A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> This paper presents a new technique for the automatic model-based segmentation of three-dimensional (3-D) objects from volumetric image data. The development closely follows the seminal work of Taylor and Cootes on active shape models, but is based on a hierarchical parametric object description rather than a point distribution model. The segmentation system includes both the building of statistical models and the automatic segmentation of new image data sets via a restricted elastic deformation of shape models. Geometric models are derived from a sample set of image data which have been segmented by experts. The surfaces of these binary objects are converted into parametric surface representations, which are normalized to get an invariant object-centered coordinate system. Surface representations are expanded into series of spherical harmonics which provide parametric descriptions of object shapes. It is shown that invariant object surface parametrization provides a good approximation to automatically determine object homology in terms of sets of corresponding sets of surface points. Gray-level information near object boundaries is represented by 1-D intensity profiles normal to the surface. Considering automatic segmentation of brain structures as their driving application, the authors' choice of coordinates for object alignment was the well-accepted stereotactic coordinate system. Major variation of object shapes around the mean shape, also referred to as shape eigenmodes, are calculated in shape parameter space rather than the feature space of point coordinates. Segmentation makes use of the object shape statistics by restricting possible elastic deformations into the range of the training shapes. The mean shapes are initialized in a new data set by specifying the landmarks of the stereotactic coordinate system. The model elastically deforms, driven by the displacement forces across the object's surface, which are generated by matching local intensity profiles. Elastical deformations are limited by setting bounds for the maximum variations in eigenmode space. The technique has been applied to automatically segment left and right hippocampus, thalamus, putamen, and globus pallidus from volumetric magnetic resonance scans taken from schizophrenia studies. The results have been validated by comparison of automatic segmentation with the results obtained by interactive expert segmentation. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> Standard practice in quantitative structural neuroimaging is a segmentation into bram tissue, subcortical structures, fluid space and lesions followed by volume calculations of gross structures. On the other hand, it is evident that object characterization by size does only capture one of multiple aspects of a full structural characterization. Desirable parameters are local and global parameters like length, elongation, bending, width, complexity, bumpiness and many more. In neuroimaging research there is increasing evidence that shape analysis of brain structures provides new information which is not available by conventional volumetric measurements. This motivates development of novel morphometry analysis techniques answering clinical research questions which have been asked for a long time but which remained unanswered due to the lack of appropriate measurement tools. Challenges are the choice of biologically meaningful shape representations, robustness to noise and small perturbations, and the ability to capture the shape properties of populations that represent natural biological shape variation. This paper describes experiments with two different shape representation schemes, a fine-scale, global surface characterization using spherical harmonics, and a coarsely sampled medial representation (3D skeleton). Driving applications are the detection of group differences of amhygdala-hippocampal shapes in schizophrenia and the analysis of ventricular shape similarity in a mono/dizygotic twin study. The results clearly demonstrate that shape captures information on structural similarity or difference which is not accessible by volume analysis. Improved global and local structure characterization as proposed herein might help to explain pathological changes in neurodevelopment/neurodegeneration in terms of their biological meaning. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> Shape analysis has become of increasing interest to the neuroimaging community due to its potential to precisely locate morphological changes between healthy and pathological structures. This manuscript presents a comprehensive set of tools for the computation of 3D structural statistical shape analysis. It has been applied in several studies on brain morphometry, but can potentially be employed in other 3D shape problems. Its main limitations is the necessity of spherical topology. ::: ::: The input of the proposed shape analysis is a set of binary segmentation of a single brain structure, such as the hippocampus or caudate. These segmentations are converted into a corresponding spherical harmonic description (SPHARM), which is then sampled into a triangulated surfaces (SPHARM-PDM). After alignment, differences between groups of surfaces are computed using the Hotelling T2 two sample metric. Statistical p-values, both raw and corrected for multiple comparisons, result in significance maps. Additional visualization of the group tests are provided via mean difference magnitude and vector maps, as well as maps of the group covariance information. ::: ::: The correction for multiple comparisons is performed via two separate methods that each have a distinct view of the problem. The first one aims to control the family-wise error rate (FWER) or false-positives via the extrema histogram of non-parametric permutations. The second method controls the false discovery rate and results in a less conservative estimate of the false-negatives. