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Hexagonal tight frame filter banks with idealized high-pass filters

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Abstract

This paper studies the construction of hexagonal tight wavelet frame filter banks which contain three “idealized” high-pass filters. These three high-pass filters are suitable spatial shifts and frequency modulations of the associated low-pass filter, and they are used by Simoncelli and Adelson in (Proc IEEE 78:652–664, 1990) for the design of hexagonal filter banks and by Riemenschneider and Shen in (Approximation Theory and Functional Analysis, pp. 133–149, Academic Press, Boston 1991; J. Approx Theory 71:18–38 1992) for the construction of 2-dimensional orthogonal filter banks. For an idealized low-pass filter, these three associated high-pass filters separate high frequency components of a hexagonal image in 3 different directions in the frequency domain. In this paper we show that an idealized tight frame, a frame generated by a tight frame filter bank containing the “idealized” high-pass filters, has at least 7 frame generators. We provide an approach to construct such tight frames based on the method by Lai and Stöckler in (Appl Comput Harmon Anal 21:324–348, 2006) to decompose non-negative trigonometric polynomials as the summations of the absolute squares of other trigonometric polynomials. In particular, we show that if the non-negative trigonometric polynomial associated with the low-pass filter p can be written as the summation of the absolute squares of other 3 or less than 3 trigonometric polynomials, then the idealized tight frame associated with p requires exact 7 frame generators. We also discuss the symmetry of frame filters. In addition, we present in this paper several examples, including that with the scaling functions to be the Courant element B 111 and the box-spline B 222. The tight frames constructed in this paper will have potential applications to hexagonal image processing.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Missouri–St. Louis, St. Louis, MO, 63121, USA

    Qingtang Jiang

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  1. Qingtang Jiang
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Correspondence to Qingtang Jiang.

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Communicated by Lixin Shen and Yuesheng Xu.

Research supported by UM Research Board 10/05 and UMSL Research Award 10/06.

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Jiang, Q. Hexagonal tight frame filter banks with idealized high-pass filters. Adv Comput Math 31, 215–236 (2009). https://doi.org/10.1007/s10444-008-9085-4

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Mathematics Subject Classifications (2000)