A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> A model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives. Metrics are described that compute an object representation's prior probability of local geometry by reflecting variabilities in the net's node and link parameter values, and that compute a likelihood function measuring the degree of match of an image to that object representation. A paradigm for image analysis of deforming such a model to optimize a posteriori probability is described, and this paradigm is shown to be usable as a uniform approach for object definition, object-based registration between images of the same or different imaging modalities, and measurement of shape variation of an abnormal anatomical object, compared with a normal anatomical object. Examples of applications of these methods in radiotherapy, surgery, and psychiatry are given. <s> BIB001 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> This paper presents a multiscale framework based on a medial representation for the segmentation and shape characterization of anatomical objects in medical imagery. The segmentation procedure is based on a Bayesian deformable templates methodology in which the prior information about the geometry and shape of anatomical objects is incorporated via the construction of exemplary templates. The anatomical variability is accommodated in the Bayesian framework by defining probabilistic transformations on these templates. The transformations, thus, defined are parameterized directly in terms of natural shape operations, such as growth and bending, and their locations. A preliminary validation study of the segmentation procedure is presented. We also present a novel statistical shape analysis approach based on the medial descriptions that examines shape via separate intuitive categories, such as global variability at the coarse scale and localized variability at the fine scale. We show that the method can be used to statistically describe shape variability in intuitive terms such as growing and bending. <s> BIB002 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> We describe a novel method for searching and comparing 3D objects. The method encodes the geometric and topological information in the form of a skeletal graph and uses graph matching techniques to match the skeletons and to compare them. The skeletal graphs can be manually annotated to refine or restructure the search. This helps in choosing between a topological similarity and a geometric (shape) similarity. A feature of skeletal matching is the ability to perform part-matching, and its inherent intuitiveness, which helps in defining the search and in visualizing the results. Also, the matching results, which are presented in a per-node basis can be used for driving a number of registration algorithms, most of which require a good initial guess to perform registration. We also describe a visualization tool to aid in the selection and specification of the matched objects. <s> BIB003 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> Principal component analysis has proven to be useful for understanding geometric variability in populations of parameterized objects. The statistical framework is well understood when the parameters of the objects are elements of a Euclidean vector space. This is certainly the case when the objects are described via landmarks or as a dense collection of boundary points. We have been developing representations of geometry based on the medial axis description or m-rep. Although this description has proven to be effective, the medial parameters are not naturally elements of a Euclidean space. In this paper we show that medial descriptions are in fact elements of a Lie group. We develop methodology based on Lie groups for the statistical analysis of medially-defined anatomical objects. <s> BIB004 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> A primary goal of statistical shape analysis is to describe the variability of a population of geometric objects. A standard technique for computing such descriptions is principal component analysis. However, principal component analysis is limited in that it only works for data lying in a Euclidean vector space. While this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points, it does not handle more complex representations of shape. We have been developing representations of geometry based on the medial axis description or m-rep. While the medial representation provides a rich language for variability in terms of bending, twisting, and widening, the medial parameters are not elements of a Euclidean vector space. They are in fact elements of a nonlinear Riemannian symmetric space. In this paper, we develop the method of principal geodesic analysis, a generalization of principal component analysis to the manifold setting. We demonstrate its use in describing the variability of medially-defined anatomical objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented. <s> BIB005 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <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> BIB006 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> Deformable shape models (DSMs) comprise a general approach that shows great promise for automatic image segmentation. Published studies by others and our own research results strongly suggest that segmentation of a normal or near-normal object from 3D medical images will be most successful when the DSM approach uses (1) knowledge of the geometry of not only the target anatomic object but also the ensemble of objects providing context for the target object and (2) knowledge of the image intensities to be expected relative to the geometry of the target and contextual objects. The segmentation will be most efficient when the deformation operates at multiple object-related scales and uses deformations that include not just local translations but the biologically important transformations of bending and twisting, i.e., local rotation, and local magnification. In computer vision an important class of DSM methods uses