Translation-invariant Multiwavelets for Image De-noising

Over the past decade wavelet transforms have received a lot of attention from researchers in many diierent areas. Both discrete and continuous wavelet transforms have shown great promises in such diverse elds as image compression, image De-noising, signal processing, computer graphics, and pattern recognition, to name a few. Most of the work has been done on scalar wavelets, i.e., wavelets generated by one scalar function. Scalar wavelets, however, cannot possess all the important properties needed such as short support, regularity, orthogonality, symmetry, and high order vanishing moments. Therefore, multiwavelets have been developed by using translates and dilates of more than one mother wavelet functions. Strela et al have done some experiments by applying non-translation-invariant multiwavelet to image De-noising and they get very good results. Translation-invariant(TI) scalar wavelet De-noising has superior performance than the non-TI approach as claimed by Coifman and Donoho. In this paper, we are going to incorporate multiwavelts into the TI De-noising scheme, and apply it to image De-noising. The complexity of our TI multiwavelet algorithm is O((n log n) 2), where the image dimension is nn. We conduct our experiments by using diierent threshold values for both hard and soft thresholding. It is possible that we can develop directional-invariant De-noising by transform the original image into polar coordinates. We can also extend the TI multiwavelet De-noising scheme into higher dimensional datasets in 3D.