Multiresolution modeling techniques in CAD
Introduction
Current CAD systems provide several methods for object modeling. For example, non-uniform rational B-splines (NURBS) are used in surface models, solid models are represented with constructive solid geometry (CSG) and boundary representation (B-rep) models represent the bounding surfaces and topology. Several editing operations like translation, rotation, scale, reflection, shear and control point manipulation are employed to generate complicated objects.
However, with only these basic operations, it is difficult to accomplish more complex editing and deformation. Several researchers have worked on improving the object modeling and editing process. Sederberg and Parry [1] presented a volume deformation method called free-form deformation (FFD) for global editing. The geometric model is embedded in a parallelepiped lattice of control points. The deformations of the FFD lattice are then passed onto the model. Coquillart [2] later extended this method to include non-parallelepiped lattices for more general deformations. Interfaces that allow the user to directly manipulate the curve or surface have also been developed [3], [4]. Direct manipulation may include moving a single point to a target point, specifying tangent values and normal vectors. Based on these specifications, constraints are developed and a least-squares type of solution for the new control points is obtained. The finite element method has also been applied to generate primitives that build continuous deformable shapes [5]. The primitives autonomously deform to minimize an energy functional, subject to user controlled geometric constraints and loads. Variational surface modeling is another technique that allows direct manipulation for surface editing by defining a set of constraints, such as a surface passing through a curve [6]. The new surface is a solution that extremizes integrals subject to constraints.
Finkelstein and Salesin [7] suggested multiresolution curve representation based on B-spline wavelets to support a variety of editing operations. With multiresolution editing, the curve may be smoothed and the overall form of the curve may be changed while preserving its details (sweep editing). The curve may be edited at any continuous level of detail (fractional editing). Additionally, the curve's character may be changed without affecting its overall sweep. The multiresolution framework is valuable because the higher resolution details and the lower resolution (smoothed) curves are simultaneously available. Hierarchical B-splines [8] can also edit the overall form of a surface while maintaining its details. This formulation either requires the user to design an explicit hierarchy into the system or recursively fit hierarchical surfaces, which is similar to the filter bank process in multiresolution analysis (MRA) as described in Section 2.3. However, there are an infinite number of possible representations for the same surface, whereas with MRA there is one unique solution.
Multiresolution editing can be useful in CAD systems. An object may be edited at different scales for different effects. Detail features may be shown or hidden at will to ease the editing process. In this paper, multiresolution editing is extended from earlier work on multiresolution curves [7] and surfaces [9], [10], [11]. Two new methods called synthesis editing and detail blending are added to previous sweep editing and fractional editing capabilities. Synthesis editing is a way to increase the editing resolution of the surface so that detail features may be conveniently added. Detail blending makes the task of adding new features to existing features easier. Existing features are filtered out, new details are added to the smoothed surface and then the previously filtered features are added back.
The end-point interpolating B-spline wavelet transformation developed in [9], [10], [11] is confined to a single tensor-product B-spline patch. In this paper, we investigate techniques for applying wavelet transformation to multiple patches so that they can be more useful in CAD systems. For the simple case of two patches, least-squares data fitting is employed to merge the two separate patches into a single patch. For more complex patches, each patch is assigned a different priority and a G0 continuity constraint is enforced according to the priority of adjacent patches at each phase of multiresolution transformation. The user is thereby presented with a G0 continuous model at each phase and all the previously mentioned wavelets deformation methods can be applied. After all editing operations are done, a global G1 optimization based on non-linear least squares of the error vectors along the seams of adjoining surfaces can be applied to obtain a final smooth model [12].
To demonstrate the editing and modeling capabilities of wavelet based methods, B-spline patch models are obtained from reverse engineered data and through a layer based modeling interface. While wavelet transformation provides the means for multiresolution editing, three-dimensional (3D) editing and modeling remain a problem of current CAD systems, since the screen is only two-dimensional (2D). Many methods have been proposed to implement editing at a 3D level by providing shape and depth cues. These methods include shadow widgets [13], stereo imaging [14], semi-transparent volume cursor [15], and a semi-transparent moving cursor plane [16]. The moving cursor plane is a semi-transparent plane that intersects the object. The part of the object in front of and behind the plane are each shaded differently. The intersection contour provides shape cues, and position cues are provided by moving the cursor plane perpendicular to the screen.