explicit geometric models in a Bayesian statistical framework to provide a priori information used in posterior optimization to match the DSM against a target image. In this approach a DSM of the object to be segmented is placed in the target image data and undergoes a series of rigid and nonrigid transformations that deform the model to closely match the target object. The deformation process is driven by optimizing an objective function that has terms for the geometric typicality and model-to-image match for each instance of the deformed model. The success of this approach depends strongly on the object representation, i.e., the structural details and parameter set for the DSM, which in turn determines the analytic form of the objective function. This paper describes a form of DSM called m-reps that has or allows these properties, and a method of segmentation consisting of large to small scale posterior optimization of m-reps. Segmentation by deformable m-reps, together with the appropriate data representations, visualizations, and user interface, has been implemented in software that accomplishes 3D segmentations in a few minutes. Software for building and training models has also been developed. The methods underlying this software and its abilities are the subject of this paper. <s> BIB007 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> Abstract In magnetic resonance imaging (MRI) research, significant attention has been paid to the analysis of the hippocampus (HC) within the medial temporal lobe because of its importance in memory and learning, and its role in neurodegenerative diseases. Manual segmentation protocols have established a volume decline in the HC in conjunction with Alzheimer's disease, epilepsy, post-traumatic stress disorder, and depression. Furthermore, recent studies have investigated age-related changes of HC volume which show an interaction with gender; in early adulthood, volume reduction of the HC is found in men but not in women. In this paper, we investigated gender differences in normal subjects in young adulthood by employing a shape analysis of the HC using medial surfaces. For each subject, the most prominent medial manifold of the HC was extracted and flattened. The flattened sheets were then registered using both a rigid and a non-rigid alignment technique, and the medial surface radius was expressed as a height function over them. This allowed for an investigation of the association between subject variables and the local width of the HC. With regard to the effects of age and gender, it could be shown that the previously observed gender differences were mostly due to volume loss in males in the lateral areas of the HC head and tail. We suggest that the analysis of HC shape using medial surfaces might thus serve as a complimentary technique to investigate group differences to the established segmentation protocols for volume quantification in MRI. <s> BIB008 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> We consider the use of medial surfaces to represent symmetries of 3-D objects. This allows for a qualitative abstraction based on a directed acyclic graph of components and also a degree of invariance to a variety of transformations including the articulation of parts. We demonstrate the use of this representation for 3-D object model retrieval. Our formulation uses the geometric information associated with each node along with an eigenvalue labeling of the adjacency matrix of the subgraph rooted at that node. We present comparative retrieval results against the techniques of shape distributions (Osada et al.) and harmonic spheres (Kazhdan et al.) on 425 models from the McGill Shape Benchmark, representing 19 object classes. For objects with articulating parts, the precision vs recall curves using our method are consistently above and to the right of those of the other two techniques, demonstrating superior retrieval performance. For objects that are rigid, our method gives results that compare favorably with these methods. <s> BIB009 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> A circuit for controlling a display panel identifying malfunctions in an engine generator receives a plurality of electrical signals from the engine generator, each of which identifies a particular trouble. The electrical signal may be produced by closing a switch. It is caused to operate a latch that lights a light associated with the particular malfunction. Indications of other malfunctions are suppressed until the circuit is reset. A manual reset tests all lights and then leaves them off ready to respond. A power-up reset does not test lights but leaves all lights off ready to respond. The circuit is rendered especially appropriate for military use by hardening against radiation and against pulses of electromagnetic interference. <s> BIB010 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> A new approach for constructing deformable continuous medial models for anatomical structures is presented. Medial models describe geometrical objects by first specifying the skeleton of the object and then deriving the boundary surface corresponding to the skeleton. However, an arbitrary specification of a skeleton will not be “Valid” unless a certain set of sufficient conditions is satisfied. The most challenging of these is the non-linear equality constraint that must hold along the boundaries of the manifolds forming the skeleton. The main contribution of this paper is to leverage the biharmonic partial differential equation as a mapping from a codimension-0 subset of Euclidean space to the space of skeletons that satisfy the equality constraint. The PDE supports robust numerical solution on freeform triangular meshes, providing additional flexibility for shape modeling. The approach is evaluated by generating continuous medial models for a large dataset of hippocampus shapes. Generalizations to modeling more complex shapes and to representing branching skeletons are demonstrated. <s> BIB011 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> In this paper we propose a multilinear model