Medial Spheres for Shape Representation - CIM - McGill University

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1.1.2 Implicit/Interior Representations In this type of shape representation, the object interior is represented using volumetric primitives. Among the different types of primitives that we will discuss are voxels, spheres, and metaballs. This representation is less flexible than the boundary mesh representation, in that a substantial number of volumetric primitives may be required to capture small features of the boundary. On the other hand, the surface of an object represented implicitly is non self-intersecting by definition. This is in contrast with parametric representations, where care must be taken to ensure such correctness, especially as objects deform. Further, efficient point-location queries can be made by checking if the point lies in any of the volumetric regions. For points outside the object, the distance to the object represented implicitly is the minimum of the distances to each of the volumetric primitives. When the distance between a point and a volumetric primitive can be computed efficiently and when the number of volumetric primitives is small, implicit representations offer substantial effi- ciency improvements over parametric representations for distance computations. Further, the given partition into volumetric primitives can serve to provide a meaningful part-based description of the object for such applications as registration, morphing, and matching. Obtaining a part-based object description for objects described parametrically can be dif- ficult. However, rendering implicit representations is usually done by either conversion to a polygon mesh representation of potentially very large size or by ray tracing, and tends to be slow. Cellular Subdivision Representations When space is subdivided into regular-size cubes, each such cube is called a voxel. Voxel representations identify those cubes in a regular partition of space into cubes that 5
epresent a shape. Octrees use axis-aligned cells of mixed size: coarse cells containing geometry are subdivided until a desired resolution is reached. Because each voxel has at most 26 voxel neighbours, voxel data sets can be analyzed by exploring these digital neighbourhoods. Storage requirements of high resolution voxel data sets are large. Octrees have smaller storage requirements than voxel representations when a small number of cells in a volume is ‘on’. Union of Balls Representations Given a set of balls B = {(ci, ri)|ci ∈ R 3 , ri ∈ R} with centres ci and radii ri, the surface of the union of balls is the zero level set of the function F : R 3 → R, x ↦→ min i {d(x, ci) − ri}. This shape representation is advocated in the work of Alain Fournier [97] as a supe- rior shape representation to triangle meshes because of its stability, as shapes represented using the same or similar sets of spheres can be described by completely different triangle meshes in terms of the location of the triangle vertices and their adjacency, and because of the relative invariance of union of spheres representations to noise in the input. The generation of union of balls representations that offer advantages over alterna- tive representations is not trivial. Chapter 7, Section 7.1, surveys existing approaches for approximating a shape with balls. Union of balls shape representations can provide advantages against triangle mesh representations in those applications that do not need to represent small-scale, sharp, and 6
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 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(
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
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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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