Generation of anisotropic-smoothness regularization filters for EIT

Abstract

Abstract-In the inverse conductivity problem, as in any illposed inverse problem, regularization techniques are necessary in order to stabilize inversion. A common way to implement regularization in electrical impedance tomography is to use Tikhonov regularization. The inverse problem is formulated as a minimization of two terms: the mismatch of the measurements against the model, and the regularization functional. Most commonly, differential operators are used as regularization functionals, leading to smooth solutions. Whenever the imaged region presents discontinuities in the conductivity distribution, such as interorgan boundaries, the smoothness prior is not consistent with the actual situation. In these cases, the reconstruction is enhanced by relaxing the smoothness constraints in the direction normal to the discontinuity. In this paper, we derive a method for generating Gaussian anisotropic regularization filters. The filters are generated on the basis of the prior structural information, allowing a better reconstruction of conductivity profiles matching these priors. When incorporating prior information into a reconstruction algorithm, the risk is of biasing the inverse solutions toward the assumed distributions. Simulations show that, with a careful selection of the regularization parameters, the reconstruction algorithm is still able to detect conductivities patterns that violate the prior information. A generalized singular-value decomposition analysis of the effects of the anisotropic filters on regularization is presented in the last sections of the paper.

Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid MRI-guided near-infrared spectral tomography

“…Significant improvement in the stability and accuracy of the reconstruction process can be obtained through MRI segmented anatomical information as an input to the NIR parameter estimation problem. However, the way in which the prior information is used is critical (17)(18)(19)(20)(21)(22). Misguided constraints can lead to errors that are detrimental to the image outcome.…”

Magnetic Resonance-Based Electrical Property Tomography (MR-EPT) for Prostate Cancer Grade Imaging

“…Our approach also enables incorporating prior structural information, so that preferential directions of change of EP can be embedded into the regularization (e.g. [18], [19]). In general we believe that the approach developed here has advantages over algorithms based directly on (2) since it does not require differentiation of .…”

“…The inverse formulation developed here enables one to select regularization functionals that are appropriate for the problem at hand. Besides TV and similar edge-preserving techniques it is possible to envision using ad-hoc functionals that incorporate prior structural information extracted from other anatomic MR images [18], [19].…”

The reconstruction problem

“…As constant conductivity values are easily obtained in EIT the null space does not diminish the regularizing properties of this choice of G. Similarly one could choose a first order differential operator for L [152]. Other smooth choices of G include the inverse of a Gaussian smoothing filter [16], effectively an infinite order differential operator. In these cases where G is smooth and for α large enough the Hessian of f will be positive definite, we can then deduce that f is a convex function [160, Ch 2], so that a critical point will be a strict local minimum, guaranteeing the success of smooth optimization methods.…”

“…Although LU factorization would be one alternative, perhaps a better choice is to use the Generalized Singular Value Decomposition GSVD [72], which allows the regularized solution to be calculated efficiently for any value of α. The GSVD is now a standard tool for understanding the effect of the choice of the regularization matrix L in a linear ill-conditioned problem, and has been applied to linearized EIT [152,16]. The use of a single linearized Tikhonov regularized solution is widespread in medical industrial and geophysical EIT, the NOSER algorithm [35] being a well known example.…”