Figure 1: The notion of shape difference defined in this paper provides a way to compare deformations between shape pairs. This allows us to recognize similar expressions of shape A (top row) to those of shape B (bottom row), without correspondences between A and B and without any prior learning process. AbstractWe develop a novel formulation for the notion of shape differences, aimed at providing detailed information about the location and nature of the differences or distortions between the two shapes being compared. Our difference operator, derived from a shape map, is much more informative than just a scalar global shape similarity score, rendering it useful in a variety of applications where more refined shape comparisons are necessary. The approach is intrinsic and is based on a linear algebraic framework, allowing the use of many common linear algebra tools (e.g, SVD, PCA) for studying a matrix representation of the operator. Remarkably, the formulation allows us not only to localize shape differences on the shapes involved, but also to compare shape differences across pairs of shapes, and to analyze the variability in entire shape collections based on the differences between the shapes. Moreover, while we use a map or correspondence to define each shape difference, consistent correspondences between the shapes are not necessary for comparing shape differences, although they can be exploited if available. We give a number of applications of shape differences, including parameterizing the intrinsic variability in a shape collection, exploring shape collections using local variability at different scales, performing shape analogies, and aligning shape collections.

Finding an informative, structure-preserving map between two shapes has been a long-standing problem in geometry processing, involving a variety of solution approaches and applications. However, in many cases, we are given not only two related shapes, but a collection of them, and considering each pairwise map independently does not take full advantage of all existing information. For example, a notorious problem with computing shape maps is the ambiguity introduced by the symmetry problem -for two similar shapes which have reflectional symmetry there exist two maps which are equally favorable, and no intrinsic mapping algorithm can distinguish between them based on these two shapes alone. Another prominent issue with shape mapping algorithms is their relative sensitivity to how "similar" two shapes are -good maps are much easier to obtain when shapes are very similar. Given the context of additional shape maps connecting our collection, we propose to add the constraint of global map consistency, requiring that any composition of maps between two shapes should be independent of the path chosen in the network. This requirement can help us choose among the equally good symmetric alternatives, or help us replace a "bad" pairwise map with the composition of a few "good" maps between shapes that in some sense interpolate the original ones. We show how, given a collection of pairwise shape maps, to define an optimization problem whose output is a set of alternative maps, compositions of those given, which are consistent, and individually at times much better than the original. Our method is general, and can work on any collection of shapes, as long as a seed set of good pairwise maps is provided. We demonstrate the effectiveness of our method for improving maps generated by state-of-the-art mapping methods on various shape databases.

We present an efficient method to conformally parameterize 3D mesh data sets to the plane. The idea behind our method is to concentrate all the 3D curvature at a small number of select mesh vertices, called cone singularities, and then cut the mesh through those singular vertices to obtain disk topology. The singular vertices are chosen automatically. As opposed to most previous methods, our flattening process involves only the solution of linear systems of Poisson equations, thus is very efficient. Our method is shown to be faster than existing methods, yet generates parameterizations having comparable quasi-conformal distortion.

Figure 1: Using our framework various vector field design goals can be easily posed as linear constraints. Here, given three symmetry maps: rotational (S1), bilateral (S2) and front/back (S3), we can generate a symmetric vector field using only S1 (left), S1 + S2 (center) and S1 + S2 + S3 (right). The top row shows the front of the 3D model, and the bottom row its back. AbstractIn this paper, we introduce a novel coordinate-free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace-Beltrami and symmetry operators, as well as to a straight-forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.

We present a novel representation of maps between pairs of shapes that allows for efficient inference and manipulation. Key to our approach is a generalization of the notion of map that puts in correspondence real-valued functions rather than points on the shapes. By choosing a multi-scale basis for the function space on each shape, such as the eigenfunctions of its Laplace-Beltrami operator, we obtain a representation of a map that is very compact, yet fully suitable for global inference. Perhaps more remarkably, most natural constraints on a map, such as descriptor preservation, landmark correspondences, part preservation and operator commutativity become linear in this formulation. Moreover, the representation naturally supports certain algebraic operations such as map sum, difference and composition, and enables a number of applications, such as function or annotation transfer without establishing point-to-point correspondences. We exploit these properties to devise an efficient shape matching method, at the core of which is a single linear solve. The new method achieves state-of-the-art results on an isometric shape matching benchmark. We also show how this representation can be used to improve the quality of maps produced by existing shape matching methods, and illustrate its usefulness in segmentation transfer and joint analysis of shape collections.

Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data val-

A space deformation is a mapping from a source region to a target region within Euclidean space, which best satisfies some userspecified constraints. It can be used to deform shapes embedded in the ambient space and represented in various forms -polygon meshes, point clouds or volumetric data. For a space deformation method to be useful, it should possess some natural properties: e.g. detail preservation, smoothness and intuitive control. A harmonic map from a domain Ω ⊂ R d to R d is a mapping whose d components are harmonic functions. Harmonic mappings are smooth and regular, and if their components are coupled in some special way, the mapping can be detail-preserving, making it a natural choice for space deformation applications. The challenge is to find a harmonic mapping of the domain, which will satisfy constraints specified by the user, yet also be detail-preserving, and intuitive to control. We generate harmonic mappings as a linear combination of a set of harmonic basis functions, which have a closed-form expression when the source region boundary is piecewise linear. This is done by defining an energy functional of the mapping, and minimizing it within the linear span of these basis functions. The resulting mapping is harmonic, and a natural "As-Rigid-As-Possible" deformation of the source region. Unlike other space deformation methods, our approach does not require an explicit discretization of the domain. It is shown to be much more efficient, yet generate comparable deformations to state-ofthe-art methods. We describe an optimization algorithm to minimize the deformation energy, which is robust, provably convergent, and easy to implement.

Abstract