of human pose and body shape which is estimated from a database of registered 3D body scans in different poses. The model is generated by factorizing the measurements into pose and shape dependent components. By combining it with an ICP based registration method, we are able to estimate pose and body shape of dressed subjects from single images. If several images of the subject are available, shape and poses can be optimized simultaneously for all input images. Additionally, while estimating pose and shape, we use the model as a virtual calibration pattern and also recover the parameters of the perspective camera model the images were created with. <s> BIB012 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> One goal of statistical shape analysis is the discrimination between two populations of objects. Whereas traditional shape analysis was mostly concerned with single objects, analysis of multi-object complexes presents new challenges related to alignment and pose. In this paper, we present a methodology for discriminant analysis of multiple objects represented by sampled medial manifolds. Non-Euclidean metrics that describe geodesic distances between sets of sampled representations are used for alignment and discrimination. Our choice of discriminant method is the distance-weighted discriminant because of its generalization ability in high-dimensional, low sample size settings. Using an unbiased, soft discrimination score, we associate a statistical hypothesis test with the discrimination results. We explore the effectiveness of different choices of features as input to the discriminant analysis, using measures like volume, pose, shape, and the combination of pose and shape. Our method is applied to a longitudinal pediatric autism study with 10 subcortical brain structures in a population of 70 subjects. It is shown that the choices of type of global alignment and of intrinsic versus extrinsic shape features, the latter being sensitive to relative pose, are crucial factors for group discrimination and also for explaining the nature of shape change in terms of the application domain. <s> BIB013 </s> A Survey on Non-rigid 3D Shape Analysis <s> Deformation models <s> Three-dimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods. <s> BIB014
The deformation model P encodes different deformations that a triangular face X i k ∈ X i undergoes with respect to its corresponding triangle T k on T . These can be rotations due to articulated motion (e.g. bending an arm of a human body), stretching due to shape differences between say two individuals in the same pose, or pose induced deformations (e.g. muscle stretch due to arm bending). SCAPE BIB006 used two deformation models, one acting on a template mesh and another on an articulated skeleton (which captures articulated motion). This greatly simplifies the mathematical formulation; A given mesh X i is represented with three sets of transformation matrices: Using separable deformation models assumes that intrinsic shape deformations and pose variations are independent. This, however, prevents one from capturing phenomena where there is a strong correlation between body pose and local deformations (e.g. near the joints or at the muscles). For example, the surface deformation generated by the motion performed by an athletic person exhibits different properties than the same motion carried out by a person with less pronounced muscles BIB010 . Separable models cannot capture such correlation between pose and body shape. To overcome this limitation, Hasler et al. BIB010 , BIB012  The matrices R i k and S i k , which represent the pose and shape components, respectively, are affine transforms applied to the template T . Note that, while SCAPE requires a set of 3D shapes of the same subject in different poses to learn the space of pose deformations, Hasler et al. BIB010 , BIB012 learn both spaces using a random mix of 3D models from different subjects in various poses. Both methods have shown detailed representation of the human body shape. However, using strong priors on a human body model, i.e. manually defined skeleton and body parts, makes these methods difficult to extend to other, especially free-form, object categories without redesigning the representation BIB012 . Another property of the representations above is that they have redundant degrees of freedom (DOF). For instance, Hasler et al. model deformations with 15 DoF and a non-linear encoding of triangle deformations that captures dependencies between pose and shape. Freifeld and Black BIB014 showed that triangle deformations lie in a 6 dimensional nonlinear manifold. More precisely, where O k is a rotation that maps the template triangle T k to a canonical planar triangle. The scaling S Finally, medial representations (M-reps), which are an extension of curve skeletons BIB003 and medial surfaces BIB009 , have been introduced by Blum  as a tool for representing the geometry and structure of anatomical shapes BIB001 , BIB002 , BIB004 , BIB005 . Unlike the deformation models described above, which act on the shape's surface, a medial surface is formed by the centers of all spheres that are interior to the object and tangent to the object boundary in two or more points. Each medial point, called medial atom, is a 4-tuple m = {x, r, n 0 , n 1 }, where x ∈ R 3 is the location of the atom, r ∈ R + is the radius of the maximum sphere that is tangent to the surface in two or more points, n 0 and n 1 ∈ S 2 are two unit spoke directions. A medial atom m is an element of the manifold Fletcher et al. BIB005 showed that by representing the atoms with the deformation, i.e. translation, scaling, and rotation, of a canonical atom, e.g. (0, 0, 1), they become elements of S(1) , which is a Lie group. A grid of l medial atoms representing a 3D object is an element of the Lie group S = S(l), the product of l copies of S(1). Since their introduction, M-reps have been used in various medical image processing tasks, including segmentation BIB007 ,  , registration, shape discrimination BIB008 , and brain-structure analysis BIB011 , BIB013 .