The Best Sensitivity Matrix by Matrix Transformation in Electrical Impendence Tomography

Yuwei Zhao; Shihong Yue; Shiyuan Zhu

doi:10.20944/preprints202606.0187.v1

Submitted:

01 June 2026

Posted:

02 June 2026

You are already at the latest version

Abstract

As an advanced visualization technique, the electrical impendence tomography (EIT) can reconstruct the distribution of electrical parameter and thereby visually show the object distribution within a detection field. The EIT reconstruction quality greatly depends on a selected sensitivity matrix, while various matrixes can lead to very different EIT reconstruction results. A large number of efforts to improve the sensitivity matrix have been made to enhance EIT reconstruction quality, but how to construct and select the best matrix in a generally and feasibly way remains unsolved to date. In this paper, we use the matrix transformation method to address the issue, and two types of symmetric and diagonally dominant matrixes are optimally selected to multiply the sensitivity matrix on left and right, respectively. Therefore, the EIT reconstruction quality can generally be improved. The optimality and generalization of the proposed method have been theoretically demonstrated when specially using either Gaussian kernel-based or power function-based matrices, respectively. Experiments validate the proposed method by the EIT reconstruction quality.

Keywords:

electrical impendence tomography

;

sensitivity matrix

;

matrix transformation

;

reconstruction quality

Subject:

Engineering - Industrial and Manufacturing Engineering

I. Introduction

Electrical impedance tomography (EIT) [1] can reconstruct the distribution of electrical parameters and thereby visually show the distribution of the object within a detection field. EIT has increasingly been used in most application fields [2,3,4,5]. However, the reconstruction quality of EIT is constrained by its inherent ill-posedness and soft-field effects [6,7]. In the past decades, developing high-quality EIT reconstruction algorithms has been a research focus.

The EIT reconstruction quality depends on the selected algorithm and the available measurements. The existing EIT algorithms include direct and indirect types [8]. The direct include the linear back projection (LBP) [9], algebraic computation [10], D-bar [11], and so on. The indirect usually rely on optimization techniques and related prior parameters, such as conjugate gradient (CG) [12], Tikhonov regularization (TR) [13], and the Landweber iteration (LW) [14]. But the EIT reconstruction quality depends heavily on the selected sensitivity matrix, and various matrices can lead to very different EIT reconstruction results [15]. To improve the ERT reconstruction quality, various EIT algorithms use different matrix transformations on the left or right side of the original sensitivity matrix as follows.

1) Left multiplication of a matrix. The typical example is the changed LBP algorithm that multiples a filtering matrix to the sensitivity matrix on the left. The one-step TR algorithm multiplies an invertible matrix on the left of the sensitivity matrix, while Ding et al. [16] described a nonlinear second-order sensitivity matrix and added one nonlinear item on the left of the sensitivity matrix. Liu et al. [17] began with the perspective of compressed sensing and represented the distribution of electrical parameters using a Fourier basis and reconstructed the sensitivity matrix. Generally, almost all EIT methods need to normalize a sensitivity matrix, which naturally multiplies the corresponding transformation matrix to the existing sensitivity matrix on the left [18].

Recently, Yue et al. [19] demonstrated that most existing EIT algorithms have a unified form, and their differences result from multiplying different transformation matrixes to the existing sensitivity matrix on the left. Therefore, the left multiplication of a selected matrix to the existing sensitivity matrix has become a useful way to improve EIT reconstruction quality.

2) Right multiplication by a matrix. Moura et al. [20] proposed a sparse reconstruction scheme using a redundant sensitivity matrix. The scheme utilized K-singular value decomposition and nonlinear simulation to develop a dictionary learning model with real data. Dong et al. [21] used a nonlinear symmetrical matrix to multiply the existing sensitivity matrix on the right, and the new sensitivity matrix not only addresses the limitations of the existing one but also is easily realized and controlled in practice via a kernel-width parameter. Barber et al. [22] originally proposed a filtering LBP algorithm that select an additional matrix to improve the sensitivity matrix.

The above progresses are credited to using the matrix transformation on the existing sensitivity matrix. However, given a group of EIT measurements, how to find the best matrix to the existing sensitivity matrix on the left or right remains unsolved to date, and lacks theoretical and practical guidelines. To address the issue, in this paper, we firstly demonstrate that the left and right multiplication of an existing sensitivity matrix by a selected diagonal matrix can reduce the condition number of the matrix, while the smaller condition number can increase the optimality and robustness of the EIT reconstruction results in the mathematical sense. Then, two types of transformation matrixes, Gaussian kernel and power function, are introduced to improve the EIT reconstruction quality and the robustness of the solution. The optimality and generalization of our proposed method have been verified by typical simulations.

