A new image denoising technique using orthogonal complex wavelets

The complex wavelet Transforms CWTs are known for their excellent edge preserving together with nearly shift invariant features. They are implemented as two real DWTs connected in parallel. These two DWTs are designed such that their wavelet coefficients form a nearly Hilbert transform pairs at every decomposition level. This paper, presents a new orthogonal filter design for these CWT Hilbert transform pairs. The proposed design satisfies in a least squares sense, the Hilbert constraints over the filter's pass-band. In the meantime, the half band properties of the orthogonal filter, are guaranteed. Simulation results show that the designed filter is nearly shift invariant. Next, the designed filter was used in image de-noising. In this respect, the bivariate shrinkage algorithm is used to threshold the magnitudes of the CWT wavelet coefficients. Unlike earlier designs that suffer from excessive processing time, a simple model is proposed to model the dependence between the magnitudes of the wavelet coefficient and its parent at adjacent sub band. This allows the derivation of a closed form expression for the thresholded magnitudes. Subsequently, a fast estimation of the clean wavelet coefficient, at every pixel and every sub band is obtained. Several illustrative examples are given to verify the superior de-noising performance and their nearly shift invariance features.