Mathematical concepts of multiscale smoothing

The starting point for this paper is the well-known equivalence between convolution filtering with a rescaled Gaussian and the solution of the heat equation. In the first sections we analyze the equivalence between multiscale convolution filtering, linear smoothing methods based on continuous wavelet transforms and the solutions of linear diffusion equations. This means we determine a wavelet ψ, respectively a convolution filter φ, which is associated with a given linear diffusion equation ∂u∂t=Pu and vice versa. This approach has an extension to non-linear smoothing techniques. The main result of this paper is the derivation of a differential equation, whose solution is equivalent to non-linear multiscale smoothing based on soft shrinkage methods applied to Fourier or continuous wavelet transforms.