A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouve (1995) in which two images I 0, I 1 are given and connected via the diffeomorphic change of coordinates I 0???1=I 1 where ?=?1 is the end point at t= 1 of curve ? t , t?[0, 1] satisfying .? t =v t (? t ), t? [0,1] with ?0=id. The variational problem takes the form ::: ::: $$\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right),$$ ::: ::: where ?v t? V is an appropriate Sobolev norm on the velocity field v t(·), and the second term enforces matching of the images with ?·?L 2 representing the squared-error norm. ::: ::: In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields v t, t?[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ?0 1?v t? V dt on the geodesic shortest paths. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> We introduce the SCAPE method (Shape Completion and Animation for PEople)---a data-driven method for building a human shape model that spans variation in both subject shape and pose. The method is based on a representation that incorporates both articulated and non-rigid deformations. We learn a pose deformation model that derives the non-rigid surface deformation as a function of the pose of the articulated skeleton. We also learn a separate model of variation based on body shape. Our two models can be combined to produce 3D surface models with realistic muscle deformation for different people in different poses, when neither appear in the training set. We show how the model can be used for shape completion --- generating a complete surface mesh given a limited set of markers specifying the target shape. We present applications of shape completion to partial view completion and motion capture animation. In particular, our method is capable of constructing a high-quality animated surface model of a moving person, with realistic muscle deformation, using just a single static scan and a marker motion capture sequence of the person. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space ? d . Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in ?2 and ?3. Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and experiments with 3D curves extracted from brain cortical surfaces are presented. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> In the context of large deformations by diffeomorphisms, we propose a new diffeomorphic registration algorithm for 3D images that performs the optimization directly on the set of geodesic flows. The key contribution of this work is to provide an accurate estimation of the so-called initial momentum, which is a scalar function encoding the optimal deformation between two images through the Hamiltonian equations of geodesics. Since the initial momentum has proven to be a key tool for statistics on shape spaces, our algorithm enables more reliable statistical comparisons for 3D images. ::: ::: Our proposed algorithm is a gradient descent on the initial momentum, where the gradient is calculated using standard methods from optimal control theory. To improve the numerical efficiency of the gradient computation, we have developed an integral formulation of the adjoint equations associated with the geodesic equations. ::: ::: We then apply it successfully to the registration of 2D phantom images and 3D cerebral images. By comparing our algorithm to the standard approach of Beg et al. (Int. J. Comput. Vis. 61:139---157, 2005), we show that it provides a more reliable estimation of the initial momentum for the optimal path. In addition to promising statistical applications, we finally discuss different perspectives opened by this work, in particular in the new field of longitudinal analysis of biomedical images. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> We provide an overview of several shape space models that have been proposed in the past few years, focusing, in particular on models involving Riemannian manifolds of shapes. The discussion is organized in three stages, starting with a review of some shape representation methods, followed by shape space structures, from metric spaces to manifolds, and concluding with a short description of some of the applications that resulted from such models. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Transformation-based representations <s> In this paper, we propose a new algorithm to compute intrinsic means of organ shapes from 3D medical images. More specifically, we explore the feasibility of Karcher means in the framework of the large deformations by diffeomorphisms (LDDMM). This setting preserves the topology of the averaged shapes and has interesting properties to quantitatively describe their anatomical variability. Estimating Karcher means requires to perform multiple registrations between the averaged template image and the set of reference 3D images. Here, we use a recent algorithm based on an optimal control method to satisfy the geodesicity of the deformations at any step of each registration. We also combine this algorithm with organ specific metrics. We demonstrate the efficiency of our methodology with experimental results on different groups of anatomical 3D images. We also extensively discuss the convergence of our method and the bias due to the initial guess. A direct perspective of this work is the computation of 3D+time atlases. <s> BIB006
The approaches described in Sections 3.1 and 3.2 represent a shape as an embedding f : D → R 3 while the metric quantifies the deformations the shape undergoes. Another approach is to model transformations that 3D objects undergo rather than the transformed objects themselves. These representations derive from Grenander's pattern theory  in which a shape is not represented as such but as a deformation of another (fixed) shape, T , called template BIB005 . This paradigm has been introduced by Grenander  , developed in BIB001 , BIB003 , BIB004 , BIB006 and applied to various problems in computational anatomy and more recently to the analysis of human body shapes BIB002 . Let's assume that we are given a set of registered surfaces represented as triangulated meshes with the same graph topology as the template T . We assume that the template has m triangular faces, T = {T k , k = 1 . . . m}. Each triangular face can be represented with a matrix of its three vertices, . . m} of m triangles can be represented with a set of transformation matrices k is a 3 × 3 matrix that encodes various types of deformations that T i k can undergo. Hereinafter, we refer to P as the deformation model. Existing methods differ in the choice of (1) the template T , (2) the deformation model P , and (3) the metric used to measure distances in the space of deformations. We will discuss these aspects in the following subsections.