II. Related Work

This section introduces the ERT principle, then introduces three typical ERT algorithms which used to validate the proposed methodology in this study, finally we introduce two works about right-multiplication for a sensitivity matrix.

A. EIT Principle

The EIT techniques mainly include electrical resistance tomography (ERT) [23], electrical capacitance tomography (ECT) [24] and electrical magnetic tomography (EMT) [25], all of which have similar mathematical expressions. Thus, we only emphasize ERT.

Let Ω represents a circular ERT detection field bounded by ∂Ω, to which m measurement electrodes are attached to reconstruct a frame of ERT image, as shown in Figure 1a.

We use a typical 16-electrode ERT system with adjacent excitation adjacent measurement way, and 208 measurements are obtained [26]. Partition Ω into n units (pixels) in which σ is the normalized conductivity vector, σ =(σ₁, σ₂, …, σ_n)^T. The ERT aims at determining the conductivity distributions of n units based on all measurements on ∂Ω: u₁, u₂, …, u₂₀₈, denoting the measurement vector as U=(u₁, u₂, …, u₂₀₈)^T. Based on the finite element method [27], the linearized relationship from σ to U can nearly be expressed as

U=Sσ, s.t., U∈∂Ω, σ∈Ω

(1)

where S is called as the sensitivity matrix in the EIT imaging process, S={s_ij}∈R^240×ⁿ, which reflects a linear map of unit (pixel) from σ to U. To solve σ along with U, S is usually determined in advance.

B. EIT Algorithms

The issue to solve the variable σ in (1) usually is turned to optimize the following objective function:

min z=||Sσ-U||

(2)

where || · || is a specified norm. The LW algorithm [28] is a typical iteration solution, and is formulated as

σ_{k + 1} = σ_{k} + ω S^{T} (S σ_{k} - U), k = 1, 2, \dots

(3)

where 0≤ω≤2/||S||. The LW algorithm can converge fast and is nonparametric.

The TR is another frequently used ERT algorithm for dealing with the ill-posed problem by regularization process, which solves σ by optimizing the objective function as

\min_{σ} z = {| | U - S σ | | + λ R (σ)}

(4)

where the regularization item R(σ) often is taken as ||σ||² to obtain an analytical solution (S^TS+λI)^-1S^T, which actually left-multiples a matrix (S^TS+λI)^-1 to the transpose of S. The λ in TR is a hyperparameter, and its different values may greatly affect the imaging qualities. The L-curve method [29] often is used to select the optimal parameter λ.

The ARW algorithm expands the exponent of the regularization term to (0, 2], which can greatly enhance solution sparsity and improve model robustness [30].

The objective function with the L_p norm can be expressed as

\min z = {| | U - S σ | |_{2}^{2} + λ | | σ | |_{p}^{p}}

(5)

The calculation steps of the algorithm are as follow,

1): Initialize D=I, t=0;
2): Update σ by σ=(S^TS+λD)^-1S^TU;
3): Calculate the diagonal matrix D, where the ith diagonal elements is d_i=p/2|σ|p-2
4): t=t+1, if t=T, stop, otherwise, return to 2).

In ARW algorithm, the results obtained from the previous iteration are incorporated as prior information into the computation of the next iteration, thereby enhancing the quality of reconstructed image.

The previous study [19] has concluded and demonstrated that most of the existing EIT algorithms can be generally formulated as

σ=M(S)U, s.t.,U∈∂Ω, σ∈Ω

(6)

where M(•) stands for various matrices along different EIT algorithms and M(S)={f(s_ij)}, where f(s_ij) is the computed sensitivity coefficients, i=1, 2, …, m; j=1, 2, …, n.

C. Right-Multiplication for a Sensitivity Matrix

Compared with the left multiplication, few researches focus on the right-multiplication for a sensitivity matrix. However, there are two remarkable progresses. One uses the following Gaussian kernel function to compute the distance between any pair of units(pixels)

R

and

R'

in Ω [21,31], it is

g (R, R') = e^{- | | R - R' | | / (2 Γ^{2})}

(7)

where Γ is the kernel width of the function. After denoting G ={g(R, R’)}, it updates S by S’, i.e.,

S' = S G, s . t ., S, S' \in R^{m \times n}, G = {σ (R_{i}, R_{i}^{'})} \in R^{n \times n}

(8)

Therefore, (1) is turned to solve the equation: S’σ=U. When G is a diagonally dominant matrix, as shown in Figure 1(b), the reconstruction quality based on S’ has been demonstrated to be higher than that based on S.

