Loop, C., and DeRose, T. Generalized B-spline surfaces of arbitrary topology. Proceedings of SIGGRAPH'90 (Dallas, August 6--10, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....patches to a polygonal model has been extensively studied in computer aided geometric design. A survey of different techniques has been given in [Die93] These include algorithms that fit smooth spline surface over irregular meshes [Sar90, Pet95, Loo94, EH96] that use n sided patches for fitting [LD90] and spline approximation algorithms [dBF73, dB74] More recently, many algorithms have used subdivision surfaces for piecewise smooth reconstruction [HDD 94] 2.3 Tessellating Spline Surfaces A number of algorithms have been proposed in the literature to tessellate spline models and ....

Charles Loop and Tony DeRose. Generalized B-spline surfaces of arbitrary topology. In Forest Baskett, editor, Computer Graphics (SIGGRAPH '90 Proceedings) , volume 24, pages 347--356, August 1990.

....is inherently smooth, a parametric surface is more suitable to represent it than a triangular mesh. Various parametric surface methods have been developed for representing shapes created by meshes. They include: subdivision methods [4, 9] triangular patches [13, 20, 41] generalized Bsplines [32], triangular NURBS [43] and wavelets [35] Among these methods, wavelets 13 provide an efficient means for reconstructing a shape at multiresolution. Such representa tions are becoming increasingly important because of the need for transmission of geometric data over the Internet. In the ....

C. Loop and T. DeRose, Generalized B-spline surfaces of arbitrary topology, Computer Graphics, vol. 24, no. 4, 1990, 34356. 25

....i S inversely proportional to its distance to P. In the optimal case, if P exactly matches the seed point i S , the weight of its cost suggestion should be 1 while all other weights should be zero. This is exactly the case in the so called barycentric coordinate system proposed by Lood and DeRose [7, 8]. Calculating barycentric coordinates = m k i i p p l ) 1 p i p i m k k a a a = Consider Figure 5, where P denotes a point in the interior of the so called domain polygon and i S denotes the seed points or the vertices of the polygon. The barycentric coordinates ....

C. T. Loop, T. D. DeRose, Generalized B-spline Surfaces of Arbitrary Topology, Computer Graphics (SIGGRAPH), ACM Press, 1990, 24(4):347-356

....reproduction in any form reserved. Fig. 1. Motivating examples for our work. When the parameterization is addressed, it is often piecemeal, composed of a series of adjacent parametric patches. Finally, there are a number of classic works on completing a surface from a series of bounding curves [2,4,6,8]. This work is most closely related to the algorithm we will develop. However, in contrast to our approach, these techniques generally assume the boundary can be naturally decomposed into n faces, which can in turn be blended together. For example, schemes have been developed to complete a surface ....

Loop, Charles and Tony DeRose, Generalized B-spline surfaces of arbitrary topology, in Computer Graphics (Proceedings of SIGGRAPH 90) 24(4) 347--356.

....base points to create parameterizations of four , ve , and six sided surface patches using rational B ezier surfaces de ned over triangular domains. Setting a triangle of weights to zero at one corner of the domain triangle produces a four sided patch that is the image of the domain triangle. [13, 12] present generalizations of biquadratic and bicubic B spline surfaces that are capable representing surfaces of arbitrary topology by placing restrictions on the connectivity of the control mesh, relaxing C continuity to G (geometric) continuity, and allowing n sided nite elements. This ....

C. Loop and T. DeRose. Generalized B{spline Surfaces of Arbitrary Topology. Computer Graphics, 24(4):347-356, August 1990.

....particular representations are available. BMVC 1994: A J Stoddart, Slime. We believe that all existing methods fail on one or more of the above. The objective of this paper is to introduce a new deformable surface. We exploit a sophisticated new representation developed by Loop and De Rose [2, 3] for use in computer graphics. Our contribution is to add the dynamics that make it a deformable surface, and suggest a seeding algorithm. Deformable surfaces with fixed mesh topology are similar to elastic membranes or flexible plates. They may deform but cannot restructure their topology. The ....

