An algorithmic perspective on problems in biological imaging

The last decade has seen considerable advances in our understanding of the genome, and computational methods have played a very crucial role in these developments. Decoding the human genome posed a significant computational challenge and is undoubtedly one of our major accomplishments; today, research is intently focused on understanding the sequences at an unprecedented level of detail. This process is continuously generating a steady stream of interesting computational problems. In this dissertation, we concentrate on several such problems arising from the need to understand the structural and functional organization of the nuclear components from microscopic images. The first problem we study is related to extracting the same or similar 'foreground' from a set of image slices. The set of 2D slices correspond to images of the 3D chromosome acquired at different focal planes of the fluorescent microscope. But since the resolution in-plane (in the 2D slice) is higher than the resolution along the stack, a standard three dimensional segmentation is rather problematic. Our view of this problem is to perform figure-ground segmentation in the 2D planes individually with the additional condition that the foregrounds must be consistent w.r. t. each other. We design new based algorithms for this paired segmentation problem formulating it as a Markov Random Field (MRF) type energy function and report on encouraging results on a variety of images. The second problem we tackle is the Generalized Median Graph problem motivated from the requirement of building topological maps of chromosome organization in the nucleus. The problem belongs to a class of prototype building problems where the input class is a set of graphs; in other words, we want to build a 'model graph' for a set of graphs. Our main result is a polynomial time algorithm for this problem which yields near-optimal solutions in practice. We show that using our LP based algorithm, even in the worst case, a solution within a factor of two of the optimal solution can be obtained if the distances between the graphs are known correctly. We propose an additional algorithm based on a bi-level framework to obtain solutions arbitrarily close to the optimal but in non-polynomial time. The third problem we address is an important partial matching problem of geometric objects. The input is in the form of sets of under-sampled slices (e.g., 2D contours) of one (or more) unknown 3D objects (e.g., 3D chromatin surfaces), possibly generated by slicing planes of arbitrary orientations, the question we are interested in is whether it is 'possible' that two under-sampled sets have been taken from the same object. Since the three dimensional structure of objects of interest (e.g., chromosomes) often cannot be inferred precisely in fluorescence microscopy because of poor resolution in the third dimension, such techniques are important to establish correspondence between structures in a sequence of time-lapse images. We present efficient algorithms for addressing this question. Next, we propose new approaches for analyzing time-lapse microscopic nuclear images. Our techniques address the problem of limited spatial and temporal resolution capacities of current microscopic imaging techniques which allow acquisition of images only in time-lase mode (which leads to significant frame-to-frame information loss). We present a suite of geometric approaches for solving the problem. Our techniques provide a comprehensive solution to (1) raw image simplification (2) segmentation and (3) effective recovery of complicated motion and deformation as well as the change of intensity surfaces from pairs of images in a microscopic image sequence. These techniques are also readily applicable to other types of images for reconstructing motion and intensity surfaces of deform able objects. Finally, we propose a set of novel techniques to determine, analyze, and interpret the mobility patterns of functional sites (replication and transcription) in the nucleus. This is motivated from recent studies that have shown a link between movement of the sites and the actively expressing components of DNA. Our algorithms provide the tools to interpret the seemingly stochastic motion patterns of the functional sites within the nucleus in terms of a set of tractable 'patterns' which can then be analyzed to understand their biological significance.