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Register a new userBack-Projection based Fidelity Term for Ill-Posed Linear Inverse Problems
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Added by
Tom Tirer
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Ill-posed linear inverse problems appear in many image processing applications, such as deblurring, super-resolution and compressed sensing. Many restoration strategies involve minimizing a cost function, which is composed of fidelity and prior terms, balanced by a regularization parameter. While a vast amount of research has been focused on different prior models, the fidelity term is almost always chosen to be the least squares (LS) objective, that encourages fitting the linearly transformed optimization variable to the observations. In this paper, we examine a different fidelity term, which has been implicitly used by the recently proposed iterative denoising and backward projections (IDBP) framework. This term encourages agreement between the projection of the optimization variable onto the row space of the linear operator and the pseudo-inverse of the linear operator (back-projection) applied on the observations. We analytically examine the difference between the two fidelity terms for Tikhonov regularization and identify cases (such as a badly conditioned linear operator) where the new term has an advantage over the standard LS one. Moreover, we demonstrate empirically that the behavior of the two induced cost functions for sophisticated convex and non-convex priors, such as total-variation, BM3D, and deep generative models, correlates with the obtained theoretical analysis.
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The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from (finite-dimensional) linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems. This approach allows a user to work with functions instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods.
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