# Regularization by Functions of Bounded Variation and Applications to Image Enhancement

## Abstract

Optimization problems regularized by bounded variation seminorms are analyzed. The optimality system is obtained and finite-dimensional approximations of bounded variation function spaces as well as of the optimization problems are studied. It is demonstrated that the choice of the vector norm in the definition of the bounded variation seminorm is of special importance for approximating subspaces consisting of piecewise constant functions. Algorithms based on a primal-dual framework that exploit the structure of these nondifferentiable optimization problems are proposed. Numerical examples are given for denoising of blocky images with very high noise.

- Authors:

- Departamento de Matematica Aplicada y Ciencias de la Computacion, E.T.S.I. Industriales y de Telecomunicacion, Universidad de Cantabria, Av. Los Castros s/n, 39071 Santander (Spain)
- Institut fuer Mathematik, Universitaet Graz, Heinrichstrasse 36, A-8010 Graz (Austria)
- Departamento de Matematicas, Estadistica y Computacion, Facultad de Ciencias, Universidad de Cantabria, Av. Los Castros s/n, 39071 Santander (Spain)

- Publication Date:

- OSTI Identifier:
- 21067540

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 2; Other Information: DOI: 10.1007/s002459900124; Copyright (c) 1999 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 1999 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; APPROXIMATIONS; FUNCTIONS; IMAGES; MATHEMATICAL SPACE; OPTIMIZATION; VARIATIONS; VECTORS

### Citation Formats

```
Casas, E., Kunisch, K., and Pola, C.
```*Regularization by Functions of Bounded Variation and Applications to Image Enhancement*. United States: N. p., 1999.
Web. doi:10.1007/S002459900124.

```
Casas, E., Kunisch, K., & Pola, C.
```*Regularization by Functions of Bounded Variation and Applications to Image Enhancement*. United States. doi:10.1007/S002459900124.

```
Casas, E., Kunisch, K., and Pola, C. Wed .
"Regularization by Functions of Bounded Variation and Applications to Image Enhancement". United States. doi:10.1007/S002459900124.
```

```
@article{osti_21067540,
```

title = {Regularization by Functions of Bounded Variation and Applications to Image Enhancement},

author = {Casas, E. and Kunisch, K. and Pola, C.},

abstractNote = {Optimization problems regularized by bounded variation seminorms are analyzed. The optimality system is obtained and finite-dimensional approximations of bounded variation function spaces as well as of the optimization problems are studied. It is demonstrated that the choice of the vector norm in the definition of the bounded variation seminorm is of special importance for approximating subspaces consisting of piecewise constant functions. Algorithms based on a primal-dual framework that exploit the structure of these nondifferentiable optimization problems are proposed. Numerical examples are given for denoising of blocky images with very high noise.},

doi = {10.1007/S002459900124},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 2,

volume = 40,

place = {United States},

year = {1999},

month = {9}

}

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