A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <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> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <s> In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1---1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <s> We propose a novel method for computing a geometrically consistent and spatially dense matching between two 3D shapes. Rather than mapping points to points we match infinitesimal surface patches while preserving the geometric structures. In this spirit we consider matchings as diffeomorphisms between the objects' surfaces which are by definition geometrically consistent. Based on the observation that such diffeomorphisms can be represented as closed and continuous surfaces in the product space of the two shapes we are led to a minimal surface problem in this product space. The proposed discrete formulation describes the search space with linear constraints. Computationally, our approach leads to a binary linear program whose relaxed version can be solved efficiently in a globally optimal manner. As cost function for matching, we consider a thin shell energy, measuring the physical energy necessary to deform one shape into the other. Experimental results demonstrate that the proposed LP relaxation allows to compute highquality matchings which reliably put into correspondence articulated 3D shapes. Moreover a quantitative evaluation shows improvements over existing works. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <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> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <s> We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells modulo rigid body motions. Based on this foundation we develop a time- and space-discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <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> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> The shape space of thin shells <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> BIB007
Based on a physical deformation paradigm, a surface f can be viewed as the middle layer of a thin viscous material of some finite (albeit small) thickness δ BIB006 . Wirth et al. BIB002 , Heeren et al.  , BIB007 , Berkels et al. BIB004 , and Zhang et al. BIB006 used the deformation model of Eqn. (11), with α s = 1 and α b proportional to the squared thickness δ 2 . The stretch energy, referred to as the membrane term, is given as The log det term penalizes material compression. λ and µ are the Lamé constants of the tangential Newtonian dissipation measure. The Cauchy-Green strain tensor, 1 g 2 , where g i is the first fundamental form or metric of the surface f i , accounts for changes in the metric between the source and target surfaces. In particular, the trace of the Cauchy-Green strain tensor, tr(G), accounts for local changes in curve lengths while the determinant det(G) accounts for local changes in surface area BIB006 . Thus, this can be seen as the generalization of the stretch metric of Kilian et al. BIB001 , which only considered changes in the length of curves. The bending energy should account for out-ofplane bending and changes in curvature. Since such information is encoded in the shape operator, then the most natural choice of measure for quantifying bending is variation in the shape operator  : where · F denotes the Frobenius norm. The energy of Eqn. (13) takes into account full change in the second fundamental form. Heeren et al. BIB007 and Windheuser et al. BIB003 simplified this metric by only considering changes in the mean curvature, which is the trace of the shape operator S; This energy is also known as the Willmore energy. In the discrete setup, Heeren et al. BIB005 approximated the mean curvature by the dihedral angle, i.e. the angle between the normals of two adjacent faces on the triangulated mesh. Since this deformation model acts on the CauchyGreen tensor G and mean curvature H then one can define the preshape space as the space of all pairs (G, H). An important observation that has been made in  , BIB006 ,  is that the Hessian of the elastic deformation energy results in a proper Riemannian metric on the space of shells, modulo rigid body motions. That is, for an infinitesimal deformation is a proper metric that measures the strength of this deformation. is the second derivative of E in the direction of v.) Moreover, Heeren et al.  showed that Hess(E)(v, v) = 0 if and only if v induces an infinitesimal rigid motion. This last property is highly desirable in shape analysis where, as described in Section 2.2, one seeks a framework that is invariant to rigid motions of shapes. This properties guarantees this invariance without normalization for translation and rotation of the shapes being analyzed. Shapes, however, need to be normalized for scale.