CN103776873B - A kind of method mapped by voltage-to-current map construction current-voltage - Google Patents

Abstract

The present invention relates to a kind of building method mapped by voltage-to-current map construction current-voltage, for the sensor in electricity tomography with N number of electrode, by the parameter such as eigenwert and proper vector of calculating voltage-current matrix, by derived formula, unique calculates current-voltage matrix, constructs current-voltage and maps.The present invention gives the direct building method that current-voltage maps, can be applicable in the direct reconstruction algorithm of electricity chromatography imaging field, its explicit physical meaning, and simple.

Description

A kind of method mapped by voltage-to-current map construction current-voltage

Technical field

The present invention relates to electricity chromatography imaging field, particularly relate to a kind of method mapped by voltage-to-current map construction current-voltage.

Background technology

Electricity tomography (ElectricalTomography is called for short ET) technology is the one of chromatography imaging technique.By applying excitation to testee, and detecting the change of its boundary value, utilizing specific reconstruction algorithm to reconstruct the distribution of measurand internal electrical characterisitic parameter, thus obtaining the distribution situation of interior of articles.Compared with other chromatography imaging techniques, electricity tomography has radiationless, Noninvasive, portability, 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 through directly, independently calculating acquisition.

The building method that current-voltage maps is the important component part of the direct reconstruction algorithm of electricity tomography.For the sensor having N number of electrode, Law of Inner Product can be applied and map to construct current-voltage.But also do not map according to voltage-to-current the direct building method constructing current-voltage and map at present.

Summary of the invention

The object of the invention is to propose a kind of direct building method constructing current-voltage mapping according to voltage-to-current mapping.

Technical scheme of the present invention is:

Step one, have N number of electrode sensor electrode on current density have general expression write as matrix form to have:

$J=\frac{1}{A}C\·V---\left(1\right)$

Wherein J is current density vector, and A is the internal surface area of each electrode, and C is capacitance matrix, and V is voltage vector, and expression formula is:

$C=\left[\begin{array}{ccccc}{C}_{\mathrm{1,1}}& -C_{\mathrm{1,2}}& -C_{\mathrm{1,3}}& ...& -C_{1,N}\\ -C_{\mathrm{2,1}}& C_{\mathrm{2,2}}& -C_{\mathrm{2,3}}& ...& -C_{2,N}\\ -C_{\mathrm{3,1}}& -C_{3,2}& C_{\mathrm{3,3}}& ...& -C_{3,N}\\ .& .& .& .& .\\ .& .& .& .& .\\ .& .& .& .& .\\ -C_{N,1}& -C_{N,2}& -C_{N,3}& ...& C_{N,N}\end{array}\right],$ $\left[\begin{array}{c}{j}_1\\ j_2\\ j_3\\ .\\ .\\ .\\ j_N\end{array}\right],$ $V=\left[\begin{array}{c}{V}_1\\ V_2\\ V_3\\ .\\ .\\ .\\ V_N\end{array}\right]---\left(2\right)$

J _sthe current density on s (1≤s≤N) individual electrode inside surface, V _tthe magnitude of voltage on t electrode, C _s,tit is the capacitance between electrode s and electrode t.Self-capacitance C _s,sbe defined as the electric capacity summation between s electrode and other N-1 electrode, namely

$C_{s,s}=\underset{t\≠s}{\overset{N}{\underset{t=1}{\mathrm{\Σ}}}}{C}_{s,t}---\left(3\right)$

So have

${\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}=\frac{1}{A}C---\left(4\right)$

being number of poles is N, and specific inductive capacity is distributed as voltage-to-current mapping matrix during ε (z).

According to Kirchhoff's law, loop current sum is zero, and namely the order of current density vector J is N-1, and the order of Matrix C is also N-1, can show that the order of voltage vector is also N-1 thus, therefore make

So

$V=R_{\mathrm{\ϵ}}^{N\×N}J=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}V---\left(6\right)$

Wherein being number of poles is N, and specific inductive capacity is distributed as current-voltage mapping matrix during ε (z).

