Fuzzy Techniques Provide a Theoretical Explanation for the Heuristic l^p-Regularization of Signals and Images

Authors

Fernando Cervantes, The University of Texas at El PasoFollow
Bryan E. Usevitch, The University of Texas at El PasoFollow
Leobardo Valera, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow
Olga Kosheleva, The University of Texas at El PasoFollow

Publication Date

4-2016

Comments

Technical Report: UTEP-CS-15-97a

To appear in Proceedings of the 2016 IEEE International Conference on Fuzzy Systems FUZZ-IEEE'2016, Vancouver, Canada, July 24-29, 2016.

Abstract

One of the main techniques used to de-noise and de-blur signals and images is regularization, which is based on the fact that signals and images are usually smoother than noise. Traditional Tikhonov regularization assumes that signals and images are differentiable, but, as Mandelbrot has shown in his fractal theory, many signals and images are not differentiable. To de-noise and de-blur such images, researchers have designed a heuristic method of l^p-regularization.

l^p-regularization leads to good results, but it is not used as widely as should be, because it lacks a convincing theoretical explanation -- and thus, practitioners are often reluctant to use it, especially in critical situations. In this paper, we show that fuzzy techniques provide a theoretical explanation for the l^p-regularization.

Fuzzy techniques also enables us to come up with natural next approximations to be used when the accuracy of the l^p-based de-noising and de-blurring is not sufficient.