Directional (Quasi)-tight Wavelet Framelets for Image Processing

Abstract：

Directional representation systems can effectively capture edge singularities for many high-dimensional problems such as image processing. In this talk, we first discuss directional complex tight framelets and their applications to image/video processing. However, constructing compactly supported multivariate tight framelets is known to be a challenging problem because it is linked to sum of squares and factorization of multivariate Laurent polynomials in algebraic geometry. To circumvent this difficulty, next we introduce the notion of quasi-tight framelets, which behaves almost identical to a tight framelet. From an arbitrary compactly supported multivariate refinable function (such as refinable box splines) with a general dilation matrix, we constructively prove that we can always derive a directional compactly supported quasi-tight framelet with vanishing moments. Moreover, any 1D wavelets or framelets can be adapted into bounded intervals. Consequently, their tensor products can avoid the boundary effects and can be applied to many problems such as manifold data processing and spherical data processing.

Speaker：

Bin Han is a professor in the Department of Mathematical and Statistical Sciences at the University of Alberta. He got his bachelor's degree from Fudan University in 1991, a master's degree from the Institute of Mathematics, Academia Sinica in 1994, and his Ph.D. from the University of Alberta in 1998. His research interests include computational mathematics, applied and computational harmonic analysis, and signal and image processing. He has published over 100 papers in various top journals in applied mathematics and serves on the editorial boards of several journals, such as Applied and Computational Harmonic Analysis and the Journal of Approximation Theory.

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https://ymsc.tsinghua.edu.cn/info/1053/3773.htm