Medial Spheres for Shape Representation - CIM - McGill University

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My labmates over the years have made working in the lab a very desirable social ex- perience. I want to thank Ryan Eckbo, Peter Savadjiev, Scott McCloskey and Nils Tilton for being fun to be around and for being my friends. Over time, I was rewarded with new friends: Parya Momayyez-Siahkal and Emmanuel Piuze-Phaneuf (who also helped edit Section 1.1), who made coming to McGill thoroughly rewarding, shared my hardships and happiness, and provided constant support. I would like to thank Paul Kry for his insights, for listening to my concerns, and for offering encouragement in our collaborative work on the material in Part III, and for sharing the code that enabled the physically-based simulations. I am grateful to Nina Amenta for encouraging me to pursue the research direction that resulted in Part III of this thesis during our discussion in Barbados in 2009, when I was having doubts. I am very grateful to Patrice Koehl for sharing with me the AlphaBall software that computes the volume of a ball restricted to its power cell and thus opening up new research possibilities explored in this thesis. I would like to thanks the members of my examination committee for their efforts in reading the thesis and for providing valuable suggestions for improvement. This research was supported by an FQRNT scholarship. iii
ABSTRACT This thesis presents a particular set of spheres as new representation of the shape of a 3D solid. The spheres considered are maximal inscribed spheres in the solid and their centres are chosen in such a way that at most one sphere centre lies in a cubic region of space. The shape representation proposed is a discretization of the medial surface transform of a solid. Part I of this thesis presents algorithms for the computation of this representation given a boundary representation of a solid by approximating its medial surface transform. Properties of those medial spheres that are not detected by our algorithm in 3D are described and a complete characterization of those medial circles that are not detected by a 2D version of our algorithm is given. In Part II, recent results from differential geometry are used to compute principal cur- vatures and principal curvature directions on the boundary of the smooth solid represented using the union of medial spheres. This computation is performed using only the medial sphere centres and a pair of points on each medial sphere that lies on the surface of the solid being modeled. It is shown how the union of medial spheres allows a part-based description of the solid, with a significance measure associated with each part. In Part III, it is shown that our shape representation can offer a tight volumetric fit to a polyhedron, using a small number of spheres. The spheres used in our representation can be quickly updated as the solid undergoes a certain class of deformations. It is shown how our set of medial spheres allows efficient and accurate proximity queries between polyhedra. iv
Page 1: Medial Spheres for Shape Representa
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 50 and 51: the poles). Aichholzer et al. [5, 3
Page 52 and 53: set of disjoint convex sites, the e
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the medial surface of a polyhedron
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Figure 2-5: Arrows show ∇D, the d
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size. The algorithm we will describ
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2.4.1 Precise Medial Points Given a
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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
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
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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