D. DeCarlo and J. Gallier, "Topological Evolution of Surfaces," Proc. Graphics Interface, pp. 194-203, May 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1B would satisfy the stated approximation constraint for ffl 1 . Yet, the curves c 0 and c 1 are quite different topologically, and that information is important in models for engineering [16, 23] The self intersection of curves and surfaces has been considered by several contemporary authors [2, 7, 12, 13, 14, 17], where a representative theorem for a single segment B ezier curve is given by the following. Let C be a B ezier curve [2, 8] defined on the interval [0,1] let s; t 2 [0; 1] such that 0 t 1 Gamma s t and form the difference quotient S(s; t) n C(1 Gamma s) Gamma C(t) 1 Gamma s) ....

DeCarlo, D. and Gallier, J., Topological evolution of surfaces, Graphics Interface '96, 194-203, Toronto, Ontario, 22-24 May, 1996.

....and isosurface extraction. Malgouyres et al. 7] also recently published an excellent paper on topology preservation within digital surfaces. A detailed survey on digital topology in CVGIP # canbeseenbyKonget al... 8] and also the related research work by Bertalmio et al. 9] and DeCarlo et al. [10]. The diversity of applications of the level sets has reached into several elds. Some of them are: 1) geometry: see Angenent et al. 11] Chopp et al. 12] 13] and Sethian et al. 14] 2) grid generation: see Sethian et al. 15] 3) uid mechanics: see Mulder et al. 16] Sethian et ....

.... et al. 38] Sochen et al. 39] 12) color image segmentation: see Sapiro et al. 40] 13) 3 D reconstruction and modeling: see Caselles et al. 41] and [42] 14) surfaces and level sets: see Chopp et al. 43] and Kimmel et al. 44] 15) topological evaluations: see DeCarlo et al. [10]) and (16) 2 D and 3 D medical image segmentation: see the works by Malladi et al. 45] 46] 47] 48] 49] 50] Yezzi et al. 51] GM WM # boundary estimation byGomeset al... 52] GM WM boundary estimation with Fuzzy models by Suri et al. 53] GM WM thickness estimation byZenget al... ....

DeCarlo, D. and Gallier, J., Topological Evolution of Surfaces, Graphics Interface, pp.

....contains the topology of both models and can easily be deformed into the shape of the source models. Any interpolation scheme yields a transformation. To date the problem of mapping an arbitrary genus 0 polyhedron to sphere remains challenging. Carmel and Cohen Or use curve evolution for this task [3], but admit some implementation problems in the three dimensional case. Feature alignment in transformation of polyhedra has been considered by Lazarus and Verroust [14] who mainly align major axes. Carmel and Cohen Or use a two part transform, an affine transformation and a deformation inspired ....

....in the three dimensional case. Feature alignment in transformation of polyhedra has been considered by Lazarus and Verroust [14] who mainly align major axes. Carmel and Cohen Or use a two part transform, an affine transformation and a deformation inspired from image warping, to align features [3]. DeCarlo and Gallier need user defined correspondences to transform between shapes of different topology [3] 1.2. Overview The general idea of our approach follows the one of Kent et al. [13] We try to find a polyhedral shape that results from merging the two input shapes. This shape can ....

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Douglas DeCarlo and Jean Gallier. Topological Evolution of Surfaces. Proc. of Graphics Interface `96, 194-203, 1996

....surfacebased methods to morphing multiresolution meshes. Rossignac and Kaul [22, 34] introduce the notion of an interpolating polyhedron, an application of mathematical morphology to face sets. They also describe a user control methodology based on a Bezier curve paradigm. DeCarlo and Gallier [12] demonstrate that it is possible (with significant effort) to morph polygonal models of differing genus. Surface based methods are important because of the wealth of polygonal models available to animators, but they are burdened with the difficult task of creating a single topological structure ....

D. DeCarlo and J. Gallier. Topological evolution of surfaces. In Proceedings of Graphics Interface '96, pages 194--203, May 1996.

