Medial Spheres for Shape Representation - CIM - McGill University

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2–11 Left: The original polyhedra; Centre: the union of spheres computed with our method; Right: approximate medial points grouped into smooth sheets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2–12 Medial points computed for a solid with varying voxel side length σ. Shown are the results for voxel lengths equal to 1/10 th , 1/20 th , 1/40 th and 1/100 th of the maximum dimension of the object bounding box, from left to right, respectively. . . . . . . . . . . . . . . . . . . . . . . 59 2–13 Left: A bumpy sphere model. Centre: Its medial surface approximation when θ = 0.6 and no radius threshold is used. Right: The medial surface approximation when θ = 0.6 and a radius threshold of 10% of the maximum dimension of the bounding box is used. . . . . . . . . . . 60 2–14 Voronoi diagram of three lines cropped to a cube. . . . . . . . . . . . . . 60 3–1 When B consists of two points outside the sphere, the medial axis is shown as a dashed line. Points Φ are big dots on the sphere. . . . . . . . . . . 65 3–2 Objects of interest in the proof of Lemma 3.1. . . . . . . . . . . . . . . . 66 3–3 If a medial point m is in the convex hull of the 6 sampled points on the boundary of the dark disk with nearest boundary points A, B, and C, then the nearest boundary points to m are inside the green disk and outside the grey disks. The dashed lines are the bisectors of A, B and C. 69 4–1 ∇D(A), ∇D(B) point below the plane through A, B and C, while ∇D(C) points above this plane. The nearest boundary points to A, B and C lie on an edge of polyhedron. The medial surface does not necessarily intersect △ABC. . . . . . . . . . . . . . . . . . . . . . . . 82 4–2 Cases A (Left) and B (Right) of Lemma 4.2. . . . . . . . . . . . . . . . . 83 4–3 Situations when Algorithm 4 returns ‘Undetermined’ while a section of the medial axis passes through circle C. Medial points p and q are equidistant from points A and B on the boundary. Point p has a small object angle, while the distance d from point p to the boundary is small. 85 4–4 Illustrations for error analysis in Case 1. . . . . . . . . . . . . . . . . . . 90 4–5 Angle β is an upper bound for α, which is twice the object angle. . . . . . 92 xv
4–6 The angles of interest in Case 2. . . . . . . . . . . . . . . . . . . . . . . 94 4–7 When (p, P ) is orthogonal to (x, y) and dip = L, d = 2L, the y-coordinate of Y is ≈ −0.21L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5–1 An illustration of S a rad (v) for a given U a (based on a figure in [37]). . . . . 105 5–2 The case when κri = −proj U ∂U1 ∂v1 1 = . . . . . . . . . . . . . . . . . . . . 109 r 5–3 The medial axis of this object consists of low-object angle segments (dashed lines) and a high-object angle segment (bold line). When approximating the medial axis with a set of points, we only retain points on high-object angle segments of the medial axis. The boundary of the union of the associated medial circles approximates the original object. Medial point m is equidistant from points A and B on the boundary. Surface curvatures at points A and B may be found using the radial shape operator, while for point C a different strategy is used. . . . . . . 113 5–4 Left: Approximate medial points coloured according to principal curvatures on the solid’s boundary. Centre: Projection of curvature values at the approximate medial points to the boundary of the solid. Right: Curvature values obtained using the method in [100]. The colourmap used is shown in the bottom right corner. See the associated text for a discussion of these results. . . . . . . . . . . . . . . . . . . . . . . . . 117 5–5 The principal curvature directions recovered on the surface of a cup. . . . 118 5–6 Top: An oblate spheroid coloured by principal boundary curvature κ1 (left) and κ2 (right). Middle: Points near the medial surface of the oblate spheroid coloured by principal boundary curvature. Bottom: Respective principal curvature estimates κ ′ 1 and κ ′ 2 shown on the medial surface. The mean absolute error of the estimation of κ1 and κ2 is shown below each column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6–1 (a) A union of the set of balls corresponding to three medial sheets decomposed using the power diagram; (b) the dual of the decomposition of the set of balls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6–2 A pear model approximated using a progressively larger number of sheets of medial balls. Most of the shape is covered using 4 sheets only. . . . . 127 xvi
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 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
Page 64 and 65: 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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