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Levenberg-Marquardt algorithm for acousto-electric tomography based on the complete electrode model

12/17/2019
by   Changyou Li, et al.
0

The inverse problem in Acousto-Electric tomography concerns the reconstruction of the electric conductivity in a domain from knowledge of the power density function in the interior of the body. This interior power density results from currents prescribed at boundary electrodes (and can be obtained through electro-static boundary measurements together with auxiliary acoustic measurement. In Electrical Impedance Tomography, the complete electrode model is known to be the most accurate model for the forward modelling. In this paper, the reconstruction problem of Acousto-Electric tomography is posed using the (smooth) complete electrode model, and a Levenberg-Marquardt iteration is formulated in appropriate function spaces. This results in a system of partial differential equations to be solved in each iteration. To increase the computational efficiency and stability, a strategy based on both the complete electrode model and the continuum model with Dirichlet boundary condition is proposed. The system of equations is implemented numerically for a two dimensional scenario and the algorithm is tested on two different numerical phantoms, a heart and lung model and a human brain model. Several numerical experiments are carried out confirming the feasibility, accuracy and stability of the methods.

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1 Introduction

Electrical Impedance Tomography (EIT) is an emerging technology for obtaining the internal conductivity of a physical body from boundary measurements of currents or voltages on the surface of the body Holder2005 ; Yorkey1990 ; Calderon1980 . It is an ill-posed problem due to the fact that boundary measurements show little sensitive to (even large) changes of interior conductivity distribution Ammari2008 . Intensive research exists on this topic Uhlmann2009 ; Brown2009 ; many regularization methods have been proposed to overcome the ill-posedness and to improve the imaging quality Han1999 ; Hsiao2001 ; Chung2005 ; Knudsen2009 .

More recently it has been suggested to augment the measurement setup in EIT with an ultra-sonic device thus yielding the hybrid imaging method known as Acousto-Electric tomography (AET) Ammaribook2008 . The resulting modality has been investigated theoretically and numerically, and AET seems to have the potential to dramatically increase the contrast, resolution, and stability of the conductivity reconstruction Bal2012 ; Kuchment2012 .

The idea of AET it is to conduct a usual EIT experiment while a known focused ultrasonic wave travels through the object. The high intensity of the acoustic pressure will create a small local deformation in the physical body and thus of the electrical conductivity due to the acousto-electric effect Geselovitz1971 .

A physical body imaged by EIT is modeled as a bounded Lipschitz domain for . The changes caused by the acoustic wave can be recorded by EIT measurements on the boundary Jossinet1999 ; Jossinet2005 . The power density in is then

(1)

Here, is the conductivity and is the electrical potential inside produced by applying an electric field on the boundary . We call the power density operator. With the assumption that measurements are carried out in low temporal frequency and that contains no interior sources or sinks of charges the governing equation is the generalized Laplace equation

(2)

subject to suitable boundary conditions. Given noisy measurements of the true power densities the problem is to reconstruct We approach the problem by optimization

(3)

In AET the power density can be computed from the boundary measurements Ammari2008 ; Kuchment2012 ; Jensen2019 . The non-linear relationship between and renders the problem nonlinear Song2017 . Some methods have been developed in the literature for reconstructing from Ammari2008 ; Kuchment2012 ; Song2016 ; Song2017 ; Ammaribook2008 ; Bal2012 ; Bal2013 , but all these methods are based on the continuum model with Dirichlet or Neumann boundary conditions. See also Adesokan2018 ; Jensen2018 ; roy2018a ; gupta2019a for optimization approaches to the problem different from (3). The limited data problem was considered in Hubmer2018 .

The Complete Electrode Model (CEM) is a practical model for EIT, and it can simulate the electrical potential with a very good accuracy Somersalo1992 . In the model, electrodes are attached on boundary . A known total current injected through the -th electrode is given as

(4)

Here,

is the outward unit normal vector to the boundary

. is the -th electrode, and indicates the derivative of in the direction of the outward unit normal vector . Since there is no current flowing out through boundary regions without electrodes, one has

(5)

On the electrode the electric potential is assumed to be constant. This boundary potential consist of a part due to the interior potential and a part due to the electrode contact, and it is comprised in the model

(6)

where denotes the so-called contact impedance assumed to be constant on the -th electrode. The partial differential equation (2) with boundary conditions (4)-(6) gives the CEM. To ensure existence and uniqueness of the solution, this model also needs to include the law of charge conservation

(7)

and to determine the potential’s grounding by

(8)

The CEM problem (2)-(6) has a unique solution for any Hyvonen2017 , however the mixed boundary conditions (5)-(6) allow singularities near the edges of the electrodes. In order to increase the regularity of the electrode conductance is introduced as a function

(9a)
(9b)
(9c)

Assuming that for some then Hyvonen2017 . In particular for , and . We call (9) the smoothed CEM (SCEM).

