Keeping pace with the times: Quantifying variation of newly emerging biological shape data

One goal of this dissertation is to construct a new type of shape representation to quantify gene expression data. Advances in microscopy and techniques such as Optical Projection Tomography (OPT) allow researchers to visualize and to study 3D morphological patterns of gene expression domains. Quantitative analysis of gene expression domains and investigation of relationships between gene expression and developmental and phenotypic outcomes are central to advancing our understanding of the genotype-phenotype map. However, quantification of shape variation in gene expression domains poses particularly challenging problems, as these domains typically have no clearly defined forms, often appearing seemingly amorphous. Those properties of the gene expression domains make it difficult to analyze shape variation with the tools of landmark-based geometric morphometrics. In addition, 3D image acquisition and processing introduce many artifacts that further exacerbate the problem. To overcome these difficulties, we present a method that combines OPT scanning, a shape regularization technique and a landmark-free approach to quantify variation in the morphology of sonic hedgehog expression domains in the frontonasal ectodermal zone (FEZ) of avians and investigate relationships with embryonic craniofacial shape. The landmark-free approach quantifies variation in shape of amorphous gene expression domains, enhancing their most salient morphological characteristics and being robust to uninformative local shape variation and irregularities associated with image acquisition. The correlation analysis reveals axes in FEZ and embryonic-head morphospaces along which variation exhibits a sharp linear relationship at high statistical significance. Combined with qualitative findings, these results have the potential to benefit biologists in exploring the gene expression pathway and in understanding the underlying expression mechanisms. The techniques we used to deal with FEZ meshes should be applicable to analyses of other 3D surface-like biological structures that have ill-defined shape and are relevant to understanding developmental processes and phenotypic variation.;Existing biological shape models, such as those based on landmarks, rely on sparse landmarks on the shapes to model shape variations. However, on soft-tissue surfaces as the face there are few such landmarks. Across the cheek and forehead, for instance, there are no points that have exact biological correspondence and yet aspects of their shape contains useful biological information. The analysis based on the sparse landmarks will compromise the deep and comprehensive morphological information collected by advanced image processing technologies. Thus, instead of using only the limited number of landmarks, we propose to use the spline method to construct dense surface model which covers the entire shape. This brings another goal of this dissertation - to develop such a spline method to build a dense correspondence across all shapes. Although, spline is an active area in shape analysis and also in many other disciplines for interpolation, approximation and regression. Most results have been focused on Euclidean domain. However, data living on manifold occurs often, especially when dealing with shape surfaces, so constructing spline with manifold domain and providing effective computation method for such spline are desirable in real-life problems. To fulfill this goal, we present a general theoretical framework of spline in which the Euclidean domain can be extended to manifold domain. Additionally, we provide computationally effective algorithm to compute such spline function based on bounded rectangular domain. We demonstrate the advantages of this framework by using examples on closed and open manifold domains and by comparing performance with other spline methods. The computation framework shows comparable result with the spline directly constructed on the manifold and also shows clear improvement respect to the thin plate spline method. This manifold spline method has been applied to construct dense surface models of avian embryos shapes. Those dense surface models can establish a correspondence of thousands of points across each 3D image and provide dramatic visualization of shape variation. (Abstract shortened by UMI.).