....curves or surfaces are useful in their own right, but in vision problems B splines are often more convenient. A major advantage of the B spline is that by construction it joins together a number of Bezier curves or surfaces with C n continuity. BMVC 1994: A J Stoddart, Slime. Loop and De Rose [3] used the S patch to build a generalization of the biquadratic B spline surface that can describe arbitrary topology and is G 1 continuous everywhere. Due to space limitations it is not possible to include a full description of their surface, and the reader is referred to their papers. However ....

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C. T. Loop, T. D. DeRose, Generalized B-spline surfaces of arbitrary topology, ACM Computer Graphics, 24(4) 347-356 (1990).

....developed multiresolution representations for triangulated meshes of planar faces. 3 The Surface Representation The spline based surface that we use is a G 1 continuous arbitrary topology called a generalised biquadratic B Spline (GBBS) first developed by Loop and DeRose TO APPEAR IN ICPR 98 [5]. We have formulated a deformable surface based on this representation [7] and developed algorithms for creating valid mesh topologies and optimally positioning the control points [8] The GBBS is defined by a set of M 3D control points Q = #c m : m =1. M together with connectivity ....

C. T. Loop and T. D. DeRose. Generalized b-spline surfaces of arbitrary topology. ACM Computer Graphics, 24(4):347--356, 1990.

....spline patches to a polygonal model has been extensively studied in computer aided geometric design. A survey of different techniques has been given in [Die93] These include algorithms smooth spline surface over irregular meshes [Sar90, Pet95, Loo94, EH96] using n sided patches for fitting [LD90] and using spline approximation [dBF73, dB74] More recently, many algorithms have used subdivision surfaces for piecewise smooth reconstruction [HDD 94] 3 Overview of our approach Our approach for simplifying geometric models makes use of surface fitting algorithms. It can handle ....

Charles Loop and Tony DeRose. Generalized B-spline surfaces of arbitrary topology. Computer Graphics (SIGGRAPH 90 Proceedings),vol- ume 24, pages 347--356, August 1990.

....one of figure 1; as well as degree 1 vertices, loops, bridging edges, and multiple edges with same endpoints. An obvious way to realize a complex geometrically is to model each face as a polynomial surface patch, implicit or parametric, with suitable continuity constraints between adjacent patches [19,1]. However, in a general complex a face may have any number of sides (including 1 or 2) and may be glued in arbitrarily complex ways. Thus, in general, it is not possible to realize each face as a single polynomial surface patch of bounded degree. Therefore, instead of using the faces as ....

....(dashed) and the tile boundaries (dotted) 2.1 Modeling a tile Since every tile has only four sides, and therefore only four neighboring tiles, it is possible to realize it as a geometric object of bounded complexity. A obvious candidate for this role would be a parametric polynomial patch [19,21]. This approach would allow us in principle to obtain a truly smooth surface, with continuity of tangent plane (and possibly curvature) between adjacent tiles. However, it is not at all trivial to enforce those constraints for complexes arbitrary topology. Also, some of the energy functions we use ....

Charles Loop and Tony DeRose. Generalized B-spline surface of arbitrary topology. In Proc. SIGGRAPH'90, pages 347--356, August 1990.

....during shape optimization, and also when rendering the optimized surface. 2. 2 Tiles An obvious way to model the geometry of the surface is to model each face of the complex as a piece of polynomial surface, implicit or parametric, with continuity constraints imposed on pairs of adjacent patches [25, 2]. However, the faces of the complex may have any number of sides, which may be glued among themselves in almost any fashion. In general, it is not possible to realize such a face as a single polynomial surface patch of bounded degree. Therefore, instead of viewing the surface as the union of ....

....of the other. Visualization of cellular complexes 7 2.3 Realizing a tile Since each tile has only four sides, and therefore only four neighboring tiles, we can in principle realize it as a geometric object of bounded complexity. One obvious alternative is to use a parametric polynomial patch [25, 27]: a polynomial mapping of the unit square [0 . 1] 2 of R 2 to R 3 . The main difficulty of this approach is to guarantee a smooth join (with tangent plane continuity, and possibly curvature continuity) between adjacent tiles. Moreover, we must be careful to avoid degeneracies points where ....

