductivity distribution and unknown dielectric constant distribution can be obtained.

When the two-dimensional region of interest or the one-dimensional region of interest is targeted, the above-described conductivity estimation method is similarly applied. From this, when the above-mentioned three-dimensional region of interest is targeted, a plurality of two-dimensional regions of interest or a plurality of one-dimensional regions of interest are provided in the region, and when the two-dimensional region of interest is targeted, a plurality of A one-dimensional region of interest is provided, an algebraic equation for unknown distributions Ai and Bi is derived in each low-dimensional region of interest, and regularization using the above square norm is appropriately performed in the high-dimensional region of interest or each low-dimensional region of interest. May be given.

Further, the unknown three-dimensional distribution Ai (x, y, z)
And Bi (x, y, z) and the two-dimensional distribution Ai (x, y) and Bi (x, y) cannot be applied, but like the above-mentioned conductivity estimation method,
When applying the variational principle to one-dimensional conductivity distribution or one-dimensional permittivity distribution, instead of those one-dimensional distributions themselves,
It is also possible to apply the variational principle to the one-dimensional distribution of the one-dimensional differential of the one-dimensional conductivity and the one-dimensional permittivity.

The power value and the regularization parameter value for normalization, which are necessary when estimating the electric conductivity and the dielectric constant, are applied to them according to the above-mentioned electric conductivity distribution estimation method. It is determined from the spatial distribution of the time series spectrum of the current density vector. Finite element approximation
Basis function φ when performing (variational principle or Galerkin method)
Also follows the above.

Next, as a current density vector estimation method used for estimating the conductivity spatial distribution and the permittivity spatial distribution,
Technique 1 that can be used when the two-dimensional spatial distribution of the two-dimensional magnetic field vector can be measured and technique 2 that can be used when the three-dimensional or three-dimensional spatial distribution of the two-dimensional magnetic field vector can be measured Show.

Technique 1 (a technique used when the two-dimensional spatial distribution of the two-dimensional magnetic field vector can be measured)

Intersects with a three-dimensional space of interest in the three-dimensional space (I, J, K) at K coordinate K = K ', or includes a two-dimensional region of interest at K coordinate K = K'. , A two-dimensional distribution of three-dimensional current density vector J (I, J, K ') [= (Jx (I, J, K'), Jy (I, J, K '),
Jz (I, J, K ')) ^T ] or two-dimensional distribution of two-dimensional current density vector J (I, J, K') [= (Jx (I, J, K '), Jy (I, J,
K ')) ^T ] generates a three-dimensional magnetic field vector B (I, J, K ₀ ; K') [= in the plane z = z ₀ (≠ Z) [K = K ₀ (≠ K ')] (Bx
(I, J, K ₀ ; K '), By (I, J, K ₀ ; K'), Bz (I, J, K ₀ ;;
K ')) ^T ] two components Bx (I, J, K ₀ ; K') and By (I, J, K ₀ ;).
K ') two-dimensional distribution is measured, and a continuous Cartesian coordinate system (x, y,
z) the Biot-Savart law

[Equation 29] However, μ (x ', y', z '): Given permeability r: Consider the distance vector from (x', y ', Z) to (x, y, z ₀ ), = K ') three-dimensional current density vector J (I, J, K') [= (Jx (I,
J, K '), Jy (I, J, K'), Jz (I, J, K ')) ^T ] pseudo-two-dimensional current density vector distribution (Jx (I, J, K'), Jy (I , J,
K ')) ^T , or the two-dimensional current density vector distribution J (I, J,
Estimate K ') = (Jx (I, J, K'), Jy (I, J, K ')) ^T. However, it is assumed that the electric current is dominantly flowing in the plane (x, y, Z).

Regarding the measurement of the magnetic field vector in this case, by mechanically scanning the above-mentioned magnetic field vector detector, a sufficiently wide periphery in the two-dimensional region of interest of K coordinate K = K'and in its plane is obtained. Two-component Bx of three-dimensional magnetic field vector distribution B (I, J, K ₀ ; K ') which is considered to be generated by the current in the region
It is necessary to measure (I, J, K ₀ ; K ') and By (I, J, K ₀ ; K') as close as possible to the 2D region of interest surface.
(That is, make | K ₀ −K '| small as much as possible).

In order to reduce the magnetic field component generated by the electric current existing outside the two-dimensional region of interest with respect to the measured magnetic field vector data, 2 of size 1 in any one direction is set in the plane K = K ′. Assuming that the three-dimensional current density vector is spatially and uniformly distributed only within this two-dimensional region of interest, Biot-Savart's law [Eq. (18)] is applied to the discrete Cartesian system.
Discretize the current density vector distribution from (I, J, K) to (x / Δx, y / Δy, z / Δz) or approximate the finite element [current component Jx
(x, y, Z) and Jy (x, y, Z) are two-dimensional basis functions φ _J (I, J,
K ', x, y, Z) [the number of nodes is omitted], Σ _{I, J} φ _J (I, J,
K ', x, y, Z) Jx (I, J, K') and Σ _{I, J} φ _J (I, J, K ', x, y, Z) Jy (I,
J, K ') is interpolated. ] B = LJ (19) where B: two-dimensional component Bx (I, of three-dimensional magnetic field vector B (I, J, K ₀ ; K ')
J, K ₀ ; K ') and By (I, J, K ₀ ; K') vector J: Pseudo two-dimensional current density vector (Jx (I, J, K '), Jy
(I, J, K ')) ^T , or a two-dimensional current density vector J (I, J,
K ') vector L: B and J (I, J, K') consisting of a two-dimensional distribution of two components Jx (I, J, K ') and Jy (I, J, K') [discrete approximation] Or, it is calculated at the measured position (I, J, K ₀ ) from the matrix representing the lead field that correlates J (x, y, Z) [finite element approximation] 2
Measured magnetic field data Bx (I, J, J, K '; window function W (I, J, K ₀ ; K') expressed by absolute value of dimension magnetic field vector B (I, J, K ₀ ; K ')
Multiply by K ₀ ; K ') and By (I, J, K ₀ ; K').

Therefore, the magnetic field measurement data W (I, J, K ₀ ; K ') Bx (I, J, K ₀ ; K') multiplied by the window function and W (I, J, K ₀ ;) are obtained.
K ') By (I, J, K ₀ ; K') Vector B _W consisting of spatial distribution
Is used to estimate the vector J from B _W = LJ (20). Specifically, the estimation is stabilized by applying a regularization method when estimating based on the least squares method. That is, using the regularization parameters α _J1 , α _J2 , α _J3 (positive values), R _J (J) = || B _W -LJ || ² + α _J1 || J || ² + α _J2 || DJ || ² + α _J3 || D ^T DJ || ² (21) where D: current density vector (Jx (x, y, Z), Jy (x, y,
Z)) Finite difference approximation constant or finite element approximation constant of the two-dimensional gradient operator in the x and y directions regarding the two-dimensional distribution of the ^T component [Basis function φ _J (I, J, K ', x, y, Z) (1 Can be partially differentiated more than once)
From consisting Matrix DJ: (Jx (x, y , Z), Jy (x, y, Z)) approximation of a two-dimensional gradient of ^T D ^T D: current density vector (Jx (x, y, Z ), Jy ( x, y, Z)) ^T
2D Laplacian finite difference approximation constant or finite element approximation constant [Basis function φ _J (I, J, K ',
x, y, Z) (determined from two or more partial differentiations)] D ^T DJ: (Jx (x, y, Z), Jy (x, y, Z)) Approximation of two-dimensional Laplacian of ^T By minimizing with respect to the vector J, the K coordinate K =
Pseudo two-dimensional current density vector in the two-dimensional region of interest of K '
(Jx (I, J, K '), Jy (I, J, K')) ^T or two-dimensional current density vector distribution J (I, J, K ') [= (Jx (I, J, K') , Jy (I, J,
K ')) ^T ] is obtained.

In this way, the pseudo two-dimensional current density vector distribution (Jx (I, J, K K '), Jy (I, J, K')) ^T
Or two-dimensional current density vector distribution J (I, J, K ') [= (Jx (I,
J, K '), Jy (I, J, K')) ^T ] by estimating the pseudo two-dimensional current density vector (Jx (I, J, J,
K), Jy (I, J, K)) ^T and two-dimensional current density vector distribution J
(I, J, K) [= (Jx (I, J, K), Jy (I, J, K)) ^T ] is obtained.

Further, the pseudo two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K)) ^T and the two-dimensional current density vector distribution J (I, J, K) [= (Jx (I, J, K), Jy (I, J,
K)) ^T ] is a two-dimensional current density vector that is the same as in the case of obtaining the three-dimensional conductivity spatial distribution, and that all equations that are valid in the two-dimensional region of interest for each K coordinate in the three-dimensional region of interest are simultaneous. (Jx (I, J, K), Jy (I, J, K)) It is also possible to perform regularization and estimation by using a three-dimensional gradient operator D ′ regarding the three-dimensional spatial distribution of the ^T component. In this case, D'J and D' ^{T in} equation (21)
D′ J is the three-dimensional gradient and the three-dimensional Laplacian of the two-component three-dimensional spatial distribution of (Jx (I, J, K), Jy (I, J, K)) ^T , respectively.

