A Survey on Non-rigid 3D Shape Analysis <s> Morphable models <s> !, Model-based vision is firmly established as a robust approach to recognizing and locating known rigid objects in the presence of noise, clutter, and occlusion. It is more problematic to apply modelbased methods to images of objects whose appearance can vary, though a number of approaches based on the use of flexible templates have been proposed. The problem with existing methods is that they sacrifice model specificity in order to accommodate variability, thereby compromising robustness during image interpretation. We argue that a model should only be able to deform in ways characteristic of the class of objects it represents. We describe a method for building models by learning patterns of variability from a training set of correctly annotated images. These models can be used for image search in an iterative refinement algorithm analogous to that employed by Active Contour Models (Snakes). The key difference is that our Active Shape Models can only deform to fit the data in ways consistent with the training set. We show several practical examples where we have built such models and used them to locate partially occluded objects in noisy, cluttered images. Q 199s A&& prrss, IN. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Morphable models <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> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Morphable models <s> We describe a new method of matching statistical models of appearance to images. A set of model parameters control modes of shape and gray-level variation learned from a training set. We construct an efficient iterative matching algorithm by learning the relationship between perturbations in the model parameters and the induced image errors. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Morphable models <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> Morphable models <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> BIB005
In the case of planar objects, simply treating Kendall's representation as a vector space, i.e. assuming that the preshape space C f is Euclidean, and ·, · is an L 2 metric, results in the 2D Active Shape Models (ASM) of Cootes et al. BIB001 , BIB003 ,  . This formulation has been later used for the analysis of the shape of 3D objects that are in static poses, leading to what is now known as morphable models. Morphable models have been originally introduced for the analysis of the 3D shape of human faces  . They have been later used for the analysis of the 3D shape of human bodies BIB004 as well as the shape of objects originating from various domains including archaeology, astronomy, morphometrics, and medical diagnosis  , BIB002 ,  . Mathematically, for a given pair of vectors v 1 and 2 metric is defined as: where ·, · is the standard inner product. The geometric interpretation of this metric is as follows; A surface f 1 can be optimally deformed to another surface f 2 by simply adding to each point The optimal path F * that connects f 1 to f 2 , induced by the metric of Eqn. 8, is simply a straight line. Subsequently, the difference, or dissimilarity, between f 1 and f 2 is given by Assuming that d(·, ·) is Euclidean, and under a fixed parameterization, finding the optimal rotation that aligns one surface onto another is equivalent to finding the rotation O ∈ SO(3) that minimizes f 1 −Of 2 . This results in the Procrustean metric on the shape space S f , which can be efficiently solved using Singular Value Decomposition (SVD)  . Now, for a fixed rotation and reparameterization, i.e. the surfaces are already registered, both the mean shape of Eqn. (1) as well as the covariance matrix K have analytical solutions: Thus, efficient parameterization of the shape variability in a collection of 3D shapes can be performed using Principal Component Analysis (PCA). Let λ k , k = 1, . . . , d, be the leading eigenvalues of K, and v k their corresponding eigenvectors, called modes of variation. A new statistically feasible shape f is given where α k ∈ R are coefficients that control the contribution of each mode of variation. Arbitrary new shapes can be generated by varying the coefficient vector − → α = (α 1 , α 2 , . . . , α d ) whose probability follows a Gaussian distribution defined as (a) Linear path (b) Geodesic path (1 − τ )f1 + tf2 by SRNF inversion Fig. 2 . Linear interpolation in F vs. geodesic path by SRNF inversion (see Section 4) . Image courtesy of BIB005 . λi +const. Thus, if one wants to synthesize a plausibe shape, i.e. a shape that is similar, but not identical, to those in the collection, one only has to generate arbitrary αs and retain those that have high probability. This approach is known as morphable models and is an extension of the Active Shape Models (ASM) developed by Cootes et al. BIB001 , BIB003 ,  for the analysis of planar objects. They have been first introduced to 3D shape analysis by Blanz and Vetter in their seminal work on the synthesis of 3D faces  . Morphable models have been later generalized to characterize the space of the entire human body shapes BIB004 . Both in  and BIB004 , the objects under study are assumed to be in neutral poses and their landmarks are in a one-to-one correspondence.