The other is the filtered back projection (FBP) algorithm [22]. According to the filtering principle for all measurements, FBP reconstructs any EIT image by

σ = B {(S B)}^{- 1} U ，

(9)

The equation shows that FBP firstly multiplies a selected matrix B on the right of S to filter all measurements U. Then, this filtered data is then back-projected to form the image σ.

However, there are three problems at least. First, when a group of EIT measurements are available, how to select the best matrix to S remains unsolved to date. Second, why the Gaussian kernel can improve the imaging quality is unclear due to the lack to theoretical and practical guidelines. Finally, it is unclear that whether the left and right multiplications can together improve the EIT reconstruction. In this paper, we first demonstrate that multiplying the matrix S on the left and right by diagonal matrixes can reduce its condition number and thus improve reconstruction quality. Then, under the basic form of the matrix, we improve the form of the right-hand matrix of S by introducing two typical kernel matrixes like the Gaussian kernel and power function. Finally, the optimalization and generalization of our proposed method are verified using the two kernel matrixes through typical experiments.

III. EIT Reconstruction by Matrix Transformation

In this section, we firstly manifest the practical and theorical meanings that using a pair of diagonal matrixes to multiply the sensitivity matrix S. Then, the two symmetric and diagonally dominant matrixes are used to improve the EIT reconstruction quality.

A. The Meaning of Multiplying a Diagonal Matrix to S

Let Ω be a detection field that is typically partitioned to 812 units (pixels) after using 208 measurements of 16 electrodes.

Multiplying two diagonal matrixes to the sensitive matrix S on the left and right refer to performing transformations on the columns and rows of S, as shown in Figure 2, where D₁=diag(D)=diag{d₁, d₂, …, d₂₀₈}and D₂=diag(K)=diag{c₁, c₂, …,c₈₁₂}.

When D₁ and D₂ are multiplied to S on the left and right, the matrix transformations have clear practical and theorical meanings. Let

S = {[{\vec{p}}_{1}, {\vec{p}}_{2}, \dots, {\vec{p}}_{208}]}^{T} = [{\vec{q}}_{1}, {\vec{q}}_{2}, \dots, {\vec{q}}_{812}]

, where

{\vec{p}}_{s}

and

{\bar{q}}_{t}

are the row vector and the column vector of S, respectively. The effect of both D₁ and D₂ are explained as follows.

1) Practical meanings. Note that

\{\begin{cases} D_{1} S = {[d_{1} {\vec{p}}_{1}, d_{2} {\vec{p}}_{2}, \dots, d_{208} {\vec{p}}_{208}]}^{T} \\ S D_{2} = [c_{1} {\vec{q}}_{1}, c_{2} {\vec{q}}_{2}, \dots, c_{812} {\vec{q}}_{812}] \end{cases}

(10)

Since

{\vec{p}}_{s}

points at the sth measurement, d₁ determines the importance degree of the measurement. It has been verified [33] that various measurements have different signal-to-noise ratios, and thus have different effects to reconstruct any ERT image. Alternatively,

{\bar{q}}_{t}

can endow various weights to different pixels to stress on these interesting objects.

2) Theorical meanings. Using these matrix transformations the condition number of S can be reduced. We use the matrix balancing theory [32] for solving (1), and thereby achieve more effective solutions, as explained below.

Let the two diagonal matrixes D₁ and D₂ multiply S individually on the left and right, and obtain an equivalent equation that has the same solution with (1) as follows,

(D_{1} S D_{2}) σ = D_{1} U, s . t ., U \in \partial Ω, σ \in Ω

(11)

where D₁ is constructed based on the following steps:

1): Calculate the norm of each row (commonly L₂-norm or L_∞-norm): s_i=||S_i·||, i=1, 2, …, m, where “S_i·” refers to the norm for each row vector in S);
2): Avoid zero norm: if s_i=0, d_i=1; Otherwise, d_i=1/s_i;
3): Generate the diagonal matrix:

D₁=diag{d₁, d₂, …, d_m}∈R^m^×m

(12)

Then, the diagonal matrix D₂ is constructed as follows:

1): Calculate the norm of each column (commonly L₂-norm or L_∞-norm): t_j=||S_·j||, j=1, 2, …, n, where “S_·j” refers to the norm for each column vector in S);
2): Avoid zero norm: if t_i=0, c_i=1; Otherwise, c_i =1/ t_i;
3): Generate the diagonal matrix:

D₂=diag{c₁, c₂, …, c_n}∈Rⁿ^×n

(13)

where both D₁ and D₂ must be normalized by dividing their maximum value to decrease in the condition number of S.