In the two-dimensional region of interest at each K coordinate K = K ', the pseudo two-dimensional current density vector distribution (Jx (I, J, K'),
Jy (I, J, K ')) ^T time series or two-dimensional current density vector distribution J (I, J, K') = (Jx (I, J, K '), Jy (I, J, K') )) When estimating the time series of ^T , the above three-dimensional magnetic field vector B
_{(I, J, K 0;} K ') 2 component Bx of _{(I, J, K 0;} K')及By (I, J, K
All the equations that hold at each time when the time series of the two-dimensional distribution of ₀ ; K ') are measured are simultaneous, and each two-dimensional current density vector (Jx (I, J, K'), Jy (I, J, K ')) Regarding the time series of the ^T component distribution, the time series itself or the first-order partial derivative of the time series in the time direction or the square norm of the second-order partial derivative [Regularization parameters for each are α _J4 and α _J5 To be used]. In this case, the above two-dimensional gradient of the two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K')) ^T distribution of each time and the square norm of the two-dimensional Laplacian are used. It may be made regular.

Further, a pseudo two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K')) ^{T in} the three-dimensional region of interest or a two-dimensional current density vector J ( I, J, K ') = (Jx (I, J,
K '), Jy (I, J, K')) When estimating the time series of ^T, the above three-dimensional magnetic field vector _{B (I, J, K 0} ; 2 -component Bx (I, J of the K),
K ₀ ; K) and By (I, J, K ₀ ; K) are all simultaneous equations that hold at each time when the time series of the two-dimensional distribution is measured,
Two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K))
With respect time series of three-dimensional spatial distribution of ^{the T} component, the regularization parameter according to the square norm [each of first-order partial derivative and second-order partial differential in the time direction of the time series itself or chronological alpha _J4
And α _J5 ]]. In this case, the two-dimensional current density vector (Jx (I, J,
K ′), Jy (I, J, K ′)) The three-dimensional gradient of the three-dimensional spatial distribution of the ^T component or the square norm of the three-dimensional Laplacian may be used for regularization.

The regularization parameter α _J1 , α _J2 , α _J3 is adjusted to a large value such that the matrix relating to the distribution of the current density vector is sufficiently positive definite numerically at each time, or Accuracy of measured magnetic field vector distribution data
Adjusted by (SN ratio) (small when SN ratio is high, SN
It may be large when the ratio is low). According to this, for example, it may be inversely proportional to the SN power ratio. ).

The regularization parameters α _J4 and α _J5 are adjusted to large values such that the matrix relating to the distribution of the current density vector is sufficiently positive definite numerically at each time.
Alternatively, it may be adjusted by the accuracy (SN ratio) of the amount of change in the measured magnetic field vector distribution data in the time direction (small when the SN ratio is high and large when the SN ratio is low). According to this, for example, it may be inversely proportional to the SN power ratio. ).

Further, the regularization parameter α _J1 · α _J2 · α
_J3 may be realized as different for each distribution of the current density vector component at each time, and as a result, the matrix for the distribution of the current density vector may have a positive definite value by numerical analysis. Adjusted to a large value,
Or, it may be adjusted by the accuracy (SN ratio) of the measured two-component distribution data of the magnetic field vector (for the distribution of the current density vector component in the direction orthogonal to the direction of the magnetic field vector component distribution with a high SN ratio). It is small and large with respect to the distribution of the current density vector component in the direction orthogonal to the direction of the magnetic field vector component distribution with a low SN ratio. Sometimes.). According to this, the regularization parameter value of the component dependence evaluated from the SN power ratio of the two-component distribution data of the measured magnetic field vector and the regularization parameter value evaluated from the SN power ratio of the magnetic field vector distribution itself are calculated. It may be a value proportional to the product calculated after weighting the importance.

Further, the regularization parameters α _J4 and α _J5 may be realized as different for each distribution of the current density vector component at each time, and as a result, the matrix relating to the distribution of the current density vector may be obtained. Can be adjusted to a large value so that it becomes a positive definite value numerically, or it can be adjusted according to the accuracy (SN ratio) of the amount of change of the two-component distribution data of the measured magnetic field vector in the time direction. Some (for a time series of current density vector component distribution in the orthogonal direction,
It is small when the SN ratio of the amount of change in the magnetic field vector component distribution in the time direction is high, and is large when the SN ratio of the amount of change in the magnetic field vector component distribution in the time direction is low. It may be inversely proportional to the SN power ratio of the amount of change in the component distribution data in the time direction. ). According to this, the regularization parameter value that depends on the time and the direction evaluated from the SN power ratio of the change amount of the two-component distribution data of the measured magnetic field vector in the time direction and the change amount of the magnetic field vector distribution in the time direction are calculated. The regularization parameter value evaluated from the SN power ratio may be a value proportional to the product calculated by weighting the importance.

The regularization parameter α _J1 · α _J2 · α
_J3 may be realized as spatially varying at each time, and as a result, the local matrix of the current density vector at each interest point is numerically sufficiently large so as to be positive definite. The value is adjusted. In addition, it is adjusted according to the accuracy (SN ratio) of each measured magnetic field vector data (small when the SN ratio is high, large when the SN ratio is low, according to this, for example, the SN power ratio of the magnetic field vector data It may be inversely proportional) and may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector.
According to this, the regularization parameter α _J1 , α _J2 , α _J3 applied to the current density vector of each interest point is the distance between the regularization parameter value evaluated from the SN power ratio of each magnetic field vector data and each magnetic field vector. In some cases, the regularization parameter value determined by is weighted with importance and is proportional to the product calculated.

Further, the regularization parameters α _J4 and α _J5 may be realized as spatially varying at each time, and as a result, the local matrix applied to the current density vector of each interest point becomes a numerical value. It is adjusted to a large value so that it will be a positive definite value analytically. Also, it is adjusted by the accuracy (SN ratio) of the amount of change of each measured magnetic field vector data in the time direction (small when the SN ratio is high, large when the SN ratio is low,
According to this, for example, it may be inversely proportional to the SN power ratio. ), And may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector. According to this, the regularization parameter α _J4 · α _J5 applied to the current density vector of each interest point is the regularization parameter value and each magnetic field vector evaluated from the SN power ratio of the change amount of each magnetic field vector data in the time direction. The regularization parameter value determined by the distance between and may be a value proportional to the product calculated after weighting the importance.

Further, the regularization parameter α _J1 · α _J2 · α
_J3 may be realized as spatially varying at each time as described above and different for each current density vector component. As a result, the current density vector of each interest point The local matrix is adjusted to a large value so that it is positively definite in terms of numerical analysis, or the accuracy of the two-component data of each measured magnetic field vector (S
It may be adjusted by the (N ratio) (current density vector component in the direction orthogonal to the magnetic field vector component direction with a low SN ratio, which is smaller than the current density vector component in the direction orthogonal to the magnetic field vector component direction with a high SN ratio) Is larger than, for example, SN of each two-component data of the magnetic field vector according to this
It may be inversely proportional to the power ratio. ). According to this, the regularization parameter α _J1 · α _J2 · α _J3 for each current density vector component of each interest point is the component dependence evaluated from the SN power ratio of the two component data of each measured magnetic field vector. The regularization parameter value of, the regularization parameter value determined by the distance from each magnetic field vector, and the regularization parameter value evaluated from the SN power ratio of each magnetic field vector are calculated after weighting the importance. It may be a value proportional to the product.

Moreover, the regularization parameters α _J4 and α _J5 may be realized as spatially varying as described above at each time and different for each current density vector component. Therefore, the local matrix applied to the current density vector of each interest point is adjusted to a large value so as to be sufficiently positive definite numerically, or the measured two-component data of each magnetic field vector is changed in the time direction. It may be adjusted depending on the accuracy of the change amount (SN ratio). (For the time series of the current density vector component in the orthogonal direction, the change amount of the magnetic field vector component in the time direction is small when the SN ratio is high and the magnetic field is small. It is large when the SN ratio of the change amount of the vector component in the time direction is low, and accordingly, for example, it may be inversely proportional to the SN power ratio of the change amount of the two component data of the magnetic field vector in the time direction. There.). According to this,
The regularization parameters α _J4 and α _J5 for each current density vector component at each interest point depend on the time and direction evaluated from the SN power ratio of the change amount of the two-component data of each measured magnetic field vector in the time direction. The regularization parameter value to be determined, the regularization parameter value determined by the distance from each magnetic field vector, and the regularization parameter value evaluated from the SN power ratio of the change amount of each magnetic field vector in the time direction are weighted with importance. It may be a value proportional to the product calculated by performing the calculation.

