Medial Spheres for Shape Representation - CIM - McGill University

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6–3 A head model approximated using a progressively larger number of sheets of medial balls. Most of the shape is covered using 6 sheets only. . . . . 128 7–1 A polyhedron (Left) and two distributions of sphere centres for a spherebased approximation this object. (Centre) The subset of the Voronoi vertices of points sampled on the polyhedron boundary that lies near its medial surface. (Right) Sphere centres computed with our method, developed in Chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . 135 7–2 Consider the solid that is the union of the two grey balls, whose medial surface is the dashed red line. Increasing the voxel resolution (from blue to green) decreases the number of medial spheres generated for the solid. 136 7–3 Error and timing results for the pear (5154 triangles) and triple torus (16000 triangles) models for our method (SD) and the AMAA method (SV ). Note that true SV error may be higher than plotted. . . . . . . . 141 7–4 (a) Reference polyhedron (top) approximated using internal spheres tangent to the polyhedron (bottom). (b)-(d) This sphere set is updated to approximate deformed versions of the reference polyhedron. The fraction of the time to compute the sphere set compared to computing the reference sphere set is shown below each pose. Also shown is the range of the volumetric error normalized by polyhedron volume. . . . . 148 8–1 The envelope of the dark circles s1 and s2 is non-differentiable at point v, whose nearest point on the boundary is m. The radii of both internal circles are increased to create circles that pass through m. . . . . . . . 151 8–2 Comparison of the S + D and SV approximations for various models. (Centre) Histograms of signed distances from points sampled on the boundary of polyhedra to points on the envelopes of S + D and SV . (Right) Histograms of signed distances from points sampled on meshes ap- proximating the envelopes of S + D and SV to the boundary of polyhedra. spheres. Means of the absolute distance are shown as a dashed line for SV and a dotted line for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Stars denote SV spheres, while circles denote S + D S + D 8–3 A rectangle swept sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . 156 xvii
8–4 A fat triangle, or the convex hull of 3 spheres. The spheres contribute red patches to the hull; pairwise sphere interpolants are cylindrical sections shown in blue; the linear section of the fat triangle is shown in yellow. . 165 xviii
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 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 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
Page 62 and 63: the theory of geometric curve evolu
Page 64 and 65: point are P = B(p) and Q ′ = p+d(
Page 66 and 67: (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.
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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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