Charles Loop and Tony DeRose. Generalized B-spline surfaces of arbitrary topology. In SIGGRAPH '90 Conference Proceedings, pages 347--356, August 1990. In Computer Graphics 24 (4).

....stitching together tensor product patches. 6, 14] 3 The surface representation The spline based surface that we use is a G 1 continuous arbitrary topology surface called a generalised biquadratic B Spline (GBBS) It was first developed in the context of computer graphics by Loop and De Rose [15, 16]. It is important to note that it is not possible to maintain C 1 continuity (first order parametric derivative) over an arbitrary topology surface. Instead the concept of G 1 continuity (first order geometric) is introduced. In effect it means that the tangent plane at a point on the surface ....

C. T. Loop and T. D. DeRose. Generalized b-spline surfaces of arbitrary topology. ACM Computer Graphics, 24(4):347--356, 1990.

....2 Delta J. Peters Fig. 1. top) input polyhedron; middle) planar cut polyhedron; bottom) NURBS surface. 1. INTRODUCTION Polyhedra can be smoothed into free form surfaces using a variety of approaches such as rational blends, generalized subdivision or simplex splines (see e.g. [3], 1] 2] A major criticism leveled at these techniques is that they are incompatible, i.e. cannot be represented exactly or efficiently in the dominant patch representation, tensor product B splines. Tensor product B splines serve under the pseudonym NURBS as a standard for storage, ....

Loop, C., and DeRose, T. Generalized b-spline surfaces of arbitrary topology. Computer Graphics 24, 4 (1990), 347--356.

.... constructions for irregular meshes have been derived over the last decade [10] Compared to the standard tensor product B spline representation, these constructions either sacrifice the low degree of the surfaces (e.g. 20] 12] or depart from the standard polynomial representation (e.g. 5] [16]) However, 2 FAIR CURVES AND SURFACES the main drawback of the patching approach seems to arise from the very tool that allows it to model complex smooth surfaces; reparametrizing when crossing from one patch to the next shifts the focus from the geometry of the modeling problem to clever ....

C. Loop, T. DeRose, Generalized B-spline surfaces of arbitrary topology, Proceedings of Siggraph '90.

....parametric patches and by ensuring that proper geometric continuity is realized between patches. For instance Hoppe and Eck [EH96] use the construction scheme of Peters [J94] to build G 1 continuous surfaces of arbitrary topology while other reconstruction systems such as Loop and De Rose [LD90] rely on dioeerent patch corners such as Sabin nets. An important constraint in order to eOEciently represent general deformable models with spline patches is that the G 1 continuity equation across patches must be linear in the control points. Those continuity constraints across patches usually ....

C. Loop and T. DeRose. Generalized b-spline surfaces of arbitrary topology. In Computer Graphics (SIGGRAPH'90), pages 347356, 1990.

....and others [1] have developed multiresolution representations for triangulated meshes of planar faces. 3 The Surface Representation The spline based surface that we use is a G 1 continuous arbitrary topology called a generalised biquadratic B Spline (GBBS) first developed by Loop and DeRose [5]. We have formulated a deformable surface based on this representation [7] and developed algorithms for creating valid mesh topologies and optimally positioning the control points [8] The GBBS is defined by a set of M 3D control points Q = f c m : m = 1: Mg together with connectivity ....

C. T. Loop and T. D. DeRose. Generalized b-spline surfaces of arbitrary topology. ACM Computer Graphics, 24(4):347--356, 1990.

....redundant edges in the planarcut polyhedron. Supported by NSF National Young Investigator grant 9457806 CCR 1 Introduction Polyhedra can be smoothed into free form surfaces using a variety of approaches such as rational blends, generalized subdivision or simplex splines (see e.g. [3], 1] 2] A major criticism leveled at these techniques is that they are incompatible, i.e. cannot be represented exactly or efficiently in the dominant patch representation, tensor product B splines. Tensor product B splines serve under the pseudonym NURBS as a standard for storage, ....

