Theory of Compactly Supported Quasi-tight Framelets

题目：Theory of Compactly Supported Quasi-tight Framelets
时间：11月27日（周二）下午16:00
地点：工商楼4楼报告厅
报告人：Bin Han教授，加拿大阿尔伯塔大学

摘要：Constructing compactly supported multivariate tight or dual framelets, even without directionality or basic vanishing moments, is known to be a very challenging problem because it is linked to sum of squares and factorization of multivariate Laurent polynomials in algebraic geometry. To overcome this difficulty, we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. From an arbitrary compactly supported 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. If in addition all the coefficients of its low-pass filter are nonnegative, such a quasi-tight framelet becomes a directional tight framelet. Moreover, from an arbitrary refinable function, we can constructively derive a compactly supported quasi-tight framelet with the highest possible order of vanishing moments. Examples will be provided to illustrate our results. This talk is based on following joint work with Chenzhe Diao and Xiaosheng Zhuang:

/begin{enumerate}
/item[1.] B. Han, Framelets and Wavelets: Algorithms, Analysis, and Applications, in /emph{Applied and Numerical Harmonic Analysis}, Birkhauser/Springer, (2017), 724 pages.

/item[2.] C. Diao and B. Han,
Quasi-tight Framelets with High Vanishing Moments Derived from Arbitrary Refinable Functions, preprint, (2018).

/item[3.] C. Diao and B. Han, Generalized matrix spectral factorization and quasi-tight framelets with minimum number
of generators, preprint, (2018).

/item[4.]
B. Han, T. Li, and X. Zhuang, Directional compactly supported box spline tight framelets with simple structure,
preprint, (2017).
/end{enumerate}


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