A New Proximal Method for Joint Image Restoration and Edge Detection with the Mumford-Shah Model

In this paper, we propose an adaptation of the PAM algorithm to the minimization of a nonconvex functional designed for joint image denoising and contour detection. This new functional is based on the Ambrosio-Tortorelli approximation of the well-known Mumford-Shah functional. We motivate the proposed approximation, offering flexibility in the choice of the possibly non-smooth penalization, and we derive closed form expression for the proximal steps involved in the algorithm. We focus our attention on two types of penalization: ℓl-norm and a proposed quadratic-f. function. Numerical experiments show that the proposed method is able to detect sharp contours and to reconstruct piecewise smooth approximations with low computational cost and convergence guarantees. We also compare the results with state-of-the-art relaxations of the Mumford-Shah functional and a recent discrete formulation of the Ambrosio-Tortorelli functional.