Based on the obtained D₁ and D₂, (1) is turned to

AY=B, s.t., A=D₁SD₂; Y=σ; B=D₁U

(14)

Consequently, σ is solved by (14).

The theoretical basis on using D₁ and D₂ to reduce the conditional number of S is illustrated as follows.

Lemma 1.

When B and A satisfy (14) based on D₁ and D₂, the following inequality holds:

\{\begin{cases} λ_{\max} (B^{T} B) \leq | | D_{1} | |_{2}^{2} λ_{\max} (A^{T} A) | | D_{2} | |_{2}^{2} \\ λ_{\min} (B^{T} B) \geq σ_{\min}^{2} (D_{1}) \cdot λ_{\min} (A^{T} A) \cdot σ_{\min}^{2} (D_{2}) \end{cases}

(15)

where

λ_{\max} (\cdot)

and

λ_{\min} (\cdot)

are the maximal and minimal eigenvalue of the relative matrix, respectively;

σ_{\max} (\cdot)

and

σ_{\min} (\cdot)

are their corresponding singular values.

The key to validating this lemma is that the scaling factor in D₁ and D₂ is usually less than 1, which can be achieved by normalizing the rows and columns of S.

According to Lemma 1, the following conclusion hold.

Theorem 1.

Let cond(B) and cond(A) be individual condition number of B and A. If B and A satisfy (14) based on D₁ and D₂, cond(B) <cond(A).

The proofs of Lemma 1 and Theorem 1 are provided in the Appendix in this paper. It ensures that the solution of (14) is more stable than that of the existing (1).

B. Realization of D₁ and D₂ by Diagonally Dominant Matrix

Equation (6) has shown that most existing EIT algorithms are actually realized by constructing various D₁ from the simplest normalization of S to complex iteration algorithms. Hence, selecting various forms of D₁ can lead to different algorithms to improve the EIT reconstruction quality.

In the following, as examples, we further approximate D₂ using two classes of diagonally dominant and symmetrical matrixes. One class is the previously used Gaussian kernel as shown in (7), and the other is the newly constructed power function. For any pair of units R, R’ in Ω, D₂={k(R, R’)}, and k(R, R’) is defined as

k (R, R') = {(1 + | | d (R, R') | |^{p} / c)}^{- 1}

(16)

where c is a constant to enlarge the resolution of d(R, R’). As an example, in this paper, we take c=0.001; p is a power parameter whose various values play vital key roles in improving the sensitivity matrix S.

Table 1 shows the three-dimensional distributions of D₂ when using (7) and (16) under various values of Γ and p. The sum of each column of the sensitivity matrix SD₂ in three dimensions Table 1 also is visualized. To clearly show the differences of D₂ or SD₂ as Γ or p changes, we also calculated their differences after and before multiplying D₂ to S. Table 1.

As Γ or p is changed, the distribution of D2 can mostly manifest a diagonally dominant form, leading to a sufficient approximation for a perfectly diagonal matrix. According to various forms of D2, we turn the issue to solve (1) to

D_{1} S D_{2} σ = D_{1} U, s . t ., D_{2} = {g (R, R')} o r {k (R, R')} \in R^{n \times n}

(17)

where g(R, R’) is computed by (7) and k(R, R’) by (15). In terms of the typical EIT algorithm, (17) can be solved by minimizing the following objective function,

\min z = | | D_{1} S D_{2} σ - D_{1} U | |

(18)

Any optimization algorithm principally is applicable to solve (18). Generally, when using the a diagonally dominant matrix, the following conclusion holds.

Theorem 2.

The solution of (18) has smaller condition number than that of (1) if using the L₂ norm.

The proof the theorem is shown in Appendix. On the other hand, the two diagonally dominant matrix include the parameter Γ and p, and we will demonstrate their effective range through the experiment below.

