A Survey on Non-rigid 3D Shape Analysis <s> Statistical analysis by SRNF inversion <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> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Statistical analysis by SRNF inversion <s> Statistical shape analysis develops methods for comparisons, deformations, summarizations, and modeling of shapes in given data sets. These tasks require a fundamental tool called parallel transport of tangent vectors along arbitrary paths. This tool is essential for: (1) computation of geodesic paths using either shooting or path-straightening method, (2) transferring deformations across objects, and (3) modeling of statistical variability in shapes. Using the square-root normal field (SRNF) representation of parameterized surfaces, we present a method for transporting deformations along paths in the shape space. This is difficult despite the underlying space being a vector space because the chosen (elastic) Riemannian metric is non-standard. Using a finite-basis for representing SRNFs of shapes, we derive expressions for Christoffel symbols that enable parallel transports. We demonstrate this framework using examples from shape analysis of parameterized spherical surfaces, in the three contexts mentioned above. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Statistical analysis by SRNF inversion <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> BIB003
Computing shape summaries on non-linear shape spaces using complex elastic metrics, or deformation energies, requires computing geodesics many times. For instance, at each iteration of Algorithm 1, n geodesics should be computed where n is the number of shapes in the collection to average. This is computationally very expensive as discussed in BIB003 . SRNF representations offer an elegant and a computationally efficient alternative solution; Instead of using the pullback metric, one can simplify the iterative procedure given in Algorithm 1 by first mapping all surfaces to the SRNF space, C q , resulting in {q 1 , . . . , q n }. Then, the mean shape in SRNF space, denoted byq, is computed by iterating between coregistering all q i s toq, and subsequently updatingq. This is computationally very efficient since C q is an Euclidean space. Finally, at the last step, the mean shapef ∈ [f ] is computed by SRNF inversion, i.e. finding a shapef such that Q(f ) =q. The entire procedure is summarized in Algorithm 3 and detailed in BIB003 . Next, letq denote the SRNF of the final average shape. With a slight abuse of notation, let q i , i = 1, . . . , n denote the SRNFs of the surfaces in the sample optimally registered to this average. Then, since the SRNF space is Euclidean, the modes of variation can be computed in a standard way, i.e. using singular value decomposition (SVD) of the covariance matrix. Let u k q denote the k−th principal direction. Then, one can explore the variability in this direction around the mean usingq + λu k q , where λ ∈ R. To visualize the principal directions of variation in S f , we only need to compute, by SRNF inversion, f Finally, given a sample of observed surfaces f 1 , . . . , f n ∈ C f , one wants to fit a probability model to the data and sample from it new 3D shape instances. Estimating probability models on nonlinear spaces like S f is difficult: classical statistical approaches, developed for vector spaces, do not apply directly. By first computing the SRNF representations of the surfaces, one can use all the usual statistical tools in the vector space S q , and then map the results back to C f using the SRNF inversion algorithm. Let G(q) be the model probability distribution fitted to {q 1 , . . . , q n }. A random sample q s can be generated from G. We then find f s such that Q(f s ) = q s . Note that G can be an arbitrary distribution, parametric or nonparametric. Here, we use a wrapped truncated normal distribution, which can be learned using Principal Component Analysis on S q . We caution the reader that the distribution is defined and analyzed in SRNF space; a distribution on S q induces a distribution on C f , but it has not been derived explicitly. In terms of computational complexity, computing the mean shape only requires the inversion of one SRNF, while techniques based on the pullback metric such as BIB002 ,  , BIB001 are iterative, and require the computation of n geodesics per iteration, with n being the number of shapes to average. This is computationally very expensive especially since each geodesic is computed using the expensive pullback metric.