Reconstruction of Topology and Geometry from Digitisations

By Sabrina Tang Christensen
PhD Dissertations
November 2016
Abstract:

Elucidation of the relation between real world objects and their digitisations is an inherent challenge of image analysis. This thesis addresses the reconstruction of topological as well as geometrical features of sets in Euclidean 3-space respectively the Euclidean plane from binary images.

The first problem, the reconstruction of topology in dimension three, is approached using combinatorics of voxel reconstructions in combination with differential topology results. An improved digital reconstruction based on binary images of objects with sufficiently smooth boundary is proposed. It is shown that this reconstruction is ambient isotopic to the underlying object provided the resolution of the digitisation be sufficiently high and under certain assumptions on the classical voxel reconstruction. The exact lower bound on the resolution, related to the curvature of the boundary of the object, is given, and it is argued that this is the best possible bound for which topological equivalence is guaranteed. Making no restrictions on the classical reconstruction, similar results can be proved for slightly stronger assumptions on the resolution. It is conjectured that the lower of the bounds on the resolution also suffices in this setting.

For the reconstruction of geometry in dimension two, a practical approach employing digital algorithms is selected. The geometry of real world objects can be described by Minkowski tensors. It has been shown that these tensors can be approximated by digital algorithms, provided that the underlying object has positive reach and that a binary image of the object is available. Two such algorithms are implemented. Their suggested convergence properties are confirmed in practice via simulations on test sets, and recommendations for input arguments of the algorithms are given. Simulations imply that the accuracy of the estimators is directly related to the resolution of the digitisation, and, further, that an increase in complexity of the object necessitates higher resolutions. Finally, a third algorithm, combining and thus exploiting the most advantageous qualities of both aforementioned algorithms, is proposed.

Thesis advisor: Andrew du Plessis
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