Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 February 2021 HyEIT : Iterative Reconstruction Methods in MATLAB for Hybrid Inverse Problems in Electrical Impedance Tomography 1 A.D. Garnadi23a,b, Mokhamad Fakhrul Ulum4d, Deddy Kurniadi51, Utriweni Mukhaiyar6f, Lina Choridah7c, Nurhuda Hendra Setyawan8c, Khusnul Ain9g aInstitute for Globally Distributed Open Research and Education bIPB University, Indonesia cFaculty of Medicine, Public Health, and Nursing Universitas Gadjah Mada, Yogyakarta, Indonesia dFaculty of Veterinary Medicine, IPB University, Indonesia eInstrumentation and Control Research Division, Faculty of Industrial Technology, Institut Teknologi Bandung, Indonesia fStatistics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia gFaculty of Sciences and Technology, Airlangga University, Surabaya, Indonesia Abstract We present a MATLAB-based, open source reconstruction software for Hybrid Imaging problem arising from Current Density Impedance Imaging (CDII) and Acoustic-Electrical Impedance Tomography (AEIT), utilizing mixed finite element method solver of Darcy toolbox DarcyLite . For a formulation of hybrid inverse problems in impedance tomography the successive substitution iterative schemes are adapted and an reconstruc- tion algorithm is developed. The problem formulation is a class of hybrid 1Funded by "Riset Kolaborasi Indonesia 2019","Pengembangan Tomografi Elektrik Se- bagai 'Contrasting Agent' Magnetic Resonance Imaging" AND by "Riset Kolaborasi In- donesia 2020", 'Pengembangan Tomografi Hibrida : Electrical Impedance Tomography + Ultrasonography Dalam Bidang Biomedika' 2e-mail address:[email protected] 3home address: Gg Bungur IV 107, Perumahan Cijerah II, Melong, Cimahi Selatan, Indonesia 40534 4e-mail address: [email protected]; OrcID: http://orcid.org/0000-0003-2092-9645 5e-mail address: [email protected] 6e-mail address:[email protected] 7e-mail address: [email protected] 8e-mail address: [email protected] 9e-mail address: k [email protected] Preprint submitted to SoftwareX February 6, 2021 © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 February 2021 imaging modality which is current density impedance imaging or Acoustic- Electrical Impedance Tomography. The proposed algorithm is implemented numerically in two dimensions. Keywords: Hybrid inverse problems, Iterative reconstruction methods, Impedance tomography, Current density impedance imaging, Magnetic resonance imaging, electrical impedance tomography, Ultrasound modulated electrical impedance tomography Required Metadata Current code version Ancillary data table required for subversion of the codebase. Kindly re- place examples in right column with the correct information about your cur- rent code, and leave the left column as it is. Nr. Code metadata description Please fill in this column C1 Current code version Version 0.0 C2 Permanent link to code/repository NA used for this code version C3 Code Ocean compute capsule C4 Legal Code License CC-BY NC 4.0 International C5 Code versioning system used For example svn, git, mercurial, etc. put none if none C6 Compilation requirements, operat- MATLAB/GNU OCTAVE for the ing future C7 Compilation requirements, operat- DarcyLite ing environments & dependencies http://www.math.colostate.edu/ li- u/DarcyLite.zip C8 If available Link to developer docu- NA mentation/manual C9 Support email for questions agah.garnadi[at]gmail.com Table 1: Code metadata (mandatory) 1 The permanent link to code/repository or the zip archive should include 2 the following requirements: 3 README.txt and LICENSE.txt. 4 Source code in a src/ directory, not the root of the repository. 5 Tag corresponding with the version of the software that is reviewed. 2 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 February 2021 6 Documentation in the repository in a docs/ directory, and/or READMEs, 7 as appropriate. 8 1. Motivation and significance 9 In this note we develop and analyse an iterative reconstruction method 10 for the following hybrid inverse problem: 11 12 Reconstruct the scalar function σ from measurements of the form A = σjrujp in Ω; p = 1; 2 (1) 13 where u is the solution to the linear PDE problem r(−σru) = 0 in Ω; (2) uj@Ω = uD; n 14 Here Ω is a smooth open bounded domain in R ; n 2 f2; 3g; and σ is 15 bounded from above and from below by positive constants in Ω: We will 16 henceforth assume that jruj is uniformly bounded from below by a positive 17 constant in Ω: The problem formulation and the reconstruction methods are 18 independent of the spatial dimension, here we only works on n = 2; for pro- 19 totyping. This inverse problem can be identified as a hybrid inverse problem 20 or, equivalently, as a coupled-physics inverse problem, since it models an 21 imaging modality utilizing coupled different physical phenomena. 