A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> We present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes -- triangular meshes or more generally straight line graphs in Euclidean space -- are treated as points in a shape space. We introduce useful Riemannian metrics in this space to aid the user in design and modeling tasks, especially to explore the space of (approximately) isometric deformations of a given shape. Much of the work relies on an efficient algorithm to compute geodesics in shape spaces; to this end, we present a multi-resolution framework to solve the interpolation problem -- which amounts to solving a boundary value problem -- as well as the extrapolation problem -- an initial value problem -- in shape space. Based on these two operations, several classical concepts like parallel transport and the exponential map can be used in shape space to solve various geometric modeling and geometry processing tasks. Applications include shape morphing, shape deformation, deformation transfer, and intuitive shape exploration. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple $\ensuremath{\mathbb{L}^2}$ metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> Three-dimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two- or three-dimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is deduced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging. <s> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. <s> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> In this paper we describe how to perform Principal Components Analysis in "shell space". Thin shells are a physical model for surfaces with non-zero thickness whose deformation dissipates elastic energy. Thin shells, or their discrete counterparts, can be considered to reside in a shell space in which the notion of distance is given by the elastic energy required to deform one shape into another. It is in this setting that we show how to perform statistical analysis of a set of shapes (meshes in dense correspondence), providing a hybrid between physical and statistical shape modelling. The resulting models are better able to capture non-linear deformations, for example resulting from articulated motion, even when training data is very sparse compared to the dimensionality of the observation space. <s> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <s> Cubic splines in Euclidean space minimize the mean squared acceleration among all curves interpolating a given set of data points. We extend this observation to the Riemannian manifold of discrete shells in which the associated metric measures both bending and membrane distortion. Our generalization replaces the acceleration with the covariant derivative of the velocity. We introduce an effective time-discretization for this novel paradigm for navigating shell space. Further transferring this concept to the space of triangular surface descriptors---edge lengths, dihedral angles, and triangle areas---results in a simplified interpolation method with high computational efficiency. <s> BIB009 </s> A Survey on Non-rigid 3D Shape Analysis <s> Comparison and discussion <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> BIB010
As the reader can see, many shape spaces and metrics have been defined in the literature. To help the reader (a) Linear path (1 − t)f1 + tf2, registration computed with SRNF. (b) Geodesic path using SRNF inversion. navigate in this rich literature, we summarize in this section their properties and discuss their similarities and differences. The first aspect to consider when choosing a metric and a shape space is the invariance properties of the metric, which defines the type of pre-processing requirements. For instance, as summarized in Table 1 , the metrics of Kilian et al. BIB002 as well as those defined in the space of thin-shells  , BIB008 , BIB009 , BIB006 are invariant to translation, scale, and rotation and thus one does not need to normalize the input 3D models in a pre-processing step. On the other hand, diffeomorphisms do not act by isometries in these shape spaces under the defined metrics. As a consequence, these metrics are not invariant to re-parameterization and thus are not suitable for joint registration and geodesic computation. On the other hand, the L 2 metric on the space of square-root maps such as the SRNF BIB003 and Q-maps BIB004 are invariant to reparameterization. Thus the correspondence and registration problem can be solved jointly, using the same metric, with the geodesic computation. The SRNF requires normalization for scale while the Q-maps require normalization for translation and scale. Finally, methods based on M-reps BIB001 and Lie bodies BIB005 require normalization for all rigid transformations as well as the pre-computation of correspondences using a different approach. The other aspect to consider is the types of deformations that are quantified, and subsequently, the complexity of the metric and the representation, which affect the processing time. Ideally, the process of evaluating the metric should be fast so that geometry processing and shape analysis tasks can be performed in a computationally reasonable time. From this aspect, Euclidean shape spaces and Euclidean metrics are probably the most convenient to use, since evaluating the L 2 metric, computing short paths, and performing statistics under the metric are straightforward. Moreover, when working in Euclidean spaces one can use all the computational tools developed for vector spaces, e.g. interpolations and extrapolations, Principal Component Analysis (PCA), and probabilistic models. As such, the SRNF representation BIB003 is very convenenint since all these tasks can be performed in the SRNF space using the L 2 metric, which is equivalent to the partial elastic metric in C f , and map the results back to C f for visualization BIB010 , BIB007 .