....resolution) meshes are preserved in the lowest resolution mesh. Consequently, feature correspondences can be established in the simplest, lowest resolution mesh. As mentioned, this class of algorithm is not suited for interpolating shapes having di erent topology.Apaperby DeCarlo and Gallier [8] is the only example we know of in this class that explicitly dealt with interpolation of shapes having di erent topologies. 2. Volume based approach: This approach employs level sets of distance functions computed from the source shapes for morphing. These distance functions are interpolated to ....

DeCarlo D, Gallier J, Topological evolution of surfaces, in: Graphics Interface '96, 1996. p. 194{ 203.

....[1, Sec. 1.1] This definition is also implicit in the problems of graph isomorphism [2, p. 10] and congruence of point sets [3] Similarly, the concept of equivalent topological form is used in many application fields of computational geometry. In particular, it is used in computer graphics [4, 5], robotics [6, p. 95] image processing [7, 8, 9] computer aided geometry design (CAGD) 10] and solid modeling [11, p. 110] It is the last two of the mentioned application fields that provide the primary motivation for this paper. Other fields that use the concept of equivalent topological ....

DeCarlo, D. and Gallier, J. Topological evolution of surfaces. Graphics Interface '96, 194-203, Toronto, Ontario, 22-24 May, 1996. 20

....solids. and the property of non3 self intersection forms part of the STEP standard [9] Also, the problem of identifying the range of perturbations, for which the topological form of a geometric object remains invariant, is of crucial importance in morphing for animation [10]. Furthermore, avoiding self intersection is implicit in certain work [11] on the topologically reliable approximation of composite B ezier curves. Finally, avoidance of cusps and singularities is often important [12, 13, 14] the conditions we give for preventing intersections preclude cusps and ....

DeCarlo, D. and Gallier, J. Topological evolution of surfaces. Graphics Interface '96, 194-203, Toronto, Ontario, 22-24 May, 1996.

....the surfaces 1a and 1b, 2a and 2b, and 3 forming the three planes. deal with polyhedral objects are in general very restrictive about the type of topologies supported. In recent years, Gregory et al. 5] proposed a method to deal with objects of equivalent topologies and DeCarlo and Gallier [4] proposed a method to deal with objects of different topologies. These two methods suffered from heavy user interaction, difficult correspondence input, and complex algorithms. The method we propose for polyhedral object morphing was motivated by two primary factors. The first and foremost ....

D. DeCarlo and J. Gallier. Topological evolution of surfaces. In Graphics Interface'96), pages 194--203, 1996.

....representation on a finite grid. Since the grid is three dimensional, memory and computation costs can be prohibitive, limiting the visual fidelity of the results. The alternative is to work directly on boundary representations such as polygonal meshes or patch complexes. Methods for this approach [20, 27, 29, 22, 30, 33, 8, 9, 6, 17, 14] have to first solve the vertex correspondence problem, i.e. computing the association of vertices or triangles between the source mesh and the target mesh. Lazarus and Verroust identify this as the key problem and it forms the focus of our paper. Many approaches to the correspondence problem ....

DECARLO, D., AND GALLIER, J. Topological Evolution of Surfaces. In Graphics Interface '96, 194--203, May 1996.

....between two homeomorphic polyhedra. Initially the user selects some corresponding elements called feature pairs. Although we borrow this term from previous morphing algorithms for images or 3D volumetric models [2, 24] our concept of a feature is closer to the sparse control mesh used in [10]. Our algorithm includes a simple and intuitive user interface for feature specification and automatically generates a feature net. Based on the feature nets, the algorithm decomposes the boundary of the polyhedra into morphing patches, computes a mapping for each morphing patch to a 2D polygon, ....

....Akkouche [12] have presented an algorithm for blob metamorphosis based on Minkowski sums. Lazarus and Verroust [22] have proposed a method based on skeletal curves. Kanai et al. 18] have presented an algorithm for shape transformation of genus 0 polyhedra using harmonic maps. DeCarlo and Gallier [10] have proposed a morphing technique that establishes correspondence by allowing the user to divide the surface into triangular and quadrilateral patches which can be projected onto a plane. Our overall approach shares their theme. However, we improve upon several restrictions in their technique ....

