A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

In image and signal processing, the beta-divergence is well known as a similarity measure between two positive objects. However, it is unclear whether or not the distance-like structure of beta-divergence is preserved, if we extend the domain of the beta-divergence to the negative region. In this article, we study the domain of the beta-divergence and its connection to the Bregman-divergence associated with the convex function of Legendre type. In fact, we show that the domain of beta-divergence (and the corresponding Bregman-divergence) include negative region under the mild condition on the beta value. Additionally, through the relation between the beta-divergence and the Bregman-divergence, we can reformulate various variational models appearing in image processing problems into a unified framework, namely the Bregman variational model. This model has a strong advantage compared to the beta-divergence-based model due to the dual structure of the Bregman-divergence. As an example, we demonstrate how we can build up a convex reformulated variational model with a negative domain for the classic nonconvex problem, which usually appears in synthetic aperture radar image processing problems.