Shape from Focus and Defocus: Convexity, Quasiconvexity and Defocus-Invariant Textures

In this paper we analyze the convexity and the quasiconvexity of shape from focus/defocus and image restoration. We show that these problems are strictly quasiconvex for a family of Bregman's divergences, and in particular for least-squares. In addition to giving novel analytical insight to these problems, this study can be readily exploited to design algorithms: One can do away with global minimizers and obtain the same optimal solution by employing simple and efficient local methods. We experimentally validate this investigation by comparing two minimization algorithms: one based on a local method (gradient-flow) and another based on a global method (graph cuts). We show that both algorithms find the global optimum. Finally, we fully characterize defocus-invariant textures, a class of textures that do not allow depth recovery. We show how to decompose textures into defocus-invariant and defocus-varying components, and how this decomposition can be used to dramatically improve depth estimates.