A nonlinear primal-dual method for total variation-based image restoration

We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $|\nabla u|$ in the definition of the TV-norm before we apply a linearization technique such as N...

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Bibliographic Details
Published in:SIAM journal on scientific computing 1999, Vol.20 (6), p.1964-1977
Main Authors: CHAN, T. F, GOLUB, G. H, MULET, P
Format: Article
Language:English
Subjects:
Algorithms
Applied mathematics
Applied sciences
Artificial intelligence
Computer science
control theory
systems
Exact sciences and technology
Mathematics
Methods
Noise
Norms
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Pattern recognition. Digital image processing. Computational geometry
Sciences and techniques of general use
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Summary:We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity $|\nabla u|$ in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint, 1994] and Andersen [Ph.D. thesis, Odense University, Denmark, 1995] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent.
ISSN:1064-8275
1095-7197
DOI:10.1137/s1064827596299767