Can prove that front N-1 the eigenwert of V is-1, and last eigenwert is 0.By calculating eigenwert and the proper vector of V, can obtain:

V=P{diag([-1-1…-10]) _N×N}P ^T(7)

Wherein diag () _{n × N}represent the diagonal matrix on N rank, P ^tp=I, P=[p ₁p ₂p _n-1p _n].Defined feature vector p _ii-th (1≤i≤N-1) row of P, i.e. Vp _i=(-1) p _i, p _neigenwert 0 characteristic of correspondence vector, i.e. Vp _n=(0) p _n.Because the mean value of often going in equation (5) is all 0, therefore

$p_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×N}^T,$ So V can be written as:

$V=-I_{N\×N}+p_N{p}_N^T---\left(8\right)$

Wherein I _{n × N}it is the unit matrix on N rank.

Similar, by calculating eigenwert and proper vector, order be N-1, can be written as following form:

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

Wherein Σ _Λ=diag ([λ ₁λ ₂λ _n-10] _{n × N}) diagonal matrix that is made up of N number of 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 (1≤i≤N-1) row of Q, i.e. Rq _i=λ _iq _i, q _nproper vector 0 characteristic of correspondence vector, i.e. Rq _n=(0) q _n, $p_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×N}^T,$ Therefore have

$Q^T\mathrm{VQ}=Q^T(-I_{N\×N}+p_N{p_N}^T)Q$ $=-I_{N\×N}+Q^T\left(q_N{q_N}^T\right)Q---\left(10\right)$

$=\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}$

Step 2, by formula (7) substitute into formula (6) have:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}\right\}{P}^T=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}\right\}{P}^T$

(11)

Both members is premultiplication P simultaneously ^t, the right side takes advantage of P to obtain:

$\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}=P^T{R}_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}\right\}$

(12)

The matrix that a Matrix Multiplication obtains with diagonal matrix if easily know is still for diagonal matrix, then this matrix is also diagonal matrix.So known:

$P^T{R}_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}1& 1& ...& 1& 0\end{array}\right]\right)}_{N\×N}\right\}---\left(13\right)$

Again at formula (13) premultiplication P, P is taken advantage of on the right side ^t:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}1& 1& ...& 1& 0\end{array}\right]\right)}_{N\×N}\right\}{P}^T=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}---\left(14\right)$

Formula (7) and formula (9) are substituted into formula (14) obtain

$-V=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}=R_{\mathrm{\ϵ}}^{N\×N}Q{\mathrm{\Σ}}_{\mathrm{\Λ}}{Q}^T---\left(15\right)$

At formula (15) premultiplication Q ^t, the right side takes advantage of Q to obtain:

$-Q^T\mathrm{VQ}=Q^T{R}_{\mathrm{\ϵ}}^{N\×N}Q{\mathrm{\Σ}}_{\mathrm{\Λ}}---\left(16\right)$

Formula (10) is substituted into formula (16) and get final product:

$-\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{N\×N}={Q^{T}R}_{\mathrm{\ϵ}}^{N\×N}\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}{\mathrm{\λ}}_1& {\mathrm{\λ}}_2& ...& {\mathrm{\λ}}_{N-1}& 0\end{array}\right]\right)}_{N\×N}---\left(17\right)$

So have:

$\mathrm{diag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\λ}}_1}& \frac{1}{{\mathrm{\λ}}_2}& ...& \frac{1}{{\mathrm{\λ}}_{N-1}}& 0\end{array}\right]\right)}_{N\×N}{Q}^T{R}_{\mathrm{\ϵ}}^{N\×N}Q---\left(18\right)$

At formula (18) premultiplication Q, Q is taken advantage of on the right side ^t:

$R_{\mathrm{\ϵ}}^{N\×N}=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\λ}}_1}& \frac{1}{{\mathrm{\λ}}_2}& ...& \frac{1}{{\mathrm{\λ}}_{N-1}}& 0\end{array}\right]\right)}_{N\×N}\right\}{Q}^T---\left(19\right)$

Formula is known thus, matrix can be now uniquely determined.

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

Accompanying drawing explanation

Fig. 1 is implementing procedure figure.

Fig. 2 is embodiment isoboles.

