A Low-Dimensional Representation for Robust Partial Isometric Correspondences Computation

Intrinsic isometric shape matching has become the standard approach for pose invariant correspondence estimation among deformable shapes. Most existing approaches assume global consistency, i.e., the metric structure of the whole manifold must not change significantly. While global isometric matching is well understood, only a few heuristic solutions are known for partial matching. Partial matching is particularly important for robustness to topological noise (incomplete data and contacts), which is a common problem in real-world 3D scanner data. In this paper, we introduce a new approach to partial, intrinsic isometric matching. Our method is based on the observation that isometries are fully determined by purely local information: a map of a single point and its tangent space fixes an isometry for both global and the partial maps. From this idea, we develop a new representation for partial isometric maps based on equivalence classes of correspondences between pairs of points and their tangent spaces. From this, we derive a local propagation algorithm that find such mappings efficiently. In contrast to previous heuristics based on RANSAC or expectation maximization, our method is based on a simple and sound theoretical model and fully deterministic. We apply our approach to register partial point clouds and compare it to the state-of-the-art methods, where we obtain significant improvements over global methods for real-world data and stronger guarantees than previous heuristic partial matching algorithms.Comment: 17 pages, 12 figure

Parametric Regression on the Grassmannian

We address the problem of fitting parametric curves on the Grassmann manifold for the purpose of intrinsic parametric regression. As customary in the literature, we start from the energy minimization formulation of linear least-squares in Euclidean spaces and generalize this concept to general nonflat Riemannian manifolds, following an optimal-control point of view. We then specialize this idea to the Grassmann manifold and demonstrate that it yields a simple, extensible and easy-to-implement solution to the parametric regression problem. In fact, it allows us to extend the basic geodesic model to (1) a time-warped variant and (2) cubic splines. We demonstrate the utility of the proposed solution on different vision problems, such as shape regression as a function of age, traffic-speed estimation and crowd-counting from surveillance video clips. Most notably, these problems can be conveniently solved within the same framework without any specifically-tailored steps along the processing pipeline.Comment: 14 pages, 11 figure

Self-Assembly of Geometric Space from Random Graphs

We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice

Quantum bound to chaos and the semiclassical limit

We discuss the quantum bound on chaos in the context of the free propagation of a particle in an arbitrarily curved surface at low temperatures. The semiclassical calculation of the Lyapunov exponent can be performed in much the same way as the corresponding one for the `Loschmidt echo'.The bound appears here as the impossibility to scatter a wave, by effect of the curvature, over characteristic lengths smaller than the deBroglie wavelength.Comment: References added, some typos correcte