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Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness

Authors: Bin Han, Qingtang Jiang, Zuowei Shen and Xiaosheng Zhuang
Journal: Math. Comp. 87 (2018), 347-379
MSC (2010): Primary 42C15, 42C40; Secondary 42B99, 41A30, 41A63
Published electronically: April 28, 2017
MathSciNet review: 3716199
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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

Qingtang Jiang
Affiliation: Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076 Singapore
MR Author ID: 292105

Xiaosheng Zhuang
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong

Keywords: Quincunx tight framelets, canonical tight framelets, symmetry, linear-phase moments, vanishing moments, sum rule orders, smoothness exponents, wavelet analysis
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).
Article copyright: © Copyright 2017 American Mathematical Society