A Survey on Non-rigid 3D Shape Analysis <s> 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> 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
The fundamentals of statistical analysis of shapes have been laid by Kendall  , as early as 1977, and advanced by many others  , BIB002 ,  , BIB001 . The basic idea is to represent the shape of a 3D object using a finite set p of l ordered points, called landmarks, p = {p i ∈ R 3 , i = 1, . . . l}. These points are sampled from the shape boundary and put in correspondence across shapes. In the case of our setup where a surface f is treated as a mapping from a parameterization domain D to R 3 , Kendall's representation is equivalent to having a fixed finite sampling of the parameterization domain D. That is D = {s i , i = 1, . . . , l}. As such, the space F of all surfaces that can be represented with l landmarks is R 3l . The pre-shape space, C f , i.e. the space of surfaces that are normalized for translation and scale, is a hypersphere S 3l−4 , which is a subspace of R 3l . To make the representation invariant to rotations, Kendall defines the group action SO(3) × S 3l−4 , (O, f ) = Of whose orbits are given by The shape space S f can then be viewed as the quotient space S 3n−4 /SO(3). The rotation-invariant geodesic distance between two surfaces f 1 and f 2 is given as the geodesic distance between their orbits on S f , i.e. : where d C is the length of the arc on C f that connects f 1 to Of 2 . A geodesic path is then a great circle on C f between f 1 and O * f 2 where O * = argmin Kendall's shape space, introduced in 1977, and its subsequent developments  , BIB002 ,  , BIB001 , are considered as the foundation that lead to the major recent developments in statistical shape analysis.