Explainable bilevel optimization: an application to the Helsinki deblur challenge

In general, H is a structured matrix defined in such a way that the product Hu corresponds to a convolution between the image u and a given kernel h representing the Point Spread Function (PSF) of the imaging system employed to measure the data. The deblurring (or deconvolution) problem consists in finding an approximation of g, given the blurred image f and, possibly, some information on the system PSF. If the blurring kernel h, underlying the matrix H, is completely unknown and it has to be inferred together with g, the resulting problem is a blind deconvolution one [32]. Since the PSF h usually represents a low-pass filter, the matrix H is, at best, very ill conditioned and directly solving the inverse problem Hu = f, even when it is feasible, leads to unmeaningful solutions. On the other side, the variational approach consists in designing and solving an optimization problem whose solutions are a good approximation of the unknown image g. In general, a variational model is the set composed by the objective function, i.e., the function to be minimized, and the possible constraints. In the variational models arising in image restoration applications, the objective function, called also energy functional, encompasses different kinds of information: the nature of the noise introduced in the acquisition process, geometrical and/or analytical properties on the image content and physical constraints on the pixel values. Usually, in all image reconstruction problems, and more generally inverse problems, the energy functional, besides the data, depends on a set of parameters; they may simply reduce to tuning parameters balancing the relative weights of the different terms in the functional but can also represent more complicate structures of the functionals themselves.