A Survey on Non-rigid 3D Shape Analysis <s> Registration <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> Registration <s> In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple $\ensuremath{\mathbb{L}^2}$ metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Registration <s> Three-dimensional surface registration transforms multiple three-dimensional data sets into the same coordinate system so as to align overlapping components of these sets. Recent surveys have covered different aspects of either rigid or nonrigid registration, but seldom discuss them as a whole. Our study serves two purposes: 1) To give a comprehensive survey of both types of registration, focusing on three-dimensional point clouds and meshes and 2) to provide a better understanding of registration from the perspective of data fitting. Registration is closely related to data fitting in which it comprises three core interwoven components: model selection, correspondences and constraints, and optimization. Study of these components 1) provides a basis for comparison of the novelties of different techniques, 2) reveals the similarity of rigid and nonrigid registration in terms of problem representations, and 3) shows how overfitting arises in nonrigid registration and the reasons for increasing interest in intrinsic techniques. We further summarize some practical issues of registration which include initializations and evaluations, and discuss some of our own observations, insights and foreseeable research trends. <s> BIB003
In shape analysis, registration quality is a major factor in determining the accuracy of the subsequent analysis steps such as computing optimal deformations (geodesics) or summary statistics (e.g. mean shapes and modes of variations). It has been extensively studied in the past two decades, see  and BIB003 for comprehensive surveys and different taxonomies of the state-of-the-art. In general, the registration problem can be mathematically stated as the problem of finding the optimal transformation, T , and a reparameterization γ : is a certain measure of closeness in the space of surfaces. We have seen in Section 2.2 how to remove the variabilities due to translation and scale. In this section, we will focus on the remaining variables, which are rotations and reparameterizations. Thus, we are given two surfaces f 1 and f 2 ∈ C f and we seek to solve the following optimization problem: The optimization problem of Eqn. (19) is a complex problem that can be solved by iterating between the optimization over rotations O ∈ SO(3), i.e. finding the optimal rotation, given known correspondences (i.e. fixed γ), and then finding the optimal correspondences or reparameterization γ given fixed rotation O. If correspondences are given, finding the best O that aligns f 2 onto f 1 is straightforward. Under the L 2 metric, the best rotation O can be analytically found using Singular Value Decomposition (SVD) leading to the Procrustes analysis of shapes  , which forms the basis of 2D Active Shape Models (ASM) BIB001 and 3D morphable models  . Similarly, if one is using the partial elastic metric of Jermyn et al. BIB002 (Section 3.2.2), then one can find the optimal rotation O in the space of SRNFs and apply it to the original surface since the SRNF map of Of is exactly Oq. In practice, correspondences are unknown and better results are obtained if one solves simultaneously for the best correspondences and for the optimal rigid alignment.