CN103116101B - Adjacent excitation measurement mode based voltage-current mapping construction method - Google Patents

Abstract

The invention relates to an adjacent excitation measurement mode based voltage-current mapping construction method. Adjacent excitation measurement mode based voltage-current mapping is constructed by computing parameters of an impedance matrix, a current density matrix and the like, exclusively computing a voltage-current matrix via a derived formula and aiming at an adjacent excitation measurement mode of a sensor with N electrodes in electrical tomography. The method is simple and feasible, can be applied to direct reconstruction algorithm of the electrical tomography field, and provides a direct physical significance of voltage-current mapping.

Description

A kind of voltage-to-current map construction method based on adjacent actuators measurement pattern

Technical field

The present invention relates to electricity chromatography imaging field, particularly relate to a kind of voltage-to-current map construction method based on adjacent actuators measurement pattern newly.

Background technology

Electricity tomography (Electrical Tomography is called for short ET) technology is the one of chromatography imaging technique.It by applying excitation to testee, and detects the change of its boundary value, utilizes specific reconstruction algorithm to rebuild the distribution of measurand internal electrical characterisitic parameter, thus obtains the distribution situation of interior of articles.Compared with other chromatography imaging techniques, electricity tomography has radiationless, fast response time, the advantage such as cheap.

Electricity tomographic reconstruction algorithm generally can be divided into two classes: based on reconstruction algorithm and the direct reconstruction algorithm of sensitivity matrix.Use the former usually to need to separate ill linear equation, this just means all pixel values simultaneously must rebuilding measured zone.The latter is mapped by calculating current-voltage or voltage-to-current mapping realizes, and the gray-scale value of each pixel can directly, independently calculate.

Voltage-to-current maps the data usually obtained by adjacent actuators measurement pattern and rebuilds, and realizes on hardware because this measurement pattern is easier.But at present also not based on the direct building method that the voltage-to-current of adjacent actuators measurement pattern maps.

For the sensor having N number of electrode, can map with the invertible matrix structure current-voltage that dimension is (N-1) × (N-1), then obtain voltage-to-current mapping thus.But this method can not provide the direct physical meaning that voltage-to-current maps.

Summary of the invention

The object of the invention is to propose a kind of voltage-to-current map construction method based on adjacent actuators measurement pattern, it can provide the direct physical meaning that voltage-to-current maps.

Technical scheme of the present invention is:

Step one, when the sensor with N number of electrode is in adjacent actuators measurement pattern, if unitary current flows into tested field domain from i-th (1≤i≤N) individual electrode, the i-th+1 electrode flows out, then the electric potential difference between kth (1≤k≤N) individual electrode and kth+1 electrode meets:

$U_{i,i+1}^{k+1}-U_{i,i+1}^k=R_{i,i+1}^{k,k+1}---\left(1\right)$

that unitary current flows into tested field domain from i-th electrode, when the i-th+1 electrode flows out, i-th and i+1 electrode and the transimpedance between kth and k+1 electrode, define:

In step 2, adjacent actuators measurement pattern, current density matrix is following form:

Be the matrix of N × N with a dimension carry out mapping when approximate distribution of conductivity is σ (x, y, z):

$I_{\mathrm{adj}}={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(U\right)={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(\mathrm{BR}\right)---\left(6\right)$

If ∑ _r=diag ([λ ₁λ ₂λ _n-10]) _{n × N}the diagonal matrix be made up of n the eigenwert of R, the matrix that Q is made up of corresponding proper vector, Q ^tq=I, Q=[q ₁q ₂q _n-1q _n], proper vector q _ii-th row (1≤i≤n-1) of Q, Rq _i=λ _iq _i, q _neigenwert 0 characteristic of correspondence vector, Rq _n=(0) q _n.And the mean value of the every a line of R is 0, therefore

$q_N=\frac{1}{\sqrt{N}}{{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×N}}^T,$ And have:

Q ^TB ^TI _adjQ＝Q ^T(-I _N×N+p _Np _N ^T)Q

＝-I _N×N+Q ^T(q _Nq _N ^T)Q (7)

＝diag([-1 -1 ··· -1 0]) _N×N

By above-mentioned equation, can derive:

$Q^T{B}^T{\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\mathrm{BQ}{\mathrm{\Σ}}_R=\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}---\left(8\right)$

Therefore, the discrete approximation that voltage-to-current maps, can be written as:

${\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}={\left(B^T\right)}^{-1}Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-\frac{1}{{\mathrm{\λ}}_1}& -\frac{1}{{\mathrm{\λ}}_2}& ...& -\frac{1}{{\mathrm{\λ}}_{N-1}}& 0\end{array}\right]\right)}_{N\×N}\right\}{Q}^T{B}^{-1}---\left(9\right)$

In this formula, Q and B is invertible matrix, therefore voltage-to-current mapping matrix determination that can be unique.

Further, with reference to this method, any orthogonal set of excitation measurement pattern may be used to calculating voltage-electric current and maps.

Accompanying drawing explanation

Fig. 1 is implementing procedure figure.

Fig. 2 is embodiment isoboles.

