Beamforming Through Regularized Inverse Problems in Ultrasound Medical Imaging

In this article we present a SCILAB implementation of algebraic iterative reconstruction methods for discretisation of inverse problems in imaging. These so-called row action methods rely on semi-convergence for achieving the necessary regularisation of the problem. We implement this method using SCILAB and provide a few simplified test problems: medical tomography, seismic tomography and walnut tomography.Numerical results show the capability of this method for the original and perturbed right-hand side vector.

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Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

Advances in Mathematical Physics ◽

10.1155/2015/908251 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 1

Author(s):

Kiwoon Kwon

Keyword(s):

Inverse Problem ◽

Partial Differential Equations ◽

Medical Imaging ◽

Inverse Problems ◽

Differential Equations ◽

Optical Tomography ◽

Elliptic Partial Differential Equations ◽

Unique Determination ◽

Partial Differential ◽

Uniqueness Results

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

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Conditional Invertible Neural Networks for Medical Imaging

Journal of Imaging ◽

10.3390/jimaging7110243 ◽

2021 ◽

Vol 7 (11) ◽

pp. 243

Author(s):

Alexander Denker ◽

Maximilian Schmidt ◽

Johannes Leuschner ◽

Peter Maass

Keyword(s):

Neural Networks ◽

Medical Imaging ◽

Inverse Problems ◽

Low Dose ◽

Data Driven ◽

Resonance Imaging ◽

Point Estimates ◽

Ill Posed ◽

Low Dose Computed Tomography

Over recent years, deep learning methods have become an increasingly popular choice for solving tasks from the field of inverse problems. Many of these new data-driven methods have produced impressive results, although most only give point estimates for the reconstruction. However, especially in the analysis of ill-posed inverse problems, the study of uncertainties is essential. In our work, we apply generative flow-based models based on invertible neural networks to two challenging medical imaging tasks, i.e., low-dose computed tomography and accelerated medical resonance imaging. We test different architectures of invertible neural networks and provide extensive ablation studies. In most applications, a standard Gaussian is used as the base distribution for a flow-based model. Our results show that the choice of a radial distribution can improve the quality of reconstructions.

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