Correspondence sampling and predictive methodologies on manifolds and polyhedral surfaces
Creators: name: Tsagkrasoulis, Dimosthenis
Publisher: name: Imperial College London
Year: 2016

description: In recent years, a significant increase of interest can be observed regarding the analysis of objects with inherently complex morphological structures. This is especially true in the fields of biomedical imaging and computer vision, where the advent of new three-dimensional (3D) data capturing technologies has enabled the relatively low-cost acquisition of detailed models representing the boundaries of physical and biological structures, such as cardiac, brain and facial surfaces. Within this framework, of large importance is the development of methodologies tailored for statistical analysis and predictive modeling with complex objects being either the input or the response variables. In this thesis, we are concerned with aspects pertaining both the manipulation of complex surface objects, so as to enable their statistical analysis, as well as the construction of predictive methodologies that are suited to such data, and take advantage of the additional morphological structure to improve prediction accuracy. In more detail, we first concentrate on the problem of establishing dense point correspondence on collections of similar 3D polyhedral surfaces. This annotation is a prerequisite for any type of further data analysis. By considering these polyhedral objects as approximations of two-dimensional non-linear surfaces, a.k.a. manifolds, we are able to present a methodology for the automatic and dense point annotation, that can be accurately and efficiently employed on large datasets. We apply our algorithm on a detailed heritability and genome association study of the human face shape. Subsequently, we deal with the problem of using morphological data, extracted from surface objects, as inputs for regression analysis. In particular, we extend our previous algorithm for point correspondence to the regression setting, by incorporating an adaptive procedure that identifies and annotates more densely subregions within the surfaces that are highly predictive of a related response variable. This procedure leads to significant improvements in prediction accuracy, as evaluated on an application of age prediction from facial surfaces. Finally, we consider the opposite scenario of predictive modeling with complex objects constituting the responses. We identify as main problem the fact that the response space can no longer be considered Euclidean. To solve this, we construct a non-parametric regression methodology for manifold-valued objects. The method is versatile and can be applied in cases where the response space is a well-defined manifold, but also when such knowledge is not available. Model fitting and prediction phases only require the definition of a suitable distance function on the response space. We apply our method in a variety of image completion problems, as well as the prediction of human faces from genotypic data. Construction of suitable regression algorithms is necessary here, since most methodologies cannot deal with the issue of non-linearity in the response space. Existing manifold regression methods either require the rigorous definition of the underlying manifold or make use of kernel functions to implicitly project the responses on a very high-dimensional vectorial space, where standard regression methods can be applied. In the first case though, applicability is restricted to only very limited types of data, while in the second case, the issue of predicting on the response space can not be tackled easily or efficiently. We present a non-parametric predictive methodology for manifold-valued objects, based on our previous work on distance-based Random Forest \cite{Sim2013}. Predictions are made through a two-step approach, where we first find a point estimate on a Euclidean embedding of the responses, and then project that point back to the original space to acquire the manifold-valued prediction. The methodology can readily handle various types of data objects, since its only requirement is the definition of a meaningful distance function for the responses. Our predictive method was shown to outperform a number of different regression methods on various problems of image completion.