Hierarchical Graphical Models for Multigroup Shape Analysis using Expectation Maximization with Sampling in Kendall's Shape Space

This paper proposes a novel framework for multi-group shape analysis relying on a hierarchical graphical statistical model on shapes within a population.The framework represents individual shapes as point setsmodulo translation, rotation, and scale, following the notion in Kendall shape space.While individual shapes are derived from their group shape model, each group shape model is derived from a single population shape model. The hierarchical model follows the natural organization of population data and the top level in the hierarchy provides a common frame of reference for multigroup shape analysis, e.g. classification and hypothesis testing. Unlike typical shape-modeling approaches, the proposed model is a generative model that defines a joint distribution of object-boundary data and the shape-model variables. Furthermore, it naturally enforces optimal correspondences during the process of model fitting and thereby subsumes the so-called correspondence problem. The proposed inference scheme employs an expectation maximization (EM) algorithm that treats the individual and group shape variables as hidden random variables and integrates them out before estimating the parameters (population mean and variance and the group variances). The underpinning of the EM algorithm is the sampling of pointsets, in Kendall shape space, from their posterior distribution, for which we exploit a highly-efficient scheme based on Hamiltonian Monte Carlo simulation. Experiments in this paper use the fitted hierarchical model to perform (1) hypothesis testing for comparison between pairs of groups using permutation testing and (2) classification for image retrieval. The paper validates the proposed framework on simulated data and demonstrates results on real data.Comment: 9 pages, 7 figures, International Conference on Machine Learning 201

STOCHASTIC OPTIMIZATION FOR TROPICAL PRINCIPAL COMPONENT ANALYSIS OVER TREE SPACES

A known challenge in the rapidly growing area of phylogenomics is the lack of tools to analyze the large volume of genome data. Genomic data includes information on the evolution, structure and mapping of genomes. Phylogenetic trees are branching diagrams that show the evolutionary history of species and their genes. Gene trees show the evolutionary history of a particular gene. To analyze evolutionary history from genomic data, we reduce the dimensionality of gene trees, overcoming high dimensional analytical challenges. Through the vectorization of pairwise distances between each combination of two leaves within a phylogenetic tree, we utilize a tropical principle component analysis: a principal component analysis (PCA) in terms of a tropical metric. We project gene trees onto a two-dimensional space using a tropical PCA, a tropical convex hull that minimizes the sum of residuals between each gene tree in the dataset and its projection onto the tropical convex hull over the tree space, which is the set of all possible gene trees. Since computing a tropical PCA for the given dataset is computationally time intensive, we implement a Markov Chain Monte Carlo Metropolis-Hastings algorithm to effectively and efficiently estimate the tropical PCA. Utilizing simulation and real-world data, we implement our tropical PCA algorithm and visualize the results in two-dimensional plots, the results of which look promising and demonstrate our algorithm's strengths.http://archive.org/details/stochasticoptimi1094562731Major, United States ArmyApproved for public release; distribution is unlimited

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Studies in probabilistic methods for scene analysis

In this thesis, probabilistic methods are applied to a number of problems in computer vision. The goal is to provide means for a vision based system that is able to analyze and recognize scenes and objects in camera images and to use that information for autonomous navigation and machine learning. New methods are developed for different functions that are needed in such a system, including segmentation of images, model-based recognition of objects, robot navigation and model complexity control. The approach is based on generative probability models, and Bayesian statistical inference is used to match these models with image data. Stochastic sampling methods are applied to obtain numerical results. The self-organizing map is a neural network algorithm that has many applications in computer vision. In this thesis, the algorithm is analyzed in a probabilistic framework. A probability density model is derived and new model selection techniques are proposed, which enable complexity control for the self-organizing map. The analysis of images is discussed from the point of view of segmentation and object recognition. Segmentation aims at dividing the image into parts of different appearance, while object recognition is meant to identify objects that fulfill given criteria. These are different goals, but they complement each other. When the recognition of all objects in an image is not possible, segmentation can provide an explanation to the rest of the image. For object recognition, different two and three dimensional object models are considered and Bayesian matching techniques are applied to them. Efficient techniques for image segmentation are proposed and results are presented.Tässä väitöskirjassa sovelletaan todennäköisyyslaskennan menetelmiä eräisiin tietokonenäköongelmiin. Työn tarkoituksena on tuottaa keinoja näköön perustuvaan järjestelmään, joka voi analysoida ja tunnistaa näkymiä ja kohteita kamerakuvista ja käyttää näin saatua informaatiota itsenäiseen navigointiin ja koneoppimiseen. Työssä kehitetään uusia menetelmiä järjestelmän tarvitsemiin toimintoihin kuten kuvien segmentointiin, mallipohjaiseen kohteiden tunnistukseen, robottinavigointiin ja mallien kompleksisuuden hallintaan. Työssä käytettävä lähestymistapa perustuu generatiivisiin todennäköisyysmalleihin, ja mallit sovitetaan kuvadataan bayesiläistä tilastollista päättelyä soveltaen. Numeeristen tulosten saamiseksi käytetään stokastisia poimintamenetelmiä. Itsejärjestyvä kartta on neuroverkkoalgoritmi, jolla on useita tietokonenäköalan sovelluksia. Tässä työssä algoritmia analysoidaan todennäköisyyspohjaisesti. Algoritmin tuottamalle mallille johdetaan todennäköisyysjakaumamalli ja sille esitetään uusia mallinvalintamenetelmiä, jotka mahdollistavat itsejärjestyvän kartan kompleksisuuden hallinnan. Kuvien analysointia käsitellään sekä segmentoinnin että kohteiden tunnistuksen näkökulmasta. Segmentoinnissa kuva jaetaan erilaisilta näyttäviin osiin. Kohteiden tunnistus perustuu niiden ennalta tunnettuihin ominaisuuksiin. Tavoitteet ovat siten varsin erilaisia, mutta ne täydentävät toisiaan. Silloin kun vain osa kuvassa olevista kohteista pystytään tunnistamaan, segmentoinnilla voidaan saada kuvan muille osille selitys. Väitöskirjassa esitetään laskennallisesti tehokkaita menetelmiä kuvien segmentointiin. Kohteiden tunnistusta kaksi- ja kolmiulotteisten mallien avulla tarkastellaan bayesiläisiä menetelmiä käyttäen.reviewe