Loop, C., and DeRose, T. Generalized b-spline surfaces of arbitrary topology. Computer Graphics 24, 4 (1990), 347--356.

.... and deformable surface models is that the rough shape of the object must be known or specified in advance [Terzopoulos et al. 1987a] For spline models, this means discretizing the surface into a collection of patches with appropriate continuity conditions, which is generally a difficult problem [Loop and DeRose, 1990]. For deformable surface models, we can bypass the patch formation stage by specifying the location and interconnectivity of the point masses in the finite element approximation. In either approach, defining the model topology in advance remains a tedious process. Furthermore, it severely limits ....

....configuration. We can create any desired topology with this technique. For example, we can form a flat sheet into an object with a stem and then a handle (Figure 13) Forming such surface with traditional spline patches is a difficult problem that requires careful attention to patch continuities [Loop and DeRose, 1990]. To make this example work, we add the concept of heating the surface near the tool [Tonnesen, 1991] and only allowing the hot parts of the surface to deform and stretch. Without this modification, the extruded part of the surface has a tendency to pinch off similar to how soap bubbles pinch ....

C. Loop and T. DeRose. Generalized B-spline surfaces of arbitrary topology. Computer Graphics (SIGGRAPH'90), 24(4):347--356, August 1990. 26 10 Conclusion

....of [32, Section 3.5] to the case where n 4 patches meet at a mesh point; the representation based on three sided patches can be viewed as a special case of [21] using a well chosen mesh of quadratic boundary curves as input. A patches [1] see also [15] 8] B patches ( 30] 7] and S patches [19] provide elegant new solutions to the smoothing problem at the cost of a currently non standard patch representation. Overview. The remainder of the paper consists of the definition and the proof of properties of the first order free form surface splines. Section 2 defines the splines in terms of ....

C. T. Loop, T. DeRose, Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24,4(1990): 347--356.

....G is the entire v = 1 line. The vertex (1; 0; 0) is therefore blown up . Since G is in triangular patch form, the control points and weights of H = F ffi G will form a rational triangular representation for the surface of revolution. Composition is also a useful tool when dealing with S patches[12, 13]. S patches are rational generalizations of B ezier surfaces that admit any number n of boundary curves. In [12] it is shown how to use composition to convert an n sided S patch into an m sided S patch, for arbitrary n and m. Another problem that can be solved using composition is finding the ....

Charles T. Loop and Tony D. DeRose. Generalized B-spline surfaces of arbitrary topology. In SIGGRAPH '90 Proceedings, pages 347--356, 1990.

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Loop, C., and DeRose, T. Generalized B-spline surfaces of arbitrary topology. Proceedings of SIGGRAPH'90 (Dallas, August 6--10, 1990.

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C. Loop and T. DeRose, Generalized b-spline surfaces of arbitrary topology, Computer Graphics (SIGGRAPH '90 Proceedings) ## (1990), no. 4, 347-356.

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C. Loop and T. DeRose, Generalized B-spline surfaces of arbitrary topology, Proc. SIGGRAPH '90 1990, 347-- 356.

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Loop, Ch. and De Rose, T. (1990), Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24, 347--356.

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Loop, Ch. and De Rose, T. (1990), Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24, 347--356.

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C.T. Loop and T.D. DeRose. Generalized B-spline surfaces of arbitrary topology. In Proc. SIGGRAPH'90, pages 347--356, 1990.

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C. Loop and T. DeRose : Generalized B-spline Surfaces of Arbitrary Topology, Computer Graphics, Vol.24, No.4, 1990, pp.347-356.

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C. Loop, T. DeRose (1990), Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24(4): 347--356.

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C. Loop, T. DeRose, Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24,4(1990): 347--356.

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C. Loop, T. DeRose, Generalized B-spline surfaces of arbitrary topology, Computer Graphics 24,4(1990): 347--356.

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