BPM: Blended Piecewise Moebius Maps

We propose a novel Moebius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a smooth map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi-conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed-form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi-conformal errors

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

EXTENDING CONVOLUTION THROUGH SPATIALLY ADAPTIVE ALIGNMENT

Convolution underlies a variety of applications in computer vision and graphics, including efficient filtering, analysis, simulation, and neural networks. However, convolution has an inherent limitation: when convolving a signal with a filter, the filter orientation remains fixed as it travels over the domain, and convolution loses effectiveness in the presence of deformations that change alignment of the signal relative to the local frame. This problem metastasizes when attempting to generalize convolution to domains without a canonical orientation, such as the surfaces of 3D shapes, making it impossible to locally align signals and filters in a consistent manner. This thesis presents a unified framework for transformation-equivariant convolutions on arbitrary homogeneous spaces and 2D Riemannian manifolds called extended convolution. This approach is based on the the following observation: to achieve equivariance to an arbitrary class of transformations, we only need to consider how the positions of points as seen in the frames of their neighbors deform. By defining an equivariant frame operator at each point with which we align the filter, we correct for the change in the relative positions induced by the transformations. This construction places no constraints on the filters, making extended convolution highly descriptive. Furthermore, the framework can handle arbitrary transformation groups, including higher-dimensional non-compact groups that act non-linearly on the domain. Critically, extended convolution naturally generalizes to arbitrary 2D Riemannian manifolds as it does not need a canonical coordinate system to be applied. The power and utility of extended convolution is demonstrated in several applications. A unified framework for image and surface feature descriptors called Extended Convolution Histogram of Orientations (ECHO) is proposed, based on the optimal filters maximizing the response of the extended convolution at a given point. Using the generalization of extended convolution to surface vector fields, state-of-the-art surface convolutional neural networks (CNNs) are constructed. Last, we move beyond rotations and isometries and use extended convolution to design spherical CNNs equivariant to Mobius transformations, representing a first step toward conformally-equivariant surface networks

Diskrete Spin-Geometrie für Flächen

This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten für den Dirac Operator eng verwandt wie in der glatten Theorie. Darüber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren Krümmung

Higher-order Graph Principles towards Non-rigid Surface Registration

This report casts surface registration as the problem of finding a set of discrete correspondences through the minimization of an energy function, which is composed of geometric and appearance matching costs, as well as higher-order deformation priors. Two higher-order graph-based formulations are proposed under different deformation assumptions. The first formulation encodes isometric deformations using conformal geometry in a higher-order graph matching problem, which is solved through dual-decomposition and is able to handle partial matching. Despite the isometry assumption, this approach is able to robustly match sparse feature point sets on surfaces undergoing highly anisometric deformations. Nevertheless, its performance degrades significantly when addressing anisometric registration for a set of densely sampled points. This issue is rigorously addressed subsequently through a novel deformation model that is able to handle arbitrary diffeomorphisms between two surfaces. Such a deformation model is introduced into a higher-order Markov Random Field for dense surface registration, and is inferred using a new parallel and memory efficient algorithm. To deal with the prohibitive search space, we design an efficient way to select a number of matching candidates for each point of the source surface based on the matching results of a sparse set of points. A series of experiments demonstrate the accuracy and the efficiency of the proposed framework, notably in challenging cases of large and/or anisometric deformations, or surfaces that are partially occluded.Ce rapport formalise le problème du recalage de surfaces 3D comme la recherche d’un ensemble de correspondances discrètes par la minimisation d’une fonction d’énergie, qui est composée de fonctions de coûts mesurant des similitudes géométriques et d’apparence, et des à priori d’ordre élevé sur la déformation. Deux formulations à base de graphes d’ordre élevé sont proposées sous différentes hypothèses de déformation. La première formulation encode la déformation isométrique, à partir de géométrie conforme, dans un problème d’appariement de graphes d’ordre élevé, qui est résolu par décomposition duale et est capable de gérer les cas de correspondance partielle. Malgré l’hypothèse d’isométrie, cette approche est capable de mettre en correspondance de manière robuste deux ensembles clairsemés de points sur deux surfaces, y compris lorsque celles-ci subissent une déformation fortement anisométrique. Cependant, sa performance se dégrade de manière significative lorsqu’elle est étendue au recalage anisométrique d’un ensemble de points à forte densité. Ce problème est rigoureusement traité par la suite à travers un nouveau modèle de déformation capable de gérer des difféomorphismes arbitraires entre deux surfaces. Ce modèle de déformation est introduit dans une formulation MRF d’ordre élevé pour le recalage dense de surfaces, et être inféré en utilisant un nouvel algorithme parallèle et efficace en termes de mémoire. Pour traiter l’espace de recherche prohibitif, nous concevons une méthode efficace pour sélectionner un ensemble de correspondants potentiels pour chaque point appartenant à la surface source. Cette méthode est basée sur les résultats d’appariement d’un ensemble clairsemé de points. Notre méthode est validée au moyen d’une série d’expériences qui démontrent sa précision et son efficacité, notamment dans les cas difficiles où des déformations importantes et/ou anisométriques sont présentes, ou lorsque les maillages sont partiellement cachés