Medial Spheres for Shape Representation - CIM - McGill University

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8 Application to Proximity Queries . . . . . . . . . . . . . . . . . . . . . . . . 149 8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2 Improving Boundary Coverage by Conservative Dilation . . . . . . . 151 8.3 (σ, θ)-fat Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.4 Hierarchy Construction using Rectangle Swept Spheres . . . . . . . . 156 8.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.7 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.7.1 Faster Bottom-Up Sphere Hierarchies . . . . . . . . . . . . . . 162 8.7.2 Interruptible Collision Detection . . . . . . . . . . . . . . . . 163 8.7.3 Sphere-Based Shape Primitives . . . . . . . . . . . . . . . . . 164 8.7.4 Probabilistic Hierarchy Construction . . . . . . . . . . . . . . 166 9 Thesis Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 168 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 xi
LIST OF TABLES Table page 2–1 Statistics for the computation of medial points in Figure 2–11. . . . . . . . 58 2–2 Parameters and their definitions for the medial surface approximation and partitioning algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5–1 Mean (µ) and standard deviation (σ) of the 3 different types of curvature on a section of a cylindrical cup wall (shown in Figure 5–5) consisting of 551 vertices obtained using the method of [100] (Top) and our method (Bottom). We estimate the true positive curvature to be approximately 0.069898 and the true negative curvature to be approximately −0.0875045.118 7–1 Sphere sets SD are generated with our method; sphere sets SV are generated with the method of [23]. Timings are in seconds. . . . . . . . . . . . . 138 8–1 Timing and error results for separation distance tests for our grown spheres S + D and spheres SV of [23] for a variety of polyhedral models. Timings are shown in milliseconds. Spheres are used as bounding volumes in the SH column and RSSs are used in the RSSH column. The maximum dimension of bounding boxes for all models is 10. . . . . . . . . . . . . 160 xii
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 8 and 9: DECLARATION This thesis presents or
Page 10 and 11: I Medial Spheres - Construction 23
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
Page 58 and 59: size. The algorithm we will describ
Page 60 and 61: 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
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[24] F. Cazals, H. Kanhere, and S.
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[52] H. Federer. Curvature measures
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179 [77] G. Lavoué, F. Dupont, and
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[103] D. Shaked and A.M. Bruckstein
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Medial Spheres for Shape Representation - CIM - McGill University

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