Tight Frames of Rational Splines and Application to CAD/CAM and Computer Graphics

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Cardinal splines (i.e. splines with equally-spaced knots) provide a canonical example for demonstrating the mathematical structure of multiresolution analysis (MRA), which is a very powerful tool for the construction of orthogonal, semi-orthogonal, and bi-orthogonal wavelets. However, for practical applications, splines with arbitrary knots are often needed; and although the multi-level (ML) structure of spline subspaces is still valid, the Fourier approach to the study of cardinal splines and their corresponding wavelets no longer applies to this more general study. In addition, even for the cardinal setting, orthogonal and semi-orthogonal splines must be replaced by tight affine (or wavelet) frames when compact support (or finite time-duration) is needed for both spline-wavelet analysis and spline-wavelet synthesis. Furthermore, for CAD/CAM applications, there is also the need of rational B-splines (or NURBS) for precise representation of various geometric objects such as conic sections.This proposal is based on our recent results on tight frames of cardinal splines and on our newly developed complete ML structure of NURBS in terms of their weights. We propose to introduce non-stationary time-domain techniques to build a unified mathematical foundation of tight affine frames of compactly supported splines with arbitrary knots, and more generally, of NURBS. This will be followed by development of efficient algorithms and computational schemes associated with these new basic functions, which will be used as analysis and synthesis tools for CAD/CAM and computer graphics applications. We also propose to develop software libraries of these analysis and design tools that will be fully compatible with the industrial standards such as IGES, STEP, and PHIGS, for the CAD/CAM/CAE industries.