A Survey on Non-rigid 3D Shape Analysis <s> Classification <s> We describe our preliminary findings in applying the spherical parametrization and geometry images to the task of 3D shape matching and similarity based comparison of polygon soup models. Unlike traditional approach where multiple 2D views of the same object are required to capture the relevant geometry features, our proposed technique uses spherical parametrization and geometry images for 3D shape matching. This technique reduces the problem to the 2D space without computing 2D projections and guarantees the preservation of small details. Moreover, we take advantage from the hierarchical nature of the parametrization process, through the progressive mesh simplification, to derive a multiresolution analysis technique. Our proposed algorithm is invariant to similarity transformations such as rotation and scaling. The efficiency of this approach is discussed through a set of experiments <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Classification <s> In this paper, we describe our preliminary findings in applying the spherical parameterization and geometry images to the task of 3D shape matching. View-based techniques compare 3D objects by comparing their 2D projections. However, it is not trivial to choose the number of views and their settings. Geometry images overcome these limitations by mapping the entire object onto a spherical or planar domain. We make use of this property to derive a rotation invariant shape descriptor. Once the geometry image encoding the object’s geometric properties is computed, a 1D rotation invariant descriptor is extracted using the spherical harmonic analysis. The parameterization process guarantees the scale invariance, while its coarse-to-fine nature allows the comparison of objects at different scales. We demonstrate and discuss the efficiency of our approach on a collection of 120 three-dimensional models. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Classification <s> In this brief communication, we provide an overview of the density-based shape description framework in and the details of the runs we have submitted to the Watertight Models Track of the SHREC’07. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Classification <s> We argue that full surface correspondence (registration) and optimal deformations (geodesics) are two related problems and propose a framework that solves them simultaneously. We build on the Riemannian shape analysis of anatomical and star-shaped surfaces of Kurtek et al. and focus on articulated complex shapes that undergo elastic deformations and that may contain missing parts. Our core contribution is the re-formulation of Kurtek et al.'s approach as a constrained optimization over all possible re-parameterizations of the surfaces, using a sparse set of corresponding landmarks. We introduce a landmark-constrained basis, which we use to numerically solve this optimization and therefore establish full surface registration and geodesic deformation between two surfaces. The length of the geodesic provides a measure of dissimilarity between surfaces. The advantages of this approach are: (1) simultaneous computation of full correspondence and geodesic between two surfaces, given a sparse set of matching landmarks (2) ability to handle more comprehensive deformations than nearly isometric, and (3) the geodesics and the geodesic lengths can be further used for symmetrizing 3D shapes and for computing their statistical averages. We validate the framework on challenging cases of large isometric and elastic deformations, and on surfaces with missing parts. We also provide multiple examples of averaging and symmetrizing 3D models. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Classification <s> Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of $\mathbb{S}^2$ in $\mathbb{R}^3$, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the $\mathbb{L}^2$ metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures. <s> BIB005
The length of a geodesic, or deformation energy, between a pair of surfaces in the shape space is a measure of dissimilarity that can be used for unsupervised as well as supervised classification of 3D objects. To demonstrate this, we use the SHREC07 watertight 3D model benchmark BIB003 , which is composed of 400 watertight 3D models evenly divided into 20 shape classes. We only consider the 13 classes that are composed of genus-0, triangulated meshes. First, we compute their spherical parameterizations BIB004 , BIB001 , BIB002 , and then normalize them for translation and scale. Thus, these surfaces become elements of the preshape space C f . Next, we map the surfaces to the space of SRNFs, C q and compute their pairwise distance matrix. That is, for a given pair of surfaces f 1 and f 2 and their respective SRNF maps q 1 and q 2 , their dissimilarity is given as For comparison, we also compute the pairwise distances between the shapes using (1) the L 2 distance in C f , i.e. without registration, and (2) the L 2 metric in S f after elastic registration using the SRNF framework. Figure 7 shows the multidimensional scaling (MDS) plot of these three pairwise distance matrices. As one would expect, when using the L 2 metric in C f , the 3D models are all spread across the space (Figure 7-(a) ). Clusters start to emerge when using the L 2 metric in S f with elastic registration performed using the SRNF framework ( Figure 7-(b) ). When using the L 2 metric in S q , which is equivalent to partial elastic metric (Figure 7-(c) ), we can clearly see that the 3D models are clustered by shape classes. This suggests that the SRNF framework is suitable for 3D shape classification and clustering. We refer the reader to BIB005 for a detailed analysis of the performance of SRNFs in classification and clustering.