Geometric algorithms for three dimensional reconstruction in medical imaging

In this dissertation, we present a set of algorithms and results for several challenging problems in Medical Imaging. Medical Imaging, in a clinical context, is closely related to radiology, where the main interest is to acquire and interpret images of high diagnostic quality. Many problems in Medical Imaging can be seen as the mathematical inverse problem, e.g., obtaining three-dimensional (3D) data from 2D x-ray projection images. In this dissertation, we show that by combining techniques from computational geometry, optimization, and computer vision, we are able to achieve highly efficient solutions to several of these inverse problems. Below is a brief summary of our results. Cerebrovascular diseases, such as stroke and myocardial infarction, are among the leading causes of death in the United States; therefore, improving assessment and treatment of cerebrovascular disease could impact a large patient population. For improved assessment and treatment, we have focused on improving the extent and quality of the 3D reconstructions of vascular information determined from multiple projection images routinely acquired during interventions. The use of these multiple views could overcome some of the defficiencies of two-view and biplane techniques. We present two novel self-calibration methods: a non-linear amoeba based multiple-view technique (A-MVT), and a linear programming based multiple-view technique (LP-MVT). Our approaches generate 3D data with a quality comparable to other modalities (rotational angiography) but in a much shorter period of time (as few as 5 seconds), are reliable on large clinical studies, have low sensitivity to incorrect user indications, and produce more reliable 3D reconstructions than those obtained from biplane and two-view techniques. Computed tomography (CT), especially since the introduction of helical CT, provides excellent visualization of the internal organs of the body. As a result, CT is used routinely in the clinical arena to obtain 3D and 4D data. One popular application in CT is Cone Beam Computed Tomography (CBCT), due to its high diagnostic quality (sub-millimeter resolution), and its short scanning times (60 seconds), and is used routinely in the clinical arena to obtain 3D and 4D data. The increasing use of CT has resulted in a substantial rise in population radiation dose, which may lead to an increased incidence of cancer in the population. Methods for reducing the radiation dose while maintaining an improved image quality must address a number of open problems which include (i) the detection and segmentation of anatomic structures from a limited number of projections, (ii) the reduction of artifacts (which reduce the image quality), (iii) the reduction of computation time, and (iv) the reliable reconstruction from a limited number of projection images. For (i), we propose two new techniques for segmenting incomplete CT volumes based on Active Shape Models (ASM) and clustering algorithms. The segmentation problem is challenging since standard reconstructions from a limited number of projection images are noisy and have many artifacts. Both techniques rapidly identify a volume of interest using a limited number of projection images. In our results, we report that even in the case of volumes reconstructed from 1/3 of the projection images, objects can be segmented with an accuracy of over 90%. For (ii), we present a novel algorithm to reduce artifacts in CT reconstructions. Generally, these artifacts are caused by high contrast objects, such as metal, resulting in streaks in the 3D data, which obscure lower contrast objects. In our approach, these objects are identified and then removed in the sinogram space by using optimal computational geometry techniques. The artifacts are significantly reduced when using our approach. For (iii), we introduce GPU-based approaches to improve CT reconstruction times and investigate GPU implementations of direct inverse and iterative techniques. We show that with our techniques the Radon-based reconstruction of a 2563 volume can be reduced from 25 minutes on a CPU to 3.5 seconds on a GPU without losing any reconstruction quality. For (iv), we propose a new reconstruction approach called GOAT (Geometric Object Aware Tomography) which achieves accurate reconstructions from just tens of projection images (rather than the typical hundreds or thousands). Our approach reconstructs the 3D volume by first identifying objects-of-interest (OoIs) and their complements (C-OoIs), and then separating them in the projection space using geometry techniques so that only directly related projection rays are used to reconstruct each OoI. Comparing to existing approaches, our approach simultaneously and significantly improves patient dose (through reducing the number of projections), image quality, and computation time.