Higher-Order TV Methods—Enhancement via Bregman Iteration
- 1.1k Downloads
In this work we analyze and compare two recent variational models for image denoising and improve their reconstructions by applying a Bregman iteration strategy. One of the standard techniques in image denoising, the ROF-model (cf. Rudin et al. in Physica D 60:259–268, 1992), is well known for recovering sharp edges of a signal or image, but also for producing staircase-like artifacts. In order to overcome these model-dependent deficiencies, total variation modifications that incorporate higher-order derivatives have been proposed (cf. Chambolle and Lions in Numer. Math. 76:167–188, 1997; Bredies et al. in SIAM J. Imaging Sci. 3(3):492–526, 2010). These models reduce staircasing for reasonable parameter choices. However, the combination of derivatives of different order leads to other undesired side effects, which we shall also highlight in several examples.
The goal of this paper is to analyze capabilities and limitations of the different models and to improve their reconstructions in quality by introducing Bregman iterations. Besides general modeling and analysis we discuss efficient numerical realizations of Bregman iterations and modified versions thereof.
KeywordsTotal variation regularization Higher order methods Staircasing Exact solutions Bregman iteration
Financial support is acknowledged to the German Science Foundation (DFG) via grants SFB 656 (Subproject B2) and BU 2327/1. The third author thanks Stanley Osher for initiating his interest in variational methods for image processing.
- 3.Benning, M.: Singular regularization of inverse problems. Ph.D. thesis, University of Münster, Institute for Computational and Applied Mathematics, Einsteinstr. 62, 48149 Münster, May 2011 Google Scholar
- 4.Bertozzi, A.L., Greer, J.B., Osher, S., Vixie, K.: Nonlinear regularizations of TV based PDEs for image processing. In: Chen, G.-Q., Gasper, G., Jerome, J. (eds.) AMS Series of Contemporary Mathematics, vol. 371, pp. 29–40 (2005) Google Scholar
- 5.Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds. Dover, New York (1980) Google Scholar
- 10.Burger, M., Osher, S.: A guide to tv zoo. In: Level Set and PDE-based Reconstruction Methods. Springer, Berlin (2012, to appear) Google Scholar
- 29.Setzer, S., Steidl, G.: Variational methods with higher order derivatives in image processing. In: Neamtu, M., Schumaker, L.L. (eds.) Approximation XII. Nashboro Press, Brentwood (2008) Google Scholar