Embodiment

See Fig. 1, a kind of building method algorithm block diagram constructing current-voltage mapping according to voltage-to-current mapping.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, have N=16 electrode sensor electrode on current density have general expression write as matrix form to have:

$J=\frac{1}{A}C\·V---\left(20\right)$

Wherein J is current density vector, and A is the internal surface area of each electrode, and C is capacitance matrix, and V is voltage vector, and expression formula is:

$C=\left[\begin{array}{ccccc}{C}_{\mathrm{1,1}}& -C_{\mathrm{1,2}}& -C_{\mathrm{1,3}}& ...& -C_{1,16}\\ -C_{\mathrm{2,1}}& C_{\mathrm{2,2}}& -C_{\mathrm{2,3}}& ...& -C_{2,16}\\ -C_{\mathrm{3,1}}& -C_{3,2}& C_{\mathrm{3,3}}& ...& -C_{3,16}\\ .& .& .& .& .\\ .& .& .& .& .\\ .& .& .& .& .\\ -C_{16,1}& -C_{16,2}& -C_{16,3}& ...& C_{16,16}\end{array}\right],$ $\left[\begin{array}{c}{j}_1\\ j_2\\ j_3\\ .\\ .\\ .\\ j_{16}\end{array}\right],$ $V=\left[\begin{array}{c}{V}_1\\ V_2\\ V_3\\ .\\ .\\ .\\ V_{16}\end{array}\right]---\left(21\right)$

J _sthe current density on the individual electrode inside surface of s (1≤s≤16), V _tthe magnitude of voltage on t electrode, C _s,tit is the capacitance between electrode s and electrode t.Self-capacitance C _s,sbe defined as the electric capacity summation between s electrode and other 15 electrodes, namely

$C_{s,s}=\underset{t\≠s}{\overset{N}{\underset{t=1}{\mathrm{\Σ}}}}{C}_{s,t}---\left(22\right)$

So have

${\mathrm{\Λ}}_{\mathrm{\ϵ}}^{16\×16}=\frac{1}{A}C---\left(23\right)$

According to Kirchhoff's law, loop current sum is zero, and namely the order of current density vector J is 15, and the order of Matrix C is also 15, can show that the order of voltage vector is also 15 thus, therefore make

So

$V=R_{\mathrm{\ϵ}}^{16\×16}J=R_{\mathrm{\ϵ}}^{16\×16}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{16\×16}V---\left(25\right)$

Can prove that front 15 eigenwerts of V are-1, and last eigenwert is 0.By calculating eigenwert and the proper vector of V, can obtain:

V=P{diag([-1-1…-10]) _16×16}P ^T(26)

Wherein diag () _{16 × 16}represent the diagonal matrix on 16 rank, P ^tp=I, P=[p ₁p ₂p ₁₅p ₁₆].Defined feature vector p _ii-th (1≤i≤15) row of P, i.e. Vp _i=(-1) p _i, p ₁₆eigenwert 0 characteristic of correspondence vector, i.e. Vp ₁₆=(0) p ₁₆.Because the mean value of a line every in equation (24) is all 0, therefore $p_{16}=\frac{1}{4}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×16}^T,$

So V can be written as:

$V=-I_{16\×16}+p_{16}{p}_{16}^T---\left(27\right)$

Wherein I _{16 × 16}it is the unit matrix on 16 rank.

Similar, by calculating eigenwert and proper vector, order be 15, can be written as following form:

${\mathrm{\Λ}}_{\mathrm{\ϵ}}^{16\×16}={\mathrm{Q\Σ}}_{\mathrm{\Λ}}{Q}^T=Q\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}{\mathrm{\λ}}_1& {\mathrm{\λ}}_2& ...& {\mathrm{\λ}}_{15}& 0\end{array}\right]\right)}_{16\×16}\right\}{Q}^T---\left(28\right)$

Wherein Σ _Λ=diag ([λ ₁λ ₂λ ₁₅0] _{16 × 16}) diagonal matrix that is made up of 16 eigenwerts of R, the matrix that Q is made up of corresponding proper vector.Q ^tq=I, Q=[q ₁q ₂q ₁₅q ₁₆], proper vector q _ii-th (1≤i≤15) row of Q, i.e. Rq _i=λ _iq _i, q ₁₆proper vector 0 characteristic of correspondence vector, i.e. Rq ₁₆=(0) q ₁₆,

$q_{16}=\frac{1}{4}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×16}^T.$

From formula (24), in ring resistance network, can be calculated current matrix:

It can thus be appreciated that voltage-to-current mapping matrix is

Calculate according to formula (28)

(31)

Step 2, by formula (26) substitute into formula (25) have:

$P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{16\×16}\right\}{P}^T=R_{\mathrm{\ϵ}}^{16\×16}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{16\×16}P\left\{\mathrm{diag}{\left(\left[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\right]\right)}_{16\×16}\right\}{P}^T$ (33)

Obtain

$R_{\mathrm{\ϵ}}^{16\×16}=\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}\frac{1}{{\mathrm{\λ}}_1}& \frac{1}{{\mathrm{\λ}}_2}& ...& \frac{1}{{\mathrm{\λ}}_{15}}& 0\end{array}\right]\right)}_{16\×16}{Q}^T$

$=\mathrm{Qdiag}{\left(\left[\begin{array}{ccccc}\frac{1}{0.5+0.2071\mathrm{i\λ}}& \frac{1}{0.5-0.2071i}& ...& \frac{1}{0.5-2.5137i}& 0\end{array}\right]\right)}_{16\×16}{Q}^T---\left(34\right)$

Formula is known thus, matrix can be now uniquely determined.

