Shape Modeling and Analysis with Entropy-Based Particle Systems

Abstract

Many important fields of basic research in medicine and biology routinely employ tools for the statistical analysis of collections of similar shapes. Biologists, for example, have long relied on homologous, anatomical landmarks as shape models to characterize the growth and development of species. Increasingly, however, researchers are exploring the use of more detailed models that are derived computationally from three-dimensional images and surface descriptions. While computationally-derived models of shape are promising new tools for biomedical research, they also present some significant engineering challenges, which existing modeling methods have only begun to address.In this dissertation, I propose a new computational framework for statistical shape modeling that significantly advances the state-of-the-art by overcoming many of the limitations of existing methods. The framework uses a particle-system representation of shape, with a fast correspondence-point optimization based on information content. The optimization balances the simplicity of the model (compactness) with the accuracy of the shape representations by using two commensurate entropy metrics and no free parameters. The idea is to maximize both the geometric accuracy and the statistical simplicity of the shape model, in accordance with the principle of parsimony in model selection. The nonparametric representation allows the method to be applied to a larger class of problems than existing methods, including nonspherical surfaces, open surfaces, and sets of multiple surfaces.The relative simplicity of the surface representation and the low number of free parameters results in a framework that is easy to use and can operate directly on image segmentations. In collaboration with scientists from several important areas of biomedicine, I have demonstrated that the proposed method is indeed an e↵ective tool for scientific research.The specific research contributions of this dissertation are as follows. First, I describe a mathematical framework and a robust numerical algorithm for computing optimized correspondence-point shape models using an entropy-based optimization and particle-system technology. Second, I develop a series of extensions of the framework to more general classes of shape analysis problems, including the analysis of multiple-object complexes, the generalization to correspondence based on generic functions of position, an extension to handle surfaces with open boundaries, and shape modeling with simple regression. Third, I describe the application of statistical hypothesis testing, regression analysis, and multiple-analysis of covariance to the proposed shape models. I also introduce new techniques for visualization and interpretation of these statistics. Finally, in cooperation with biomedical researchers, I present validation of the above research contributions by their successful application to real-world research problems.

Discovering Sparse Functional Brain Networks Using Group Replicator Dynamics (GRD)

“…Also, the common core network may not necessarily be the most coherent network for every subject, potentially due to different processing strategies or simply due to different levels of noise [29]. Given w i (k), to encourage networks consisting of the same brain regions across subjects, one could adjust w i (k) such that the group entropy is minimized [39,40]. For analytic simplicity, if we assume w i (k) are instances of a normal random variable, w, then the entropy of w is given by:…”

“…where Σ is the covariance of w. To minimize the entropy, a gradient descent approach is used with w i (k) updated as follows [39,40]:…”

“…λ governs the learning rate and must be set below α to ensure numerical stability [39]. Since the number of subjects, N s , will typically be much less than the number of brain regions, (k) at the next iteration, k+1, and the same process is repeated until convergence.…”

Training models of anatomic shape variability

“…As examples, Ref. 28 uses no image match grounds, but requires a regular sampling with tight geometric distributions. Active appearance models 12,29 require both a tight geometric distribution as well as similar local image intensities.…”

Nested Sphere Statistics of Skeletal Models

“…The previous work on achieving correspondence in training populations of anatomic object models has involved shifting points on object boundaries in point-distribution models (PDMs) [3,17] (ref Twining). We find the most attractive method is that of [3,17], which minimizes a sum of two entropy terms: H(z) − α ∑ i H(x i ).…”