IV. Simulation and Evaluation

Simulations were performed using COMSOL 6.2 and MATLAB 2020a software on a PC equipped with an Intel^® Core™ i7 3.4GHz CPU and 20GB of RAM. The circular detection field Ω is divided into 812 pixels. The measurements are obtained by a 16-electrode ERT system when the excitation mode employs adjacent excitation adjacent measurement way, and 208 measurements can be obtained. The original sensitivity matrix is S∈R^208×812. Eleven simulation models with various features termed as M1~M11 are used to evaluate EIT quality, where the conductivity of blue background pixels in each model is set at 0.1 S/m, and the conductivity of red target pixels is set at 0.3 S/m.

To reduce the conditional number of S and improve the EIT imaging quality, D₁ is constructed as follows:

D_{1} = d i a g {d_{i}}, d_{i} = (1 / \sum_{j = 1}^{812} | s_{i j} |) / \max_{j} (1 / \sum_{j = 1}^{812} | s_{i j} |)

(19)

where the effect of D₁ is only the most used normalization form, while D₁ can respond to various EIT algorithms.

As the representative example of dominantly diagonal matrixes, D₂ uses the Gaussian kernel or the power function. All models are reconstructed by the three algorithms LW, TR and ARW using the same measurements, while the sensitivity matrix S is updated as D₁SD₂. Additionally, since the parameters in the TR can affect the reconstruction quality, thus we find their optimal value in the range [10⁻⁶, 10⁰], and fix the optimal value in all experiments. And the ARW has two hyperparameters which include regularization λ and norm of the regularization term, the range of regularization λ of ARW is as same as the TR and the range of the norm of regularization term p is [10⁻², 2].

Their imaging quality of all models are visually compared by target shapes, artifacts, positions, as well as two evaluation indexes RE and CC. Also, we take their best images with smallest RE and largest CC over all candidate values as comparisons.

The ERT quality is evaluated by two typical indexes [33]. One it the correlation coefficient (CC):

C C = \frac{{(Δ σ - Δ \bar{σ})}^{T} (Δ σ^{*} - Δ {\bar{σ}}^{*})}{\sqrt{| | Δ σ - Δ \bar{σ} | | \cdot | | Δ σ^{*} - Δ {\bar{σ}}^{*} | |}}

(20)

where ∆σ and ∆σ* represent the specific gray values of the reconstructed and original images, respectively. Both

Δ \bar{σ}

and

Δ {\bar{σ}}^{*}

are their respective means, respectively.

The other is the relative error (RE):

R E = | | Δ σ - Δ σ^{*} | | / | | Δ σ^{*} | |

(21)

The two indexes have typically and frequently been used to evaluate the ERT reconstruction quality.

Table 2 shows the reconstruction results along with eleven models M1~M11 and their relative CCs and REs, by LW, TR and ARW with D₁ and two different forms of D₂, respectively.

According to Table 2, all reconstructed images from the LW TR and ARW algorithms with D₁ and D₂ have higher imaging quality than those from the same algorithms with S itself, with more precise image targets and reduced artifacts, essentially for the LW algorithm, which is unable to fully reconstruct the targets for models M2, M3, M4, and M5 using the original sensitivity matrix S. However, by applying left- and right-matrix multiplication, the targets of these models can be successfully reconstructed, and image clarity has improved. For the TR algorithm, the original method exhibits significant central artifacts. The method proposed in this paper not only reduces these central artifacts but also enhances the contrast between the targets and the background. For the ARW algorithm, which is known for its superior performance in imaging small targets, there were still considerable artifacts at the edges of the targets. After employing the optimized method, the imaging of the target edges has become clearer, and the precision has been further enhanced.

As shown in the CCs and REs, the reconstructed image quality of all models using the proposed method has improved, with higher CC and lower RE. The red number in the table of each image shows its best value. With the increase in the target number and decrease in the target size, the CCs of all models decreased, while the REs increased, indicating lower imaging accuracy. However, after multiplying D₁ and D₂ on the left and right sides of S, no matter which Gaussian kernel or the power function is used, the accuracy of all the models has been significantly increased. These results show that the two types of matrices can effectively improve the quality of the reconstructed image.

Figure 3 shows the differences of condition numbers between the sensitivity matrix S and D₁SD₂ along with the eleven models. As seen, after multiplying D₁ and D₂ to S, the condition numbers of D₁SD₂ are lower than those of the original sensitivity matrix S. This result indicates that the left- and right-multiplying matrix can optimize solution stability and improve imaging accuracy from another perspective.

