## Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness

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- by Bin Han, Qingtang Jiang, Zuowei Shen and Xiaosheng Zhuang PDF
- Math. Comp.
**87**(2018), 347-379 Request permission

## Abstract:

In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets $\{\phi ; \psi _1,\psi _2,\psi _3\}$ in $L_2(\mathbb {R}^2)$ with*symmetry property*and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks $\{a;b_1,b_2,b_3\}$ having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: \[ \begin {aligned} b_1(k_1,k_2)&=(-1)^{1+k_1+k_2} a(1-k_1,-k_2),\\ b_3(k_1,k_2)&=(-1)^{1+k_1+k_2} b_2(1-k_1,-k_2), \end {aligned} \qquad \forall k_1,k_2\in \mathbb {Z}. \] Moreover, the supports of all the high-pass filters $b_1, b_2,b_3$ are no larger than that of the low-pass filter $a$. For a low-pass filter $a$ which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank $\{a;b_1,\ldots ,b_L\}$ with $b_1$ taking the above canonical form must have $L\ge 3$ high-pass filters. Thus, our family of double canonical quincunx tight framelets with symmetry property has the minimum number of generators. Numerical calculation indicates that this family of double canonical quincunx tight framelets with symmetry property can be arbitrarily smooth. Using one-dimensional filters having linear-phase moments, in this paper we also provide a second approach to construct multiple canonical quincunx tight framelets with symmetry property. In particular, the second approach yields a family of $6$-multiple canonical real-valued quincunx tight framelets in $L_2(\mathbb {R}^2)$ and a family of double canonical complex-valued quincunx tight framelets in $L_2(\mathbb {R}^2)$ such that both of them have symmetry property and arbitrarily increasing orders of smoothness and vanishing moments. Several examples are provided to illustrate our general construction and theoretical results on canonical quincunx tight framelets in $L_2(\mathbb {R}^2)$ with symmetry property, high vanishing moments, and smoothness. Quincunx tight framelets with symmetry property constructed by both approaches in this paper are of particular interest for their applications in computer graphics and image processing due to their polynomial preserving property, full symmetry property, short support, and high smoothness and vanishing moments.

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## Additional Information

**Bin Han**- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 610426
- Email: bhan@ualberta.ca
**Qingtang Jiang**- Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121
- Email: jiangq@umsl.edu
**Zuowei Shen**- Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076 Singapore
- MR Author ID: 292105
- Email: matzuows@nus.edu.sg
**Xiaosheng Zhuang**- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong
- Email: xzhuang7@cityu.edu.hk
- Received by editor(s): August 19, 2015
- Received by editor(s) in revised form: August 16, 2016
- Published electronically: April 28, 2017
- Additional Notes: The research of the first author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant 05865.

The research of the third author was supported by several grants from Singapore.

The research of the fourth author was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11304414) and grants from City University of Hong Kong (Project Nos. 7200462 and 7004445). - © Copyright 2017 American Mathematical Society
- Journal: Math. Comp.
**87**(2018), 347-379 - MSC (2010): Primary 42C15, 42C40; Secondary 42B99, 41A30, 41A63
- DOI: https://doi.org/10.1090/mcom/3205
- MathSciNet review: 3716199