Conglin Lu

Principal geodesic analysis for the study of nonlinear statistics of shape

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.

Deformable M-Reps for 3D Medical Image Segmentation

M-reps (formerly called DSLs) are a multiscale medial means for modeling and rendering 3D solid geometry. They are particularly well suited to model anatomic objects and in particular to capture prior geometric information effectively in deformable models segmentation approaches. The representation is based on figural models, which define objects at coarse scale by a hierarchy of figures—each figure generally a slab representing a solid region and its boundary simultaneously. This paper focuses on the use of single figure models to segment objects of relatively simple structure.A single figure is a sheet of medial atoms, which is interpolated from the model formed by a net, i.e., a mesh or chain, of medial atoms (hence the name m-reps), each atom modeling a solid region via not only a position and a width but also a local figural frame giving figural directions and an object angle between opposing, corresponding positions on the boundary implied by the m-rep. The special capability of an m-rep is to provide spatial and orientational correspondence between an object in two different states of deformation. This ability is central to effective measurement of both geometric typicality and geometry to image match, the two terms of the objective function optimized in segmentation by deformable models. The other ability of m-reps central to effective segmentation is their ability to support segmentation at multiple levels of scale, with successively finer precision. Objects modeled by single figures are segmented first by a similarity transform augmented by object elongation, then by adjustment of each medial atom, and finally by displacing a dense sampling of the m-rep implied boundary. While these models and approaches also exist in 2D, we focus on 3D objects.The segmentation of the kidney from CT and the hippocampus from MRI serve as the major examples in this paper. The accuracy of segmentation as compared to manual, slice-by-slice segmentation is reported.

Statistics of shape via principal geodesic analysis on Lie groups

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.

Gaussian Distributions on Lie Groups and Their Application to Statistical Shape Analysis

The Gaussian distribution is the basis for many methods used in the statistical analysis of shape. One such method is principal component analysis, which has proven to be a powerful technique for describing the geometric variability of a population of objects. The Gaussian framework is well understood when the data being studied are elements of a Euclidean vector space. This is the case for geometric objects that are described by landmarks or dense collections of boundary points. We have been using medial representations, or m-reps, for modelling the geometry of anatomical objects. The medial parameters are not elements of a Euclidean space, and thus standard PCA is not applicable. In our previous work we have shown that the m-rep model parameters are instead elements of a Lie group. In this paper we develop the notion of a Gaussian distribution on this Lie group. We then derive the maximum likelihood estimates of the mean and the covariance of this distribution. Analogous to principal component analysis of covariance in Euclidean spaces, we define principal geodesic analysis on Lie groups for the study of anatomical variability in medially-defined objects. Results of applying this framework on a population of hippocampi in a schizophrenia study are presented.

Statistical Multi-Object Shape Models

The shape of a population of geometric entities is characterized by both the common geometry of the population and the variability among instances. In the deformable model approach, it is described by a probabilistic model on the deformations that transform a common template into various instances. To capture shape features at various scale levels, we have been developing an object-based multi-scale framework, in which a geometric entity is represented by a series of deformations with different locations and degrees of locality. Each deformation describes a residue from the information provided by previous steps. In this paper, we present how to build statistical shape models of multi-object complexes with such properties based on medial representations and explain how this may lead to more effective shape descriptions as well as more efficient statistical training procedures. We illustrate these ideas with a statistical shape model for a pair of pubic bones and show some preliminary results on using it as a prior in medical image segmentation.

Representing multifigure anatomical objects

We use multifigure m-reps to represent anatomical objects, such as human livers, with named components. Each component is represented as a single figure m-rep. These figures of the object are connected via hinge geometry. A smooth object surface is then computed using our technique based on the subdivision method applied to a single figure m-rep. This novel representation allows us to represent and analyze a complex anatomical object by its individual components, by relations among its components, and by the object itself as a whole entity. Using our representation, some preliminary results on statistical analysis of multifigure anatomical objects are presented.

A Markov random field approach to multi-scale shape analysis

With a mind towards achieving means of image comprehension by computer, we intend to convey the benefits of (1) characterizing the geometry of object complexes in the real world as contrasted with the geometric conformation of their images, and (2) describing populations of object complexes probabilistically. We show how a multi-scale description of inter-scale residues of geometric features provides a set of efficiently trainable probability distributions via a Markov random field approach, and specifies the location and scale of geometric differences between populations. These ideas and methods are illustrated using medial representations for 3D objects, depending on their properties (1) that local descriptors have an associated coordinate frame and distance metric, and (2) that continuous geometric random variables can be used to describe all members of a population of object complexes with a common structure and the variation among those members. We demonstrate with respect to the following object-complex-relative discrete scale levels: a whole object complex, individual objects, various object parts and sections, and fine boundary details. Using this illustrative framework, we show how to build Markov random field (MRF) models on the geometry scale space based on the statistics of shape residues across scales and between neighboring geometric entities at the level of locality given by its scale. In this paper, we present how to design and estimate MRF models on two scale levels, namely boundary displacement and object sections.