Medial Spheres for Shape Representation - CIM - McGill University

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DECLARATION This thesis presents original scholarship by the author. The results reported in Chap- ters 2-8 reflect work done largely by the author, with assistance offered by Dr. Kaleem Siddiqi, Dr. Sue Whitesides, and Dr. Paul Kry, as is reflected in the co-authorship of publications arising from this thesis described in Section 1.3.3. The primary contributor to the development of algorithms, the analysis of their correctness, and their implementations, as well as the technical descriptions, was the author of this thesis. vii
TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv ABRÉGÉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Shape Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Parametric/Boundary Representations . . . . . . . . . . . . . 4 1.1.2 Implicit/Interior Representations . . . . . . . . . . . . . . . . 5 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Medial Surface Transform . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Space Partitions . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.3.1 Objective and Outline . . . . . . . . . . . . . . . . . . . . . . 18 1.3.2 List of Novel Contributions . . . . . . . . . . . . . . . . . . . 19 1.3.3 Publications Arising from This Thesis . . . . . . . . . . . . . 21 PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 viii
Page 1: Medial Spheres for Shape Representa
Page 4 and 5: My labmates over the years have mad
Page 6 and 7: ABRÉGÉ Cette thèse présente une
Page 10 and 11: I Medial Spheres - Construction 23
Page 12 and 13: 8 Application to Proximity Queries
Page 14 and 15: LIST OF FIGURES Figure page 1-1 A
Page 16 and 17: 2-11 Left: The original polyhedra;
Page 18 and 19: 6-3 A head model approximated using
Page 20 and 21: Chapter 1 Introduction This chapter
Page 22 and 23: Figure 1-1: A ‘bunny’ shape rep
Page 24 and 25: 1.1.2 Implicit/Interior Representat
Page 26 and 27: narrow boundary features, as prohib
Page 28 and 29: and a line segment cannot be a soli
Page 30 and 31: Figure 1-4: Top: The medial axis (b
Page 32 and 33: Figure 1-6: Different classes of po
Page 34 and 35: Voronoi Diagram The Voronoi diagram
Page 36 and 37: (a) (b) (c) (d) Figure 1-8: (a) A u
Page 38 and 39: Part II investigates the descriptiv
Page 40 and 41: can be quickly updated and how the
Page 42 and 43: Part I Medial Spheres - Constructio
Page 44 and 45: Figure 2-1: A polyhedron (left) and
Page 46 and 47: Figure 2-3: Left: The medial surfac
Page 48 and 49: Figure 2-4: The edges Voronoi diagr
Page 50 and 51: the poles). Aichholzer et al. [5, 3
Page 52 and 53: set of disjoint convex sites, the e
Page 54 and 55: the medial surface of a polyhedron
Page 56 and 57: Figure 2-5: Arrows show ∇D, the d
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size. The algorithm we will describ
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2.4.1 Precise Medial Points Given a
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the theory of geometric curve evolu
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point are P = B(p) and Q ′ = p+d(
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(a, b) that cross B, and further, i
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Algorithm 2 SALIENTMEDIALPOINT(Φ,
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Consider a medial sphere with radiu
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(left) shows 20 spheres whose envel
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Figure 2-10: An example of an objec
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Figure 2-11: Left: The original pol
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object bounding box. The sampling r
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lists these parameters, along with
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Chapter 3 Theoretical Analysis: 3D
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Figure 3-1: When B consists of two
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of the line l with π1. Since ∠Cp
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Figure 3-3: If a medial point m is
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other point in Θ is outside of V (
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When computing the full foam, consi
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We will show that either the distan
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of the medial points that are not d
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Chapter 4 Theoretical Analysis: 2D
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Let l(a, b) denote the line through
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Figure 4-2: Cases A (Left) and B (R
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Algorithm DECIDEMA performs O(N) di
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Let N be big enough such that L < 2
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Thus, ∠uwv is not maximum when w
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Taking the derivative with respect
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Let X, Y and P be the nearest point
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In this case, the y-coordinate of Y
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disks, we ensure that if a medial p
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Part II Quality of Shape Descriptio
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5.1 Differential Geometric Shape Op
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are sampled on a particular kind of
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Figure 5-1: An illustration of S a
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medial surface is approximated by a
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When working with approximate media
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Let v2 = NMS(x0) × v1 , (5.11) NMS
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it may happen that all the approxim
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Proof. Equation 5.7 can be rewritte
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Figure 5-4: Left: Approximate media
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Figure 5-6 presents results for ano
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However, existing discrete differen
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Chapter 6 A Significance Measure fo
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of a sheet is also related to the o
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1 sheet 2 sheets 4 sheets 38 sheets
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6.4.2 Shape Indexing and Retrieval
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Chapter 7 Fast and Tight Spheres In
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We now overview methods that seek a
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Figure 7-1: A polyhedron (Left) and
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A be the complement of a set A. The
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The method of Bradshaw and O’Sull
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Figure 7-3: Error and timing result
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Incorrect normal estimates or local
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vol(Ω) − vol(U) + 2(vol(U) −
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a poorer estimate to a medial spher
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Chapter 8 Application to Proximity
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computational problem from collisio
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measurement is outside the polyhedr
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e the side length of the voxels use
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8.5 Experimental Results We evaluat
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our method, a small value of σ nee
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spheres. However, we must note that
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dual complex of the associated ball
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Figure 8-4: A fat triangle, or the
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and the power diagram of balls, we
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the medial surface. For a finite sa
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Listing of Future Work Throughout t
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REFERENCES [1] CGAL, Computational
Page 194 and 195:
[24] F. Cazals, H. Kanhere, and S.
Page 196 and 197:
[52] H. Federer. Curvature measures
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179 [77] G. Lavoué, F. Dupont, and
Page 200 and 201:
[103] D. Shaked and A.M. Bruckstein
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Medial Spheres for Shape Representation - CIM - McGill University

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