D. DeCarlo and J. Gallier. Topological Evolution of Surfaces. Prof. of Graphics Interface'96, pp. 229-235, 1996.

....[1, Sec. 1.1] This definition is also implicit in the problems of graph isomorphism [2, p. 10] and congruence of point sets [3] Similarly, the concept of equivalent topological form is used in many application fields of computational geometry. In particular, it is used in computer graphics [4, 5], robotics [6, p. 95] image processing [7, 8, 9] computer aided geometry design (CAGD) 10] and solid modeling [11, p. 110] It is the last two of the mentioned application fields that provide the primary motivation for this paper. Other fields that use the concept of equivalent topological ....

DeCarlo, D. and Gallier, J. Topological evolution of surfaces. Graphics Interface '96, 194-203, Toronto, Ontario, 22-24 May, 1996.

....instantiations: one for each source and target object. This single mesh can be obtained by merging the two object meshes as in [BU89, KPC91, KPC92] or by creating a new common mesh as in [LV97] The correspondence problem remains a difficult step and some authors [HMTT88, BU89, CP89, Par92, DG96, KHSK98, ST98] focus on the correspondence problem independently of the interpolation. According to the above remark the existence of a common mesh for the source and target object implies that they have the same (global) topology. This forbid the transformation of a sphere into a torus. To get ....

....problem independently of the interpolation. According to the above remark the existence of a common mesh for the source and target object implies that they have the same (global) topology. This forbid the transformation of a sphere into a torus. To get round this constraint DeCarlo and Gallier [DG96] propose to use degenerate geometric instantiations of the common mesh where an edge or a face can be embedded onto a single point or edge. Once the correspondence is established, most of the papers use a linear or an Hermite spline interpolation between the corresponding vertices to compute the ....

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D. DeCarlo and J. Gallier. Topological evolution of surfaces. In Graphics Interface '96, pages 194--203, 1996.

....specifications, mathematical techniques should be used to compute the transformation for any point in the warping domain. Shape blending algorithms usually refer to techniques operating on a combinatorial representation. Most of the work done here has dealt with polygonal or polyhedral objects [18, 29, 31, 33, 24, 19, 9, 10, 6, 16]. The general method of these algorithms is to displace the vertices, edges and faces of S over time to coincide in position with the vertices, edges and faces of T . This usually requires solving the vertex correspondence problem, which determines which vertex travels to which vertex during the ....

....approach is that no correspondence between the models geometric primitives (e.g. vertices) needs to be established. Moreover, the input objects do not necessarily needs to have the same representation or the same topological genus, and can have a different number of connectivity components (see [9]) Although there is no explicit definition of what the in between objects (a) b) c) d) Figure 11: Comparative results of morphing algorithms. a) Sederberg et al. b) Shapira and Rappoport. c) Goldstein and Gotsman. d) DF Morphing. The anchor points are marked on the two key figures. ....

D. DeCarlo and J. Gallier. Topological evolution of surfaces. In Graphics Interface'96, pages 194--203, 1996.

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D. DeCarlo and J. Gallier, "Topological Evolution of Surfaces," Proc. Graphics Interface, pp. 194-203, May 1996.

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Douglas DeCarlo and Jean Gallier. Topological evolution of surfaces. Graphics Interface '96, pages 194--203, May 1996.

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D. DeCarlo and J. Gallier. Topological evolution of surfaces. In Graphics Interface'96, pages 194--203, Toronto, Ontario, Canada, 1996. Canadian Information Processing Society.

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D. DeCarlo and J. Gallier, "Topological evolution of surfaces", Graphics Interface '96, 194-203, Toronto, Ontario, 22-24 May, 1996.

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D. DeCarlo and J. Gallier, Topological Evolution of Surfaces. In Proceedings of Graphics Interface'96. pages 194-203, 1996.

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