Technique 2 (a technique used when the three-dimensional or three-dimensional spatial distribution of the two-dimensional magnetic field vector can be measured)

Technique 2-1: Directly targeting a 3D space of interest
And when 3D space of interest (I, J, K ') in 3D space (I, J, K')
K ') and its space are wide enough to extend to the surface of the object
As a current density vector distribution that can exist in space, a cubic
Original current density vector J (I, J, K ') [= (Jx (I, J, K'), Jy
 (I, J, K '), Jz (I, J, K'))^T] Three-dimensional spatial distribution,
Is the two-dimensional current density vector J (I, J, K ') [= (Jx (I, J,
K '), Jy (I, J, K'))^T], 3D spatial distribution
z coordinate z = z ₀(≠ Z) [K = K₀(≠ K ')] generated in three dimensions
Magnetic field vector B (I, J, K₀) [= (Bx (I, J, K₀), By (I, J,
K₀), Bz (I, J, K₀))^T3D spatial distribution of
Min Bx (I, J, K₀) And By (I, J, K₀) 3D distribution is measured
Then, the biot represented by the continuous Cartesian coordinate system (x, y, z)
Savart's law

[Equation 30] However, μ (x ', y', z '): Given permeability r: Considering the distance vector from (x', y ', z') to (x, y, z ₀ ) 3D current density Vector J (I, J, K ') [= (Jx
(I, J, K '), Jy (I, J, K'), Jz (I, J, K ')) ^T ], or
The pseudo two-dimensional current density vector (Jx (I, J, K '), Jy (I,
J, K ')) ^T , or two-dimensional current density vector J (I, J, K')
Estimate the three-dimensional spatial distribution of [= (Jx (I, J, K '), Jy (I, J, K')) ^T ]. Basically, it is assumed that the current is dominant in the plane (x, y, Z).

Regarding the measurement of the magnetic field vector in this case, it is generated by the current in the three-dimensional space of interest and in a sufficiently wide space including the space by mechanically scanning the magnetic field vector detector described above. Three-dimensional magnetic field vector B (I, J, K ₀ ) [= (Bx (I, J, K ₀ ), By (I, J,
Three-dimensional spatial distribution of K ₀ ), Bz (I, J, K ₀ )) ^T ], or 2
It is necessary to measure the three-dimensional spatial distribution of the components Bx (I, J, K ₀ ) and By (I, J, K ₀ ) at the shortest distance to the three-dimensional space of interest as much as possible (ie, Minimize | K ₀ −K '| as much as possible.)

With respect to the measured magnetic field vector data, in order to reduce the magnetic field component generated by the electric current existing outside the three-dimensional space of interest, the magnitude 1 in any one direction is reduced.
Assuming that the three-dimensional current density vector of is uniformly distributed only in the three-dimensional space of interest, Biot-Savart's law [Equation (22)] is applied to the discrete Cartesian coordinate system (I, J, K) ~ (x /
Δx, y / Δy, z / Δz) Discretization or finite element approximation about the current density vector [current component Jx (x, y, Z), Jy (x,
y, Z) and Jz (x, y, Z) are three-dimensional basis functions φ _J (I, J, K ',
x, y, Z) [Number of nodes is omitted], Σ _{I, J, K '} φ _J (I, J,
K ', x, y, Z) Jx (I, J, K'), Σ _{I, J, K '} φ _J (I, J, K', x, y, Z) Jy
(I, J, K ') and Σ _{I, J, K'} φ _J (I, J, K ', x, y, Z) Jz (I, J,
K ') is interpolated. ] B = LJ (23) where B: three-dimensional magnetic field vector B (I, J, K ₀ ) Bx (I, J,
K ₀ ), By (I, J, K ₀ ), and Bz (I, J, K ₀ ), or the two-component Bx (I, J, K ₀ ) of the three-dimensional magnetic field vector B (I, J, K ₀ ). ) And By
Vector J consisting of the three-dimensional spatial distribution of (I, J, K ₀ ): Three-dimensional current density vector J (I, J, K ') with three components Jx
A vector consisting of the three-dimensional spatial distribution of (I, J, K '), Jy (I, J, K') and Jz (I, J, K '), or its pseudo two-dimensional current density vector (Jx (I , J, K '), Jy (I, J, K')) ^T or two-dimensional component Jx (I, J, K ') of the two-dimensional current density vector J (I, J, K')
Vector L: B and J (I, J, K ') [discrete approximation] or J (x, y, Z) [finite element] consisting of three-dimensional spatial distribution of K') and Jy (I, J, K ') The window representing the absolute value of the magnetic field vector B (I, J, K ₀ ) calculated at the measured position (I, J, K ₀ ) from the matrix representing the lead field Function W (I,
J, K ₀₎ to measure the magnetic field data _{Bx (I, J, K 0} ), By (I, J,
Multiply by K ₀ ), Bz (I, J, K ₀ ).

Therefore, the three magnetic field measurements multiplied by the window function
Constant data W (I, J, K₀) Bx (I, J, K₀), W (I, J, K₀) By
(I, J, K₀) And W (I, J, K₀) Bz (I, J, K₀), Or 2
Magnetic field measurement data W (I, J, K₀) Bx (I, J, K₀) And W (I,
 J, K₀) By (I, J, K₀) Vector B consisting of the spatial distribution of_W
Using, B_W = L J (24) Estimate the vector J from. Specifically, the least squares method
By applying the regularization method when estimating based on
Stabilize the estimation. That is, the regularization parameter
α_J1, Α_J2, Α_J3(Positive value) R_J(J) = || B_W-LJ ||²+ α_J1|| J ||²+ α_J2|| D'J ||²+ α_J3|| D'D'J ||²  (twenty five) However, D ': Vector J is three-dimensional current density vector J (x, y, Z) 3 components of Jx (x, y, Z), Jy (x, y, Z) and Jz (x, y, Z)
If the three-dimensional spatial distribution (I, J, K ') of
Finite difference of 3D gradient operator in x, y, z directions with respect to minutes
Approximation constant or finite element approximation constant [Basis function φ_J(I, J, K ', x,
y, Z) (which can be partially differentiated more than once)]
Vector J is the three-dimensional current density vector J (x, y, Z)
Pseudo two-dimensional current density vector (Jx (x, y, Z), Jy (x, y,
 Z))^TOr the two-component Jx of the two-dimensional current density vector J (x, y, Z)
Three-dimensional spatial distribution (I, J, K ') of (x, y, Z) and Jy (x, y, Z)
In the x, y, z directions for these two components
Finite difference approximation constant or finite element approximation of three-dimensional gradient operator
Constant [basis function φ_J(I, J, K ', x, y, Z) (Partial differentiation is possible more than once)
Matrix obtained from D '^TD ': Vector J is three-dimensional current density vector J (x, y,
Z) three components Jx (x, y, Z), Jy (x, y, Z) and Jz (x, y, Z)
If the three-dimensional spatial distribution (I, J, K ') of
Finite-Difference Approximation of 3-D Laplacian Operators for Minutes
Number or finite element approximation constant [basis function φ_J(I, J, K ', x, y, Z)
Matrix obtained from (can be partially differentiated more than once), or
Vector J is a pseudo of three-dimensional current density vector J (x, y, Z)
Two-dimensional current density vector (Jx (x, y, Z), Jy (x, y, Z))
^TOr two-dimensional current density vector J (x, y, Z) two component Jx (x,
y, Z) and Jy (x, y, Z) three-dimensional spatial distribution (I, J, K ')
3D Laplacian for these two components
Finite difference approximation constant of operator or finite element approximation constant [base
Function φ_JCalculated from (I, J, K ', x, y, Z) (partially differentiable more than once)
To minimize the matrix of
From the three-dimensional space of interest, the three-dimensional current density vector J
(I, J, K ') [= (Jx (I, J, K'), Jy (I, J, K '), Jz (I,
J, K '))^T] Or its pseudo two-dimensional current density vector (Jx
 (I, J, K '), Jy (I, J, K'))^T, Or two-dimensional current density
Cutle J (I, J, K ') [= (Jx (I, J, K'), Jy (I, J,
K ')) ^T] The three-dimensional spatial distribution of

Further, the three-dimensional current density vector J (I, J, K ') [= (Jx (I, J, K'), Jy (I, J,
K '), Jz (I, J, K')) ^T ] time series or its pseudo two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K')) Time series of ^T or two-dimensional current density vector J (I, J, K ')
When estimating the time series of [= (Jx (I, J, K '), Jy (I, J, K')) ^T ], the 3D magnetic field vector B (I, J, K ₀ ) of Three components Bx (I, J, K ₀ ), By (I, J, K ₀ ), and Bz (I, J,
K ₀ ), or its two components Bx (I, J, K ₀ ), and By (I, J, K ₀ ).
All the equations that hold at each time when the time series of the three-dimensional spatial distribution of are measured are simultaneous, and each three-dimensional current density vector (Jx (I, J, K), Jy (I, J, K), Jz ( I, J, K)) ^T component or each two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K))
With respect time series of three-dimensional spatial distribution of ^{the T} component, the regularization parameter according to the square norm [each of first-order partial derivative and second-order partial differential in the time direction of the time series itself or chronological alpha _J4
And α _J5 ]]. In this case, three-dimensional current density vector (Jx (I, J, K),
Three-dimensional of Jy (I, J, K), Jz (I, J, K)) ^T component and two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K)) ^T component The three-dimensional gradient of the spatial distribution or the square norm of the three-dimensional Laplacian may be used for regularization.

The regularization parameter α _J1 , α _J2 , α _J3 is adjusted to a large value such that the matrix relating to the spatial distribution of the current density vector is sufficiently positive definite numerically at each time, or , It may be adjusted according to the accuracy (SN ratio) of the measured magnetic field vector spatial distribution data (small when the SN ratio is high, large when the SN ratio is low, according to this, for example, the SN power ratio May be inversely proportional to.).

The regularization parameters α _J4 and α _J5 are adjusted to a large value or measured at each time so that the matrix relating to the spatial distribution of the current density vector is sufficiently positive definite numerically. The magnetic field vector space distribution may be adjusted by the accuracy of the amount of change in the time direction (SN ratio) (small when the SN ratio is high, large when the SN ratio is low, and according to this, for example, It may be inversely proportional to the SN power ratio.).