The main aim of this paper is to develop an iterative method for reconstructing from Inspired by Bal2012 the method will be based on the Levenberg-Marquardt method for solving the least squares problem (3). The main novelty here being the use the SCEM to accurately and stably model electrodes and the electric current in the forward problem. In addition, a reconstruction strategy based on a combined use of both CEM and Continuum model with Dirichlet boundary condition (DCM) is proposed to increase the computational efficiency.

The outline of the paper is as follows. In Section 2, the Levenberg-Marquardt method is briefly introduced. The non-linear problem is linearized, and the adjoint problem is setup. In Section 3, the iterative reconstruction method is developed based on CEM and Levenberg-Marquardt iteration. A linear system is built to calculate the updating step for each iteration. The algorithm for increasing efficiency by exploiting DCM is also introduced in this section. These algorithms are implemented and applied to reconstruct the conductivity distribution of several phantoms in Section 4, numerical performances of different algorithms are discussed in detail. The conclusion of the presented work is given in Section 5.

2 Reconstruction algorithm

In this section we will first recap the Levenberg-Marquardt Algorithm for solving (non-linear) optimization problems. Then we will for the particular problem in AET with CEM derive the necessary ingredients, that is the Fréchet derivative of and it’s adjoint.

2.1 The Levenberg-Marquardt Algorithm

Let be a (possibly non-linear) operator between Hilbert spaces and For some with the problem is to solve at least approximately the equation and often the minimization problem

(10)

is considered. If is (Fréchet) differentiable, the linear approximation

yields the iterative scheme

solved by the Newton algorithm

This is applicable only when is left-invertible. In general one can instead minimize the following Tikhonov functional Kaltenbacher2008

(11)

where is the regularization parameter; this minimization problem is solved by the Levenberg-Marquardt Algorithm (LMA)

(12)

Here, is the adjoint of and is identity operator. When the operator satisfies a certain non-linarity condition, a proper choice of the parameter and the initial guess sufficiently close to the desired solution, the LMA converges to a solution of (10).

The LMA can be thought of as a combination of steepest descent and Gauss-Newton method. When the current solution is far from the correct one, a large value is assigned to , and LMA behaves like a steepest-descent method which converges slowly. When the current solution is close to the correct solution, a small is used, and LMA behaves like a Gauss-Newton method, which has faster convergence.

2.2 The Fréchet derivative and its adjoint

In the present work we consider the operator in (1). In order to apply (12) we therefore need to calculate the Fréchet derivative of the power density operator at and its adjoint

We start with the Fréchet derivative of i.e. the operator The Fréchet derivative can be calculated (see the general approach in Lechleiter2008a ) in the following way:For a given with compact support inside the difference is approximated by a function that is linear in Indeed, define as the solution to the modified CEM problem

(13a)
(13b)
(13c)

With the grounding (13) has a unique weak solution which obviously is linear with respect to . Moreover,

showing that is indeed the Fréchet derivative of at

The Fréchet derivative of in the direction is obtained as in Bal2013 by

(14)

We now compute the adjoint first as an operator in Consider for some

(15)

We focus on the second part of the integral (the first part is self-adjoint). Introduce the pair and defined by the weak PDE form (for all )

(16)

The strong form reads

Then insert the choice in (16) to calculate the latter term in the RHS of (15)

where the last equality follows from the weak form of (13). Thus we find the form

that is

(17)

To get to the adjoint in higher order Sobolev spaces for the chosen (e.g.  in 2D) we lift the operator to higher order spaces, i.e. we solve for

This is a fourth order PDE problem when With an embedding operator , can be written for as

(18)
(19)

where is the -adjoint of .