22 The functions (1) model measurements from a couple of hybrid inverse 23 problems of Electrical Impedance Tomography, i.e. 24 1. In the case p = 1; the functions (1) model measurements of the internal 25 electrical current density and the tomographic method is known as 26 Current Density Impedance Imaging (CDII); the same model appears 27 in certain formulations of Magnetic Resonance Electrical Impedance 28 Tomography (MREIT). [12], [13],[14] 29 2. While for p = 2; coming from AEIT (Acoustics Induce Electric Impedance 30 Tomography). [15], [16] [17], [18] 31 The only package we are aware of with fully fledged capabilities was [21], 32 implemented in PDE suite FENICS, written in C++. 33 2. Software description 34 In this section we derive an iterative reconstruction algorithms for the pre- 35 sented inverse problem. It is common feature of all the presented algorithms, 3 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 February 2021 36 that they are based on a transformation of a non-linear problem into a series 37 of linear problems for which the corresponding solutions, under appropriate 38 conditions, converge to a solution to the original non-linear problem. In this 39 section, we explain how to use the successive iterative scheme to reformulate 40 the inverse problem, develop a general iterative reconstruction algorithm and 41 implement it numerically. 42 A way to reformulate (1)-(2) is to recast the set of generalized Laplace 43 problems (2) as non-linear PDE problems. The unknown function σ can be 44 eliminated from the PDEs using the internal measurements, since A σ = ; p = 1; 2: jrujp 45 This gives the non-linear PDE problems r − A ru = 0 in Ω; p = 1; 2; jrujp (3) uj@Ω = uD; 46 In this setting a procedure for solving the inverse problem would be to 47 first solve the non-linear problems for u and then determine σ by (1). One of 48 the simplest approaches to approximate solutions to such a non-linear PDE 49 problem is to use a fixed-point iteration or successive iterative scheme, also 50 known as the method of successive substitutions [4]. The idea behind the 51 successive substitution scheme is basically to make an initial guess on the 52 non-linear coefficient in the PDE problems (3); then the now linear PDE 53 problem is solved, and we get an approximation of u which is used to up- 54 date the estimate of the coefficient. Let the initial guess be denoted by σ0: 55 Throughout this paper, we use a superscript to denote the iteration num- 56 ber. The successive substitution scheme, applied to (3), generates a series of 57 solving linear problems of the type r(−σkruk) = 0 in Ω; k = 0; 1; :::; k (4) u j@Ω = uD; k k+1 58 which are solved successively for (u ): The coefficient σ is defined by 59 the relation A σk+1 = jrukjp 60 It is important to notice that the transformation of the non-linear problem 61 into a series of linear problems is not a result of any algebraic or symbolic 62 manipulations; it simply a consequence of a substitution made directly in the 63 non-linear PDE. The choice of stopping criteria for the successive iterations 64 depends on the application. For problems like ours, where the objective is 4 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 8 February 2021 65 to estimate the PDE coefficient σ; and not to the functions u; it is natural k+1 k 66 to choose a stopping criteria based on the tolerance kσ − σ k in some 67 suitable function space and the iteration number k: 68 The performance of the successive substitution relies on an initial guess 69 that provides a sufficiently good starting point, such that the successive ap- 70 proximation of the nonlinear coefficients is improved in each iteration. The 71 convergence properties of the successive substitution scheme can be studied 72 using contraction maps and the application of the Banach fixed point theo- 73 rem, however the practicality of applying this theory on a general non-linear 74 PDE problem seems rather limited [4]. Nonetheless, for our case, the con- 75 vergence properties of the successive substitution have been analysed [2, 6]. 76 The successive substitution scheme was the main idea behind one of the 77 first reconstruction algorithms for the hybrid inverse problems expressed in 78 the form of the non-linear PDE problems given by (3). The resulting al- 79 gorithm was denoted the J-substitution algorithm, due to the fact that the 80 functions A models measurements of the magnitude of the current density 81 vector field, often denoted by A [3]. Note that the original formulation of 82 the J-substitution algorithm considered the corresponding Neumann prob- 83 lem. A similar implementation of the Dirichlet problem for a single internal 84 measurement has also been used to solve the equivalent inverse problem from 85 CDII [5, 6]. 86 2.1. Software Architecture 87 The successive iterative scheme can be applied directly to the non-linear 88 PDE formulation of the inverse problem to produce the following algorithm: 89 - Algorithm.