Further, the regularization parameter α _J1 · α _J2 · α
_J3 may be realized as different for each spatial distribution of the current density vector component at each time, and as a result, the matrix relating to the spatial distribution of the current density vector has a numerically sufficiently positive definite value. It may be adjusted to a large value, or may be adjusted depending on the accuracy (SN ratio) of the measured direction component spatial distribution data of the magnetic field vector (direction of the magnetic field vector component spatial distribution with a high SN ratio. It is small for the current density vector component spatial distribution in the orthogonal direction and large for the current density vector component spatial distribution in the direction orthogonal to the direction of the magnetic field vector component spatial distribution with a low SN ratio. Each of the above 3 in vector
It may be inversely proportional to the SN power ratio of the component or each two component spatial distribution data. Therefore, when the three-dimensional magnetic field vector data is measured, the two-dimensional magnetic field vector data is measured as the product of the values determined from the SN ratios of the two magnetic field vector component spatial distributions for the three current density vector component spatial distributions. In case of current density vector, the x and y component spatial distributions are determined by the SN ratio of one magnetic field vector component spatial distribution, and the z component spatial distribution is determined by the SN ratio of two magnetic field vector component spatial distributions. The product of the values
It will be set to a proportional value. ). According to this, the regularization parameter value of the component dependence evaluated from the SN power ratio of the above three component or two component spatial distribution data of the measured magnetic field vector and the magnetic field vector spatial distribution itself
The regularization parameter value evaluated from the SN power ratio may be a value proportional to the product calculated by weighting the importance.

Further, the regularization parameters α _J4 and α _J5 may be realized as being different for each spatial distribution of the current density vector component at each time, and as a result, the spatial distribution of the current density vector may be changed. The matrix is adjusted to a large value so that it is positively definite numerically.
Or, it may be adjusted by the accuracy of the amount of change in the time direction of the spatial distribution data of each direction component of the measured magnetic field vector (SN ratio) (for the time series of the current density vector component distribution in the orthogonal direction, It is small when the SN ratio of the amount of change in the magnetic field vector component distribution in the time direction is high, and large when the SN ratio of the amount of change in the magnetic field vector component distribution in the time direction is low. It may be inversely proportional to the SN power ratio of the amount of change in the time direction of the three component or each two component spatial distribution data.). In accordance with this, the regularization parameter value and the magnetic field vector distribution depending on the time and direction evaluated from the SN power ratio of the amount of change in the time direction of the spatial distribution data of the above-mentioned three or two components of the measured magnetic field vector The regularization parameter value evaluated from the SN power ratio of the amount of change in the time direction may be a value proportional to the product calculated after weighting the importance.

The regularization parameter α _J1 · α _J2 · α
_J3 may be realized as spatially varying at each time, and as a result, the local matrix of the current density vector at each interest point is numerically sufficiently large so as to be positive definite. The value is adjusted. In addition, it is adjusted according to the accuracy (SN ratio) of each measured magnetic field vector data (small when the SN ratio is high, large when the SN ratio is low, according to this, for example, the SN power ratio of the magnetic field vector data It may be inversely proportional) and may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector.
According to this, the regularization parameter α _J1 , α _J2 , α _J3 applied to the current density vector of each interest point is the distance between the regularization parameter value evaluated from the SN power ratio of each magnetic field vector data and each magnetic field vector. In some cases, the regularization parameter value determined by is weighted with importance and is proportional to the product calculated.

Further, the regularization parameters α _J4 and α _J5 may be realized as spatially varying at each time, and as a result, the local matrix applied to the current density vector of each interest point becomes a numerical value. It is adjusted to a large value so that it will be a positive definite value analytically. Also, it is adjusted by the accuracy (SN ratio) of the amount of change of each measured magnetic field vector data in the time direction (small when the SN ratio is high, large when the SN ratio is low,
According to this, for example, it may be inversely proportional to the SN power ratio. ), And may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector. According to this, the regularization parameters α _J4 and α _J5 applied to the current density vector of each interest point are the regularization parameter value and each magnetic field vector evaluated from the SN power ratio of the change amount data of each magnetic field vector in the time direction. The regularization parameter value determined by the distance between and may be a value proportional to the product calculated after weighting the importance.

Further, the regularization parameter α _J1 · α _J2 · α
_J3 may be realized as spatially varying as described above at each time and different for each current density vector component, and as a result, the current density vector of each interest point The local matrix may be adjusted to a large value so as to be sufficiently positive definite numerically, or may be adjusted according to the accuracy (SN ratio) of each direction component data of each measured magnetic field vector (SN ratio). The ratio is small for the current density vector component in the direction orthogonal to the direction of the magnetic field vector component, and the SN ratio is large for the current density vector component in the direction orthogonal to the direction of the magnetic field vector component, and in accordance with this, for example, , It may be inversely proportional to the SN power ratio of each of the above-mentioned three-component or each two-component data of the magnetic field vector.Thus, when three-dimensional magnetic field vector data is measured, three current densities For the vector component, the product of the values determined from the SN ratio of the two magnetic field vector components is used to calculate the x and y components of the current density vector when the two-dimensional magnetic field vector data is measured.
The value determined from the SN ratio and the z component are set to a value proportional to the product of the values determined from the SN ratios of the two magnetic field vector components. ). According to this, the regularization parameters α _J1 , α _J2 , α _J3 for each current density vector component of each interest point are evaluated from the SN power ratio of the measured three-component or two-component data of each magnetic field vector. The importance of the component-dependent regularization parameter value, the regularization parameter value determined by the distance to each magnetic field vector, and the regularization parameter value evaluated from the SN power ratio of each magnetic field vector were weighted. It may be a value proportional to the product calculated above.

The regularization parameters α _J4 and α _J5 may be realized as spatially varying at each time as described above and different for each current density vector component. Therefore, the local matrix applied to the current density vector of each interest point is adjusted to a large value so that it is sufficiently positive definite numerically, or the time direction of each direction component data of each measured magnetic field vector is adjusted. May be adjusted by the accuracy of the amount of change (SN ratio) (to the time series of the current density vector component in the orthogonal direction, the SN ratio of the amount of change in the time direction of the magnetic field vector component is small, Of the amount of change in the time direction of the magnetic field vector component
When the SN ratio is low, it is large, and in accordance therewith, for example, it may be inversely proportional to the SN power ratio of the amount of change in the time direction of each of the three component data or each two component data of the magnetic field vector. ). According to this, the regularization parameter α _J4 · α _J5 for each current density vector component of each interest point is calculated from the SN power ratio of the change amount of the measured three-component or two-component data of each magnetic field vector in the time direction. The regularization parameter value that depends on the time and direction to be evaluated, the regularization parameter value that is determined by the distance to each magnetic field vector, and the regularization parameter value that is evaluated from the SN power ratio of the change amount of each magnetic field vector in the time direction And may be values proportional to the product calculated after weighting the importance.

Technique 2-2: Directly target two-dimensional region of interest
And when 3D space of interest and K constellation in 3D space (I, J, K ')
Cross at mark K = K ', or K coordinate in 2D region of interest
Current density vector that can exist in the plane including K = K '
As the distribution, the two-dimensional distribution J (I,
J, K ') [= Jx (I, J, K'), Jy (I, J, K '), Jz (I, J,
K ')]^T], Or the two-dimensional distribution J of the two-dimensional current density vector
(I, J, K ') [= (Jx (I, J, K'), Jy (I, J, K '))^T]
, Plane z = z ₀(≠ Z) [K = K₀3 generated in (≠ K ')]
Dimensional magnetic field vector B (I, J, K₀; K ') [= (Bx (I, J, K₀;
K '), By (I, J, K₀; K '), Bz (I, J, K₀; K '))^T] Third
Original spatial distribution or its two components Bx (I, J, K₀; K ') and By (I,
J, K₀K ') 3D distribution is measured and continuous Cartesian coordinates
Biot-Savart law represented by the system (x, y, z) [Equation (18)]
In the two-dimensional region of interest in this plane (K = K ')
3D current density vector J (I, J, K ') [= (Jx (I, J,
K '), Jy (I, J, K'), Jz (I, J, K '))^T], Or its pseudo
Similar two-dimensional current density vector (Jx (I, J, K '), Jy (I, J,
K '))^T, Or two-dimensional current density vector J (I, J, K ') =
(Jx (I, J, K '), Jy (I, J, K'))^TThe two-dimensional distribution of
To do. However, the current flows dominantly in the plane (x, y, Z).
It is assumed that

Regarding the measurement of the magnetic field vector in this case, by mechanically scanning the above-mentioned magnetic field vector detector, etc., in the two-dimensional region of interest of K coordinate K = K'and in the peripheral region in its plane. Three-dimensional magnetic field vector distribution B (I, J, K ₀ ; K ') [= (Bx (I, J, K ₀ ;;
K '), By (I, J, K ₀ ; K')) ^T ] three-dimensional spatial distribution, or
It is necessary to measure the 3-dimensional spatial distribution of the two components Bx (I, J, K ₀ ) and By (I, J, K ₀ ) at the shortest distance position to the 2-dimensional region of interest as much as possible ( That is, make | K ₀ −K '| as small as possible.)

