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Abstract

This thesis is motivated by the problem of segmenting extremely noisy images of geometric objects. To this end, it combines randomized combinatorial set cover optimization with a statistical model of object interaction. The set cover approach provides stability and applicability in cases in which many traditional methods of segmentation fail due to noise and imperfect data. The statistical model provides additional information that is not directly supplied by the image, and leads to a more realistic depiction of physical object properties in the resulting segmentation. This dissertation is divided into three parts: The first covers topics of randomized combinatorial optimization. This includes improving bounds of convergence and establishing a method of parallelization for an existing approach, as well as linking solutions to different combinatorial problems, such as geometric set cover and a general linear program. Part two is concerned with constructing a point process model of object interaction that fits later applications, and exploring some theoretical and practical pitfalls in its simulation, estimation, and coupling with a combinatorial approach. Part three compares previously discussed methods empirically, and demonstrates the performance of the established combination of randomized optimization and statistical model on microscopic cell images and 3D μCT scans of fiber reinforced materials.