Exploiting the scaling indetermination of bi-linear models in inverse problems

Many inverse problems in imaging require estimating the parameters of a bi-linear model, e.g., the crisp image and the blur in blind deconvolution. In all these models, there is a scaling indetermination: multiplication of one term by an arbitrary factor can be compensated for by dividing the other by the same factor. To solve such inverse problems and identify each term of the bi-linear model, reconstruction methods rely on prior models that enforce some form of regularity. If these regularization terms verify a homogeneity property, the optimal scaling with respect to the regularization functions can be determined. This has two benefits: hyper-parameter tuning is simplified (a single parameter needs to be chosen) and the computation of the maximum a posteriori estimate is more efficient. Illustrations on a blind deconvolution problem are given with an unsupervised strategy to tune the hyper-parameter.