A Fast Method for Finding the Global Solution of the Regularized Structured Total Least Squares Problem for Image Deblurring

Given a linear system ${\bf A} {\bf x} \approx {\bf b}$ over the real or complex field, where both ${\bf A}$ and ${\bf b}$ are subject to noise, the total least squares (TLS) problem seeks to find a correction matrix and a correction right-hand side vector of minimal norm which makes the linear syst...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 30; no. 1; pp. 419 - 443
Main Authors: Beck, Amir, Ben-Tal, Aharon, Kanzow, Christian
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2008
Subjects:
Algebra
Algebraic geometry
Algorithms
Boundary conditions
Decomposition
Exact sciences and technology
Fourier transforms
Industrial engineering
Linear and multilinear algebra, matrix theory
Mathematics
Noise
Nonlinear algebraic and transcendental equations
Norms
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Numerical linear algebra
Optimization
Regularization methods
Sciences and techniques of general use
Noise
Optimization method
Duality
nonconvex optimization
Boundary condition
Image
Optimization
simultaneously diagonalizable matrices
Least squares method
Matrices
Ill posed problem
Vector
unimodal functions
Least squares problem
Global optimum
image deblurring
90C26
Algorithm
Global solution
15A29
Regularization method
structured total least squares
Numerical analysis
Linear system
Linear algebra
Objective function
Regularization
Corrections
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary:Given a linear system ${\bf A} {\bf x} \approx {\bf b}$ over the real or complex field, where both ${\bf A}$ and ${\bf b}$ are subject to noise, the total least squares (TLS) problem seeks to find a correction matrix and a correction right-hand side vector of minimal norm which makes the linear system feasible. To avoid ill posedness, a regularization term is added to the objective function; this leads to the so-called regularized TLS problem. A further complication arises when the matrix ${\bf A}$ and correspondingly the correction matrix must have a specific structure. This is modeled by the regularized structured TLS (RSTLS) problem. In general this problem is nonconvex and hence difficult to solve. However, the RSTLS problem arising from image deblurring applications under reflexive or periodic boundary conditions possesses a special structure where all relevant matrices are simultaneously diagonalizable (SD). In this paper we introduce an algorithm for finding the global optimum of the RSTLS problem with this SD structure. The devised method is based on decomposing the problem into single variable problems and then transforming them into one-dimensional unimodal real-valued minimization problems which can be solved globally. Based on the uniqueness and attainment properties of the RSTLS solution we show that a constrained version of the problem possesses a strong duality result and can thus be solved via a sequence of RSTLS problems.
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ISSN:0895-4798
1095-7162
DOI:10.1137/070709013