A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric. In this SRV representation, the elastic metric simplifies to the IL(2) metric, the reparameterization group acts by isometries, and the space of unit length curves becomes the unit sphere. The shape space of closed curves is the quotient space of (a submanifold of) the unit sphere, modulo rotation, and reparameterization groups, and we find geodesics in that space using a path straightening approach. These geodesics and geodesic distances provide a framework for optimally matching, deforming, and comparing shapes. These ideas are demonstrated using: 1) shape analysis of cylindrical helices for studying protein structure, 2) shape analysis of facial curves for recognizing faces, 3) a wrapped probability distribution for capturing shapes of planar closed curves, and 4) parallel transport of deformations for predicting shapes from novel poses. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> We consider 3-D brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized. Past comparisons of such surfaces involve either volume deformations or nonrigid matching under fixed parameterizations of surfaces. We propose a new mathematical representation of surfaces, called q-maps, such that L² distances between such maps are invariant to re-parameterizations. This property allows for removing the parameterization variability by optimizing over the re-parameterization group, resulting in a proper parameterization-invariant distance between shapes of surfaces. We demonstrate this method in shape analysis of multiple brain structures, for 34 subjects in the Detroit Fetal Alcohol and Drug Exposure Cohort study, which results in a 91% classification rate for attention deficit hyperactivity disorder cases and controls. This method outperforms some existing techniques such as spherical harmonic point distribution model (SPHARM-PDM) or iterative closest point (ICP). <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> We consider the task of computing shape statistics and classification of 3D anatomical structures (as continuous, parameterized surfaces). This requires a Riemannian metric that allows re-parameterizations of surfaces by isometries, and computations of geodesics. This allows computing Karcher means and covariances of surfaces, which involves optimal re-parameterizations of surfaces and results in a superior alignment of geometric features across surfaces. The resulting means and covariances are better representatives of the original data and lead to parsimonious shape models. These two moments specify a normal probability model on shape classes, which are used for classifying test shapes into control and disease groups.We demonstrate the success of this model through improved random sampling and a higher classification performance. We study brain structures and present classification results for Attention Deficit Hyperactivity Disorder. Using the mean and covariance structure of the data, we are able to attain an 88% classification rate. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <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> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> The shapes of plant leaves are of great importance to plant biologists and botanists, as they can help to distinguish plant species and measure their health. In this paper, we study the performance of the Squared Root Velocity Function (SRVF) representation of closed planar curves in the analysis of plant-leaf shapes. We show that it provides a joint framework for computing geodesics (registration) and similarities between plant leaves, which we use for their automatic classification. We evaluate its performance using standard databases and show that it outperforms significantly the state-of-the-art descriptor-based techniques. Additionally, we show that it enables the computation of shape statistics, such as the average shape of a leaf population and its principal directions of variation, suggesting that the representation is suitable for building generative models of plant- leaf shapes. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <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> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> We introduce a framework for analyzing symmetry of 3D anatomical structures using elastic deformations of their boundaries (surfaces). The basic idea is to define a space of parameterized surfaces and to compute geodesic paths between the objects and their arbitrary reflections using a Riemannian structure. Elastic matching, based on optimal (non-linear) re-parameterizations (grid deformations) of surfaces, provides a better registration of points across shapes, as compared to the commonly-used linear registrations. A crucial step of orientation alignment, akin to finding planes of symmetry, is performed as a search for shortest geodesic paths. This framework is fully automatic and provides a measure of symmetry, the nearest symmetric shape and the optimal deformation to make an object symmetric. We demonstrate this framework through multiple toy examples on simple and complicated surfaces. We also explore the use of symmetry analysis in differentiating between healthy and subjects with Attention Deficit Hyper-activity Disorder. <s> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <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> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <s> The shapes of plant leaves are important features to biologists, as they can help in distinguishing plant species, measuring their health, analyzing their growth patterns, and understanding relations between various species. Most of the methods that have been developed in the past focus on comparing the shape of individual leaves using either descriptors or finite sets of landmarks. However, descriptor-based representations are not invertible and thus it is often hard to map descriptor variability into shape variability. On the other hand, landmark-based techniques require automatic detection and registration of the landmarks, which is very challenging in the case of plant leaves that exhibit high variability within and across species. In this paper, we propose a statistical model based on the Squared Root Velocity Function (SRVF) representation and the Riemannian elastic metric of Srivastava et al. (2011) to model the observed