In order to reduce the magnetic field component generated by the electric current existing outside the two-dimensional region of interest with respect to the measured magnetic field vector data, 2 of size 1 in any one direction is set in the plane K = K '. Assuming that the three-dimensional current density vector is spatially and uniformly distributed only within this two-dimensional region of interest, Biot-Savart's law [Eq. (18)] is applied to the discrete Cartesian system.
Discretization or finite element approximation of the current density vector from (I, J, K) to (x / Δx, y / Δy, z / Δz) [current component Jx (x, y,
Z), Jy (x, y, Z), and Jz (x, y, Z) are two-dimensional basis functions φ
_{Using J} (I, J, K ', x, y, Z) [the number of nodes is omitted], Σ _{I, J} φ
_J (I, J, K ', x, y, Z) Jx (I, J, K'), Σ _{I, J} φ _J (I, J, K ', x, y, Z) J
y (I, J, K ') and Σ _{I, J} φ _J (I, J, K', x, y, Z) Jz (I, J, K ')
Is interpolated. ] B = LJ (26) where B: three-dimensional magnetic field vector B (I, J, K ₀ ; K ') Bx (I,
J, K ₀ ; K '), By (I, J, K ₀ ; K') and Bz (I, J, K ₀ ;).
K '), or the two components of the three-dimensional magnetic field vector B (I, J, K ₀ ; K') Bx (I, J, K ₀ ; K ') and By (I, J, K ₀ ; K') , A vector J consisting of the three-dimensional spatial distribution of: the three-component Jx of the three-dimensional current density vector J (I, J, K ')
(I, J, K '), Jy (I, J, K'), Jz (I, J, K ') two-dimensional distribution vector or its pseudo two-dimensional current density vector (Jx (I, K') J, K '), Jy (I, J, K')) ^T or two-component Jx (I, J, K ') and Jy of the two-dimensional current density vector J (I, J, K')
Associates the vector L: B and J (I, J, K ') [discrete approximation] or J (x, y, Z) [finite element approximation] of two-dimensional distribution of (I, J, K') Window function represented by the absolute value of magnetic field vector B (I, J, K ₀ ; K ') calculated at the measured position (I, J, K ₀ ) from the matrix representing the lead field
W (I, J, K ₀ ; K ') measured magnetic field data Bx (I, J, K ₀ ;
K '), By (I, J, K ₀ ; K') and Bz (I, J, K ₀ ; K ').

Therefore, the three magnetic field measurement data W (I, J, K ₀ ) Bx (I, J, K ₀ ; K '), W (I, J, K ₀ ) multiplied by the window function
_{By (I, J, K 0} ; K ') and _{W (I, J, K 0} ) Bz (I, J, K 0; K'),
Or, two magnetic field measurement data W (I, J, K ₀ ) Bx (I, J,
Using the vector B _W consisting of the spatial distribution of K ₀ ; K ') and W (I, J, K ₀ ) By (I, J, K ₀ ; K'), we obtain a vector from B _W = LJ (27) Estimate J. Specifically, the estimation is stabilized by applying a regularization method when estimating based on the least squares method. That is, using the regularization parameters α _J1 , α _J2 , α _J3 (positive value), R _J (J) = || B _W -LJ || ² + α _J1 || J || ² + α _J2 || DJ || ² + α _J3 || D ^T DJ || ² (28) where D: vector J is the three-dimensional current density vector J (x,
y, Z) three components Jx (x, y, Z), Jy (x, y, Z) and Jz (x,
y, Z) consisting of a two-dimensional distribution (I, J, K '), a finite difference approximation constant or a finite element approximation constant of the two-dimensional gradient operator in the x and y directions for these three components [base function φ _J (I , J, K ', x, y, Z)
Matrix obtained from (can be partially differentiated more than once)], or
If the vector J is a three-dimensional current density vector J (I, J, K '), a pseudo two-dimensional current density vector (Jx (x, y, Z), Jy (x, y,
Z)) ^T or two component Jx of two-dimensional current density vector J (x, y, Z)
Two-dimensional distribution (I, J, K ') of (x, y, Z) and Jy (x, y, Z)
Finite difference approximation constant or finite element approximation constant of the two-dimensional gradient operator in the x and y directions for these two components
The matrix D ^T D consisting of [basic function φ _J (I, J, K ', x, y, Z) (which can be partially differentiated more than once)] is a three-dimensional current density vector J (x, y , Z) three components Jx (x, y, Z), Jy (x, y,
Z) and Jz (x, y, Z) two-dimensional distribution (I, J, K '),
A finite difference approximation constant or a finite element approximation constant of a two-dimensional Laplacian operator for these three components [basis function φ _J (I, J,
K ', x, y, Z) (determined from two or more partial differentiations)], or the vector J is a three-dimensional current density vector J (I, J,
K ') pseudo two-dimensional current density vector (Jx (x, y, Z), Jy
(x, y, Z)) ^T or 2 of the two-dimensional current density vector J (x, y, Z)
Two-dimensional distribution of the components Jx (x, y, Z) and Jy (x, y, Z) (I,
J, K '), a finite difference approximation constant or a finite element approximation constant of a two-dimensional Laplacian operator for these two components [basic function φ _J (I, J, K', x, y, Z) (more than twice ## EQU1 ##] is minimized with respect to the vector J, so that 3 in the 2D region of interest with K coordinate K = K '
Dimensional current density vector J (I, J, K ') [= (Jx (I, J, K'),
Jy (I, J, K '), Jz (I, J, K')) ^T ] or its pseudo two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K) ')) ^T , or two-dimensional current density vector J (I, J, K') [= (Jx (I,
A two-dimensional distribution of J, K '), Jy (I, J, K')) ^T ] is obtained.

As in the technique 1, the three-dimensional space of interest
Within the two-dimensional region of interest at each K coordinate K = K'in (I, J, K)
And the three-dimensional current density vector J (I, J, K ') [= (Jx (I, J,
 K '), Jy (I, J, K'), Jz (I, J, K '))^T], Or its pseudo
Similar two-dimensional current density vector (Jx (I, J, K '), Jy (I, J,
K '))^T, Or two-dimensional current density vector J (I, J, K ') [=
 (Jx (I, J, K '), Jy (I, J, K'))^T] By estimating
The three-dimensional current density vector J (I,
J, K) [= (Jx (I, J, K), Jy (I, J, K), Jz (I, J,
K)) ^T] Or its pseudo two-dimensional current density vector (Jx (I,
J, K), Jy (I, J, K))^T, Or two-dimensional current density vector
J (I, J, K) [= (Jx (I, J, K), Jy (I, J, K))^TEstimate
You can also do it.

Also, the three-dimensional current density vector J (I, J, K) [= (Jx (I, J, K), Jy (I, J, K), J in the three-dimensional space of interest)
z (I, J, K)) ^T ] or its pseudo two-dimensional current density vector
(Jx (I, J, K), Jy (I, J, K)) ^T and two-dimensional current density vector distribution J (I, J, K) [= (Jx (I, J, K), Jy (I , J, K)) ^T ] is
All of the equations that hold in the two-dimensional region of interest at each K coordinate in the three-dimensional region of interest are combined, and the three-dimensional current density vector J (I, J, K) [= (Jx (I, J, K ), Jy (I, J, K), Jz
(I, J, K)) ^T ], or its pseudo two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K)) ^T and two-dimensional current density vector J (I, J, K) [= (Jx (I, J, K), Jy (I, J, K))
Three-dimensional gradient operator for two-dimensional three-dimensional spatial distribution of ^T ]
It is also possible to regularize and estimate using D '[(28)
formula].

In the two-dimensional region of interest at each K coordinate K = K ', the three-dimensional current density vector distribution J (I, J, K') [= (Jx
(I, J, K '), Jy (I, J, K'), Jz (I, J, K ')) ^T ] time series, or its pseudo two-dimensional current density vector distribution (Jx
(I, J, K '), Jy (I, J, K')) ^T time series or two-dimensional current density vector distribution J (I, J, K ') = (Jx (I, J, K''), J
y (I, J, K ' )) When estimating the time series of ^T, the above three-dimensional magnetic field vector _{B (I, J, K 0} ; K') of the three components _{Bx (I, J, K 0} ;
K '), By (I, J, K ₀ ; K') and Bz (I, J, K ₀ ; K '), or
The two components Bx (I, J, K ₀ ; K ') and By (I, J, K ₀ ; K')
All three-dimensional current density vectors (J
x (I, J, K '), Jy (I, J, K'), Jz (I, J, K ')) ^T component or each two-dimensional current density vector (Jx (I, J, K'), Jy (I, J,
K ')) Regarding the time series of the ^T component distribution, the time series itself or the first-order partial derivative or the square norm of the second-order partial derivative in the time direction of the time series [Regularization parameters for each are α _J4 and α _J5 To be used]. in this case,
Three-dimensional current density vector (Jx (I, J, K '), Jy at each time
(I, J, K '), Jz (I, J, K')) ^T component or two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K')) ^T component The above two-dimensional gradient of the distribution or the square norm of the two-dimensional Laplacian may be used for regularization.

