Medial Spheres for Shape Representation - CIM - McGill University

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LIST OF FIGURES Figure page 1–1 A ‘bunny’ shape represented using (left) a polygon mesh having 5110 triangle faces, and (centre) a union of 1559 spheres. Also shown is the detail of the bunny’s knee in the union of spheres representation. . . . . 3 1–2 Several implicit surfaces obtained by varying the distance between pairs of generating primitives. Image adapted from [15]. . . . . . . . . . . . 7 1–3 An example of a non-solid. . . . . . . . . . . . . . . . . . . . . . . . . . 9 1–4 Top: The medial axis (black) of a 2D solid (orange), overlayed. Bottom: A 3D solid (left) and its medial surface (right). . . . . . . . . . . . . . 11 1–5 Left: Both the interior and the exterior medial axis, cropped to a rectangle. Right: An illustration of Property 1.1. . . . . . . . . . . . . . . . . . . 12 1–6 Different classes of points that compose the medial surface of a smooth 3D object [58]. A 2 1 points are smooth medial points, A 3 1 are junction medial points and A3 are edge or rim medial points. Adapted from [106]. 13 1–7 The edges of the Voronoi diagram of set of points (black) in the plane are shown in grey. They dual of the Voronoi diagram is the Delaunay Triangulation, whose edges are shown in blue. . . . . . . . . . . . . . . 16 1–8 (a) A union of a set of disks; (b) its power diagram overlayed; (c) the union of the set of disks decomposed using the power diagram; (d) the dual of this decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2–1 A polyhedron (left) and its medial surface (middle). The medial sheets that do not touch the boundary are shown on the right. Figure adapted from [36]. Some corresponding points on the polyhedron and its medial surface are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 xiii
2–2 Medial axes, in black, for two solids with red boundaries. The boundary on the right is obtained by smoothing the boundary on the left. Produced with [84]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2–3 Left: The medial surface of a box, with each sheet shown in a different colour. The object angle θ at the selected medial point is π/4 and is half the angle between the two spoke vectors shown with black arrows. The object angle for points on the maroon sheet is π/2. Right: In this 2D example, the boundary of a 2D solid is shown in black and its medial axis in red. The object angle of medial point m1 (∠A1m1B1) is greater than that of medial point m2 (∠A2m2B2). . . . . . . . . . . . . . . . . 27 2–4 The edges Voronoi diagram of a set of blue points is shown in grey. The dark edges are the subset of the Voronoi diagram that approximates the inner medial axis. Produced with [84]. . . . . . . . . . . . . . . . . . . 29 2–5 Arrows show ∇D, the directions to nearest locations on the boundary to points on R. In this example, the medial axis intersects the line segment (b, opp(b)) because ∇D(b) = ∇D(opp(b)). . . . . . . . . . . . . . . . 37 2–6 The objects of interest in the proof of Lemma 2.4. . . . . . . . . . . . . . 45 2–7 The boundary of a 2D solid is shown in black and its medial axis in red. In this case (a, b = a + γ∇D(a)) intersects the boundary and although ∇D(a) = ∇D(b), a medial point lies on (a, b). . . . . . . . . . . . . . 47 2–8 A dragon polyhedron (left) approximated with 1324 spheres (centre) and 4102 spheres (right). Medial points were found in voxels intersected by the boundary B on the right, but not on the left. . . . . . . . . . . . . . 52 2–9 (a) Sampled points on the red sphere, except A, are medial points of the envelope of the black circles shown. (b) The medial axis of a regular polygon with low object angle medial points shown in black and a high object angle medial point M shown in red. . . . . . . . . . . . . . . . 53 2–10 An example of an object with boundary B whose medial surface is a single medial sheet MS where neighbouring smooth medial points have different surface normals (two surface normals are shown). Adopted from [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 xiv
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 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
Page 62 and 63: 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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