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> The goal of our work is to develop an efficient, automatic algorithm for discovering point correspondences between surfaces that are approximately and/or partially isometric. Our approach is based on three observations. First, isometries are a subset of the Mobius group, which has low-dimensionality -- six degrees of freedom for topological spheres, and three for topological discs. Second, computing the Mobius transformation that interpolates any three points can be computed in closed-form after a mid-edge flattening to the complex plane. Third, deviations from isometry can be modeled by a transportation-type distance between corresponding points in that plane. Motivated by these observations, we have developed a Mobius Voting algorithm that iteratively: 1) samples a triplet of three random points from each of two point sets, 2) uses the Mobius transformations defined by those triplets to map both point sets into a canonical coordinate frame on the complex plane, and 3) produces "votes" for predicted correspondences between the mutually closest points with magnitude representing their estimated deviation from isometry. The result of this process is a fuzzy correspondence matrix, which is converted to a permutation matrix with simple matrix operations and output as a discrete set of point correspondences with confidence values. The main advantage of this algorithm is that it can find intrinsic point correspondences in cases of extreme deformation. During experiments with a variety of data sets, we find that it is able to find dozens of point correspondences between different object types in different poses fully automatically. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> In this paper, we propose a high-order graph matching formulation to address non-rigid surface matching. The singleton terms capture the geometric and appearance similarities (e.g., curvature and texture) while the high-order terms model the intrinsic embedding energy. The novelty of this paper includes: 1) casting 3D surface registration into a graph matching problem that combines both geometric and appearance similarities and intrinsic embedding information, 2) the first implementation of high-order graph matching algorithm that solves a non-convex optimization problem, and 3) an efficient two-stage optimization approach to constrain the search space for dense surface registration. Our method is validated through a series of experiments demonstrating its accuracy and efficiency, notably in challenging cases of large and/or non-isometric deformations, or meshes that are partially occluded. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> This paper describes a fully automatic pipeline for finding an intrinsic map between two non-isometric, genus zero surfaces. Our approach is based on the observation that efficient methods exist to search for nearly isometric maps (e.g., Mobius Voting or Heat Kernel Maps), but no single solution found with these methods provides low-distortion everywhere for pairs of surfaces differing by large deformations. To address this problem, we suggest using a weighted combination of these maps to produce a "blended map." This approach enables algorithms that leverage efficient search procedures, yet can provide the flexibility to handle large deformations. The main challenges of this approach lie in finding a set of candidate maps {mi} and their associated blending weights {bi(p)} for every point p on the surface. We address these challenges specifically for conformal maps by making the following contributions. First, we provide a way to blend maps, defining the image of p as the weighted geodesic centroid of mi(p). Second, we provide a definition for smooth blending weights at every point p that are proportional to the area preservation of mi at p. Third, we solve a global optimization problem that selects candidate maps based both on their area preservation and consistency with other selected maps. During experiments with these methods, we find that our algorithm produces blended maps that align semantic features better than alternative approaches over a variety of data sets. <s> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> We consider 3-D brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized. Past comparisons of such surfaces involve either volume deformations or nonrigid matching under fixed parameterizations of surfaces. We propose a new mathematical representation of surfaces, called q-maps, such that L² distances between such maps are invariant to re-parameterizations. This property allows for removing the parameterization variability by optimizing over the re-parameterization group, resulting in a proper parameterization-invariant distance between shapes of surfaces. We demonstrate this method in shape analysis of multiple brain structures, for 34 subjects in the Detroit Fetal Alcohol and Drug Exposure Cohort study, which results in a 91% classification rate for attention deficit hyperactivity disorder cases and controls. This method outperforms some existing techniques such as spherical harmonic point distribution model (SPHARM-PDM) or iterative closest point (ICP). <s> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> We consider the task of computing shape statistics and classification of 3D anatomical structures (as continuous, parameterized surfaces). This requires a Riemannian metric that allows re-parameterizations of surfaces by isometries, and computations of geodesics. This allows computing Karcher means and covariances of surfaces, which involves optimal re-parameterizations of surfaces and results in a superior alignment of geometric features across surfaces. The resulting means and covariances are better representatives of the original data and lead to parsimonious shape models. These two moments specify a normal probability model on shape classes, which are used for classifying test shapes into control and disease groups.We demonstrate the success of this model through improved random sampling and a higher classification performance. We study brain structures and present classification results for Attention Deficit Hyperactivity Disorder. Using the mean and covariance structure of the data, we are able to attain an 88% classification rate. <s> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <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> BIB009 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <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> BIB010 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> We introduce a framework for analyzing symmetry of 3D anatomical structures using elastic deformations of their boundaries (surfaces). The basic idea is to define a space of parameterized surfaces and to compute geodesic paths between the objects and their arbitrary reflections using a Riemannian structure. Elastic matching, based on optimal (non-linear) re-parameterizations (grid deformations) of surfaces, provides a better registration of points across shapes, as compared to the commonly-used linear registrations. A crucial step of orientation alignment, akin to finding planes of symmetry, is performed as a search for shortest geodesic paths. This framework is fully automatic and provides a measure of symmetry, the nearest symmetric shape and the optimal deformation to make an object symmetric. We