Among all models, the value of λ has been fixed at its optimal value. At the optimal λ, we multiply D₂ with the Gaussian or power functions to S. Various values of Γ or p can affect their relative EIT reconstruction quality. Figure 4 shows the range of Γ or p by which the three algorithms LW, TR, and ARW with D₁SD₂ can obtain higher CC value than them with S. Specially, the range nearly tends infinity in most models when using the power function to D₂. These results indicate that improving the condition number of S can greatly enhance the EIT reconstruction quality.

V. Conclusion

Most of the exiting EIT reconstruction algorithms are equivalent to multiply the existing sensitivity matrix on the left and right by two diagonal matrixes, thus these algorithms can be uniformly characterized by the matrix transformation. Note that the stableness and quality of the EIT reconstruction can be represented by the condition number of the sensitivity matrix, we demonstrate that the left and right multiplication of the sensitivity matrix by their respective diagonal matrix can reduce the condition number. Then, the EIT reconstruction quality can be effectively improved after multiplying the two diagonal matrixes. As examples, a Gaussian kernel-based symmetric matrix and a power function-based matrix are used to validate the proposed method. The experimental results further prove the potential of the new sensitivity matrix proposed in this study, which can provide a new theoretical and practical strategy for EIT reconstruction.

In the future, two issues must be focused on. One is how to obtain the better diagonally dominant matrix to reduce the condition number. The other is how to use the better matrix transformation for improving the EIT reconstruction accuracy and reliability. Such matrix transformations have a lot available, and we believe that their various forms can have potentials for further arising EIT reconstruction quality.

Acknowledgments

This work is supported by the National Science Foundation of China (No. 61973232).

Data Availability Statement

The data and code are available from the corresponding author upon reasonable request.

Declaration of Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Authors’ Contributions

First author: Visualization, Validation, Software. Second author: Writing – original draft, Methodology, review & editing, Visualization, Funding acquisition, Conceptualization. Third author: Validation, Data Curation.

Appendix

Lemma 1.

When B and A satisfy (14), the following inequality holds:

λ_{\max} (B^{T} B) \leq | | D_{1} | |_{2}^{2} λ_{\max} (A^{T} A) | | D_{2} | |_{2}^{2}, s . t ., B = D_{1} A D_{2}

(22)

Proof:

Since

B^{T} = {(D_{1} A D_{2})}^{T} = D_{2}^{T} A^{T} D_{1}^{T}

, thus

\begin{array}{l} B^{T} B = {(D_{1} A D_{2})}^{T} (D_{1} A D_{2}) = D_{2}^{T} A^{T} D_{1}^{T} D_{1} A D_{2} \\ = D_{2} A^{T} (D_{1} D_{1}) A D_{2} = D_{2} (A^{T} D_{1}^{2} A) D_{2} \end{array}

On the other hand, for semi-positive definite matrices S and arbitrary matrices T, their eigenvalues satisfy

λ_{\max} (T S T^{T}) \leq | | T | |_{2}^{2} λ_{\max} (S)

(23)

Denote the eigenvalue decomposition of S as

S = Q Λ Q^{T} s . t, Λ = d i a g {τ_{1}, τ_{2}, \dots, τ_{n}}

(24)

then

T S T^{T} = T (Q Λ Q^{T}) T^{T} = (T Q) Λ {(T Q)}^{T}

(25)

For any unit vector X, since

X^{T} (T S T^{T}) X = Y^{T} Λ Y, s . t, Y = T^{T} X,

(26)

thus

Y^{T} Λ Y \leq λ_{\max} (S) | | Y | |_{2}^{2} = λ_{\max} (S) | | T^{T} X | |_{2}^{2} \leq λ_{\max} (S) | | T^{T} | |_{2}^{2} | | X | |_{2}^{2}

(27)

Since ||T^T||₂=||T||₂, thus

X^{T} (T S T^{T}) X \leq λ_{\max} (S) | | T | |_{2}^{2} \Rightarrow λ_{\max} (T S T^{T}) \leq λ_{\max} (S) | | T | |_{2}^{2}

(28)

