Hyperparameter selection for Discrete Mumford–Shah

This work focuses on a parameter-free joint piecewise smooth image denoising and contour detection. Formulated as the minimization of discrete Mumford–Shah functional estimated via theoretically grounded alternating scheme, bottleneck such variational approach lies in need to fine-tune their hyperparameters, while not having access ground truth data. To that aim, Stein-like strategy providing optimal hyperparameters is designed, based an unbiased estimate quadratic risk. Efficient automated risk crucially relies gradient with respect hyperparameters. Its practical implementation performed using forward differentiation scheme minimizing functional, requiring exact proximity operators involved. Intensive numerical experiments are synthetic images different geometry noise levels, assessing accuracy robustness proposed procedure. The resulting piecewise-smooth estimation detection procedure, prior processing expertise nor annotated data, can then be applied real-world images.