We develop a rigorous and general framework for constructing information-theoretic divergences that subsume both $f$-divergences and integral probability metrics (IPMs), such as the $1$-Wasserstein distance. We prove under which assumptions these divergences, hereafter referred to as $(f,\Gamma)$-divergences, provide a notion of `distance' between probability measures and show that they can be expressed as a two-stage mass-redistribution/mass-transport process. The $(f,\Gamma)$-divergences inherit features from IPMs, such as the ability to compare distributions which are not absolutely continuous, as well as from $f$-divergences, namely the strict concavity of their variational representations and the ability to control heavy-tailed distributions for particular choices of $f$. When combined, these features establish a divergence with improved properties for estimation, statistical learning, and uncertainty quantification applications. Using statistical learning as an example, we demonstrate their advantage in training generative adversarial networks (GANs) for heavy-tailed, not-absolutely continuous sample distributions. We also show improved performance and stability over gradient-penalized Wasserstein GAN in image generation.

A global algorithm for reconstructing shape from shading is described. This algorithm incorporates an earlier local algorithm that has been shown to be capable of fast, robust surface reconstruction for general surfaces if a small amount of information on the surface is provided. The new algorithm is capable of determining this information automatically, and thus can reconstruct a general surface from shading with no a priori information on the surface. In experimental tests on complex synthetic images, this algorithm has produced good surface reconstructions over most of the image. For 128 /spl times/ 128 images, the reconstruction took less than 30 s on a DECstation 5000. The algorithm appears noise resistant, giving good reconstructions even with an added pixel noise of /spl plusmn/10%.>

An approach to shape-from-shading that is based on a connection with a calculus of variations/optimal control problem is proposed. An explicit representation corresponding to a shaded image is given for the surface; uniqueness of the surface (under suitable conditions) is an immediate consequence. The approach leads naturally to an algorithm for shape reconstruction that is simple, fast, provably convergent (in many cases, provably convergent to the correct solution), and does not require regularization. Given a continuous image, the algorithm can be proved to converge to the continuous surface solution as the image sampling frequency is taken to infinity. Experimental results are presented for synthetic and real images.>