A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <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> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <s> The paper discusses the initial results obtained for the generation of canonical 3D models of anatomical parts, built on real patient data. 3D canonical models of anatomy are key elements in a computer-assisted diagnosis; for instance, they can support pathology detection, semantic annotation of patient-specific 3D reconstructions, quantification of pathological markers. Our approach is focused on carpal bones and on the elastic analysis of 3D reconstructions of these bones, which are segmented from MRI scans, represented as 0-genus triangle meshes, and parameterized on the sphere. The original method [8] relies on a set of sparse correspondences, defined as matching vertices. For medical applications, it is desirable to constrain the mean shape generation to set-up the correspondences among a larger set of anatomical landmarks, including vertices, lines, and areas. Preliminary results are discussed and future development directions are sketched. <s> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <s> Shape morphometry of subcortical surfaces plays an important role in analyzing normal developmental changes as well as in quantifying disease-related effects in the human brain. We present a geometric approach for joint registration, deformation, and statistical analysis of shapes of subcortical surfaces. Here, subcortical surfaces are mathematically represented by vector-fields, termed square-root normal vector fields (SRNFs), on the spherical domain. The SRNF representation allows modeling of the action of the re-parameterization group to by isometries under the standard Hilbert norm, and thus enables an elastic shape analysis. This elastic analysis results in optimal deformations between shapes and helps to quantify shape differences using geodesic lengths. Importantly, the joint registration of observed shapes in a shape class removes nuisance variability due to mis-registration in the shape data and results in parsimonious statistical shape models with improved inferences for characterizing population-based subcortical structural variability. We demonstrate the ideas for shape matching and statistical analysis of hippocampal shapes for (N=120) subjects from the Alzheimer's disease Neuroimaging Initiative (ADNI) dataset. Specifically, we present results for assessing group differences both by using global descriptors such as principal components and local descriptors such as deformations induced by the tangent vectors from the mean shapes to the individuals. <s> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> Other applications <s> Statistical shape analysis plays a key role in various medical imaging applications. Such methods provide tools for registering, deforming, comparing, averaging, and modeling anatomical shapes. In this work, we focus on the application of a recent method for statistical shape analysis of parameterized surfaces to simulation of endometrial tissue shapes. The clinical data contains magnetic resonance imaging (MRI) endometrial tissue surfaces, which are used to learn a generative shape model. We generate random tissue shapes from this model, and apply elastic semi-synthetic deformations to them. This provides two types of simulated data: (1) MRI-type (without deformation) and (2) transvaginal ultrasound (TVUS)-type, which undergo an additional deformation due to the transducer's pressure. The proposed models can be used for validation of automatic, multimodal image registration, which is a crucial step in diagnosing endometriosis. <s> BIB009
Elastic registration, geodesics and statistical summaries are building blocs to many applications in computer graphics, computer vision, and medical image analysis. The methods presented in this chapter are able to perform most of these tasks. As a consequence, the framework has been used in various applications. In medical applications, Kurtek et Ten arbitrary 3D human body shapes automatically synthesized by sampling from a Gaussian distribution fitted to a collection of human body shapes in the SRNF space. al. BIB002 , BIB003 used the Euclidean distance in the space of Q-maps for shape-based analysis of 3D subcortical structures in the brain in order to detect attentiondeficit hyperactivity disorder (ADHD). Joshi et al. BIB008 , on the other hand, used the SRNF representation for shape-based hippocampal modeling in Alzheimer's disease. In particular, Joshi et al. showed that the elastic metric preserves important anatomical features when matching hippocampal structures and thus is potentially useful for accurately capturing the biological homology. The approach naturally provides a statistical framework for computing means and covariances in the shape space, which can be used for modeling the biological variability of subcortical structures. They observed a slight overlap of the statistical group difference results, obtained using SRNF, when comparing with the SPHARM approach BIB001 . Joshi et al.'s results, however, reveal additional effects due to disease or progression in different subregions of the hippocampal anatomy. One possible explanation is that the deformations generated using re-parameterization-invariant elastic metrics also account for the variability due to parameterizations, and thus may contain a richer description about the shape variability of the anatomical structures. Samir et al. BIB006 used the SRNF framework for elastic shape analysis of cylindrical surfaces and applied it to 3D/2D registration in endometrial tissue characterization and also to simulate the deformation of such tissues BIB009 . Elastic shape analysis, in particular SRM-based elastic shape analysis  , BIB004 has been later extended to include landmark constraints in the analysis process BIB005 . The landmarks can be used either to guide registration in the presence of large and complex elastic deformations BIB005 , or as anatomical features that one needs to preserve when generating patientspecific atlas of anatomical organs such as carpal bones BIB007 .