Denote T=D₂ and

S = A^{T} D_{1}^{2} A \Rightarrow B^{T} B = T S T^{T}

, thus B^TB satisfies

λ_{\max} (B^{T} B) = λ_{\max} (T S T^{T}) \leq | | D_{2} | |_{2}^{2} \cdot λ_{\max} (A^{T} D_{1}^{2} A)

(29)

On the other hand, since

A^{T} D_{1}^{2} A = {(D_{1} A)}^{T} (D_{1} A)

, its maximum eigenvalue is equal to the square of the maximum singular value of D₁A, i.e.,

λ_{\max} (A^{T} D_{1}^{2} A) = | | D_{1} A | |_{2}^{2}

(30)

Due to the multiplicative property of matrix norm, it is

| | D_{1} A | |_{2}^{2} \leq | | D_{1} | |_{2} | | A | |_{2}

(31)

it leads to

λ_{\max} (A^{T} D_{1}^{2} A) = | | D_{1} A | |_{2}^{2} \leq {(| | D_{1} | |_{2} | | A | |_{2})}^{2} = | | D_{1} | |_{2}^{2} \cdot | | A | |_{2}^{2}

(32)

But

| | A | |_{2}^{2} = λ_{\max} (A^{T} A),

thus

λ_{\max} (A^{T} D_{1}^{2} A) \leq | | D_{1} | |_{2}^{2} \cdot λ_{\max} (A^{T} A)

(33)

Integrating the two equations, it is

λ_{\max} (B^{T} B) \leq | | D_{2} | |_{2}^{2}

, thus

λ_{\max} (| | D_{1} | |_{2}^{2} \cdot λ_{\max} (A^{T} A)) = | | D_{1} | |_{2}^{2} \cdot λ_{\max} (A^{T} A) \cdot | | D_{1} | |_{2}^{2}

(34)

In the same way, we can obtain

λ_{\min} (B^{T} B) \geq σ_{\min}^{2} (D_{1}) \cdot λ_{\min} (A^{T} A) \cdot σ_{\min}^{2} (D_{2})

(35)

where

σ_{\min} (D_{i})

is the minimum singular value of D_i that is the minimum absolute value of diagonal elements.

Theorem 1

: Let cond(B) and cond(A) be individual condition number of B and A. If B and A satisfy (14) based on D₁ and D₂, cond(B) <cond(A).

Proof:

According to Lemma 1, we obtain

λ_{\max} (B^{T} B) \leq | | D_{1} | |_{2}^{2} λ_{\max} (A^{T} A) | | D_{2} | |_{2}^{2}, s . t ., B = D_{1} A D_{2}

(36)

and

λ_{\min} (B^{T} B) \geq σ_{\min}^{2} (D_{1}) \cdot λ_{\min} (A^{T} A) \cdot σ_{\min}^{2} (D_{2})

(37)

According to the two equations, we obtain

\begin{array}{l} c o n d (B) = \frac{λ_{\max} (B^{T} B)}{λ_{\min} (B^{T} B)} \leq \frac{| | D_{1} | |_{2}^{2} | | D_{2} | |_{2}^{2}}{σ_{\min}^{2} (D_{1}) \cdot σ_{\min}^{2} (D_{2})} \frac{λ_{\max} (A^{T} A)}{λ_{\min} (A^{T} A)} \\ = ω (D_{1}, D_{2}) \frac{λ_{\max} (A^{T} A)}{λ_{\min} (A^{T} A)} \end{array}

(38)

After choosing D₁ and D₂ such that

ω (D_{1}, D_{2})

is smaller than 1, the conclusion holds.

Theorem 2.

The solution of (18) has smaller condition number than that of (1) if using the L₂ norm.

Proof :

Consider the following optimization issue:

\min z = | | S σ - U | |_{2} .

(39)

If using the L₂-norm, this is a least squares problem, and has a direct solution as

σ^{*} = S^{+} U

(40)

where S⁺ is the Moore-Penrose inverse of S. If K is a symmetric matrix, the values near the diagonal are large, and the elements on the main diagonal are all the same and are the largest in each row,

\min z = | | S K σ - U | |_{2},

(41)

the minimum norm least squares solution is

σ_{1}^{*} = {(S K)}^{+} U

(42)

It is easy to know

S^{T} σ = 0 \Leftrightarrow S S^{T} σ = 0 \Leftrightarrow S K^{2} S^{T} σ = 0,

(43)

this indicates that if matrix S has r singular values, then matrix SK will also have r singular values.

