Canonical quincunx tight framelets with symmetry and high vanishing moments

Bin Han; Qingtang Jiang; Zuowei Shen; Xiaosheng Zhuang

doi:10.1090/mcom/3205

Canonical quincunx tight framelets with symmetry and high vanishing moments

Bin Han, Qingtang Jiang, Zuowei Shen, Xiaosheng Zhuang

University of Missouri-St. Louis

Research output: Contribution to journal › Article › peer-review

Abstract

<div class="line" id="line-7"> In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets  <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img1.gif"/>  in  <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img2.gif"/>  with  <i> symmetry property </i>  and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks  <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img3.gif"/>  having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties:</div><div class="line" id="line-93"> <br/></div><div class="line" id="line-96"> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img4.gif"/></div><div class="line" id="line-101"> <br/></div><div class="line" id="line-104"> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'> Moreover, the supports of all the high-pass filters  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img5.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  are no larger than that of the low-pass filter  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img6.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'> . For a low-pass filter  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img7.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img8.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  with  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img9.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  taking the above canonical form must have  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img10.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  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  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img11.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'> -multiple canonical real-valued quincunx tight framelets in  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img12.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  and a family of double canonical complex-valued quincunx tight framelets in  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img13.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  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  </span> <img src="https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/images/img14.gif"/> <span style='color: rgb(37, 40, 43); font-family: "Open Sans", sans-serif; font-size: 14px;'>  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. </span></div>

Original language	American English
Journal	Mathematics of Computation
DOIs	https://doi.org/10.1090/mcom/3205
State	Published - 2017

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Physical Sciences and Mathematics

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@article{e6a64ecdab784c4298d489dd202425e4,

title = "Canonical quincunx tight framelets with symmetry and high vanishing moments",

abstract = " In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets   in   with  symmetry property  and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks   having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: Moreover, the supports of all the high-pass filters   are no larger than that of the low-pass filter  . For a low-pass filter   which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank   with   taking the above canonical form must have   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  -multiple canonical real-valued quincunx tight framelets in   and a family of double canonical complex-valued quincunx tight framelets in   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   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. ",

author = "Bin Han and Qingtang Jiang and Zuowei Shen and Xiaosheng Zhuang",

note = "Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness Authors: Bin Han, Qingtang Jiang, Zuowei Shen and Xiaosheng Zhuang Journal: Math. Comp.",

year = "2017",

doi = "10.1090/mcom/3205",

language = "American English",

journal = "Mathematics of Computation",

}

TY - JOUR

T1 - Canonical quincunx tight framelets with symmetry and high vanishing moments

AU - Han, Bin

AU - Jiang, Qingtang

AU - Shen, Zuowei

AU - Zhuang, Xiaosheng

N1 - Symmetric canonical quincunx tight framelets with high vanishing moments and smoothness Authors: Bin Han, Qingtang Jiang, Zuowei Shen and Xiaosheng Zhuang Journal: Math. Comp.

PY - 2017

Y1 - 2017

N2 - In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets   in   with  symmetry property  and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks   having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: Moreover, the supports of all the high-pass filters   are no larger than that of the low-pass filter  . For a low-pass filter   which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank   with   taking the above canonical form must have   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  -multiple canonical real-valued quincunx tight framelets in   and a family of double canonical complex-valued quincunx tight framelets in   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   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.

AB - In this paper, we propose an approach to construct a family of two-dimensional compactly supported real-valued quincunx tight framelets   in   with  symmetry property  and arbitrarily high orders of vanishing moments. Such quincunx tight framelets are associated with quincunx tight framelet filter banks   having increasing orders of vanishing moments, possessing symmetry property, and enjoying the additional double canonical properties: Moreover, the supports of all the high-pass filters   are no larger than that of the low-pass filter  . For a low-pass filter   which is not a quincunx orthogonal wavelet filter, we show that a quincunx tight framelet filter bank   with   taking the above canonical form must have   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  -multiple canonical real-valued quincunx tight framelets in   and a family of double canonical complex-valued quincunx tight framelets in   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   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.

UR - https://www.ams.org/journals/mcom/2018-87-309/S0025-5718-2017-03205-1/

U2 - 10.1090/mcom/3205

DO - 10.1090/mcom/3205

M3 - Article

JO - Mathematics of Computation

JF - Mathematics of Computation

ER -

Canonical quincunx tight framelets with symmetry and high vanishing moments

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