Total variation (TV) regularization method is a typical method to preserve the discontinuities structure in EIT. Isotropic TV and anisotropic TV are two well-known variants of TV. The main differences between them are that the latter tends to distort the reconstructed internal inhomogeneities along the coordinate axis. In this article, we adopt the alternating direction method of multipliers (ADMM) to overcome the non-differentiability of the anisotropic TV and verify the characteristics of anisotropic TV regularization by the tank experiments.
Electrical Impedance Tomography (EIT) is a method of reconstructing interior conductivity distribution of the imaging target. In EIT, a series of low frequency current is injected through the electrodes attached around the boundary and we measure the induced voltages. Compared with the widely used imaging methods, such as computed tomography (CT), magnetic resonance imaging (MRI) and ultrasound imaging, EIT has a promising clinical application due to its advantages of non-invasive, no radicalization, high time resolution, etc. EIT also has very important applications in other fields, for example, to locate high flow drilling [
However, EIT is a typical ill-posed problem. To be precise, the voltage response due to the conductivity changes decreases rapidly with the distance between the sources of conductivity changes increases [
Depending on the prior information of the target conductivity distribution, the researchers proposed Tikhonov regularization [
There exists difficulty in using the TV regularization due to its non-differentiability structure [
There are two common variants of TV, namely the isotropic TV and the anisotropic TV [
In this article, we use an iteration scheme based on the alternating direction method of multipliers (ADMM) to overcome the non-differentiability of anisotropic TV regularization method and do the tank experiment to verify its property of distortion.
To simplify EIT imaging process, we assume that we use 16-channel EIT system and adopt adjacent current injection and measurement pattern. To be specific, low-frequency current with magnitude I is injected through j-th neighboring electrodes pair ( E j , E j + 1 ) for j = 1 , ⋯ , 16 around the boundary of imaging target Ω , where we denote E 16 + 1 = E 1 . Then the potential distribution of j-th injection u j is governed by the following equations [
{ ∇ ⋅ ( σ ∇ u j ) = 0, in Ω σ ∇ u j ⋅ n = 0, on ∂ Ω \ ∪ i = 1 16 E i ∫ E j σ ∂ u j ∂ n d s = I = − ∫ E j + 1 σ ∂ u j ∂ n d s u j | E i = constant , i = 1,2, ⋯ ,16 ∫ E i σ ∂ u j ∂ n d s = 0, i ∈ { 1,2, ⋯ , E } \ { j , j + 1 } .
Here, n is the outward unit vector to ∂ Ω , ds is the surface element. The voltage between ( E i , E i + 1 ) subject to the j-th injection can be measured:
V j i [ σ ] = u j | E i − u j | E i + 1 .
Using EIT scanner we can measure the following datum:
V = [ V 1 3 , ⋯ , V 1 15 , V 2 4 , ⋯ , V 2 16 , ⋯ , V 16 2 , ⋯ , V 16 14 ] T ∈ R 16 ( 16 − 3 ) .
where we neglect the measurement near the driving electrode to minimize the measurement error.
The relation between σ and V j i can be expressed approximately by the reciprocity principle
V j i [ σ ] = V i j [ σ ] = 1 I ∫ Ω σ ( r ) ∇ u i ( r ) ⋅ ∇ u j ( r ) d r , (1)
where r = ( x , y , z ) is a position inside Ω . The EIT problem is to reconstruct the conductivity distribution σ using the measured voltage datum V and the relation (1). However, the above equation is nonlinear. We consider to linearize the above equation. To be specific, assuming σ ( r ) = σ 0 + δ σ ( r ) , we can approximate δ σ by replacing u j by u j 0 :
δ V j i [ σ ] : = V j i [ σ ] − V j i [ σ 0 ] ≈ 1 I ∫ Ω δ σ ( r ) ∇ u i 0 ( r ) ⋅ ∇ u j 0 ( r ) d r , (2)
where u j 0 the potential computed under the reference conductivity σ 0 . For the purpose of computerized reconstruction, we discretize the imaging target into finite element elements ( Δ k , k = 1 , 2 , ⋯ , N ). Assume that on each element Δ k the conductivity is a constant. Let σ k denote the value of σ on the kth element. Then σ can be approximated by σ : = ( σ 1 , σ 2 , ⋯ , σ N ) T ∈ R N . Thus, (2) can be written as
S δ σ = δ V ,
where S is the sensitivity matrix (or jacobian matrix) given by
S = ( ⋮ ⋯ S k ⋯ ⋮ ) ∈ R 208 × N
with S k : = 1 I ∫ Δ k s ( r ) d r , s ( r ) : = [ s 1,3 ( r ) , s 1,4 ( r ) , ⋯ , s 16,14 ( r ) ] T , s i , j = ∇ u i 0 ( r ) ⋅ u j 0 ( r ) .
In practice, the number of measured data 16 × ( 16 − 13 ) is fewer than N (the total number of elements for δ σ ). We can find an estimate of δ σ by minimizing the following least square problem:
δ σ * = arg min δ σ ∈ R N ‖ S δ σ − δ V ‖ L 2 ( Ω ) 2 .
Since the above problem is ill-posed, the regularization technique is widely used to deal with this difficulty. The most common used regularization method is Tikhonov regularization (TR), which is to solve
δ σ λ * = arg min δ σ ∈ R N { 1 2 ‖ S δ σ − δ V ‖ 2 + λ ‖ K ( δ σ − δ σ 0 ) ‖ 2 } ,
where K is the regularization matrix, λ is the regularization parameter. However, it has an excessively smooth effect on the solution, which will blur the edge of reconstructed image.