We now restrict ourselves to i.e.  and consider the presence of noise. Since noise can not be assumed to be differentiable . Since is a bounded operator, its Hilbert adjoint is bounded as well Kreyszig1978 . With an embedding operator , can be written as

(20)
(21)

where is the -adjoint of . .

Instead of directly calculating in , the form of in is firstly found, which is given as

(22)
(23)

Here, with be a linear operator defined by

(24a)
(24b)
(24c)
(24d)

3 Iterative reconstruction algorithm based on LMA

According to the Levenberg-Marquardt iteration given in (12), the formulation for calculating the -th updating step for the presented problem is explicitly given as

(25)

with

(26)

Here, is easily obtained with (14), (21), and (23). After computing from (25), the conductivity map obtained from the -th iteration is updated by for a new iteration. All PDEs are coupled and collected into the PDE system

(27a)
(27b)
(27c)
(27d)
(27e)
(27f)
(27g)
(27h)
(27i)
(27j)
(27k)
(27l)
(27m)
(27n)
(27o)
(27p)
(27q)

with , , , , , and . With the number of measurements , the system is formulated with and . Since equations (27g)-(27o) need to be solved for each measurement, one additional measurement will need 3 additional partial differential equations.

It may be possible to simplify the above PDE system if the interior potential near the measurement boundary and the measured potential both turn out to converge significantly faster (with the number of iterations of the solution procedure) than the conductivity estimate

. In this case, we would expect and to hold early in the iteration. From (27h), we would then get that the change in the current at any resulting from a change in is approximately zero after only a few iterations, and conditions (27h) and (27i) might justifiably be substituted with the much simpler Dirichlet boundary condition on .

The iterative reconstruction method based on the above system is here named as LM-SCEM which is demonstrated in Algorithm 1 for a single measurement. The measured data of is simulated with SCEM in this paper. The relative error is given by where and denotes the true and reconstructed conductivities. The parameter should theoretically be updated according to the value of . If leads to a reduction of the relative error in , is decreased and is accepted. Otherwise, is discarded and is increased. Since can not be determined in practice, a relatively large value is asisgned to in this paper, and is slowly decreased to ensure convergence. The iteration is stopped when the -norm of is smaller than a given value or the maximum number of iteration is achieved.

Data: The measured power density and an initial guess
Result: The reconstructed conductivity map with a relative error
1 : the maximum number of iterations;
2 ;
3 ;
4 ;
5 ;
6 while  and  do
7       Update from with CEM;
8       Compute from ;
9       Compute Compute with the linear system defined by (27);
10       ;
11       ;
12       ;
13       Update ;
14      
15 end while
Algorithm 1 The LM-SCEM algorithm for reconstructing the conductivity map from single measurement of power density.

3.1 The LM-DCM method and the mixed reconstruction algorithm

If DCM is considered instead of SCEM, the system for reconstructing can be built by a similar calculation Bal2013 . The resulted system for computing remains the same as the one for LM-SCEM except that the boundary conditions (27h) and (27i) need to be replaced with on . The computation of in LM-CEM is actually expensive because of the additional unknowns in . To obtain a good reconstruction accuracy of LM-SCEM, multiple measurements are usually considered, but the computational efficiency will decrease quickly with the increasing number of measurements. Our investigation on the convergence of boundary potential and the conductivity shows that the boundary potential converges much faster, examples are given in numerical experiments. This is mainly because EIT measurement is not very sensitive to the internal change of the conductivity distribution Ammari2008 . This property renders EIT an ill-posed problem, but it will be taken as the foundation here to build a faster reconstruction approach by mixing LM-SCEM and LM-DCM, which is abbreviated by LM-SCEM-DCM.

