Bernhard Schmitzer

A Sparse Multiscale Algorithm for Dense Optimal Transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport, we provide a framework to verify global optimality of a discrete transport plan locally. This allows the construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multiscale scheme. Any existing discrete solver can be used as internal black-box. We explicitly describe how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance. Significant reductions in run-time and memory requirements have been observed.

An Interpolating Distance Between Optimal Transport and Fisher–Rao Metrics

This paper defines a new transport metric over the space of nonnegative measures. This metric interpolates between the quadratic Wasserstein and the Fisher–Rao metrics and generalizes optimal transport to measures with different masses. It is defined as a generalization of the dynamical formulation of optimal transport of Benamou and Brenier, by introducing a source term in the continuity equation. The influence of this source term is measured using the Fisher–Rao metric and is averaged with the transportation term. This gives rise to a convex variational problem defining the new metric. Our first contribution is a proof of the existence of geodesics (i.e., solutions to this variational problem). We then show that (generalized) optimal transport and Hellinger metrics are obtained as limiting cases of our metric. Our last theoretical contribution is a proof that geodesics between mixtures of sufficiently close Dirac measures are made of translating mixtures of Dirac masses. Lastly, we propose a numerical scheme making use of first-order proximal splitting methods and we show an application of this new distance to image interpolation.

Image Labeling by Assignment

We introduce a novel geometric approach to the image labeling problem. Abstracting from specific labeling applications, a general objective function is defined on a manifold of stochastic matrices, whose elements assign prior data that are given in any metric space, to observed image measurements. The corresponding Riemannian gradient flow entails a set of replicator equations, one for each data point, that are spatially coupled by geometric averaging on the manifold. Starting from uniform assignments at the barycenter as natural initialization, the flow terminates at some global maximum, each of which corresponds to an image labeling that uniquely assigns the prior data. Our geometric variational approach constitutes a smooth non-convex inner approximation of the general image labeling problem, implemented with sparse interior-point numerics in terms of parallel multiplicative updates that converge efficiently.

Object Segmentation by Shape Matching with Wasserstein Modes

We gradually develop a novel functional for joint variational object segmentation and shape matching. The formulation, based on the Wasserstein distance, allows modelling of local object appearance, statistical shape variations and geometric invariance in a uniform way. For learning of class typical shape variations we adopt a recently presented approach and extend it to support inference of deformations during segmentation of new query images. The resulting way of describing and fitting trained shape variations is in style reminiscent of contour-based variational shape priors, but does not require an intricate conversion between the contour and the region representation of shapes. A well-founded hierarchical branch-and-bound scheme, based on local adaptive convex relaxation, is presented, that provably finds the global minimum of the functional.

Modelling Convex Shape Priors and Matching Based on the Gromov-Wasserstein Distance

We present a novel convex shape prior functional with potential for application in variational image segmentation. Starting point is the Gromov-Wasserstein Distance which is successfully applied in shape recognition and classification tasks but involves solving a non-convex optimization problem and which is non-convex as a function of the involved shape representations. In two steps we derive a convex approximation which takes the form of a modified transport problem and inherits the ability to incorporate vast classes of geometric invariances beyond rigid isometries. We propose ways to counterbalance the loss of descriptiveness induced by the required approximations and to process additional (non-geometric) feature information. We demonstrate combination with a linear appearance term and show that the resulting functional can be minimized by standard linear programming methods and yields a bijective registration between a given template shape and the segmented foreground image region. Key aspects of the approach are illustrated by numerical experiments.

A Hierarchical Approach to Optimal Transport

A significant class of variational models in connection with matching general data structures and comparison of metric measure spaces, lead to computationally intensive dense linear assignment and mass transportation problems. To accelerate the computation we present an extension of the auction algorithm that exploits the regularity of the otherwise arbitrary cost function. The algorithm only takes into account a sparse subset of possible assignment pairs while still guaranteeing global optimality of the solution. These subsets are determined by a multiscale approach together with a hierarchical consistency check in order to solve problems at successively finer scales. While the theoretical worst-case complexity is limited, the average-case complexity observed for a variety of realistic experimental scenarios yields a significant gain in computation time that increases with the problem size.

Weakly convex coupling continuous cuts and shape priors

We introduce a novel approach to variational image segmentation with shape priors. Key properties are convexity of the joint energy functional and weak coupling of convex models from different domains by mapping corresponding solutions to a common space. Specifically, we combine total variation based continuous cuts for image segmentation and convex relaxations of Markov Random Field based shape priors learned from shape databases. A convergent algorithm amenable to large-scale convex programming is presented. Numerical experiments demonstrate promising synergistic performance of convex continuous cuts and convex variational shape priors under image distortions related to noise, occlusions and clutter.

Pulses of chaos synchronization in coupled map chains with delayed transmission.

Pulses of synchronization in chaotic coupled map lattices are discussed in the context of transmission of information. Synchronization and desynchronization propagate along the chain with different velocities which are calculated analytically from the spectrum of convective Lyapunov exponents. Since the front of synchronization travels slower than the front of desynchronization, the maximal possible chain length for which information can be transmitted by modulating the first unit of the chain is bounded.

Optimal Transport for Manifold-Valued Images

We introduce optimal transport-type distances for manifold-valued images. To do so we lift the initial data to measures on the product space of image domain and signal space, where they can then be compared by optimal transport with a transport cost that combines spatial distance and signal discrepancy. Applying recently introduced ‘unbalanced’ optimal transport models leads to more natural results. We illustrate the benefit of the lifting with numerical examples for interpolation of color images and classification of handwritten digits.

A sparse algorithm for dense optimal transport

Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport we provide a framework to verify global optimality of a discrete transport plan locally. This allows construction of a new sparse algorithm to solve large dense problems by considering a sequence of sparse problems instead. Any existing discrete solver can be used as internal black-box. The case of noisy squared Euclidean distance is explicitly detailed. We observe a significant reduction of run-time and memory requirements.