Smoothness estimates for soft - threshold denoising via translationinvariant wavelet transforms

In this paper we study a generalization of the Donoho-Johnstone denoising model for the case of the translation invariant wavelet transform. Instead of soft-thresholding coeecients of the classical orthogonal discrete wavelet transform, we study soft-thresholding of the co-eecients of the translation invariant discrete wavelet transform. This latter transform is not an orthogonal transformation. As a rst step we construct a level-dependent threshold to remove all the noise in the wavelet domain. Subsequently, we use the theory of interpolating wavelet transforms to characterize the smoothness of an estimated denoised function. Based on the fact that the inverse of the translation invariant discrete transform includes averaging over all shifts, we use smoother autocorrelation functions in the representation of the estimated denoised function in place of Daubechies scaling functions.