Hyperparameter selection for the Discrete Mumford-Shah functional

This work focuses on joint piecewise smooth image reconstruction and contour detection, formulated as the minimization of a discrete Mumford-Shah functional, performed via a theoretically grounded alternating minimization scheme. The bottleneck of such variational approaches lies in the need to finetune their hyperparameters, while not having access to ground truth data. To that aim, a Stein-like strategy providing optimal hyperparameters is designed, based on the minimization of an unbiased estimate of the quadratic risk. Efficient and automated minimization of the estimate of the risk crucially relies on an unbiased estimate of the gradient of the risk with respect to hyperparameters, whose practical implementation is performed thanks to a forward differentiation of the alternating scheme minimizing the Mumford-Shah functional, requiring exact differentiation of the proximity operators involved. Intensive numerical experiments are performed on synthetic images with different geometries and noise levels, assessing the accuracy and the robustness of the proposed procedure. The resulting parameterfree piecewise-smooth reconstruction and contour detection procedure, not requiring prior image processing expertise, is thus amenable to real-world applications.