continuous variability in the shape of plant leaves. We treat plant species as random variables on a non-linear shape manifold and thus statistical summaries, such as means and covariances, can be computed. One can then study the principal modes of variations and characterize the observed shapes using probability density models, such as Gaussians or Mixture of Gaussians. We demonstrate the usage of such statistical model for (1) efficient classification of individual leaves, (2) the exploration of the space of plant leaf shapes, which is important in the study of population-specific variations, and (3) comparing entire plant species, which is fundamental to the study of evolutionary relationships in plants. Our approach does not require descriptors or landmarks but automatically solves for the optimal registration that aligns a pair of shapes. We evaluate the performance of the proposed framework on publicly available benchmarks such as the Flavia, the Swedish, and the ImageCLEF2011 plant leaf datasets. <s> BIB009 </s> A Survey on Non-rigid 3D Shape Analysis <s> 3.2.2 <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
The shape space of square-root representations Instead of using the Hessian of the deformation energy, Jermyn et al. BIB004 showed that by representing a surface f using the pair (g,n), one can define a full elastic metric on the product space of metrics (i.e. first fundamental forms) g and unit normal fieldsn as follows: The first term of Eqn. (15) accounts for changes in the shape of a local patch that preserve patch area. The second term quantifies changes in the area of the patches. Thus, the first two terms capture full stretch. The last term measures changes in the direction of the surface normal vectors and thus captures bending. The parameters α, β, and µ are positive weights that one can use to penalize the different types of deformations, e.g. if one wants to favor stretch over bending then the third term should be given a higher weight. As such, instead of working in the original space, e.g. C f and S f , one can define a mapping Q such that Q(f ) = (g,n) where g(s) is the the first fundamental form of f at s andn(s) is the unit normal vector to the surface at s. The preshape space C q is then the product space of metrics and unit normal vectors. The shape space is the quotient space S q = C q /(SO(3)×Γ), via the action of the rotation and re-parameterization groups. Jermyn et al. BIB004 defined a new representation of surfaces called the square-root normal fields (SRNF), which are essentially surface normals scaled by the square root of the local area: The space of SRNFs, hereinafter denoted by C q , has very nice properties that are relevant for shape analysis. In particular, Jermyn et al. BIB004 showed that the L 2 metric on C q is a special case of the full elastic metric given by Eqn.  . It corresponds to the case where α = 0, β = 1 4 , and µ = 1: Since |g| is just |n|, the Eqn. (16) is equivalent to Thus, the L 2 metric in the space of SRNFs is equivalent to a weighted sum of surface area change and surface bending. This property of SRNFs makes them very promising as a representation of surfaces for elastic shape analysis. If geodesics, mean shapes, PCA, etc. could be computed in C q under the L 2 metric, and then mapped back to F or C f then there would be large gains in computational efficiency with respect to other metrics such as those defined on the space of thin-shells. Unfortunately, there is no analytical expression for Q −1 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 always pull the L 2 metric back to C f under Q and perform computations there, as in BIB008 , but this is computationally expensive, and rather defeats the purpose of having an L 2 metric in the first place. Although an analytical solution to the inversion problem remains an open problem, Laga et al. BIB010 showed that one can always estimate, for a given q ∈ C q a surface f ∈ C f whose SRNF representation is as close as possible q. SRNF representation is a generalization of the Square-Root Velocity Function (SRVF) BIB001 proposed for elastic analysis of planar curves and used in many applications including biology and botany BIB005 , BIB009 . It is also a special case of the family of square-root maps (SRM). Another example of SRM is the Q-maps introduced by Kurtek et al.  , BIB002 , BIB003 , BIB006 , BIB007 , which, similar to SRNF maps, is used for the analysis of the shapes of parameterized surfaces using the L 2 metric. It is defined a mapping Q : F → L 2 , where Properties. SRNF and Q-maps share very nice mathematical properties that make them suitable for shape analysis. In particular, both SRNFs and Q-maps are a subset of L 2 (S 2 , R 3 ). This means that one can use the standard L 2 metric in the space of SRNFs or Qmaps for the analysis of shapes instead of the complex and expensive pullback elastic metrics. Furthermore, if a surface is re-parameterized according to f → f •γ, γ ∈ Γ, its corresponding Q-map or SRNF is given by (q, γ) = (q • γ) J γ , where J γ is the determinant of the Jacobian of γ. Another main motivation behind the use of Q-maps and SRNFs in surface analysis is that the action of Γ is by isometries under the L 2 metric, i.e. q 1 − q 2 = (q 1 , γ) − (q 2 , γ) , ∀γ ∈ Γ. This property is very important for joint rotation and re-parameterizationinvariant comparison, and thus elastic registration, of surfaces as will be discussed in Section 4. As such, Qmaps and SRNF maps have been used for optimal reparameterization, and thus registration, of spherical and quadrilateral surfaces BIB010 , BIB004 , BIB002 , BIB003 that undergo isometric as well as elastic deformations, and surfaces that contain missing parts. They have been also used for computing geodesic paths between such surfaces BIB010 , BIB006 . The strength of these two approaches is that shape comparison, registration and geodesic computations are performed under the same elastic Riemannian metric. The main difference between Q-maps and SRNFs is that (1) the former lacks a physical interpretation; it does not have a clear relationship to an underlying elastic metric. The representation was solely devised for convenience of being able to compare the shapes of parameterized surfaces using the L 2 metric, and (2) unlike Q-maps, SRNF representation is invariant to translation since it is computed using the surface normals.