The PP-TSVD algorithm for image restoration problems

Abstract

Abstract. The PP-TSVD algorithm is a regularization algorithm based on the truncated singular value decomposition (TSVD) that computes piecewise polynomial (PP) solutions without any a priori information about, the locations of the break points. Here we describe an extension of this algorithm designed for two-dimensional inverse problems based on a Kronecker-product formulation. We illustrate its use in connection with deblurring of digital images with sharp edges, and we discuss its relations to total variation regularization. IntroductionIn this work we focus on discretizations of linear inverse problems in the form of square systems A z = b or overdetermined least squares systems min tIA z-bII2.These systems, which we denote discrete ill-posed problems, represent a wealth of applications of inverse problems; see, e.g., [6, §1.2]. Our main "tool" from linear algebra for analysis as well as computations is the singular value decomposition (SVD) of the coefficient matrix A. If we assume that A is m x n with m > n, then the SVD takes the formwhere ui and vi are the left, and right singular vectors which are orthonormal, and Gi are the singular values which are nonnegative and appearing in non-increasing order. In terms of the SVD, discrete ill-posed problems are characterized by having a coefficient matrix A whose singular values decay gradually to zero (in practice: until they hit a level determined by the machine precision).Standard methods for regularization of discrete ill-posed problems are Tikhonor regularization ~2 ilzll2 } (2) and truncated SVD (TSVD) minllzll2 subject to IIAk z -bll = min,

A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

“…Another interesting aspect of this particular regularization approach is that it produces very smooth solutions, which was also noted in [21], and this causes for example the resulting regular grid to miss the true deepest point located approximately at coordinates (207, 247), and yields a deepest point located at approximately (205, 245) for Figure 8. It is quite remarkable though how the known features of the lake are clearly present in this image.…”

Optimization and geophysical inverse problems

“…A variety of objective functions are encountered, linear, quadratic, and more complicated forms. The objective function is generally smooth, although there is increasing interest in problems where it is not smooth, as in the total variation method (Rudin et al, 1992;Vogel and Oman, 1996) or the piecewise polynomial modification of truncated singular value decomposition (Hansen et al, 2000). The constraints, both equality and inequality, can be linear or nonlinear.…”

Multilevel Approach For Signal Restoration Problems With Toeplitz Matrices

“…It is well known in the signal processing literature that one can restore edge information by using, for example, a Tikhonov-based approach where the regularization functional is a total-variation (TV) operator [23]. Another edge-preserving approach is the PPTSVD approach [12]. These approaches, although they can work very well, can be quite computationally expensive.…”