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Geometric Tomography With Topological Guarantees

Pooran Memari 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The purpose of this doctoral thesis is to provide a method to reconstruct three dimensional shapes from cross-sections. The principal motivation is 3D reconstruction of organs that is widely considered to be an important diagnostic aid in the medical world. However, the actual simulation results, namely in 3D ultrasonic simulation, are not reliable to be used in diagnosis. This thesis is the first geometric analysis of the reliability and the validity of reconstruction methods from cross-sectional data. Even in the case of parallel sections, no formal analysis and guarantees had been obtained before this thesis. More formally, we consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in R3 from its cross-sections with a given set of cutting planes having arbitrary orientations. The first Chapter of this manuscript is devoted to analyzing a method presented by Liu et al. in 2008. We prove that under appropriate sampling conditions, the resulting reconstructed object is homeomorphic (and isotopic) to the original object. In the second chapter, we present a second reconstruction method that makes use of the Voronoi diagram of the sections. This method performs more connections between the sections comparing to the first method. Increasing the connectivity between the sections is motivated by reconstructing tree-like structures from sparse sectional data. The provided sampling conditions, leading to topological guarantees, are adapted to tree-like structures: Indeed, we show that if the cutting planes are sufficiently transversal to the surface we want to reconstruct, then the method can handle complex branching structures. Finally, in the third chapter, we show how the Voronoi-Delaunay duality allows us to perform still more connections between the sections comparing to the two first methods. The preliminary experimental results are quite promising, e.g., regarding the practicality of the approach to reconstruct complex cross-sectional branching situations such as the coronary arterial tree. The hope is that the theoretical studies provided in this thesis will be a first step to provide solid foundations and theoretical guarantees for medical diagnostic software.
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Pooran Memari. Geometric Tomography With Topological Guarantees. Computer Science [cs]. Université Nice Sophia Antipolis, 2010. English. ⟨tel-00560010⟩

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