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Examples of these operators are shown in Figure 2.4. One of advantages to use CSG is that topological changes can be handled easily. For example, a hole can be created on an implicit volume using the difference operator with another implicit volume. Nevertheless, CSG creates discontinuity between two scalar fields, which results in undesirable artifacts when blending (Section 2.1.3) is applied. There are several approaches for applying CSG operators while preserving continuity. For example, R-Functions in FRep construct C n continuous CSG operators [40]. Barthe proposed CSG operators which can preserve G 1 continuous scalar fields [5]. Figure 2.4: Two implicit spheres to which CSG operators are applied and the result scalar fields. Union operator (a), Intersection operator (b), and Difference operator (c). 2.1.3 Blending Blending is another modeling advantage of implicit surfaces in addition to CSG operators. This operator can create smooth transitions between two objects. It is quite difficult to apply blending to other shape representations such as point based representations and parametric representations while blending can be trivially computed in implicit surface representations. A blending operator fA+B(p) between two scalar 11
fields fA(p) and fB(p) can be simply defined as follows: fA+B(p) = fA(p) + fB(p) (2.5) Nevertheless, this operator does not provide any manipulation of blending. To achieve more flexible blending, Ricci introduced another blending operator fA♦B(p) by pro- viding a parameter n [42]. This blending is referred to as Ricci blend while Equation 2.5 is called summation blend. Ricci blend is defined as follows: 12 fA♦B(p) = (fA(p) n + fB(p) n ) 1 n (2.6) By changing n, the amount of blending can be manipulated (Figure 2.5). As one property, Ricci blend can represent another operator by substituting a particular number for n. For example, fA♦B(p) = fA∪B(p) where n = ∞, fA♦B(p) = fA∩B(p) where n = −∞, and fA♦B(p) = fA+B(p) where n = 1. Figure 2.5: Two implicit spheres to which Ricci blend is applied and the result scalar fields. Blending parameter n = 1 (a), n = 2 (b), and n = 4 (c). 2.2 Rendering Implicit surfaces are rendered by evaluating field values of a scalar field. A set of points which store the iso-value is visualized as the iso-surface. Rendering implicit surfaces are not straightforward compared to polygon meshes because they are not defined
Page 1 and 2: Free-Form Deformation for Implicit
Page 3 and 4: Supervisory Committee Dr. Brian Wyv
Page 5 and 6: 3.1.1 Lattice-Based FFD . . . . . .
Page 7 and 8: List of Figures Figure 1.1 Deformat
Page 9 and 10: Figure 4.2 The rigidity of the surf
Page 11 and 12: Figure 5.11An example deformation p
Page 13 and 14: ACKNOWLEDGEMENTS First I would like
Page 15 and 16: of lattice manipulation is that man
Page 17 and 18: Figure 1.1: Deformation for implici
Page 19 and 20: Chapter 2 Introduction to Implicit
Page 21 and 22: Figure 2.1: Skeletons and skeletal
Page 23: Figure 2.3: Variational interpolati
Page 27 and 28: Figure 2.6: Voxelization. A set of
Page 29 and 30: model, even novice users can create
Page 31 and 32: Chapter 3 Background and Previous W
Page 33 and 34: Figure 3.1: Lattice-based FFD. The
Page 35 and 36: Figure 3.2: An implicit model befor
Page 37 and 38: into several systems. Chameleon [22
Page 39 and 40: as follows: • ShapeShop enables t
Page 41 and 42: Chapter 4 Deformation Interface Thi
Page 43 and 44: 4.2 Interface Design This deformati
Page 45 and 46: Figure 4.4: Grid and curve visualiz
Page 47 and 48: global deformation according to the
Page 49 and 50: Chapter 5 Deformation Algorithm Thi
Page 51 and 52: 5.2 Voxelization The user draws a c
Page 53 and 54: Figure 5.4: An example of distance
Page 55 and 56: Figure 5.7: When the handle vertex
Page 57 and 58: y the field value and then summed.
Page 59 and 60: 5.4 Scalar Field Re-construction Wh
Page 61 and 62: 5.4.2 Variational Warping Variation
Page 63 and 64: outside the deformation field is ev
Page 65 and 66: Chapter 6 Results and Conclusion Th
Page 67 and 68: Figure 6.1: The dragon was deformed
Page 69 and 70: Figure 6.3: The horse model was tra
Page 71 and 72: 6.2 Limitations There are a few lim
Page 73 and 74: twisted scalar field cannot be re-c
Page 75 and 76:
Bibliography [1] Adzhiev, V., Cartw
Page 77 and 78:
[19] Gascuel, M.-P. An implicit for
Page 79 and 80:
[40] Pasko, A., Adzhiev, V., Sourin
Page 81:
[60] Wyvill, B., Guy, A., and Galin
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