Browsing by Subject "Statistical shape analysis"

In this dissertation, we investigate several problems in shape analysis. We start by discussing the shape matching problem. Given that homeomorphisms of shapes are computed in practice by interpolating sparse correspondence, we give an algorithm to refine pairwise mappings in a collection by employing a simple metric condition to obtain partial correspondences of points chosen in a manner that outlines the shapes of interest in a relatively small number of points. We then use this mapping algorithm in two separate applications. First, we investigate the extent to which classical assumptions and methods in statistical shape analysis hold for near continuous discretizations of surfaces spanning different species and genus groups. We find that these methods yield biologically meaningful information, and that resulting operations with these correspondences, including averaging and linear interpolation, yield biologically meaningful surfaces even for distinct geometries. As a second application, we discuss the problem of dictionary learning on shapes in an effort to find sparse decompositions of shapes in a collection. To this end, we define a construction of wavelet-like and ridgelet-like objects that are easily computable at the level of the discretization, both of which have natural interpretation in the smooth case. We then use these in tandem with feature points to create a sparse dictionary, and show that standard sparsification practices still retain biological information.

A general goal of biology is to understand how two or more sets of traits in an organism are related - for example, disease state and genetics, physiology and behavior, or phenotypic variation and gene function. Many of the early advancements in statistical analysis dealt with relating measured traits when one could be represented as a single number. However, many traits are inherently multi-dimensional, and technologies are advancing for rapidly measuring many types of such highly complex traits. Making efficient use of these new, larger datasets requires new statistical models for to biological inference. In this thesis, I develop a method for relating two very different types of traits in sea urchin larvae: morphological shape, and developmental gene expression. In particular, I develop an approach for regression modeling using shape as a response variable. I use this method to address the question of whether variation in the expression of regulatory genes during development predicts later morphological variation in the larvae. I propose a hierarchical random effects factor regression model with shape as a response variable for relating morphology and gene expression when the individuals in each dataset are related, but not identical. I fit an approximation to the general model by breaking it into three discrete steps. I find that gene expression can explain ~25% of mean symmetric form variation among cultures of related larvae, and identify several groups of related genes that are correlated with aspects of morphological variation.