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

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.