Spline-Wavelet Frames in Computer Graphics and other Applications

University Of Missouri-Saint Louis, Saint Louis MO

Investigators

Abstract

Proposal 0098331 Charles K Chui, Wenjie He, and Joachim Stoeckler U of Missouri, Saint Louis Abstract: Tight frames with scaling factor 2, generated by the standard affine operations of dilation and translation of two compactly supported cardinal splines, called frame generators, can be easily constructed for any spline order m (or degree m-1), by applying matrix extension techniques. However, regardless of the number of (spline) frame generators being used, at least one of them has only one vanishing moment, when the matrix extension approach is followed. In our recent work, we introduced the notion of "vanishing-moment recovery" Laurent polynomial factors S(z) is introduced to show that the maximum number m of vanishing moments can be achieved by both compactly supported tight frame generators, for any order m. Furthermore, the Laurent polynomials S(z) can be formulated explicitly when tight frames are relaxed to be sibling frames; that is, both frame generators, together with their corresponding duals, are compactly supported cardinal splines of the same order m. These additional vanishing moments are essential for effective use of the wavelet coefficients for feature extraction, noise removal, etc. Cardinal splines are spline functions with an equally spaced knot sequence extending from. However, in most practical applications, the intervals of interest are bounded and data samples may not be uniformly distributed. Hence, mth order splines with arbitrary knots, or at least with m stacked knots at one or both end-points of the interval of interest, are needed. This new research project is concerned with formulation of the matrix equivalent Sk of the Laurent polynomials S(z), construction of Sk and the corresponding tight (and more generally sibling) frame generators of mth order compactly supported splines with arbitrary knots and with m vanishing moments, achievement of such important features as inter-orthogonality for sibling frames, development and integration of the associated frame algorithms with the existing spline tools, investigation of spline-wavelet frame tools for adding sparsification and editing fearures for applications in computer graphics, and development of a portable software library.