Medial Spheres for Shape Representation - CIM - McGill University

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Part II investigates the descriptive power of the set of well-spaced medial spheres shape representation. When spoke vector estimates are given for each medial point, Chap- ter 5 shows how boundary curvature can be recovered from this representation. We also show that the proposed shape representation allows us to easily evaluate the significance and simplify the parts of a solid in Chapter 6. Part III shows that the set of well-spaced medial spheres provides a fast and tight volumetric approximation to a solid in Chapter 7 and also studies advantages offered by this representation when the solid is deformable. Chapter 8 demonstrates how this shape representation can offer improvement over existing union of spheres shape representations for performing proximity queries. Each of the three parts includes an overview of related previous work, as well as a description of future work arising from those parts. 1.3.2 List of Novel Contributions The novel contributions of this thesis are as follows: • Section 2.4 describes an algorithm to obtain a dense set of approximate medial points within a user-specified distance of a solid’s medial surface, along with the spoke vector estimates for the medial points. In Section 2.5, we show how points that lie near distinct smooth medial sheets can be grouped by studying the spoke vector estimates at each approximate medial point. • Chapter 3 initiates the theoretical analysis of the completeness of this algorithm: we make progress towards a characterization of those medial points that are undetected by our algorithm. Section 3.2 discusses the locations of query points to be con- sidered by the algorithm. Section 3.3 establishes the locations of nearest boundary 19
points to medial points that are not detected by our algorithm for a finite sampling rate. Section 3.4 shows how the quality of the undetected medial points can be measured under certain conditions. • Chapter 4 looks at the significantly easier 2D case and considers theoretical prop- erties of an algorithm for computing the medial axis of a 2D solid, based on tools used in the 3D case. We develop additional tools to the ones used in the 3D case for detecting medial points in Section 4.1. Section 4.3 gives a complete geometric description of those medial points that are undetected by these tools. • Section 5.4 presents a numerical method that allows us to estimate the principal curvatures and principal curvature directions at the two implied boundary patches on either side of each approximate medial point, given a dense sampling of approximate medial points and their spoke vector estimates. • Sections 6.2 and 6.3 present a method by which the individual sheets of the discrete medial surface approximation may be ordered by significance. This allows us to substantially reduce the size of the resulting part-based representation and to simplify the shape of the solid. • Chapter 7 looks at the application of our shape representation to the problem of quickly generating a well-fitting sphere-based approximation to a polyhedron. We show how volumetric error of sphere-based shape representations can be evaluated in Section 7.3. Compared to a state-of-the-art method for approximating polyhedra with spheres, we show that our method is significantly faster and provides a tighter fit in terms of volumetric error in Section 7.3. When a model undergoes local feature size preserving deformation, we show in Section 7.4 how the sphere approximation 20
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 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
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
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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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