One technique to preserve the discontinuous boundary is the Total Variation (TV) regularization. It was first introduced in [
δ σ λ * = arg min δ σ ∈ R N { 1 2 ‖ S δ σ − δ V ‖ 2 + λ ‖ | D x δ σ | 2 + | D y δ σ | 2 ‖ l 1 } , (3)
and anisotropic TV
δ σ λ * = arg min δ σ ∈ R N { 1 2 ‖ S δ σ − δ V ‖ 2 + λ ( ‖ D x δ σ ‖ l 1 + ‖ D y δ σ ‖ l 1 ) } , (4)
where D x , D y ∈ R N × N are the first order discrete partial derivative operators in the horizontal direction and the vertical direction respectively.
To solve the non-differentiability problem (3) and (4), an auxillary smoothing parameter β > 0 was introduced in the time marching method for the corresponding Euler-Lagrange equation [
{ δ σ λ * = arg min δ σ { 1 2 ‖ S δ σ − δ V ‖ 2 + λ ‖ u ‖ l 1 } , s .t . u = D δ σ . (5)
The augmented Lagrangian functional for (5) is defined as:
L α ( δ σ , u ; v ) : = 1 2 ‖ S δ σ − δ V ‖ 2 + λ ‖ u ‖ l 1 + α 2 ‖ D δ σ − u ‖ l 2 2 + v T ( D δ σ − u ) ,
where the variable v ∈ R 2 N is the Lagrangian multiplier, α > 0 is a penalty parameter.
By updating δ σ and u in an alternate order, we can get its ADMM iteration scheme:
{ δ σ n + 1 = arg min δ σ L α ( δ σ , u n , v n ) ; u n + 1 = arg min u L α ( δ σ n + 1 , u , v n ) ; v n + 1 = v n + α ( D δ σ n + 1 − u n + 1 ) .
The detail of solving the above equations can refer to [
In this section, we compare the performance of isotropic TV using PDIPM and anisotropic TV using ADMM by tank experiments. The results in this article are carried out by EIDORS [
In order to quantitatively evaluate the regularization method, two indicators are introduced, respectively R E ( n ) and P S N R ( n ) . Since we don’t know the true conductivity in the tank experiments, we use the conductivity reconstructed by TV regularization as a reference value. We also compare the time spent in the reconstruction process.
We did two groups of tank experiments using the Sciospec 16-channel EIT system [
In the first group of experiments, carrot, cucumber and bean curd are placed inside the tank, separately. In the second group of tests, carrot, cucumber and bean curd are placed inside the tank in pairs. The parameters for the anisotropic TV method are set to λ = 1 × 10 − 8 , α = 1 × 10 − 7 . The parameters of TR and isotropic TV are set to be optimal empirically. The reconstructed images of single inclusion using Tikhonov regularization (TR), isotropic TV and anisotropic TV are shown in
There are several observations from the results. All the three regularization methods can well capture the main feature of the inner object, including position and shape. Numerical indicators show that using ADMM to solve the anisotropic
| TR | anisotropic TV | |
|---|---|---|
| RE | ||
| bean curd | 0.1216 | 0.0359 |
| carrot | 0.0779 | 0.0395 |
| cucumber | 0.0273 | 0.0363 |
| PSNR | ||
| bean curd | 31.02 | 40.76 |
| carrot | 32.96 | 39.08 |
| cucumber | 37.82 | 35.49 |
| TR | anisotropic TV | |
|---|---|---|
| RE | ||
| bean curd and carrot | 0.0950 | 0.0495 |
| bean curd and cucumber | 0.0870 | 0.0475 |
| cucumber and carrot | 0.0497 | 0.0404 |
| PSNR | ||
| bean curd and carrot | 30.67 | 36.85 |
| bean curd and cucumber | 31.49 | 37.12 |
| cucumber and carrot | 33.81 | 35.65 |
| TR | isotropic TV | anisotropic TV | |
|---|---|---|---|
| bean curd | 3.38 | 3.122 | 1.021 |
| carrot | 3.49 | 3.141 | 1.064 |
| cucumber | 3.32 | 3.148 | 1.122 |
| bean curd and carrot | 2.61 | 3.173 | 1.034 |
| bean curd and cucumber | 3.45 | 3.213 | 1.022 |
| cucumber and carrot | 3.42 | 3.165 | 1.036 |
TV regularization problem can obtain an accurate image and take less time than using PDIPM. However, the images reconstructed by isotropic TV have obvious ladder effect. As a result, there exists pseudo edge. As expected, the edges of the reconstructed images using the anisotropic TV distort along the coordinate axes, whether it has one or two inclusions.
In this article, we use experimental data to verify the difference between isotropic TV and anisotropic TV. Experimental results demonstrate that the reconstructed image with anisotropic TV regularization will cause geometric distortions along the coordinate axis. This proves that it is the model itself that causes the distortion. In the future work, we will focus on a method that can avoid the distortions along the coordinate axis and do not depend on the selection of regularization parameters.
The author would like to thank editor and referees for their valuable advice for the improvement of this article.
The author declares no conflicts of interest regarding the publication of this paper.
Wang, Y.Y. (2022) Anisotropic TV Regularization in Electrical Impedance Tomography: An Experimental Study. Engineering, 14, 138-146. https://doi.org/10.4236/eng.2022.143013