Medial Spheres for Shape Representation - CIM - McGill University

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Figure 1–4: Top: The medial axis (black) of a 2D solid (orange), overlayed. Bottom: A 3D solid (left) and its medial surface (right). Consider the set of all medial spheres of Ω. These spheres have the following very important property [17, 18]: Property 1.1. The envelope of the medial spheres of Ω is the boundary B of Ω. Figure 1–5(Right) provides an example of a finite set of medial spheres such that its envelope approximates the boundary of the 2D solid. Consider the following definition: Definition 1.6. The medial surface transform of Ω is the set of all medial spheres of Ω. The medial surface transform provides an alternative description of the shape of a solid that captures its local width with respect to its ‘skeleton’: a lower-dimensional representation of the volume of the solid that makes explicit its symmetries. The medial 11
Figure 1–5: Left: Both the interior and the exterior medial axis, cropped to a rectangle. Right: An illustration of Property 1.1. surface transform was proposed in the 1960s by Harry Blum [17, 18] as a shape descrip- tor that makes explicit perceptually significant shape properties. It has since been studied extensively in the computer vision community as it allows abstraction sufficient to guide shape matching, in robotics as it offers a collision avoiding path, in computer graphics to generate deformations of models, in medical imaging to capture local shape variation, in geography to simplify the shape of rivers, among some examples. See [106, Ch. 11] for a detailed overview of applications of the medial surface transform. For all solids (except those that are disjoint balls) the medial surface transform consists of an infinite set of medial spheres. The shape representation proposed in this thesis is a union of a finite set of medial spheres and is thus a discretization of the medial surface transform. 12
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 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
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
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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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