Integrodifferential equations for continuous multiscale wavelet shrinkage

The relations between wavelet shrinkage and nonlinear diffusion for discontinuity-preserving signal denoising are fairly well-understood for single-scale wavelet shrinkage, but not for the practically relevant multiscale case. In this paper we show that 1-D multiscale continuous wavelet shrinkage can be linked to novel integrodifferential equations. They differ from nonlinear diffusion filtering and corresponding regularisation methods by the fact that they involve smoothed derivative operators and perform a weighted averaging over all scales. Moreover, by expressing the convolution-based smoothed derivative operators by power series of differential operators, we show that multiscale wavelet shrinkage can also be regarded as averaging over pseudodifferential equations.