# On convergent finite difference schemes for variational–PDE-based image processing

## Abstract

We study an adaptive anisotropic Huber functional-based image restoration scheme. Using a combination of L2–L1 regularization functions, an adaptive Huber functional-based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman (In: Goldstein and Osher, SIAM J Imaging Sci 2(2):323–343, 2009) algorithm, to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrates that the best convergence rates are obtained for our algorithm.

- Authors:

- University of Missouri, Department of Computer Science (United States)
- University of Beira Interior, Department of Computer Science (Portugal)

- Publication Date:

- OSTI Identifier:
- 22769336

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 2; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; ANISOTROPY; CONVERGENCE; IMAGE PROCESSING; MINIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; VARIATIONAL METHODS

### Citation Formats

```
Prasath, V. B. Surya,, and Moreno, Juan C., E-mail: jcmb@ubi.pt.
```*On convergent finite difference schemes for variational–PDE-based image processing*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0414-9.

```
Prasath, V. B. Surya,, & Moreno, Juan C., E-mail: jcmb@ubi.pt.
```*On convergent finite difference schemes for variational–PDE-based image processing*. United States. doi:10.1007/S40314-016-0414-9.

```
Prasath, V. B. Surya,, and Moreno, Juan C., E-mail: jcmb@ubi.pt. Tue .
"On convergent finite difference schemes for variational–PDE-based image processing". United States. doi:10.1007/S40314-016-0414-9.
```

```
@article{osti_22769336,
```

title = {On convergent finite difference schemes for variational–PDE-based image processing},

author = {Prasath, V. B. Surya, and Moreno, Juan C., E-mail: jcmb@ubi.pt},

abstractNote = {We study an adaptive anisotropic Huber functional-based image restoration scheme. Using a combination of L2–L1 regularization functions, an adaptive Huber functional-based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman (In: Goldstein and Osher, SIAM J Imaging Sci 2(2):323–343, 2009) algorithm, to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrates that the best convergence rates are obtained for our algorithm.},

doi = {10.1007/S40314-016-0414-9},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 2,

volume = 37,

place = {United States},

year = {2018},

month = {5}

}