Enhanced and restored signals as a generalized solution for shock filter models. Part I—existence and uniqueness result of the Cauchy problem

Abstract Signal enhancement and restoration is one of the fields that make extensive use of PDE theory. More specifically, some authors have proposed successive improved shock filters based on non-linear hyperbolic equations. These models yield satisfactory results; however, a wider range of degrees of freedom when handling the model parameters (coefficients and components) would be of great interest because it would increase the model's efficiency and facilitate adaptation to specific situations. Naturally, the key challenge in proceeding thus is to ensure that the problem remains well-posed. In this paper, we propose a more general shock filter that introduces new parameters to control the shock speed. Interpreting the proposed model in a framework of generalized functions algebra, we prove existence and uniqueness solution results.