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Relationship of d-dimensional continuous multi-scale wavelet shrinkage with integro-differential equations

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  • Liu, Guojun
  • Feng, Xiangchu
  • Li, Min

Abstract

The goal of this paper is to extend the results of Didas and Weickert [Didas, S, Weickert, J. Integrodifferential equations for continuous multi-scale wavelet shrinkage. Inverse Prob Imag 2007;1:47–62.] to d-dimensional (d⩾1) case. Firstly, we relate a d-dimensional continuous mother wavelet ψ(x) with a fast decay and n vanishing moments to the sum of the order partial derivative of a group of functions θk(x)(∣k∣=n) with fast decay, which also makes wavelet transform equal to a sum of smoothed partial derivative operators. Moreover, d-dimensional continuous wavelet transform can be explained as a weighted average of pseudo-differential equations, too. For d=1, our results are completely same as Didas and Weickert (2007), but for d>1, it is different from the type of one variable. Finally, we exploit the reason with an example of 2-dimensional and 3-dimensional Mexican hat wavelet.

Suggested Citation

  • Liu, Guojun & Feng, Xiangchu & Li, Min, 2009. "Relationship of d-dimensional continuous multi-scale wavelet shrinkage with integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1118-1126.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:3:p:1118-1126
    DOI: 10.1016/j.chaos.2007.08.066
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    References listed on IDEAS

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    1. Yousefi, Shahriar & Weinreich, Ilona & Reinarz, Dominik, 2005. "Wavelet-based prediction of oil prices," Chaos, Solitons & Fractals, Elsevier, vol. 25(2), pages 265-275.
    2. Huang, Yongdong & Cheng, Zhengxing & Yang, Jianwei, 2007. "Design of compactly supported trivariate orthogonal wavelets," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1440-1449.
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