  This work focuses on a parameter-free joint piecewise smooth image denoising
and contour detection. Formulated as the minimization of a discrete
Mumford-Shah functional and estimated via a theoretically grounded alternating
minimization scheme, the bottleneck of such a variational approach lies in the
need to fine-tune 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. Its practical implementation is performed
using 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 geometry and noise levels, assessing the accuracy and the
robustness of the proposed procedure. The resulting parameter-free
piecewise-smooth estimation and contour detection procedure, not requiring
prior image processing expertise nor annotated data, can then be applied to
real-world images.
