A Survey on Non-rigid 3D Shape Analysis <s> Geodesics using pullback metrics <s> This is a method for determining numerically local minima of differentiable functions of several variables. In the process of locating each minimum, a matrix which characterizes the behavior of the function about the minimum is determined. For a region in which the function depends quadratically on the variables, no more than N iterations are required, where N is the number of variables. By suitable choice of starting values, and without modification of the procedure, linear constraints can be imposed upon the variables. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics using pullback metrics <s> This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics using pullback metrics <s> We study the problem of joint registration and deformation analysis of endometrial tissue using 3D magnetic resonance imaging (MRI) and 2D trans-vaginal ultrasound (TVUS) measurements. In addition to the different imaging techniques involved in the two modalities, this problem is complicated due to: 1) different patient pose during MRI and TVUS observations, 2) the 3D nature of MRI and 2D nature of TVUS measurements, 3) the unknown intersecting plane for TVUS in MRI volume, and 4) the potential deformation of endometrial tissue during TVUS measurement process. Focusing on the shape of the tissue, we use expert manual segmentation of its boundaries in the two modalities and apply, with modification, recent developments in shape analysis of parametric surfaces to this problem. First, we extend the 2D TVUS curves to generalized cylindrical surfaces through replication, and then we compare them with MRI surfaces using elastic shape analysis. This shape analysis provides a simultaneous registration (optimal reparameterization) and deformation (geodesic) between any two parametrized surfaces. Specifically, it provides optimal curves on MRI surfaces that match with the original TVUS curves. This framework results in an accurate quantification and localization of the deformable endometrial cells for radiologists, and growth characterization for gynecologists and obstetricians. We present experimental results using semi-synthetic data and real data from patients to illustrate these ideas. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Geodesics using pullback metrics <s> In this paper we describe how to perform Principal Components Analysis in "shell space". Thin shells are a physical model for surfaces with non-zero thickness whose deformation dissipates elastic energy. Thin shells, or their discrete counterparts, can be considered to reside in a shell space in which the notion of distance is given by the elastic energy required to deform one shape into another. It is in this setting that we show how to perform statistical analysis of a set of shapes (meshes in dense correspondence), providing a hybrid between physical and statistical shape modelling. The resulting models are better able to capture non-linear deformations, for example resulting from articulated motion, even when training data is very sparse compared to the dimensionality of the observation space. <s> BIB004
The first class of approaches find geodesics by solving the optimization problem of Eqn. (6) using one of the elastic metrics defined in Section 3.2. In the simplest case, we are given two surfaces f 1 and f 2 and we seek to find another surface f * on the geodesic path F and which is exactly at the mid-point between f 1 anf f 2 , i.e. f * = F (0.5). Mathematically, f * is the minimizer of the discrete path energy: which corresponds to solving for f the EulerLagrange equation Heeren et al.  solve this non-linear optimization problem using the Quasi-Newton method. That is, one can start with an initial estimate f 0 of f , e.g. , and then iteratively update the estimate via a Newton iteration with step size control: The optimal step size can be determined using the David-Fletcher-Powell (DFP) formula BIB001 . Using this formulation, one can solve for the entire timediscrete geodesic path recursively; once the mid-point f between f 1 and f 2 is computed, one can apply the same procedure to solve for the mid-point between f 1 and f , and between f and f 2 , etc.  . Alternatively, one can directly find the entire discrete path This is called a path straightening procedure. The optimization can start with an initial path, which is then iteratively straightened, i.e. updated so that its length with respect to the elastic metric is minimized. In the space of shells  , BIB004 this is done by solving the set of Euler-Lagrange equations: Instead of solving a non-linear optimization problem, Kurtek et al. BIB002 , BIB003 define the energy of a continuous path F as: The shortest path is then found by minimizing the path energy using a gradient descent approach where the current estimate of the geodesic path is updated according to F = F − ∇E path . Let A 0 be the set of all smooth paths that start at f 1 and end at f 2 and let T F (A 0 ) be the tangent space to A 0 at a given point F . T F (A 0 ) corresponds to perturbations of the path F . Kurtek at al. BIB002 describes the space of path perturbations using orthonormal basis B = {α 1 , · · · , α B } so that the gradient of the path energy can be written as We refer the reader to BIB002 , BIB003 for the full derivation of the directional derivatives ∇E path (δα i ) as well as the procedure for constructing the orthonormal basis B.