Four-Point Wavelets and Their Applications

Multiresolution analysis (MRA) and wavelets provide useful and efficient tools for representing functions at multiple levels of details. Wavelet representations have been used in a broad range of applications, including image compression, physical simulation and numerical analysis. In this paper, the authors construct a new class of wavelets, calledfour-point wavelets, based on an interpolatory four-point subdivision scheme. They are of local support, symmetric and stable. The analysis and synthesis algorithms have linear time complexity. Depending on different weight parametersw, the scaling functions and wavelets generated by the four-point subdivision scheme are of different degrees of smoothness. Therefore the user can select better wavelets relevant to the practice among the classes of wavelets. The authors apply the four-point wavelets in signal compression. The results show that the four-point wavelets behave much better than B-spline wavelets in many situations. This work is supported by NKBRSF on Mathematical Mechanics (Grant No.G1998030600), the National Natural Science Foundation of China (Grant No. 19971087) and a grant for Distinguished Young Teachers by the State Education Commission of China. WEI Guofu is currently a Ph.D. candidate in the Mathematical Department of the University of Science and Technology of China. He received his B.S. degree from the University of Science and Technology of China in 1997. His research interests include computer aided geometric design and computer graphics. CHEN Falai is currently a professor in the Mathematical Department of the University of Science and Technology of China. He received his B.S. (1987), M.S. (1989) and Ph.D. degrees (1994) from the University of Science and Technology of China. He visited Brigham Young University, USA, from April 1994 to August 1995 and from July 1999 to April 2000, respectively. His research interests include computer aided geometric design and computer graphics.