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

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