Medial Spheres for Shape Representation - CIM - McGill University

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the poles). Aichholzer et al. [5, 3] simplify the set of polar balls by growing all balls by a given parameter and then applying a heuristic to solve the set covering problem of find- ing the minimum size subset of balls that continues to cover the sampled surface points. Miklos et al. [60, 85] simplify the set of polar balls by growing each polar ball by a multi- plicative factor s, computing the medial surface of the grown spheres, and scaling the balls back by 1/s. When a set of points sampled on the surface and normal vectors to each sampled point are given, Ma et al. [81] find approximate medial points that lie on a face of the Voronoi diagram of the sample points by iteratively shrinking potential maximal spheres, with parallel computations carried out on the GPU. When the surface is given explicitly, Turkiyyah et al. [117] explain how the locations of the Voronoi vertices of points sampled on the surface can be refined to provide a better approximation to the medial surface, using the explicit knowledge of the boundary. The strengths of Voronoi methods are the substantial body of work on their theoretical properties and the existence of robust software that facilitates computation. However, the density of Voronoi vertices varies depending on the density of the sampled boundary points and the curvature of the boundary, producing dense clusters of Voronoi vertices in certain regions and sparse collections of Voronoi vertices in others. The method we will develop for medial surface approximation in this chapter will generate a set of medial points that are distributed relatively uniformly on the medial surface. The advantage of this distribution for applications in computer graphics will be discussed in Chapters 7 and 8. 31
2.2.2 Tracing Methods Computing the medial surface by tracing involves identifying points on junction curves of medial sheets and following these to discover the complete set of junctions of medial sheets. When the medial surface of a polyhedron must be computed exactly, Cul- ver et al. [36] use exact arithmetic to accurately compute the junction curves of the medial surface by tracing. Because of the high computational cost of using exact arithmetic, this method is only feasible for polyhedra having a small number of faces. Leymarie and Kimia [78] consider as input a cloud of points and trace medial sheet junctions from sources detected using exact bisector computations between clusters of surface points. In- teracting clusters are found quickly by taking advantage of spatial relationships between clusters. Seams are traced in the direction of increasing medial sphere radius. Simplifica- tion is achieved by removing junctions close to the boundary. Tracing methods provide a rich description of the medial surface by explicitly locating sheet junctions in an efficient manner. However, these descriptions are sensitive to noise or require sophisticated regularization steps [28] to be useable for the application of shape matching. 2.2.3 Spatial Subdivision Methods These types of algorithms partition space into disjoint cells and study the nearest boundary elements to these cells to locate the medial surface. Many compute the generalized Voronoi diagram of a solid, rather than the medial surface of a solid. For a polyhedron, the generalized Voronoi diagram is a superset of the medial surface that additionally con- tains bisectors of faces and their incident reflex edges and vertices. When the solid is a 32
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 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 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
Page 68 and 69: Algorithm 2 SALIENTMEDIALPOINT(Φ,
Page 70 and 71: Consider a medial sphere with radiu
Page 72 and 73: (left) shows 20 spheres whose envel
Page 74 and 75: Figure 2-10: An example of an objec
Page 76 and 77: Figure 2-11: Left: The original pol
Page 78 and 79: object bounding box. The sampling r
Page 80 and 81: lists these parameters, along with
Page 82 and 83: Chapter 3 Theoretical Analysis: 3D
Page 84 and 85: Figure 3-1: When B consists of two
Page 86 and 87: of the line l with π1. Since ∠Cp
Page 88 and 89: Figure 3-3: If a medial point m is
Page 90 and 91: other point in Θ is outside of V (
Page 92 and 93: When computing the full foam, consi
Page 94 and 95: We will show that either the distan
Page 96 and 97: of the medial points that are not d
Page 98 and 99: Chapter 4 Theoretical Analysis: 2D
Page 100 and 101:
Let l(a, b) denote the line through
Page 102 and 103:
Figure 4-2: Cases A (Left) and B (R
Page 104 and 105:
Algorithm DECIDEMA performs O(N) di
Page 106 and 107:
Let N be big enough such that L < 2
Page 108 and 109:
Thus, ∠uwv is not maximum when w
Page 110 and 111:
Taking the derivative with respect
Page 112 and 113:
Let X, Y and P be the nearest point
Page 114 and 115:
In this case, the y-coordinate of Y
Page 116 and 117:
disks, we ensure that if a medial p
Page 118 and 119:
Part II Quality of Shape Descriptio
Page 120 and 121:
5.1 Differential Geometric Shape Op
Page 122 and 123:
are sampled on a particular kind of
Page 124 and 125:
Figure 5-1: An illustration of S a
Page 126 and 127:
medial surface is approximated by a
Page 128 and 129:
When working with approximate media
Page 130 and 131:
Let v2 = NMS(x0) × v1 , (5.11) NMS
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it may happen that all the approxim
Page 134 and 135:
Proof. Equation 5.7 can be rewritte
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Figure 5-4: Left: Approximate media
Page 138 and 139:
Figure 5-6 presents results for ano
Page 140 and 141:
However, existing discrete differen
Page 142 and 143:
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
Page 148 and 149:
6.4.2 Shape Indexing and Retrieval
Page 150 and 151:
Chapter 7 Fast and Tight Spheres In
Page 152 and 153:
We now overview methods that seek a
Page 154 and 155:
Figure 7-1: A polyhedron (Left) and
Page 156 and 157:
A be the complement of a set A. The
Page 158 and 159:
The method of Bradshaw and O’Sull
Page 160 and 161:
Figure 7-3: Error and timing result
Page 162 and 163:
Incorrect normal estimates or local
Page 164 and 165:
vol(Ω) − vol(U) + 2(vol(U) −
Page 166 and 167:
a poorer estimate to a medial spher
Page 168 and 169:
Chapter 8 Application to Proximity
Page 170 and 171:
computational problem from collisio
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measurement is outside the polyhedr
Page 174 and 175:
e the side length of the voxels use
Page 176 and 177:
8.5 Experimental Results We evaluat
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our method, a small value of σ nee
Page 180 and 181:
spheres. However, we must note that
Page 182 and 183:
dual complex of the associated ball
Page 184 and 185:
Figure 8-4: A fat triangle, or the
Page 186 and 187:
and the power diagram of balls, we
Page 188 and 189:
the medial surface. For a finite sa
Page 190 and 191:
Listing of Future Work Throughout t
Page 192 and 193:
REFERENCES [1] CGAL, Computational
Page 194 and 195:
[24] F. Cazals, H. Kanhere, and S.
Page 196 and 197:
[52] H. Federer. Curvature measures
Page 198 and 199:
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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