Regularizing a linearized EIT reconstruction method using a sensitivity-based factorization method

Abstract

For electrical impedance tomography (EIT), most practical reconstruction methods are based on linearizing the underlying non-linear inverse problem. Recently, it has been shown that the linearized problem still contains the exact shape information. However, the stable reconstruction of shape information from measurements of finite accuracy on a limited number of electrodes remains a challenge.In this work we propose to regularize the standard linearized reconstruction method (LM) for EIT using a non-iterative shape reconstruction method (the factorization method). Our main tool is a discrete sensitivity-based variant of the factorization method (herein called S-FM) which allows us to formulate and combine both methods in terms of the sensitivity matrix. We give a heuristic motivation for this new method and show numerical examples that indicate its good performance in the localization of anomalies and the alleviation of ringing artifacts.

A Learning-Based Method for Solving Ill-Posed Nonlinear Inverse Problems: A Simulation Study of Lung EIT

“…. , N } to learn two functions (called encoder and decoder) Φ : R d → R k and Ψ : R k → R d from a class of functions AE described by a deep learning network by minimizing (14) (Ψ, Φ) = argmin Choosing k << d, one can interpret the encoder's output h = Φ(γ) as a compressed latent representation, whose dimensionality is much less than the original size of the imageγ. The decoder Ψ converts h to an image similar to the original input (15) Ψ • Φ(γ) ≈γ.…”

“…Definitely, the autoencoder approach aims to fulfill (P1) by minimizing the reconstruction loss of (14). However, the second property (P2), as shown in Fig.…”

“…To find Ψ, note that for all imagesγ, the concatenation Ψ(Φ(γ)) is now a random vector. Since we can also interpretγ as a random vector which always takes the same value, we could ensure the desired property (P1) by simply minimizing (14) with · now denoting the energy distance between two random vectors. But this trivial approach would obviously still prefer a deterministic encoder, i.e., Φ me will be the encoder function from the standard autoencoder approach, and Φ std ≡ 0.…”

“…Hence, in order to ensure variations in the latent space to achieve (P2), we additionally enforce that the distribution of the encoder output is close to a normal distribution. We thus minimize (14) we minimize (14) with an additional term that penalizes the Kullback-Leibler (KL) divergence loss between N (µ n , Σ n ) and N (0, I)) for all n = 1, · · · , N (19) D KL (N (µ n , Σ n ) N (0, I)) = 1 2 k j=1 (µ n (j) 2 + σ n (j) 2 − log σ n (j) − 1 .…”

Mathematical Framework for Abdominal Electrical Impedance Tomography to Assess Fatness

“…The correlation function c i,j between the column vectors can be defined by Figure 4.4, the column vectors are highly correlated. The success of the proposed least-squares method arises from taking advantage of the Tikhonov regularization [11]. The Tikhonov regularization minimizes both |Sκ − b| 2 and |κ| 2 as follows:…”