Embodiment

See Fig. 1, a kind of voltage-to-current map construction algorithm block diagram based on adjacent actuators measurement pattern.For 16 end to end ring resistance networks shown in Fig. 2, the embodiment of this method is described.

Said method comprising the steps of:

Step one, when ring resistance network is in adjacent actuators measurement pattern, if unitary current flows into tested field domain from i-th (1≤i≤16) individual node, the i-th+1 node flows out, then the electric potential difference between kth (1≤k≤16) individual node and kth+1 node meets:

$U_{i,i+1}^{k+1}-U_{i,i+1}^k=R_{i,i+1}^{k,k+1}---\left(10\right)$

that unitary current flows into tested field domain from i-th node, when the i-th+1 node flows out, i-th and i+1 node and the transimpedance between kth and k+1 node, define:

In step 2, adjacent actuators measurement pattern, current density matrix is following form:

Be the matrix of N × N with a dimension carry out mapping when approximate distribution of conductivity is σ (x, y, z):

$I_{\mathrm{adj}}={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(U\right)={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(\mathrm{BR}\right)---\left(15\right)$

If ∑ _r=diag ([λ ₁λ ₂λ _n-10]) _{n × N}the diagonal matrix be made up of n the eigenwert of R, the matrix that Q is made up of corresponding proper vector, Q ^tq=I, Q=[q ₁q ₂q _n-1q _n], proper vector q _ii-th row (1≤i≤n-1) of Q, Rq _i=λ _iq _i, q _neigenwert 0 characteristic of correspondence vector, Rq _n=(0) q _n, the discrete approximation that voltage-to-current maps, can be written as:

${\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}={\left(B^T\right)}^{-1}Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-\frac{1}{{\mathrm{\λ}}_1}& -\frac{1}{{\mathrm{\λ}}_2}& ...& -\frac{1}{{\mathrm{\λ}}_{N-1}}& 0\end{array}\right]\right)}_{N\×N}\right\}{Q}^T{B}^{-1}$

In this formula, Q and B is invertible matrix, therefore voltage-to-current mapping matrix determination that can be unique.

For proving this conclusion, do following checking: calculate node potential matrix

Can draw:

Above formula meets voltage-to-current mapping equation, can verify this method thus.

A kind of described voltage-to-current map construction method based on adjacent actuators measurement pattern, give the direct physical meaning of voltage-to-current mapping and a kind of new computing method, with reference to this method, any orthogonal set of excitation measurement pattern may be used to calculating voltage-electric current and maps.

Above to the description of the present invention and embodiment thereof, being not limited thereto, is only one of embodiments of the present invention shown in accompanying drawing.When not departing from the invention aim, designing the structure similar with this technical scheme or embodiment without creation, all belonging to scope.

Claims (1)

1., based on a voltage-to-current map construction method for adjacent actuators measurement pattern, for realizing electricity tomography through image method for reconstructing, it is characterized in that, the method comprises the steps:

Step one, for the sensor with N number of electrode, if unitary current flows into tested field domain from i-th electrode, the i-th+1 electrode flows out, wherein 1≤i≤N, the then electromotive force of a kth electrode with the electromotive force of kth+1 electrode between difference meet equation:

$U_{i,i+1}^{k+1}-U_{i,i+1}^k=R_{i,i+1}^{k,k+1}---\left(1\right)$

Wherein 1≤k≤N, that unitary current flows into tested field domain from i-th electrode, when the i-th+1 electrode flows out, i-th and i+1 electrode and the transimpedance between kth and k+1 electrode, define:

Wherein, B is parameter matrix, and R is mutual resistance matrix, and U is electrode voltage matrix;

In step 2, adjacent actuators measurement pattern, current density matrix I _adjfor following form:

Be the matrix of N × N with a dimension carry out mapping when approximate distribution of conductivity is σ (x, y, z):

$I_{\mathrm{adj}}={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(U\right)={\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}\left(\mathrm{BR}\right)---\left(6\right)$

If Σ _r=diag ([λ ₁λ ₂λ _n-10]) _{n × N}the diagonal matrix be made up of n the eigenwert of R, the matrix that Q is made up of corresponding proper vector, Q ^tq=I, Q=[q ₁q ₂q _n-1q _n], proper vector q _jthe jth row of Q, and 1≤j≤n-1, Rq _j=λ _jq _j, q _neigenwert 0 characteristic of correspondence vector, Rq _n=(0) q _n, the discrete approximation that voltage-to-current maps, can be written as:

${\mathrm{\Λ}}_{\mathrm{\σ}}^{N\×N}={\left(B^T\right)}^{-1}Q\left\{\mathrm{diag}{\left(\begin{array}{c}\left[\begin{array}{ccccc}-\frac{1}{{\mathrm{\λ}}_1}& -\frac{1}{{\mathrm{\λ}}_2}& ...& -\frac{1}{{\mathrm{\λ}}_{N-1}}& 0\end{array}\right]\end{array}\right)}_{N\×N}\right\}{Q}^T{B}^{-1}---\left(7\right)$

In this formula, Q and B is invertible matrix, therefore voltage-to-current mapping matrix can determine uniquely.