A Survey on Non-rigid 3D Shape Analysis <s> Registration and geodesic deformations <s> An efficient algorithm for isometry-invariant matching of surfaces is presented. The key idea is computing the minimum-distortion mapping between two surfaces. For this purpose, we introduce the generalized multidimensional scaling, a computationally efficient continuous optimization algorithm for finding the least distortion embedding of one surface into another. The generalized multidimensional scaling algorithm allows for both full and partial surface matching. As an example, it is applied to the problem of expression-invariant three-dimensional face recognition. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Registration and geodesic deformations <s> Deformable objects are ubiquitous in the world surrounding us, on all levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. In recent years, non-rigid shapes have attracted growing interest, which has led to rapid development of the field, where state-of-the-art results from very different sciences - theoretical and numerical geometry, optimization, linear algebra, graph theory, machine learning and computer graphics, to mention several - are applied to find solutions. This book gives an overview of the current state of science in analysis and synthesis of non-rigid shapes. Everyday examples are used to explain concepts and to illustrate different techniques. The presentation unfolds systematically and numerous figures enrich the engaging exposition. Practice problems follow at the end of each chapter, with detailed solutions to selected problems in the appendix. A gallery of colored images enhances the text. This book will be of interest to graduate students, researchers and professionals in different fields of mathematics, computer science and engineering. It may be used for courses in computer vision, numerical geometry and geometric modeling and computer graphics or for self-study. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Registration and geodesic deformations <s> One of the biggest challenges in non-rigid shape retrieval and comparison is the design of a shape descriptor that would maintain invariance under a wide class of transformations the shape can undergo. Recently, heat kernel signature was introduced as an intrinsic local shape descriptor based on diffusion scale-space analysis. In this paper, we develop a scale-invariant version of the heat kernel descriptor. Our construction is based on a logarithmically sampled scale-space in which shape scaling corresponds, up to a multiplicative constant, to a translation. This translation is undone using the magnitude of the Fourier transform. The proposed scale-invariant local descriptors can be used in the bag-of-features framework for shape retrieval in the presence of transformations such as isometric deformations, missing data, topological noise, and global and local scaling. We get significant performance improvement over state-of-the-art algorithms on recently established non-rigid shape retrieval benchmarks. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Registration and geodesic deformations <s> 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. <s> BIB004
The first example we consider is the joint registration and geodesic deformation of 3D objects that undergo complex elastic deformations. Here, we would like to emphasize for the reader that elastic (or non-rigid) deformation is composed of bending and stretching. Works based on (generalized) multi-dimensional scaling BIB001 and (heat-)diffusion geometry BIB002 , BIB003 only consider bending, which preserves intrinsic properties of surfaces. Here, we consider full elastic deformations. In Figure 4 , we consider two human body shapes of the same subject but in significantly different poses and compute a joint registration and geodesic deformation. We first compare the quality of the registration when computed with a re-parameterization invariant elastic metric (Figure 4-(b) and Figure 4 -(c)) and when computed with other methods, e.g. functional maps of BIB004 . As one can see in Figure 4 -(a), the later method does not produce correct one-toone correspondences compared to the methods that formulate registration as a re-parameterization problem. Next, we compare geodesics computed using the L 2 metric in C f vs. geodesics computed using elastic metrics by SRNF inversion. This is shown in Figures 4-(b) , (c) and Figure 5-(a), (b) , where shapes undergo complex bending and stretching.