Algorithmic tools for biomedical image analysis

Rapid development of image acquisition and processing techniques has provided researchers in the Biomedical domain with an access to high volume of data suitable for fully or semi-automated analysis. The available volume of generated data demands sophisticated, robust and high-throughput image analysis methods. We investigate and propose solutions to a set of challenging geometric and optimization problems arising in Biomedical Imaging. First chapter is devoted to a problem of extracting terrain-like surfaces in volumetric images. We propose and implement a novel and highly efficient graph-theoretical iterative method with bi-criteria of global optimality and smoothness for both single and multiple surfaces. To the best of our knowledge, this is the first method to successfully incorporate curvature constraint into the global solution. To evaluate the convergence and performance of our method, we test it on a set of 14 3 D Optical Coherence Images of human retina. We were able to report all seven surfaces, defining six layers, on each 3-D composite image dataset. Experimental results suggest that the proposed method yields the optimal (or almost optimal) solutions in 3 to 5 iterations. Comparing to the best existing approaches, our method has a much improved running time, yields almost the same global optimality but with much better smoothness, which makes it especially suitable for segmenting highly noisy images. Second chapter discusses two novel tools for learning association patterns of chromosome territories in population of cells. Obtaining accurate and robust CT patterns from cell nucleus images will significantly facilitate analysis and improve overall understanding of molecular processes in cell nucleus. In the first half of the chapter, we formulate the problem as a mixed linear integer program and show how to solve it efficiently for medium size data sets (up to 45 graphs). In the second part of chapter, we introduce a more sophisticated technique based on iterative rounding of an alternative linear program formulation of the problem, which is capable of handling much larger-sized input datasets. The central theme of the third chapter is the study of "average" geometric model for chromosome territories within the cell nucleus. Initial motivation for study comes from cell nucleus imagery data. For the purpose of our problem, we associate each chromosome territory with a unique color. In normal cells, each chromosome territory has two copies or homologs, thus usually, each scene has two chromosome territories of the same color. Next, we approximate each chromosome territory with its center of gravity (or a sphere with equivalent radius). Given a family of cells, the goal is to build a geometric structural pattern (consisting of colored points in 3-D) that can be aligned with every point sets derived from the input family of cells such that the function that quantifies the color-preserving match is optimized. To model this problem, we formally define so-called Alignment of Chromatic Point Sets Problem and show how to efficiently solve it using semidefinite programming formulation. In the final chapter, we describe development and implementation of Algorithmic Toolbox. Toolbox's primary intention is comprehensive study of 3-D spatial positioning of objects of interest in cell nucleus (e.g. chromosome territories, probes inside a DNA fiber, etc.). Yhe ultimate objective of our work is to provide a tool that takes full advantage of modern microscopy imaging and labeling techniques, and draws from the state-of-the-art image analysis methods to yield robust quantitative information. Primarily, we focus on developing tools for two problems: (1) study of spatial positioning of fluorescent probes in cell nucleus (so-called Topology Analysis of DNA strands) and (2) learning a common structural representative for chromosome territories within population of cells at pairwise association level. The obtained algorithmic and biological results will be used to study and compare the difference of various cancer and normal cells in topological structures and associations of chromosome territories and genes. This could potentially lead to significant biological discoveries and help better understanding the mechanism of diseases (such as cancers) and their relationship to chromosome structures and associations. (Abstract shortened by UMI.)