A Survey on Non-rigid 3D Shape Analysis <s> The non-linear nature of Kendall's shape space <s> Preliminaries: Size Measures and Shape Coordinates. Preliminaries: Planar Procrustes Analysis. Shape Space and Distance. General Procrustes Methods. Shape Models for Two Dimensional Data. Tangent Space Inference. Size--and--Shape. Distributions for Higher Dimensions. Deformations and Describing Shape Change. Shape in Images. Additional Topics. References and Author Index. Index. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> The non-linear nature of Kendall's shape space <s> Shapes and Shape Spaces. The Global Structure of Shape Spaces. Computing the Homology of Cell Complexes. A Chain Complex for Shape Spaces. The Homology Groups of Shape Spaces. Geodesics in Shape Spaces. The Riemannian Structure of Shape Spaces. Induced Shape--Measures. Mean Shapes and the Shape of the Means. Visualising the Higher Dimensional Shape Spaces. General Shape Spaces. Appendix. Bibliography. Index. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> The non-linear nature of Kendall's shape space <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> BIB003
Although the L 2 metric on Kendall's shape space has been extensively used in a wide range of applications, e.g. archaeology, astronomy, morphometrics, medical diagnosis  , BIB001 ,  , it is only suitable for computing geodesics, and thus statistics, between shapes that undergo small deformations, e.g. anatomical shapes, 3D faces, or human shapes in a neutral pose. This main limitation is illustrated in Figure 2 using two synthetic cylinders. The example shows optimal deformation paths between surfaces f 1 and f 2 , where f 1 is a straight cylinder and f 2 is a bent cylinder. Figure 2 -(a) shows the linear path between f 1 and f 2 in S f (i.e. after full registration) obtained using the L 2 metric by connecting each pair of corresponding points with a straight line. The intermediate shapes along this path shrink unnaturally. Figure 2 -(b) shows a natural geodesic path computed with a metric that captures the shape deformations (here using SRNF inversion, which will be described later). Kilian et al. BIB003 build on Kendall's representation a new metric that does not suffer from the shrinkage issue of the L 2 metric. The idea is to define the distance between a surface f 1 and f 2 as the strength of a vector field u ∈ T f1 (F) that penalizes elastic deformations, enforcing as rigid-as-possible and as-isometric-as-possible deformations along geodesic paths. That is, given a shape f ∈ F = R 3n , define the inner product between two vectors u and v in T f (F) as: BIB002 where λ = 0.001 and a(s) is the local Voronoi area at the vertex s. The first term penalizes non-rigid deformations. To define it, consider first that the deformation vector u can be written as u = u +u where u is only composed of rigid transformations and u is only composed of non-rigid deformations. To have a metric that is invariant to rigid transformations while quantifying non-rigid ones then one can set u, u R = u , u . As such, u, u R = 0 if and only if u is a rigid motion. This term ensures invariance to translation, scale and rotation without normalization in a pre-processing step. The second term of Eqn. (10) favours bending over stretching. It is defined as the strength of the deformations that remain after discarding isometric motions, i.e. motions that preserve geodesic distances along the surface of f . In other words, it can be seen as a measure of stretch in terms changes in the length of curves along the surface f . Since Kilian et al. BIB003 's approach works on triangulated meshes then where s and t are two adjacent vertices on the triangulated mesh that represents the surface f . Thus, u, u I = 0 if and only if u just bends the shape f . In other terms, when deforming the straight cylinder of Figure 2 , the metric will favor bending over stretching and thus it will minimize the shrinkage that is observed when using the standard L 2 metric. The last term is a regularizer that makes the metric Riemannian. It also ensures that the geodesic between two isometric shapes is unique. The main limitation of this metric is the choice of the parameter λ in order to ensures that the metric is Riemannian. Large values of λ will give high weight to the Euclidean term of Eqn. BIB002 . Thus, the metric may suffer from the shrinkage issue observed in morphable models (see Figure 2 ). Note that Kilian et al. empirically set this parameter to 0.001.