In addition, the three-dimensional current density vector J (I, J, K ') [= (Jx (I, J, K'), Jy (I, J, in the three-dimensional region of interest)
K '), Jz (I, J, K')) ^T ] time series or its pseudo two-dimensional current density vector (Jx (I, J, K '), Jy (I, J, K')) Time series of ^T or two-dimensional current density vector J (I, J, K ')
When estimating the time series of [= (Jx (I, J, K '), Jy (I, J, K')) ^T ], the 3D magnetic field vector B (I, J, K ₀ ) of Three components Bx (I, J, K ₀ ), By (I, J, K ₀ ), and Bz (I, J,
K ₀ ), or its two components Bx (I, J, K ₀ ), and By (I, J, K ₀ ).
All the equations that hold at each time when the time series of the three-dimensional spatial distribution of are measured are simultaneous, and each three-dimensional current density vector (Jx (I, J, K), Jy (I, J, K), Jz ( I, J, K)) ^T component or each two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K))
With respect time series of three-dimensional spatial distribution of ^{the T} component, the regularization parameter according to the square norm [each of first-order partial derivative and second-order partial differential in the time direction of the time series itself or chronological alpha _J4
And α _J5 ]]. In this case, three-dimensional current density vector (Jx (I, J, K),
Three-dimensional of Jy (I, J, K), Jz (I, J, K)) ^T component and two-dimensional current density vector (Jx (I, J, K), Jy (I, J, K)) ^T component The three-dimensional gradient of the spatial distribution or the square norm of the three-dimensional Laplacian may be used for regularization.

The regularization parameter α _J1 , α _J2 , α _J3 is adjusted to a large value so that the matrix relating to the distribution of the current density vector is sufficiently positive definite numerically at each time, or It may be adjusted by the accuracy of the measured magnetic field vector spatial distribution data (SN ratio) (small when the SN ratio is high, large when the SN ratio is low, according to this,
For example, it may be inversely proportional to the SN power ratio. ).

The regularization parameters α _J4 and α _J5 are adjusted to a large value so that the matrix relating to the distribution of the current density vector is sufficiently positive definite numerically at each time.
Alternatively, it may be adjusted according to the accuracy (SN ratio) of the variation data of the measured magnetic field vector space distribution in the time direction (S
When the N ratio is high, it is small, and when the SN ratio is low, it is large, and accordingly, for example, it may be inversely proportional to the SN power ratio. ).

Further, the regularization parameters α _J1 · α _J2 · α _J3
May be realized as different for each current density vector component distribution at each time, and as a result, a large value is obtained so that the matrix related to the current density vector distribution becomes sufficiently positive definite numerically. Adjusted to, or
It may be adjusted by the accuracy (SN ratio) of each direction component spatial distribution data of the measured magnetic field vector (for the current density vector component distribution in the direction orthogonal to the direction of the magnetic field vector component spatial distribution with a high SN ratio) It is small and large with respect to the current density vector component distribution in the direction orthogonal to the direction of the magnetic field vector component spatial distribution with a low SN ratio, and in accordance with this, for example, the above three components of the magnetic field vector or each two component spatial distribution data It may be inversely proportional to the SN power ratio, so when three-dimensional magnetic field vector data is measured, the three current density vector component distributions are the product of the values determined from the SN ratios of the two magnetic field vector component spatial distributions. When two-dimensional magnetic field vector data is measured, the x and y component distributions of the current density vector are determined from the SN ratio of one magnetic field vector component spatial distribution. To the value, the product value with respect to the z-component distribution determined from the SN ratio of the two magnetic field vector components spatial distribution, will be set to a value proportional.). According to this, it is evaluated from the component-dependent regularization parameter value evaluated from the SN power ratio of the measured three-component or two-component spatial distribution data of the magnetic field vector and the SN power ratio of the magnetic field vector spatial distribution itself. The regularization parameter value may be a value proportional to the product calculated after weighting the importance.

The regularization parameters α _J4 , α _J5 , α _J6
May be realized as different for each current density vector component distribution at each time, and as a result, a large value is obtained so that the matrix related to the current density vector distribution becomes sufficiently positive definite numerically. Adjusted to, or
It may be adjusted depending on the accuracy (SN ratio) of the change amount in the time direction of the spatial distribution data of each direction component of the measured magnetic field vector (for the time series of the current density vector component distribution in the orthogonal direction, the magnetic field vector It is small when the SN ratio of the variation of the component distribution in the time direction is high, and is large when the SN ratio of the variation of the magnetic field vector component in the time direction is low. Or SN of the amount of change in the time direction of each two-component spatial distribution data
It may be inversely proportional to the power ratio. ). In accordance with this, the regularization parameter value and the magnetic field vector distribution depending on the time and direction evaluated from the SN power ratio of the amount of change in the time direction of the spatial distribution data of the above-mentioned three or two components of the measured magnetic field vector The regularization parameter value evaluated from the SN power ratio of the amount of change in the time direction may be a value proportional to the product calculated after weighting the importance.

Further, the regularization parameters α _J1 · α _J2 · α _J3
May be realized as spatially varying at each time, and as a result, a large value is obtained so that the local matrix applied to the current density vector of each interest point is sufficiently positive definite numerically. Is adjusted to. In addition, it is adjusted according to the accuracy (SN ratio) of each measured magnetic field vector data (small when the SN ratio is high, large when the SN ratio is low, according to this, for example, the SN power ratio of the magnetic field vector data It may be inversely proportional) and may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector. According to this, the regularization parameter α _J1 , α _J2 , α _J3 applied to the current density vector of each interest point is the distance between the regularization parameter value evaluated from the SN power ratio of each magnetic field vector data and each magnetic field vector. In some cases, the regularization parameter value determined by is weighted with importance and is proportional to the product calculated.

Further, the regularization parameters α _J4 and α _J5 may be realized as spatially varying at each time, and as a result, the local matrix applied to the current density vector of each interest point becomes a numerical value. It is adjusted to a large value so that it will be a positive definite value analytically. Also, it is adjusted according to the accuracy (SN ratio) of the amount of change of each measured magnetic field vector data in the time direction (small when the SN ratio is high and large when the SN ratio is low, and according to this, for example, It may be inversely proportional to the power ratio) and may be proportional to the fourth power of the distance between the current density vector and each magnetic field vector. According to this, the regularization parameter α _J4 · α _J5 applied to the current density vector of each interest point is the regularization parameter value and each magnetic field vector evaluated from the SN power ratio of the change amount of each magnetic field vector data in the time direction. The regularization parameter value determined by the distance between and may be a value proportional to the product calculated after weighting the importance.

Also, the regularization parameters α _J1 · α _J2 · α _J3
May be realized as spatially varying at each time as described above and different for each of the current density vector components, and as a result, the local of the current density vector of each interest point The matrix is adjusted to a large value so that it is numerically sufficiently positive definite, or
It may be adjusted depending on the accuracy (SN ratio) of each direction component data of each measured magnetic field vector (SNR is small for the current density vector component in the direction orthogonal to the direction of the high magnetic field vector component, and the SN ratio is It is large with respect to the current density vector component in the direction orthogonal to the direction of the low magnetic field vector component, and accordingly, for example, it may be inversely proportional to the SN power ratio of each of the above three component data or each two component data of the magnetic field vector. Therefore, when three-dimensional magnetic field vector data is measured, for three current density vector components, the product of the values determined from the SN ratios of two magnetic field vector components is used to calculate the current when two-dimensional magnetic field vector data is measured. Regarding the x and y components of the density vector, one magnetic field vector component
The value determined from the SN ratio and the z component are set to a value proportional to the product of the values determined from the SN ratios of the two magnetic field vector components. ). According to this, the regularization parameters α _J1 , α _J2 , α _J3 for each current density vector component of each interest point are evaluated from the SN power ratio of the measured three-component or two-component data of each magnetic field vector. The importance of the component-dependent regularization parameter value, the regularization parameter value determined by the distance to each magnetic field vector, and the regularization parameter value evaluated from the SN power ratio of each magnetic field vector were weighted. It may be a value proportional to the product calculated above.

The regularization parameters α _J4 and α _J5 may be realized as spatially varying at each time as described above and different for each current density vector component. Therefore, the local matrix applied to the current density vector of each interest point is adjusted to a large value so that it is sufficiently positive definite numerically, or the time direction of each direction component data of each measured magnetic field vector is adjusted. May be adjusted by the accuracy of the amount of change (SN ratio) (to the time series of the current density vector component in the orthogonal direction, the SN ratio of the amount of change in the time direction of the magnetic field vector component is small, Of the amount of change in the time direction of the magnetic field vector component
When the SN ratio is low, it is large, and in accordance therewith, for example, it may be inversely proportional to the SN power ratio of the amount of change in the time direction of each of the three component data or each two component data of the magnetic field vector. ). According to this, the regularization parameter α _J4 · α _J5 for each current density vector component of each interest point is calculated from the SN power ratio of the change amount of the measured three-component or two-component data of each magnetic field vector in the time direction. The regularization parameter value that depends on the time and direction to be evaluated, the regularization parameter value that is determined by the distance to each magnetic field vector, and the regularization parameter value that is evaluated from the SN power ratio of the change amount of each magnetic field vector in the time direction And may be values proportional to the product calculated after weighting the importance.