For proving this conclusion, doing following checking: under adjacent actuators pattern, calculating node potential matrix

With setting V be consistent, can this method be verified thus.

Described a kind ofly construct by voltage-to-current the method that current-voltage maps, give a kind of computing method that current-voltage maps, the method explicit physical meaning, simple.With reference to this method, any orthogonal set of excitation measurement pattern may be used to calculating current-voltage 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. map the direct building method constructing current-voltage and map according to voltage-to-current, it is characterized in that, the method comprises the steps:

Step one, the current density had on each electrode of sensor of N number of electrode have general expression wherein A is the internal surface area of each electrode, and C is capacitance matrix, and V is voltage vector, so voltage-to-current mapping matrix when having specific inductive capacity to be distributed as ε (z)

${\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}=\frac{1}{A}C---\left(1\right)$

According to Kirchhoff's law, loop current sum is zero, and namely the order of current density vector J is N-1, and the order of Matrix C is also N-1, can show that the order of voltage vector is also N-1 thus, therefore make

So

$V=R_{\mathrm{\ϵ}}^{N\×N}J=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}V---\left(3\right)$

Calculate eigenwert and the proper vector of V, can obtain:

V＝P{diag([-1-1…-10]) _N×N}P ^T(4)

Wherein diag () _{n × N}represent the diagonal matrix on N rank, the matrix that P is made up of corresponding proper vector, P ^tp=I, P=[p ₁p ₂p _n-1p _n], proper vector p _ii-th row of P, wherein 1≤i≤N-1, i.e. Vp _i=(-1) p _i, p _neigenwert 0 characteristic of correspondence vector, i.e. Vp _n=(0) p _n, $p_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×N}^T,$ Namely V can be written as:

$V=-I_{N\×N}+p_N{p}_N^T---\left(5\right)$

Wherein I _{n × N}the unit matrix on N rank,

Similar, by calculating eigenwert and proper vector, order be N-1, following form can be written as:

${\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}={\mathrm{Q\Σ}}_{\mathrm{\Λ}}{Q}^T=Q\left\{diag{(\[\begin{array}{ccccc}{\mathrm{\λ}}_1& {\mathrm{\λ}}_2& ...& {\mathrm{\λ}}_{N-1}& 0\end{array}\])}_{N\×N}\right\}{Q}^T---\left(6\right)$

Wherein Σ _Λ=diag ([λ ₁λ ₂λ _n-10] _{n × N}) diagonal matrix that is made up of N number of 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 of Q, wherein 1≤i≤N-1, i.e. Rq _i=λ _iq _i, q _nproper vector 0 characteristic of correspondence vector, i.e. Rq _n=(0) q _n, $q_N=\frac{1}{\sqrt{N}}{\left[\begin{array}{ccccc}1& 1& ...& 1& 1\end{array}\right]}_{1\×N}^T,$ So have

$\begin{array}{c}{Q}^{T}VQ=Q^T(-I_{N\×N}+p_N{p_N}^T)Q\\ =-I_{N\×N}+Q^T\left(q_N{q_N}^T\right)Q\\ diag{(\[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\])}_{N\×N}\end{array}---\left(7\right)$

Step 2, by formula (4) substitute into formula (3) have:

$P\left\{diag{(\[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\])}_{N\×N}\right\}{P}^T=R_{\mathrm{\ϵ}}^{N\×N}{\mathrm{\Λ}}_{\mathrm{\ϵ}}^{N\×N}P\left\{diag{(\[\begin{array}{ccccc}-1& -1& ...& -1& 0\end{array}\])}_{N\×N}\right)\}{P}^T---\left(8\right)$

Then through deriving, the matrix that current-voltage mapping pair is answered can be written as:

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

Formula is known thus, matrix can be now uniquely determined.