This mixed reconstruction method is illustrated in Algorithm 2 for a single measurement. Here, , which is a vector composed of the voltages on the electrodes. The true values can be measured, which is produced when simulating the power density with CEM, therefore no additional computation is required. The information of is here exploited to define a stopping criteria for reconstructing the boundary potential with LM-SCEM. With a relative error given by , the value of is checked in each iteration, and LM-SCEM is terminated when an expected relative error is achieved. Since the regularity of potentials computed from CEM is not so good on , a smaller region is defined with a small which can “smooth” out the possible irregularity of close to . The boundary potential is here defined on and used for the reconstruction with LM-DCM. The power density in is firstly reconstructed from . This can be achieved with the method introduced in Ammari2008 but will be simulated with DCM here. Because there are no additional unknowns in LM-DCM, the computation will be more efficient, especially for the computation with multiple measurements. Meanwhile, the conductivity map produced by LM-SCEM is used as the initial guess for LM-DCM. This good initial guess will also help LM-DCM converge faster. Therefore, the mixed reconstruction method can provide a practical and efficient computational model for AET.

Data: The measured power density and the voltage vector on electrodes. An initial guess and an expected relative error for
Result: The reconstructed conductivity map with a relative error
1 : the maximum number of iteration for LM-CEM;
2 : the maximum number of iteration for LM-DCM;
3 ;
4 ;
5 ;
6 while  and  do
7       Update and from with CEM;
8       ;
9       if  then
10             ;
11             ;
12             break;
13            
14       end if
15      Update with LM-CEM;
16       ;
17       ;
18      
19 end while
20Reconstruct from in domain ;
21 ;
22 ;
23 ;
24 while  and  do
25       Update from with DCM;
26       Update with LM-DCM;
27       ;
28       ;
29      
30 end while
Algorithm 2 The LM-CEM-DCM algorithm for reconstructing the conductivity map of a domain from a single measurement of power density.

4 Numerical investigation

4.1 Phantom preparation and numerical setup

The variational forms of the linear systems defined by equations (27) in Section 3 can easily be obtained through integration by parts. They are solved with a mixed finite element method Boffi2013 which is implemented using FEniCS AlnaesBlechta2015a . To illustrate the stability and accuracy of the presented approaches we the focus on numerical examples in a 2-dimensional (2D) problem. Two phantoms are considered.

The first example is a heart-lung model Zlochiver2003 , see Figure 0(a). The considered three tissues are heart (red, ), lung (cyan, ), and soft-tissues (blue, ). The model is placed into a circular region with a background material (white, ) and a radius .

(a) Heart-lung model
(b) Human brain model
Figure 1: The 2D (a) heart-lung model and (b) human brain model embedded in a background material with electrodes (red squares) attached to the boundary (the solid black line). Different regions are marked with different colors. In (a), there are heart (red, ), lung (cyan, ), soft-tissues (blue, ), and background material (white, ). In (b), there are scalp (green, ), skull (blue, ), cerebro-spinal fluid (red, ), gray matter (yellow, ), white matter (cyan, ) and the background mateiral (white,

). All electrodes are uniformly distributed with the same corresponding central section angle. It is assumed that the electrical conductivity

in the region close to boundary (between solid and dashed black lines) is known.

The second example is the human brain model shown in Figure 0(b). The considered tissues in this model include scalp (green, ), skull (blue, ), cerebro-spinal fluid (red, ), gray matter (yellow, ) and white matter (cyan, ). Refer to Andreuccetti1997 for conductivities of different tissues. The shape of this model is close to an ellipse whose semi-major and semi-minor axes are and . The model is placed in an ellipse region with a background material (white, ). The semi-minor and semi-major axes of the region are and , respectively. The conductivity maps for the phantoms are piece-wise constant functions which will be mollified with

(28)

to produce with for 2D problem. The constant is selected so that . The value of is and for heart-lung model and human-brain model, respectively. The true smoothed distributions of for the two models are shown in Figure 2.

(a) Heart-lung model
(b) Human-brain model
Figure 2: The true distribution of (a) for the 2D heart-lung model shown in Figure 0(a) and (b) for the 2D human brain model shown in Figure 0(b)

To work with CEM and LM-SCEM, 16 electrodes (red rectangles shown in Figure 1) are uniformly attached on the boundary (solid black lines). The section occupied by each electrode has the same central angle in both the heart-lung and human-brain models. The following computations assume that the conductivity in a small region close to boundary (between solid and dashed black lines) is known. The distance between the solid and dashed lines is given by . This known region helps to improve the convergence of the algorithm. Three current patterns based on Fourier basis functions are used in the computations, which are for , and . The regularization parameter is chosen to decrease exponentially, and with . In what follows, a relatively large value is given to , and a value close to 1 is given to for a slow decreasing of to ensure the convergence of the iterations.

