Fast variational design of multiresolution curves and surfaces with B-spline wavelets
Introduction
Multiresolution modeling provides a powerful tool for complex shape editing. Traditionally, free-form curves or surfaces can be sculptured by dragging a set of control points [1], [2], [3]. In the case of the multiresolution curve or surface, modifying control points becomes more complicated since users have to edit the control points at different resolution levels. To achieve a better control of deformations and a more intuitive interface, variational principles have been used in such multiresolution models [4], [5], [6], [7].
However, in the existing variational design methods [6], [7], [8], as the data sets of the shape grow and the constraints imposed at different resolution levels increase, the number of unknown coefficients in the optimization system becomes very large. This causes the computational time to increase drastically, and hence cannot meet the requirements of real-time interactive design. Furthermore, it is often the case that the input model has high frequency geometric details across multiple resolutions, which need to be preserved during global deformations of its shape. Traditional energy functions used in variational modeling tend to smooth out not only the global shape of the object, but the high frequency details as well [9].
In 1994, Finkelstein and Salesin [10] introduced a wavelet-based multiresolution representation of B-spline curves, and demonstrated that such representation supports a variety of display and editing operations in a simple manner. Although the operations enable users to modify the overall shape of the curve while preserving its details, it is still difficult to design the overall shape by explicitly manipulating its control points. In 1995, Gortler and Cohen [4] applied the variational modeling technique, which was proposed by Welch and Witkin [8], to the design of multiresolution curves and surfaces. In their method, the wavelet basis was used to accelerate the computation of optimal curves and surfaces based on a thin plate functional. However, as enough constraints were presented, his methods no longer offered an advantage over the traditional B-spline basis. In order to limit wavelets to an interval, he constructed bi-orthogonal B-spline wavelets to represent curves and surfaces, which caused the first derivative vectors to be zero at the boundaries. Furthermore, their method did not consider multiresolution constraints and cannot smooth out the global shapes without affecting the details. In 1997, Takahashi [6], [7] developed a wavelet-based framework for variational design of curves and surfaces that accommodates linear constraints at multiresolution levels. He aimed to modify the details effectively while preserving the overall shapes and reconcile different energy measures computed at different resolution levels. He also tested several combinations of energy functions. However, in his method, the control vectors at each resolution were solved simultaneously, which means that the problem size of the optimization system can be quite large, and hence rendered the method computationally expensive. In 2001, Elber [11] presented a scheme that integrated multiresolution control with linear constraints into one framework, allowing one to perform multiresolution manipulation of planar nonuniform B-spline curves, while satisfying various linear constraints on the curves. However, the nonuniform B-spline wavelet transformation needs more operations than the quasi-uniform one. Furthermore, he employed the QR factorization to solve the under-determined system, which is computationally expensive.
In this paper, a fast approach for interactive variational design of multiresolution models is presented. The endpoint-interpolating B-splines and their corresponding wavelets are used as the underlying representation of the curve or surface. Like that in Ref. [6], this method takes advantage of the multiresolution hierarchy to allow editing with constraints at different resolution levels. However, instead of preparing control vectors for each resolution level, we determine a target level according to the number of the constraints and find the control vectors at the target resolution level. By projecting all constraints imposed at different resolution levels to the target level, a constrained optimization model can be obtained. Since the target level is usually lower than the finest level, the unknown control vectors to be solved in the optimization system are significantly reduced. This makes the computation more efficient and allows users to design in real-time. For the energy functional at the target level, we minimize energy of the deformations of the shape instead of the deformed shape, which is motivated by the method used in Ref. [9]. This can preserve high frequency geometric details during global deformations of the original shape. Finally, synthesis filters are used to reconstruct the deformed shape.
The rest of the paper is organized as follows. Section 2 briefly introduces the endpoint-interpolating B-splines and their corresponding wavelets used to represent curves and surfaces. The variational techniques for designing curves and surfaces are introduced in Section 3. Section 4 describes the details of the fast variational method for multiresolution curve and surface design. Section 5 shows several results of the variational design of multiresolution curves and surfaces. Finally, Section 6 gives conclusions and future work.
Section snippets
Multiresolution representations of endpoint-interpolating B-spline curves
A kth degree endpoint-interpolating B-spline curve with 2L+k control points CiL is defined as [1]where {Bi,kL(u)} are the B-spline basis functions of level L defined on the knot sequences UL
The functions B0,kL(u),…, and B2L+k−1,kL(u) form the bases of the space of kth piecewise polynomials of level L. This space is referred to as VL, where L denotes the times in which the vector space, knot vector in this
Variational modeling
Variational modeling has been an area actively investigated in recent years and has become a powerful tool for free-form geometric design [8], [14]. In general, after being given the representations of curves and surfaces, there are three key steps in the standard variational modeling procedure which include [8]: (1) how to select the fairness measure of curves and surfaces to meet the global requirements on physical and geometry properties of the shape, (2) how to attach the geometric
Variational modeling of multiresolution shapes
Gortler [4] and Takahashi [6] proposed variational optimization techniques to multiresolution B-spline curves and surfaces. In Gortler's method, he used hierarchical basis functions as a preconditioner. Although Gortler's method solved system (20) efficiently, his method no longer offers an advantage over the B-spline method when enough constraints are presented. In order to limit wavelets to an interval, he used bi-orthogonal B-spline wavelets to represent curves and surfaces. This cause the
Experimental results
We have tested the proposed algorithm over several curves and surfaces in an experimental CAD system. Fig. 4 shows the variational design of a pipe-shaped curve. The multiresolution constraints include point constraints and tangent vector constraints at different resolution levels. The finest resolution level of the stem curve is the sixth level, whilst the target level is the fourth level. The details of the stem curve are preserved in the deformed curve.
Fig. 5 shows another multiresolution
Conclusions and future work
In variational modeling of curves and surfaces, one of the major problems is that the final optimization system is computationally expensive when the number of control points increases. This problem becomes more prominent for multiresolution models. In this paper, we have presented a fast variational approach for the interactive design of multiresolution curves and surfaces. By converting the constraints imposed at different resolution levels to the target level, which is lower than the finest
Gang Zhao is currently a Post-Doctoral Research Fellow at Institute of High Performance Computing, Singapore. He received his BS and a PhD in Aerospace Manufacturing Engineering from Beijing University of Aeronautics and Astronautics, China, in 1994 and 2001, respectively. His research interests include CAD/CAM, geometric modeling and virtual reality.
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Gang Zhao is currently a Post-Doctoral Research Fellow at Institute of High Performance Computing, Singapore. He received his BS and a PhD in Aerospace Manufacturing Engineering from Beijing University of Aeronautics and Astronautics, China, in 1994 and 2001, respectively. His research interests include CAD/CAM, geometric modeling and virtual reality.
Shuhong Xu is a Senior Research Engineer at Institute of High Performance Computing, Singapore. He received his PhD in mechanical engineering from Zhejiang University, China, in 1998. His research interests include geometric modeling, virtual reality and grid visualization.
Weishi Li is currently a Post-Doctoral Research Fellow at Institute of High Performance Computing, Singapore. He received his BS (1992) from Harbin Engineering University, MS (1995) from Hefei University of Technology, and PhD (2002) from Zhejiang University, China all in Mechanical Engineering. His research interests include digital geometry processing, reverse engineering and geometric modeling.
Eng Teo Ong is currently a research fellow in the Institute of High Performance Computing, Singapore. He received his BEng (Hons) and PhD in Mechanical Engineering from the National University of Singapore in 1998 and 2004, respectively. His research interests are in the areas of numerical methods.