Consider the singular value decomposition of S^T, it is

S^{T} = V [\begin{matrix} \tilde{S} & 0 \\ 0 & 0 \end{matrix}] U_{H},

(44)

where

\tilde{S} = d i a g (s 1, \cdot \cdot \cdot, s r, 0 \cdot \cdot \cdot, 0)

is the singular value of S, s₁ ≥ ··· ≥ s_r > 0, and let V = [β₁, ···, β_m], U=[α₁,···, α_n]. Then it can be known from the theory of singular values that

\{\begin{cases} S α j = s j β j, j = 1, \cdot \cdot \cdot, r, \\ S α r + 1 = \cdot \cdot \cdot = S α n = 0 . \end{cases}

(45)

Especially, the solution space of S^Tσ= 0 is

span \{α_{r + 1}, \dots, α_{n}\}

, then the solution space of the equation

S K^{2} S^{T} G = 0

is also

span \{α_{r + 1}, \dots, α_{n}\}

. Note that SK²S^T is positive definite, if

S K^{2} S^{T} y = η y, η \neq 0

, then for

j \in \{r + 1, \dots, n\},

SK²S^T satisfies

< n y, α_{j} > = < S K^{2} S^{T} y, α_{j} > = < y, S K^{2} S^{T} α_{j} > = 0,

(46)

so

y \in span \{α_{1}, \dots, α_{r}\}

Since K² is positive definite, we assume its minimum eigenvalue is η₀, which is better than 1 Thus, the minimum non-zero eigenvalue p₀ of SK²S^T satisfies,

p_{0} = \min_{σ \in span \{α_{1}, \dots, α_{r}\}} < S K^{2} S^{T} σ, σ > / < σ, σ >

(47)

If

x \in span \{α_{1}, \dots, α_{r}\}

, then

S^{T} σ \in s p a n {α_{1}, α_{r}}

, therefore

σ^{T} S K^{2} S^{T} σ \geq η_{n}^{2} • σ^{T} S S^{T} σ \geq η_{0}^{2} S_{r}^{2} | | σ | |^{2},

(48)

that is

\min_{σ \in span \{α_{1}, \dots, α_{r}\}} σ^{T} S Λ^{2} S^{T} σ / | | σ {| |}^{2} \geq η_{0}^{2} s_{r}^{2}

(49)

This indicates that the minimum eigenvalue p₀ of the matrix S^TK²S satisfies

p₀ ≥ η₀²s_r²

(50)

Thus, the minimum singular value of SK is larger than η₀s_r, and thus the condition number of SK is smaller than that of S.

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Figure 1. EIT measuring principle.

Figure 2. Illustration of left- and right- multiplication of S.

Figure 3. Compared condition numbers in 11 models.

Figure 4. Valid range of Γ or p by which the three algorithms have higher CC than the existing ones.

Table 1. AND SD₂ ALONG WITH Γ or p CHANGES.

Table 2. RECONSTRUCTION IMAGES BY BOTH LW AND TR ALGORITHMS WITH THE NEW SENSITIVITY MATRIX.

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The Best Sensitivity Matrix by Matrix Transformation in Electrical Impendence Tomography

Abstract

Keywords:

Subject:

I. Introduction

II. Related Work

A. EIT Principle

B. EIT Algorithms

C. Right-Multiplication for a Sensitivity Matrix

III. EIT Reconstruction by Matrix Transformation

A. The Meaning of Multiplying a Diagonal Matrix to S

B. Realization of D₁ and D₂ by Diagonally Dominant Matrix

IV. Simulation and Evaluation

V. Conclusion

Acknowledgments

Data Availability Statement

Declaration of Interests

Authors’ Contributions

Appendix

References

MDPI Initiatives

Important Links

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The Best Sensitivity Matrix by Matrix Transformation in Electrical Impendence Tomography

Abstract

Keywords:

Subject:

I. Introduction

II. Related Work

A. EIT Principle

B. EIT Algorithms

C. Right-Multiplication for a Sensitivity Matrix

III. EIT Reconstruction by Matrix Transformation

A. The Meaning of Multiplying a Diagonal Matrix to S

B. Realization of D1 and D2 by Diagonally Dominant Matrix

IV. Simulation and Evaluation

V. Conclusion

Acknowledgments

Data Availability Statement

Declaration of Interests

Authors’ Contributions

Appendix

References

MDPI Initiatives

Important Links

Subscribe

B. Realization of D₁ and D₂ by Diagonally Dominant Matrix