The variety of wavelet editing techniques together with layer based modeling and reverse engineered patches, and 3D interactive editing are useful complements to current CAD systems as more powerful abilities of modeling and deformation are provided. In this paper an interactive editing and modeling system is developed which includes:
(I) Patch acquisition by
- 1.1.
Reverse engineering
- 1.2.
2D layer based modeling
- 1.1.
(II) 3D interactive editing using moving cursor plane
The rest of the paper is organized as follows. MRA and endpoint interpolating B-splines wavelets used for representing curves and surfaces are first introduced in Section 2. In Section 3, wavelet modeling techniques including sweep editing, fractional editing are discussed and new techniques of synthesis editing and detail blending are developed. The deformation of multiple patches is addressed next in Section 4. A simple implementation to demonstrate the wavelet deformation capability is presented with design examples in Section 5. Conclusions and future research directions are in Section 6.
Section snippets
Wavelets and multiresolution analysis
Wavelets were first developed for the purpose of approximation by Daubechies [17]. Mallat [18] proposed a framework for wavelet-based MRA and expanded its application to signal processing. Recently a lot of wavelet-based research has been done in the field of computer graphics [19], [20]. This section provides a brief background on the theory of MRA and endpoint interpolating B-spline wavelets. Readers could refer to Ref. [21] for a general introduction to wavelets and to [17], [18], [19], [20]
Multiresolution editing techniques
Wavelet theory provides techniques for 2D/3D editing of CAD models. Sweep editing and fractional editing of B-spline curves and surface patches have been introduced by Finkelstein and Salesin [7] and Stollnitz, Derose and Salesin [9], [10], [11]. In this section, we extend wavelet deformation techniques by providing synthesis editing and detail blending. We also extend the application of wavelet based editing from a single patch to multiple patches. Each of these editing techniques is explained
Multiple patch transformation
The existing end-point interpolating B-spline wavelet editing techniques are limited in its application to a single tensor-product B-spline patch. It needs to be extended to more complex models that are composed of multiple B-spline patches. In this section new techniques are investigated on the application of wavelets transformation in complex topology with multiple patches.
CAD interface
A simple CAD interface is implemented to show B-spline patch acquisition and wavelet deformation in a 3D editing environment. The B-spline patch is acquired either by reverse engineering or through a layer based modeling interface. The model is limited to a single patch in this implementation. A moving cursor plane is implemented for 3D editing, viewing and positioning.
Conclusion and future work
In this paper we have investigated methods for object deformation based on wavelet MRA. Synthesis editing and detail blending methods have been introduced to extend previous work on sweep editing and fractional editing. Sweep editing is used for the sweep shape modification of a model. Fractional editing provides a way for continuous editing between two multiresolution scales. Synthesis editing provides a method to increase the editing resolution and add detail features. Detail blending makes
Acknowledgements
Support from NSF DMII Grant NO. 9970083 is gratefully acknowledged.
Ranga Narayanaswami is an Assistant Professor in the Department of Industrial and Manufacturing Systems Engineering at Iowa State University. His research interests are in computer aided design, geometric modeling, computer graphics and applications in manufacturing. He holds a Ph.D. in mechanical engineering from the University of California at Berkeley and a BS from the Indian Institute of Technology at Madras.
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Ranga Narayanaswami is an Assistant Professor in the Department of Industrial and Manufacturing Systems Engineering at Iowa State University. His research interests are in computer aided design, geometric modeling, computer graphics and applications in manufacturing. He holds a Ph.D. in mechanical engineering from the University of California at Berkeley and a BS from the Indian Institute of Technology at Madras.
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