A Survey on Non-rigid 3D Shape Analysis <s> Statistical analysis using non-Euclidean metrics <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> Statistical analysis using non-Euclidean metrics <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> BIB002
When using non-Euclidean metrics, Eqn. (2), which is equivalent to Eqn. (1), cannot be solved analytically. Several previous works have proposed gradient-based approaches for finding the Karcher mean of Eqn. (1), see for example BIB001 ,  . For convenience, the general approach is repeated here as Algorithm 1. In addition to computing the mean shape of a collection of 3D models, we are usually interested in understanding (and modelling) how the shape varies within the collection. Let {f 1 , · · · , f n } be a collection of shapes andf be their mean. For simplicity of presentation, we assume that all the surfaces have been registered tof . Let α i be the geodesic path, with respect to the chosen metric, betweenf and f i .  where O * and γ * are the optimal rotation and reparameterization, respectively. 4: For each i = 1, . . . , n, compute the shape space geodesic path α f i betweenf j and f * i . 5: Compute the average shooting direction as 6: If v < 2 , stop and returnf j . Otherwise, update the mean usingf j+1 =f j + 1 v and normalizē f j+1 with respect to scale and translation. 7: Return to step 3. Fletcher et al. BIB002 defined the following concepts that are essential for representing variability of data that lie on general manifolds: • Variance. Flechet  defined variance as the expected value of the squared Riemannian distance from the mean. • Geodesic subspaces. The lower-dimensional subspaces in PCA are linear subspaces. For general manifolds, it is natural to use geodesic curves as one-dimensional subspace, which is equivalent to principal directions in PCA. Flechet  showed that submanifold geodesics at the mean are the generalization of the linear subspaces of PCA. • Projection. The projection of a point f ∈ C f onto a geodesic submanifold H is defined as the point on H that is the nearest to f in terms of the Riemannian distance. Thus, the projection operator π H (f ) : C f → H can be defined as: Given the non-linearity of the shape space, the modes of variation can be efficiently computed by first projecting the data to the tangent space to C f at the Karcher mean using the exp map, performing standard PCA in that linear space, and then mapping the results back to C f using the Log map. Algorithm 2 summarizes this procedure.