Skeletons, Object Shape, Statistics.

Objects and object complexes in 3D, as well as those in 2D, have many possible representations. Among them skeletal representations have special advantages and some limitations. For the special form of skeletal representation called "s-reps," these advantages include strong suitability for representing slabular object populations and statistical applications on these populations. Accomplishing these statistical applications is best if one recognizes that s-reps live on a curved shape space. Here we will lay out the definition of s-reps, their advantages and limitations, their mathematical properties, methods for fitting s-reps to single- and multi-object boundaries, methods for measuring the statistics of these object and multi-object representations, and examples of such applications involving statistics. While the basic theory, ideas, and programs for the methods are described in this paper and while many applications with evaluations have been produced, there remain many interesting open opportunities for research on comparisons to other shape representations, new areas of application and further methodological developments, many of which are explicitly discussed here.

Fitting unbranching skeletal structures to objects.

Representing an object by a skeletal structure can be powerful for statistical shape analysis if there is good correspondence of the representations within a population. Many anatomic objects have a genus-zero boundary and can be represented by a smooth unbranching skeletal structure that can be discretely approximated. We describe how to compute such a discrete skeletal structure ("d-s-rep") for an individual 3D shape with the desired correspondence across cases. The method involves fitting a d-s-rep to an input representation of an object's boundary. A good fit is taken to be one whose skeletally implied boundary well approximates the target surface in terms of low order geometric boundary properties: (1) positions, (2) tangent fields, (3) various curvatures. Our method involves a two-stage framework that first, roughly yet consistently fits a skeletal structure to each object and second, refines the skeletal structure such that the shape of the implied boundary well approximates that of the object. The first stage uses a stratified diffeomorphism to produce topologically non-self-overlapping, smooth and unbranching skeletal structures for each object of a population. The second stage uses loss terms that measure geometric disagreement between the skeletally implied boundary and the target boundary and avoid self-overlaps in the boundary. By minimizing the total loss, we end up with a good d-s-rep for each individual shape. We demonstrate such d-s-reps for various human brain structures. The framework is accessible and extensible by clinical users, researchers and developers as an extension of SlicerSALT, which is based on 3D Slicer.

Entropy-based Correspondence Improvement of Interpolated Skeletal Models.

Statistical analysis of shape representations relies on having good correspondence across a population. Improving correspondence yields improved statistics. Point distribution models (PDMs) are often used to represent object boundaries. Skeletal representations (s-reps) model object widths and boundary directions as well as boundary positions, so they should yield better correspondence. We present two methods: one for continuously interpolating a discretely-sampled skeletal model and one for improving correspondence by using this interpolation to shift skeletal samples to new positions. The interpolation operates by an extension of the mathematics of medial structures. As with Cates' boundary-based method, we evaluate correspondence in terms of regularity and shape-feature population entropies. Evaluation on both synthetic and real data shows that our method both improves correspondence of s-rep models fit to segmented lateral ventricles and that the combined boundary-and-skeletal PDMs implied by these optimized s-reps have better correspondence than optimized boundary PDMs.

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.

Geometrically proper models in statistical training.

In deformable model segmentation, the geometric training process plays a crucial role in providing shape statistical priors and appearance statistics that are used as likelihoods. Also, the geometric training process plays a crucial role in providing shape probability distributions in methods finding significant differences between classes. The quality of the training seriously affects the final results of segmentation or of significant difference finding between classes. However, the lack of shape priors in the training stage itself makes it difficult to enforce shape legality, i.e., making the model free of local self-intersection or creases. Shape legality not only yields proper shape statistics but also increases the consistency of parameterization of the object volume and thus proper appearance statistics. In this paper we propose a method incorporating explicit legality constraints in training process. The method is mathematically sound and has proved in practice to lead to shape probability distributions over only proper objects and most importantly to better segmentation results.