This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. By exploiting alternate Bregman projections, we develop the alternate scaling algorithm and non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. We further enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers, penalties and dimensions, as well as real-world data for a pattern recognition application to audio scene classification.

Abstract We present intrinsic methods to address the fundamental problem of segmenting a mesh into a specified number of patches with a uniform size and a controllable overlap. Although never addressed in the literature, such a segmentation is useful for a wide range of processing operations where patches represent local regions and overlaps regularize solutions in neighbour patches. Further, we propose a symmetry‐aware distance measure and symmetric modification to furthest‐point sampling, so that our methods can operate on semantically symmetric meshes. We introduce quantitative measures of patch size uniformity and symmetry, and show that our segmentation outperforms state‐of‐the‐art alternatives in experiments on a well‐known dataset. We also use our segmentation in illustrative applications to texture stitching and synthesis where we improve results over state‐of‐the‐art approaches.

In this paper we present a method for computing dense correspondence between a set of 3D face meshes using functional maps. The functional maps paradigm brings with it a number of advantages for face correspondence. First, it allows us to combine various notions of correspondence. We do so by proposing a number of face-specific functions, suited to either within-or between-subject correspondence. Second, we propose a groupwise variant of the method allowing us to compute cycle-consistent functional maps between all faces in a training set. Since functional maps are of much lower dimension than point-to-point correspondences, this is feasible even when the input meshes are very high resolution. Finally, we show how a functional map provides a geometric constraint that can be used to filter feature matches between non-rigidly deforming surfaces.

We present an approach to modeling ear-to-ear, high-quality texture from one or more partial views of a face with possibly poor resolution and noise. Our approach is example-based in that we reconstruct texture with patches from a database composed of previously seen faces. A 3D morphable model is used to establish shape correspondence between the observed data across views and training faces. The database is built on the mesh surface by segmenting it into uniform overlapping patches. Texture patches are selected by belief propagation so as to be consistent with neighbors and with observations in an appropriate image formation model. We also develop a variant that is insensitive to light and camera parameters, and incorporate soft symmetry constraints. We obtain textures of higher quality for degraded views as small as 10 pixels wide, than a standard model fitted to non-degraded data. We further show applications to super-resolution where we substantially improve quality compared to a state-of-the-art algorithm, and to texture completion where we fill in missing regions and remove facial clutter in a photorealistic manner.

In this paper, we propose a novel approach to seamless texture stitching on a 3D mesh. The main idea is to blend the sampled images by least-angle selection of gradients in overlapping patches. This is in contrast to previous works which focus on vertex- or face-based strategies with additional heuristics for robustness. Patches are obtained by growing a uniform mesh segmentation via geodesic projections. Blending is achieved by formulating a screened Poisson equation using discrete differential operators. The processing pipeline further includes an optional step of color transformation for calibration and correction. This is applied to a zippered mesh based on range scans and a morphable model fitted to face photographs.

We present a generic approach to real-time audio segmentation in the framework of information geometry for exponential families. The proposed system detects changes by monitoring the information rate of the signals as they arrive in time. We also address shortcomings of traditional cumulative sum approaches to change detection, which assume known parameters before change. This is done by considering exact generalized likelihood ratio test statistics, with a complete estimation of the unknown parameters in the respective hypotheses. We derive an efficient sequential scheme to compute these statistics through convex duality. We finally provide results for speech segmentation in speakers, and polyphonic music segmentation in note slices.