demonstrate this framework through multiple toy examples on simple and complicated surfaces. We also explore the use of symmetry analysis in differentiating between healthy and subjects with Attention Deficit Hyper-activity Disorder. <s> BIB011 </s> A Survey on Non-rigid 3D Shape Analysis <s> Elastic registration as a reparameterization problem <s> We argue that full surface correspondence (registration) and optimal deformations (geodesics) are two related problems and propose a framework that solves them simultaneously. We build on the Riemannian shape analysis of anatomical and star-shaped surfaces of Kurtek et al. and focus on articulated complex shapes that undergo elastic deformations and that may contain missing parts. Our core contribution is the re-formulation of Kurtek et al.'s approach as a constrained optimization over all possible re-parameterizations of the surfaces, using a sparse set of corresponding landmarks. We introduce a landmark-constrained basis, which we use to numerically solve this optimization and therefore establish full surface registration and geodesic deformation between two surfaces. The length of the geodesic provides a measure of dissimilarity between surfaces. The advantages of this approach are: (1) simultaneous computation of full correspondence and geodesic between two surfaces, given a sparse set of matching landmarks (2) ability to handle more comprehensive deformations than nearly isometric, and (3) the geodesics and the geodesic lengths can be further used for symmetrizing 3D shapes and for computing their statistical averages. We validate the framework on challenging cases of large isometric and elastic deformations, and on surfaces with missing parts. We also provide multiple examples of averaging and symmetrizing 3D models. <s> BIB012
Instead of using landmarks, some previous works tried to find the optimal diffeomorphism γ * that puts f 2 in full correspondence with f 1 . Since the search space for optimal re-parameterizations is large, some of the previous works such as SPHARM  , BIB001 and SPHARM-PDM BIB002 , BIB003 , used a fixed parameterization that is analogous to the arc-length parameterization on curves. Others restricted the type of re-parameterizations to a specific subset of the space of diffeomorphisms. Lipman et al. BIB004 , for example, showed that the optimal correspondence between nearly-isometric genus-0 surfaces, conformally embedded onto a sphere, can be found by searching for the best Möbius transform that aligns them. The approach is, however, restricted to nearly isometric surfaces. In addition, it only finds a sparse set of correspondences since complex surfaces cannot be conformally embedded onto a sphere with minimal distortion. Zeng et al. BIB005 extended Lipman et al.'s approach and finds a dense matching in the presence of large elastic deformations. Kim et al. BIB006 showed that by blending multiple maps, each one is a Möbius transform, one can match surfaces under relatively large elastic deformations. Nevertheless, the correspondences with these methods are not one-toone. The inter-surface maps they produce are not guaranteed to be a diffeomorphism. Thus they are not suitable for statistical shape analysis tasks. In addition, these methods are limited to surfaces that can be conformally parameterized onto a sphere. Although, in theory, any closed genus-0 surface can be conformally embedded into a sphere, in practice, however, the distortion that results from conformal embedding of complex surfaces that have elongated extruding parts, e.g. human or animal body shapes, is often very large. As a consequence, several important shape parts are being under-sampled and thus cannot be correctly matched across surfaces. Using a fixed set of landmarks or a fixed parameterization of the surfaces being analyzed imposes constraints on the correspondences that can be found and often may lead to undesirable results especially in the presence of large elastic deformations. Fig. 3-(a) shows an example where fixed parameterization fails. Better results can be achieved by treating the boundaries of objects as continuous surfaces, rather than discretizing them into point sets at the outset, and solving directly Eqn.  for finding the optimal sampling (i.e. reparameterization γ) that better matches features across shapes as proposed in BIB009 ,  , BIB007 , BIB008 , BIB010 , BIB011 , BIB012 , see Fig. 3-(b) for a 2D example. Continuous representations make it possible to elastically model and optimize correspondences. The process of matching two 3D shapes is, in fact, reduced to the problem of optimally re-parameterizing one shape so that it is as close as possible to the other. Thus, correspondence becomes a re-parameterization problem, which can be mathematically formulated using diffeomorphisms. In order to do that, however, the action of Γ on the space of surfaces should be by isometries, under the metric d C (·, ·) i.e. Kurtek et al. BIB009 ,  , BIB007 , BIB008 , BIB010 , BIB011 showed that when using the L 2 metric in the space of surfaces C f , the condition of Eqn. (20) is only valid when γ is an area-preserving diffeomorphism. This is, however, too restrictive especially when dealing with elastic registration because elastic deformations correspond to changes in surface area. Kurtek et al.  and Jermyn et al. BIB009 showed that in the space of Square Root Maps (SRM) and in the space of Square-Root Normal Fields (SRNF), respectively, the action of Γ is by isometries under the L 2 metric. Given that the L 2 metric in the space of SRNFs is equivalent to the partial elastic metric in the space of C f , one can • Option 1: perform the analysis in C f using the pullback of the partial elastic metric since the pullback metric inherits all the properties of the metric, or • Option 2: map the surfaces being analyzed to the space of SRNFs, perform the analysis there, e.g. registration and geodesic computation, and finally map, if needed, the results back to the space of surfaces C f . In practice, option 2 is computationally very efficient; We are given two surfaces f 1 and f 2 , and their SRNF representations q 1 = Q(f 1 ) and q 2 = Q(f 2 ). We perform elastic registration by solving for the optimal rotation O and optimal re-parameterization γ such that the L 2 distance between q 1 and O(q 2 , γ) is minimized. If O * and γ * is the solution to this optimization problem, thenf 2 = O * (f 2 • γ * ) and its corresponding SRNF is given byq 2 = O * (q 2 , γ * ). Since this process uses the L 2 metric, geodesics correspond to straight lines in the space of SRNFs and thus one can use all the standard computational tools available when dealing with Euclidean spaces.