4.2 Performance of LM-SCEM

Numerical experiments on heart-lung model are carried out here to investigate the performance of LM-SCEM. A mesh of the circular domain with 77101 triangles is used for the reconstruction. The power density for each current pattern is simulated with CEM, and Gaussian white noise is added to avoid an inverse crime. Here, the noise level is measure with signal to noise ratio (SNR)

, where is a Gaussian white noise distribution. The Levenberg-Marquardt iteration is stopped when or total number of iterations greater than 15. These values were chosen to balance the quality in the reconstructions versus the computational speed. The reconstruction with different current patterns and different level of noise are considered to check the convergence of and the stability of LM-SCEM. The parameters for the reconstruction are given as , , and . These parameters are chosen to ensure that the reconstruction with current pattern converges. Other measurements are taken into the reconstruction without changing the parameters.

To simulate the power density , SCEM is used to yield better regularity of the electrical potential. The electrode conductance of is chosen as

(29)

Here, is the length of the electrode, which is proportional to the central angle corresponding to the electrode. is then scaled to have the required maximum value. The distribution of for the calculation in this paper is given in Figure 2(a) with a maximum value 1. The power density simulated with SCEM is given in Figure 2(b) for a region near one electrode. Here, the current pattern is . A smooth distribution of near the edge of the electrode is observed. For the same region, the power density simulated from CEM is also given in Figure 2(c) with for . Two singularity points are observed at the edges of the electrode. These singularities will cause instability in the reconstruction, and therefore SCEM is used in LM-SCEM.

(a) The distribution of
(b) from SCEM
(c) from CEM
Figure 3: (a) The distribution of on each electrode. (b) The power density calculated from SCEM with the distribution of given in (a). (c) The power density calculated from CEM with

The LM-SCEM algorithm is used to reconstruct the distribution of . The initial guess is given as which is the value of the background tissue. The conductivity in the region for is supposed to be known. The values of are truncated to only update for with . The discontinuity caused by this truncation can be removed by applying (28) properly (either by mollification or simply by replacing the discontinuous values), but it does not cause any numerical problems since is small, so no special treatment was done in the following computation. The values of for the first 15 iterations are shown in Figure 3(a). The relative error is also given in Figure 3(b). With noise, the reconstruction with uniformly converges to with 15 iterations. The reconstructed is shown in Figure 4(a). To achieve a level of , it takes more than 40 iterations. A reconstruction with and is then carried out, but a similar speed of convergence and relative error is observed, result is in Figure 4(b). When the current pattern is further considered into the reconstruction, an obvious improvement of convergence is seen, and a relative error level is achieved with 14 iterations, the conductivity map is shown in Figure 4(c). Therefore, the convergence of LM-SCEM depends not only on the regularization parameter and the scaling parameter , but also on the current patterns for the measurements. Since indicates 0.1% of Gaussian noise in the simulated , the reconstructed result with is already good, and more iterations will not improve the result. To verify, a reconstruction with , and is carried out with (1% noise). Relative errors are shown in Figure 3(b), the reconstruction converges to within 7 iterations, and more iterations did not bring any improvements. The reconstructed result is given in Figure 4(d).

(a)
(b)
Figure 4: (a) The variation of . (b) The relative error of the reconstruction. With noise, a uniform convergence is observed. With noise, the reconstruction almost uniformly converges to with 7 steps.
(a) , SNR =
(b) , SNR =
(c) , SNR =
(d) , SNR =
Figure 5: The conductivity map reconstructed from LM-SCEM with current pattern (a) with noise, (b) with noise, (c) with noise, and (d) with noise.

