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

Guojun Liu, Xiangchu Feng and Min Li

Chaos, Solitons & Fractals, 2009, vol. 40, issue 3, 1118-1126

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.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:40:y:2009:i:3:p:1118-1126

DOI: 10.1016/j.chaos.2007.08.066

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