A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> A model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives. Metrics are described that compute an object representation's prior probability of local geometry by reflecting variabilities in the net's node and link parameter values, and that compute a likelihood function measuring the degree of match of an image to that object representation. A paradigm for image analysis of deforming such a model to optimize a posteriori probability is described, and this paradigm is shown to be usable as a uniform approach for object definition, object-based registration between images of the same or different imaging modalities, and measurement of shape variation of an abnormal anatomical object, compared with a normal anatomical object. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery. The segmentation procedure is based on a Bayesian deformable templates methodology in which the prior information about the geometry and shape of anatomical objects is incorporated via the construction of exemplary templates. The anatomical variability is accommodated in the Bayesian framework by defining probabilistic transformations on these templates. The transformations, thus, defined are parameterized directly in terms of natural shape operations, such as growth and bending, and their locations. A preliminary validation study of the segmentation procedure is presented. We also present a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localized variability at the fine scale. We show that the method can be used to statistically describe shape variability in intuitive terms such as growing and bending. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> Principal component analysis has proven to be useful for understanding geometric variability in populations of parameterized objects. The statistical framework is well understood when the parameters of the objects are elements of a Euclidean vector space. This is certainly the case when the objects are described via landmarks or as a dense collection of boundary points. We have been developing representations of geometry based on the medial axis description or m-rep. Although this description has proven to be effective, the medial parameters are not naturally elements of a Euclidean space. In this paper we show that medial descriptions are in fact elements of a Lie group. We develop methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> We develop a novel method for fitting high-resolution template meshes to detailed human body range scans with sparse 3D markers. We formulate an optimization problem in which the degrees of freedom are an affine transformation at each template vertex. The objective function is a weighted combination of three measures: proximity of transformed vertices to the range data, similarity between neighboring transformations, and proximity of sparse markers at corresponding locations on the template and target surface. We solve for the transformations with a non-linear optimizer, run at two resolutions to speed convergence. We demonstrate reconstruction and consistent parameterization of 250 human body models. With this parameterized set, we explore a variety of applications for human body modeling, including: morphing, texture transfer, statistical analysis of shape, model fitting from sparse markers, feature analysis to modify multiple correlated parameters (such as the weight and height of an individual), and transfer of surface detail and animation controls from a template to fitted models. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> We introduce the SCAPE method (Shape Completion and Animation for PEople)---a data-driven method for building a human shape model that spans variation in both subject shape and pose. The method is based on a representation that incorporates both articulated and non-rigid deformations. We learn a pose deformation model that derives the non-rigid surface deformation as a function of the pose of the articulated skeleton. We also learn a separate model of variation based on body shape. Our two models can be combined to produce 3D surface models with realistic muscle deformation for different people in different poses, when neither appear in the training set. We show how the model can be used for shape completion --- generating a complete surface mesh given a limited set of markers specifying the target shape. We present applications of shape completion to partial view completion and motion capture animation. In particular, our method is capable of constructing a high-quality animated surface model of a moving person, with realistic muscle deformation, using just a single static scan and a marker motion capture sequence of the person. <s> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB009 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB010 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB011 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB012 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB013 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB014 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <s> The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. <s> BIB015 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB016 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB017 </s> A Survey on Non-rigid 3D Shape Analysis <s> Metrics on the space of deformations <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> BIB018
Representing shapes with the deformations of a template leads to shape spaces that are non Euclidean (rotations, for example, are elements of SO (3)). Some methods typically use a Euclidean representation of deformations and measure distances using the L 2 metric, ignoring the geometry of the space of deformations. The SCAPE model BIB006 , for instance, learned a shape model using the Euclidean distance and PCA on the deformation matrices S using only the subjects that are in the same neutral pose. Similarly, Hasler et al. BIB006 learned, separately, two low-dimensional models of shape and pose, and then combined them at a later stage. Unlike these works, both M-reps BIB001 , BIB002 , BIB003 , BIB005 and Lie bodies BIB009 exploit the manifold structure of the space of deformations where geodesics and statistics are computed using proper Riemannian metrics. An important property that makes these methods of practical interest is that, despite the non-linearity of the metric as well as the manifold, geodesic distances and geodesic paths between a pair of shapes have a closed-form formula, unlike other methods which require expensive optimization schemes, see BIB009 and BIB003 , BIB005 for the mathematical details. This significantly simplifies the process of comparing shapes and computing geodesics and summary statistics. Note that these representations require that all the shapes being analyzed to have the same connectivity and to be in one-to-one correspondence with each other and with the template. Such correspondences need to be computed using a different approach, and more importantly, using an optimality criteria that is often different from the one used for computing geodesics and statistics. Trans. Scale Rot. Reg. Morphable models  , BIB004 f ∈ R 3l yes yes yes Thin shells  , BIB016 , BIB017 , BIB013 f ∈ R 3l no no no yes R 3l Hess(E) Q-maps  , BIB007 , BIB008 , BIB010 , BIB011  SRNFs BIB018 , BIB012 , BIB014 , BIB015  yes yes yes M-reps BIB001 , BIB002 , BIB003 , BIB005 f ∈ R 3l yes yes yes 