Multi-scale 3-D Deformable Model Segmentation Based on Medial Description

Maximum a posteriori estimation of linear shape variation with application to vertebra and cartilage modeling

The estimation of covariance matrices is a crucial step in several statistical tasks. Especially when using few samples of a high dimensional representation of shapes, the standard maximum likelihood estimation (ML) of the covariance matrix can be far from the truth, is often rank deficient, and may lead to unreliable results. In this paper, we discuss regularization by prior knowledge using maximum a posteriori (MAP) estimates. We compare ML to MAP using a number of priors and to Tikhonov regularization. We evaluate the covariance estimates on both synthetic and real data, and we analyze the estimates' influence on a missing-data reconstruction task, where high resolution vertebra and cartilage models are reconstructed from incomplete and lower dimensional representations. Our results demonstrate that our methods outperform the traditional ML method and Tikhonov regularization.

Ball-and-socket joint motion description using spherical medial representation

The close inter-relationship between shape and functionality of osteo-articular complexes is fundamental for understanding, modeling, and imaging. It is thus desirable to put forward theoretical and methodological principles that take benefit of this relationship. In this paper, we propose a new representation of spherical motions that is applied to ball-and-socket joint such as hip and shoulder. We have been developing Spherical Medial Representation (SM-Rep) based on skeletonization over the sphere. While M-Rep provides a paradigm for 3D surface and image analysis, the resulting surface remains in a 3D vector space. Since the geometry of the unit sphere is quite different, especially geodesics, SM-Rep supplies consistent elements to overcome this issue. Example of application of this framework to hip is given.

Population-based fitting of medial shape models with correspondence optimization

A crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size.

A method and software for segmentation of anatomic object ensembles by deformable m-reps

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.

Statistical shape analysis of neuroanatomical structures based on medial models

Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates the variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is a coarse-scale sampled medial description derived from a fine-scale spherical harmonics (SPHARM) boundary description. This medial description is composed of a net of medial samples (m-rep) with fixed graph properties. The medial model is computed automatically from a predefined shape space using pruned 3D Voronoi skeletons. A new method determines the stable medial branching topology from the shape space. An intrinsic coordinate system and an implicit correspondence between shapes is defined on the medial manifold. Several studies of biological structures clearly demonstrate that the novel representation has the promise to describe shape changes in a natural and intuitive way. A new medial shape similarity study of group differences between monozygotic and dizygotic twins in lateral ventricle shape demonstrates the meaningful and powerful representation of local and global form.