4.3 Performance of the mixed reconstruction approach

Though LM-SCEM performs well for reconstructing the conductivity map, its efficiency of the computation decreases quickly with the increase of measurements. Since EIT is not very sensitive to the change of interior conductivity, the electrical potential should converge faster than the convergence of the conductivity. The human-brain model is used here for numerical experiments with LM-SCEM. The elliptic domain is characterized with major and minor axes, and the domain is meshed with 36893 triangular elements. The parameters are given as , and . The current patterns and are used in the reconstruction. With noise, the iteration is terminated when or the maximum number of iterations equals 30. The reconstruction based on LM-SCEM is shown in Figure 6(a). of each current pattern is computed with SCEM and for -th iteration, easily follows then. The variation of the relative error and are shown in Figure 5(a) and Figure 5(b), respectively. As can be seen, slowly converges to . This error is much larger than the input noise level, this is mainly caused by the complexity of the phantom and the high contrast of among different tissues. But the potentials on the boundary for both and converge fast to a level of in few iterations. Therefore, the boundary potential converges much faster. This property is exploited here to accelerate the computation by mixing LM-SCEM and LM-DCM, as demonstrated in Algorithm 2. In this computation, the LM-SCEM is stoped when for all current patterns are smaller than . The LM-DCM is performed in the region with . The potential on is computed with SCEM and the reconstructed from LM-SCEM. The power density in can be reconstructed with the method introduced by Ammari et al Ammari2008 . However, it requires the knowledge on the deformation caused by the ultrasonic waves, therefore, we compute it with DCM instead. Noise with SNR is added, and LM-DCM is used for the reconstruction. The relative error is given in Figure 5(a). The conductivity map is reconstructed with in 30 iterations, and the result is given in Figure 6(b). Here, the time required for 30 LM-DCM iterations is about 20 minutes which is approximately the time needed for one LM-SCEM iteration. So the reconstruction efficiency is greatly improved, and better results are obtained. A similar computation with noise is further considered here. As seen in Figure 5(b), increasing noise does not influence much the convergence of the boundary potential, therefore, this mixed approach can be a good way to remove noise from the measured power density. With 40dB noise in the reconstructed power density in , the distribution of obtained with LM-DCM is shown in Figure 6(d). Comparing it to the results obtained with LM-SCEM, as shown in Figure 6(c), a better noise tolerance is observed in LM-DCM.

(a)
(b)
Figure 6: (a) The relative error of different reconstructions. (b) The convergence of the boundary potential for LM-CEM with different level of noise.
(a) LM-SCEM, SNR =
(b) Mixed, SNR =
(c) LM-SCEM, SNR =
(d) Mixed, SNR =
Figure 7: The conductivity map of human-brain model reconstructed with current pattern and . The reconstruction with only LM-CEM are given in (a) and (c) for and noises. Corresponding reconstructions by mixing LM-SCEM and LM-DCM are given in (b) and (d).

5 Conclusion

The work first developed a computational approach for AET by incorporating CEM into the Levenberg-Marquardt algorithm. Since the regularity of the power density obtained with traditional CEM is limited because of the Robin-type discontinues boundary conditions, a recently proposed smoothed CEM is used in this paper. Numerical investigation shows that this iterative method can stably reconstruct the conductivity map of complex phantoms, and a good accuracy can be achieved even with a certain level of noise though this also depends on the current patterns used in the measurements.

However, the reconstruction algorithm becomes inefficient quickly when the number of measurements is increased. Since EIT is not very sensitive to the changes of the internal conductivity, the boundary potential converges to its true value much faster than the convergence of the conductivity. This fact is then used to build a mixed computational approach. LM-SCEM is used to reconstruct the boundary potentials in a few iterations, and LM-DCM is applied to reconstruct the conductivity distribution based on the obtained boundary potential. It is observed in the given example that the time for one LM-SCEM iteration is enough for the whole calculation of LM-DCM, so the reconstruction efficiency is greatly improved with the mixed strategy.

Here, reconstruction with LM-DCM requires a first step to compute the power density from the obtained boundary potential from LM-SCEM. This additional step can be considered as a step to smooth the electrical potential and to remove the noise in the measured power density, in addition to the better noise tolerance of LM-DCM compared to LM-SCEM, a better reconstruction can be obtained with the mixed algorithm.

Several numerical experiments are carried out with a heart-lung phantom and a complex human brain model. The performance of the presented reconstruction approaches are well demonstrated with different number of measurements. The proposed method applies also to 3-dimensional acousto-electric tomography and anisotropic conductivy distributions; we leave the implementation in these scenarious for future work.

Acknowledgement

The majority of the work was done while Changyou Li was postdoc at the Department of Applied Mathematics and Computer Science, Technical University of Denmark in 2018. We thank the Danish Council for Independent Research — Natural Sciences (grant 4002-00123) for financial support.

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