Existence and Uniqueness of Weak Solutions for a New Class of Convex Optimization Problems Related to Image Analysis

Abstract

This paper proposes a new anisotropic diffusion model in image restoration that is understood from a variational optimization of an energy functional. Initially, a family of new diffusion functions based on cubic Hermite spline is provided for optimal image denoising. After that, the existence and uniqueness of weak solutions for the corresponding Euler–Lagrange equation are proven in an appropriate functional space, and a consistent and stable numerical model is also shown. We complement this work by illustrating some experiments on different actual brain Magnetic Resonance Imaging (MRI) scans, showing the proposed model’s impressive results.