When the one-direction component of the magnetic field vector can be measured, the Biot-Savart law [(18), (2
2)], the tangential magnetic field component orthogonal to the direction in which the current predominantly flows is to be measured, and if it is possible to measure the three-dimensional distribution of the tangential magnetic field component, then refer to Technique 2. If it is possible to estimate the three-dimensional distribution, the two-dimensional distribution, or the one-dimensional distribution of the current, and if the measurement of the two-dimensional distribution of the tangential magnetic field component is possible, the If the dimensional distribution or the one-dimensional distribution can be estimated, or if the one-dimensional distribution of the magnetic field component can be measured, the one-dimensional distribution of the current can be similarly estimated.

Measurement of multiple magnetic field vector distributions or time series
, Then the regularization parameter α _J1・ Α_J2・ Α_J3・ Α
_J4・ Α_J5Is the magnetic field of each state (time) as described above.
Regularization parameters evaluated in vector distribution data
The value is proportional to the sum of the data values.

As a measurement result, in addition to the measured magnetic vector distribution, current density distribution, current density divergence / gradient distribution, conductivity or permittivity distribution, conductivity or permittivity gradient distribution, conductivity or permittivity, etc. Laplacian distribution, permittivity-to-conductivity ratio distribution, permittivity-to-conductivity ratio gradient distribution, permittivity-to-conductivity ratio Laplacian distribution, their frequency dispersion and their changes over time, Absolute changes (difference values), relative changes over time (values of ratios), etc. are displayed on the display unit image including those using a CRT, liquid crystal, or LED, and these are displayed at any selected position. The value is displayed.

FIG. 1 shows an overall configuration diagram of a measuring apparatus according to the first embodiment of the present invention. This apparatus is a magnetic field vector detector 1 for measuring a current density vector distribution (or a time series of current density vector distribution) in a three-dimensional space of interest, a two-dimensional space of interest, or a one-dimensional space of interest in a measurement object 4. The scanning mechanism 3 for changing the position and the detection direction, the distance adjusting means 13 for adjusting the distance between the magnetic field vector detector 1 and the measuring object 4, and the current field generating means for generating a current field in the measuring object 4. 5 and magnetic field vector detector 1
Drive device 2 for driving the device, measurement control means 6 for controlling them, and data recording means 7 for storing measurement data.
And the measured magnetic field data (or measured magnetic field time series data) from the current density vector (or current density vector distribution time series) and conductivity distribution (or conductivity distribution time series) or dielectric constant distribution (or dielectric constant distribution). The data processing means 8 estimates the time series).

FIG. 2 is a schematic diagram showing the coil portion of the magnetic field vector detecting portion. However, in FIG. 2A, one element,
FIG. 2B shows two elements, and FIG. 2C shows three elements. Further, FIGS. 3 to 7 respectively show array configurations according to the embodiments of the present invention.

The measurement control means 6, the data recording means 7, and the data processing means 8 can be realized on one device, or can be distributed to a plurality of devices. The positions of the scanning mechanism 3 and the distance adjusting means 13 can be reversed.

FIG. 8 shows an embodiment in which the scanning mechanism 3 and the distance adjusting means 13 are provided on the magnetic field vector detector 1 side. The positions of the scanning mechanism 3 and the distance adjusting means 13 can be reversed.

FIG. 9 shows an embodiment in which the first scanning mechanism 10 is provided on the magnetic field vector detector 1 side and the second scanning mechanism 11 is provided on the measurement object 4 side. In this embodiment, the distance adjusting means 13 is provided on the magnetic field vector detector side 1, but it may be provided on the measurement object 4 side or both sides. Scanning mechanism 10,
The positions of 11 and the distance adjusting means can be reversed.

Next, a technique for estimating the conductivity distribution and the permittivity distribution by estimating the current density distribution will be described with reference to the flowchart of FIG.

First, the values of conductivity and permittivity are unknown. 3
A reference area is appropriately set in a two-dimensional ROI, a two-dimensional ROI, or a one-dimensional ROI. The reference region is a region whose conductivity and dielectric constant are known.

When two or more independent current fields can be set at the time of measurement, that is, two or more independent current density vector distributions (magnetic field vector distributions), or a time series of current density vector distributions (when the magnetic field vector distributions are When it is possible to measure a series, at least one reference point is set as a reference area. The reference point is a point whose conductivity or permittivity is known.

When only one current density vector distribution can be measured at the time of measurement, the reference region is set so as to extend widely in the direction in which the current dominantly flows.

An example will be given of the case where two independent magnetic field vector distributions (or time series of magnetic field vector distributions) generated by using a current field generator are measured. A first current field (or a sinusoidally time-varying current field) is generated in the space of interest or a region of interest, and a magnetic field vector distribution (or a sinusoidally time-varying magnetic field vector distribution) around the space of interest or the region of interest is generated. Cause Subsequently, a second current field (or sinusoidally time-varying current field) independent of the first current field is generated, and similarly, a second magnetic field vector distribution (or sinusoidally time-varying magnetic field). Vector distribution).
According to the technique 1 or the technique 2, a current density vector distribution (or a sinusoidally time-varying current density vector distribution) in the space of interest or the region of interest is estimated in each case, and this is performed by the data processing means. Regarding the measurement of the magnetic field vector spatial distribution (or the time series of the magnetic field vector spatial distribution), the magnetic field vector detection signal and the position information are sampled while controlling the magnetic field vector detector driving device and the scanning mechanism by the measurement control means. Is input to the data recording means. In some cases, the scanning mechanism may not be used. Then, the data processing means ([9]
Equation, or [16a] and [16b]), the conductivity distribution, the permittivity distribution, and the time constant distribution (the distribution of the ratio of the permittivity and the conductivity), and if necessary, these time series And estimate the frequency variance.

When a current field already exists, only the magnetic field vector generated by this is measured, and the conductivity distribution and the permittivity distribution are estimated only by estimating the current vector distribution. If the measurement data of the magnetic field or current is missing,
The calculation is performed by excluding the point or region of the time from the region of interest, and after the calculation, the permittivity distribution, the conductivity distribution, the ratio of the permittivity and the conductivity of the point or region of the time excluded from the region of interest are calculated. The distribution, these changes over time and frequency dispersion are interpolated within the space-of-interest in terms of the estimated permittivity distribution, conductivity distribution, distribution of the ratio of permittivity and conductivity, and these changes over time and frequency dispersion. Or it may be evaluated by extrapolation interpolation processing.

[0152]

As described above, according to the present invention,
3D space of interest or 2D region of interest in unknown sample or 1
From the magnetic field vector distribution (or time series of magnetic field vector distribution) measured using the magnetic field vector detector around the three-dimensional region of interest, calculate the current density vector distribution in the space of interest or in the region of interest (when the current density vector distribution is It is possible to estimate the three-dimensional conductivity distribution, the three-dimensional conductivity distribution, the two-dimensional conductivity distribution, the one-dimensional conductivity distribution, and the three-dimensional distribution.
The one-dimensional permittivity distribution, the two-dimensional permittivity distribution, the one-dimensional permittivity distribution can be estimated.

[Brief description of drawings]

FIG. 1 is a block diagram showing a configuration of a measuring device according to an embodiment of the present invention.

FIG. 2 is a schematic diagram showing a coil part of a magnetic field vector detector used in the embodiment of the present invention.

FIG. 3 is an explanatory diagram showing a configuration of an array according to an exemplary embodiment of the present invention.

FIG. 4 is an explanatory diagram showing a configuration of an array according to an exemplary embodiment of the present invention.

FIG. 5 is an explanatory diagram showing a configuration of an array according to the exemplary embodiment of the present invention.

FIG. 6 is an explanatory diagram showing a configuration of an array according to an exemplary embodiment of the present invention.

FIG. 7 is an explanatory diagram showing a configuration of an array according to an exemplary embodiment of the present invention.

FIG. 8 is a block diagram showing an embodiment in which a scanning mechanism and distance adjusting means are provided on the magnetic field vector detector side.

FIG. 9 is a block diagram showing an embodiment in which a first scanning mechanism is provided on the magnetic field vector detector side and a second scanning mechanism is provided on the measurement target side.

FIG. 10 is a flowchart showing an embodiment of the present invention.

FIG. 11 is a schematic view showing a sensor unit of a conventional magnetic field detector.

