A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <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> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <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> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <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> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <s> The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <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> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics in the space of SRNFs <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> BIB006
The SRNF representation, compared to others, is very promising as a representation of surfaces for elastic shape analysis. If geodesics, mean shapes, PCA, etc. could be computed in C q and S q under the L 2 metric, and then mapped back to C f , there would be large gains in computational efficiency with respect to the optimization-based methods described in the previous sections and those introduced in e.g. BIB001 , BIB005 , BIB003 . Unfortunately, unlike the curve case, there is no analytical expression for Q −1 , the inverse of SRNF map Q, for arbitrary points in C q . Moreover, the injectivity and surjectivity of Q remain to be determined, meaning that for a given q ∈ C q , there may be no f ∈ C f such that Q(f ) = q, and if such an f does exist, it may not be unique. If one cannot invert the representation, one can transfer geodesics and statistical analyses conducted in C q back to C f by pulling the L 2 metric back to C f under and perform computations there, as in BIB003 , but this is computationally expensive, and rather defeats the purpose of having an L 2 metric in the first place. To solve this inversion problem, Xie et al. BIB004 showed that in the case of star-shaped genus surfaces, an analytical solution to the inversion problem exists. Laga et al. BIB006 , on the other hand, developed a method that, given q ∈ C q , finds an f ∈ C f such that Q(f ) = q, if one exists, or an f whose image Q(f ) is the closest, in terms of the elastic metric, to q if it does not. This is achieved by formulating SRNF inversion as an optimization problem: find an element f ∈ C f whose image Q(f ) is as close as possible to the given q ∈ C q under the L 2 norm. This optimization problem is efficiently solved using a gradient descent approach BIB006 . To avoid undesirable solutions, since gradient descent procedures converge to local minima if not initialized appropriately, Laga et al. BIB006 carefully engineered an orthonormal basis of C f , and used a spherical-wavelet based multiresolution and multiscale representation of the elements of C f . The basis can be generic, e.g. spherical harmonics, or domain-specific such as PCA basis in the presence of training datasets. With this approach, SRNF maps can be inverted with high accuracy and efficiency, with robustness to local minima, and with low computational complexity. Finally, the shapes on the geodesic between two star-shaped surfaces are not necessarily star-shaped, and thus, the analytic form may not be the correct inversion. However, one can use the analytical solution as an initial guess for the inverse, thereby better initializing the reconstruction-by-optimization problem. Comparison between correspondence and geodesic computation algorithms. In (a) correspondences are computed using functional maps BIB002 , while the deformation is computed using linear interpolation in the preshape space C f . In (b), the registration is computed using the re-parameterization invariant elastic metric in the space of SRNFs while the deformation is computed using linear interpolation in he preshape space C f . Finally, in (c), the registration is computed using the re-parameterization invariant elastic metric in the space of SRNFs while the geodesic is obtained by SRNF inversion of linear paths in S q . See BIB006 for more details. By mapping back to C f geodesics computed in S q , which are straight lines since they are equipped with the L 2 metric, the computational cost is reduced by an order of magnitude compared to methods that use path straightening under the pullback metric. See Section 6 for a few examples of geodesics between pairs of complex genus-0 surfaces.