**Update.** The latest version with some corrections and clarifications canbe found below.

**Update.** The LaTex sources of the paper and the slides are now available on GitHub.

### Abstract

This paper studies the minimization of non-convex and non-smooth composite functionsIn particular, we discuss the algorithm proposed by Ochs et al. in [1], called iPiano. Following [1], we present a global convergence result for functions satisfying the Kurdyka-Lojasiewicz property [2,3] which is based on the work by Attouch et al. [4]. Furthermore, we discuss the implementation of iPiano and apply the algorithm to image denoising and image segmentation. In contrast to [1], we use simple denoising functionals instead of Fields of Expert [5,6] to demonstrate that simple functionals may benefit from non-smooth and non-convex terms. Finally, we demonstrate the applicability of the algorithm to image segmentation based on a 2-phase fields approximation of the Mumford-Shah functional [7].

Seminar PaperPresentation Slides

### Implementation

iPiano has been implemented in C++ based on Eigen. Applications include image denoising, image segmentation (utilizing OpenCV) and compressed sensing. The implementation is vailable on GitHub:

iPiano on GitHubSome articles related to the implementation can be found here:

### References

- [1] P. Ochs, Y. Chen, T. Brox, T. Pock.
*iPiano: Inertial proximal algorithm for nonconvex optimization*., SIAM J. Imaging Sciences, 7 (2014), pp. 1388–1419. - [2] S. Lojasiewicz.
*Sur la gomtrie semi- et sous- analytique*. Annales de l’institut Fourier, 43 (1993), pp. 1575–1595. - [3] K. Kurdyka.
*On gradients of functions definable in o-minimal structures*. Annales de l’institut Fourier, 48 (1998), pp. 769–783. - [4] H. Attouch, J. Bolte, B. F. Svaiter .
*Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods*. Math. Program., 137 (2013), pp. 91–129. - [5] S. Roth and M. J. Black, Fields of experts, International Journal of Computer Vision, 82 (2009), pp. 205–229.
- [6] Y. Chen, T. Pock, R. Ranftl, H. Bischof.
*Revisiting loss-specific training of filter-based MRFs for image restoration*. in Pattern Recognition, German Conference on, Saarbr¨ucken, Germany, September 2013, pp. 271–281. - [7] J. Shen.
*Gamma-convergence approximation to piecewise constant mumford-shah segmentation*. in Advanced Concepts for Intelligent Vision Systems, International Conference on, vol. 3708 of Lecture Notes in Computer Science, Antwerpen, Belgium, September 2005, Springer, pp. 499–506.

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