[Explanation of symbols]

1 Magnetic field vector detector 2 drive 3 scanning mechanism 4 Object to be measured 5 Current field generator 6 Measurement control means 7 Data recording means 8 Data processing means 9 housing 10 First scanning mechanism 11 Second scanning mechanism 12 Current source 13 Distance adjustment means 14 Sample stand

─────────────────────────────────────────────────── ─── Continuation of front page (51) Int.Cl. ⁷ Identification code FI theme code (reference) H01L 43/00 H01L 43/00 4M113 43/06 43/06 AUZ 43/08 43/08 AUZ // A61B 5/05 A61B 5/05 A B Z H01L 39/22 H01L 39/22 DF term (reference) 2G017 AD32 AD51 AD53 AD55 2G028 CG02 CG09 FK02 GL09 LR07 2G035 AA08 AA21 AC25 AD66 2G053 AA00 CA27 CA02 CA02 CA05 DB19 4C027 AA06 AA10 CC00 EE01 GG00 HH00 KK00 KK03 4M113 AA01 AC06 AC46

Claims (19)

[Claims] 1. A current density vector estimation device for estimating a current density vector distribution in a three-dimensional region of interest or a two-dimensional region of interest of a measurement object, a current density distribution in a one-dimensional region of interest, or a time series thereof. A sample installation table on which the measurement target is installed, a magnetic field vector detection unit that detects a magnetic field vector, a housing installed so that the magnetic field vector detection unit measures the measurement target, and a magnetic field of the measurement target. In order to measure the distribution of the vector, the sample installation table and the housing, at least one of the scanning means connected to at least one of the up and down, left and right, the direction can be changed, the magnetic field vector detection means is driven, the measurement target Adjustment means for adjusting the output of the magnetic field vector detection means to be measured, and data for recording the measurement result output from the adjustment means Recording means, data processing means for estimating the current density vector distribution or its time series from the magnetic field vector data measured through the application of discrete approximation or finite element approximation to the current density vector distribution or its time series, the measurement object and the A current density vector estimation device comprising: a distance adjusting unit that adjusts a distance to a magnetic field vector detecting unit. 2. The magnetic field measured to obtain the current density vector distribution in the three-dimensional region of interest or the two-dimensional region of interest of the measurement target, the current density distribution in the one-dimensional region of interest, or a time series thereof. 2. The current density vector according to claim 1, wherein the output of the vector detecting means or the output of the adjusting means is recorded in the data recording means in association with position / time data indicating the position / time of the measurement. Estimator. 3. If necessary, at least one or more of the outside of the three-dimensional space of interest of the measurement object is provided within the space or within the two-dimensional region of interest or within the one-dimensional region of interest. 2. The current density according to claim 1, further comprising a current field generating means capable of directly flowing a current or inducing an eddy current by applying a magnetic field so that a current field can be generated. Vector estimator. 4. The magnetic field vector detecting means is a superconducting quantum interference device (SQUID), a Hall device, a Josephson junction device, a magneto-impedance device, a GMR, a CMR,
The current density vector estimation device according to claim 1, wherein SET or the like is used. 5. Magnetic field vector data is collected by using a plurality of magnetic, static magnetic field or electromagnetic wave detection elements as the magnetic field vector detection means, and the detection directions are two directions or three directions orthogonal to each other, and sometimes aperture plane synthesis is performed. The current density vector estimation device according to claim 1, wherein 6. A magnetic field, a static magnetic field, or an electromagnetic wave detecting element is used as the magnetic field vector detecting means, and the detection direction is one direction, two directions orthogonal to each other, or three directions, and one-dimensional, two-dimensional, Alternatively, the current density vector estimation device according to claim 1, wherein a three-dimensional array is configured and aperture plane synthesis is sometimes performed to collect magnetic field data. 7. The magnetic field vector detecting means is used to measure a two-dimensional spatial distribution of a two-dimensional magnetic field vector or its time series, and from this measured magnetic field vector data, a discrete approximation or a finite element is made to a current density vector or its time series. By performing a predetermined numerical analysis based on the application of the approximation, the 2
The current density vector estimation device according to claim 1, wherein the three-dimensional current density vector distribution or its time series is estimated. 8. A three-dimensional or three-dimensional spatial distribution of a two-dimensional magnetic field vector or a time series thereof is measured using the magnetic field vector detecting means, and the measured magnetic field vector data is dispersed into a current density vector or a time series thereof. A spatial distribution of a three-dimensional or two-dimensional current density vector in a three-dimensional space of interest or a time series thereof is estimated by performing a predetermined numerical analysis based on applying an approximation or a finite element approximation. Item 1. The current density vector estimation device according to Item 1. 9. The magnetic field vector detecting means is used to measure a three-dimensional or three-dimensional spatial distribution of a two-dimensional magnetic field vector or its time series, and from the measured magnetic field vector data, a current density vector or its time series is discrete. Estimate the spatial distribution of the three-dimensional or two-dimensional current density vector in the three-dimensional space of interest or the two-dimensional region of interest or its time series by performing a predetermined numerical analysis based on the application of approximation or finite element approximation The current density vector estimation device according to claim 1, wherein 10. A unidirectional component of a magnetic field vector measured by the magnetic field vector detecting means by performing a predetermined numerical analysis based on applying a discrete approximation or a finite element approximation to the current density or its time series. Of the current density vector component of the current density vector component of the magnetic field vector component is estimated from the spatial distribution or its time series data. Three-dimensional spatial distribution or two-dimensional spatial distribution or one-dimensional spatial distribution, or two-dimensional spatial distribution data of the magnetic field vector component, two-dimensional spatial distribution or one-dimensional spatial distribution of the current density vector component, or its magnetic field The feature is that the one-dimensional spatial distribution of the current density vector component is estimated from the one-dimensional spatial distribution data of the vector component. Current density vector estimating apparatus according to claim 1 that. 11. A measurement object in a three-dimensional space of interest or two
In an electric conductivity estimation device for estimating an electric conductivity distribution or a dielectric constant distribution in a one-dimensional region of interest or a one-dimensional region of interest, or a time series thereof, in the current density vector estimation device, the measurement object is installed. A sample installation table, a magnetic field vector detection means for detecting a magnetic field vector, a housing installed so that the magnetic field vector detection means measures the measurement target, and a distribution of the magnetic field vector of the measurement target or a time series thereof. A scanning unit which is connected to at least one of the sample mounting table and the housing for measurement, and which can change the vertical and horizontal directions and directions; and the magnetic field which drives the magnetic field vector detection unit and measures the measurement target. Adjusting means for adjusting the output of the vector detecting means, and a data recording means for recording the measurement results output from the adjusting means. And the current density vector distribution or its time series (or spectrum) is estimated from the magnetic field vector data or its time series data measured by applying discrete approximation or finite element approximation to the current density vector distribution or its time series (or spectrum). , If multiple current density vector distributions or their time series are estimated, each current density vector distribution is normalized using its norm (magnitude) and then finite difference approximation or finite element approximation is performed to determine the electrical conductivity. Distribution and permittivity distribution, data processing means for estimating the time series of electrical conductivity distribution and permittivity distribution and their frequency dispersion, and distance adjustment for adjusting the distance between the measurement target and the magnetic field vector detecting means. An electric conductivity estimation apparatus comprising: 12. A current density vector distribution or 1 in the three-dimensional space of interest or the two-dimensional region of interest of the measurement object.
An output of the magnetic field vector detecting means measured to obtain a current density distribution in the dimensional region of interest or a time series thereof, or an output of the adjusting means is associated with position / time data indicating the position / time of the measurement, The electric conductivity estimating apparatus according to claim 11, wherein the electric conductivity is recorded in the data recording unit. 13. Further, if necessary, at least one or more of the outside of the three-dimensional space of interest of the measurement object is provided in the space or in the two-dimensional region of interest or in the one-dimensional region of interest. The electric conduction according to claim 11, further comprising a current field generating means capable of directly flowing a current or inducing an eddy current by applying a magnetic field so that a current field can be generated. Rate estimator. 14. The magnetic field vector detecting means is a superconducting quantum interference device (SQUID), a Hall device, a Josephson junction device, a magneto-impedance device, GMR, CM.
12. Use of R, SET, etc.
The electric conductivity estimating device according to item 1. 15. Magnetic field vector data is collected by using a plurality of magnetic, static magnetic field or electromagnetic wave detection elements as the magnetic field vector detection means, and the detection directions are orthogonal to each other in two or three directions. The electrical conductivity estimation device according to claim 11, wherein 16. A magnetic field, a static magnetic field or an electromagnetic wave detecting element is used as the magnetic field vector detecting means, and the detection direction is one direction, two directions orthogonal to each other, or three directions, and one-dimensional, two-dimensional, Alternatively, the electrical conductivity estimating apparatus according to claim 11, wherein a three-dimensional array is configured and aperture plane synthesis is sometimes performed to collect magnetic field data. 17. A method for estimating an electrical conductivity distribution in a three-dimensional space of interest, a two-dimensional region of interest, or a one-dimensional region of interest, its time series, or its frequency dispersion from a current density vector or its time series. Based on applying finite difference approximation or finite element approximation (using variational method or Galerkin method) to the electric conductivity distribution or the current density vector distribution represented in the first-order spatial partial differential equation The method according to claim 1, wherein a numerical method is used.
1. The electric conductivity estimation device according to 1. 18. From the time series of the current density vector, the electric conductivity distribution or the dielectric constant distribution in the three-dimensional ROI, the two-dimensional ROI, or the one-dimensional ROI, or the electric conductivity distribution or the dielectric constant distribution. When estimating the time series and their frequency dispersions, the finite difference approximation or finite element approximation (variational variation) is applied to the conductivity distribution or the spectrum distribution of the current density vector expressed in the predetermined first-order spatial partial differential equation. Method or a Galakin method) is used, and a predetermined numerical solution method based on applying the electric conductivity estimation apparatus according to claim 11 is used. 19. The measurement result includes a measured magnetic vector distribution, a measured current density distribution, a current density divergence / gradient distribution, a conductivity or permittivity distribution, a conductivity or permittivity gradient distribution, or a conductivity or The Laplacian distribution of the permittivity, the distribution of the ratio of the permittivity to the conductivity, the gradient distribution of the ratio of the permittivity to the conductivity, the Laplacian distribution of the ratio of the permittivity to the conductivity, their frequency dispersion and their changes over time. The electrical conductivity estimating device according to claim 11, further comprising display means for displaying an absolute change over time, a relative change over time, and the like.