Medial Spheres for Shape Representation - CIM - McGill University

The number of simplices of DC(B) is O(|B| 2 ) and is much smaller for typical ball distri-
butions, such as ones that we will consider in this thesis [47]. Use of this simple formula
makes the computation of the volume of a union of balls efficient.
1.3 Thesis Overview
In this section, the main objectives of the thesis and the order in which they will be
addressed are described. We will then list the novel contributions of this thesis. We will
also specify which contributions appear in published or submitted work.
1.3.1 Objective and Outline
This thesis introduces a new shape representation: the set of well-spaced medial
spheres, that is, a union of medial spheres the coordinates of whose centres, when rounded
to the nearest integer, are unique integer triples. This distribution of medial spheres is
well-spaced because there is at most one sphere centre per cubic region of space. The
goal of this thesis is to investigate the strengths of this shape representation in terms of the
three desiderata of shape representations mentioned in Section 1.1: 1) ease of generation,
2) ability to capture meaningful shape information, and 3) efficiency of geometric opera-
tions. The thesis consists of three parts, each of which addresses the desiderata above, in
that order.
Part I introduces algorithms for converting from an explicit boundary representation
of a solid to a set of well-spaced medial spheres by approximating the medial surface of
a solid. When the boundary of the solid is a polygonal mesh, practical algorithms for the
efficient computation of the union of medial spheres shape representation are developed.
Correctness and completeness issues of the proposed algorithms are discussed in each of
the three chapters of Part I.
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