Nonlinear Shrinkage of Undecimated DWT for Noise Reduction and Data Compression

In this paper, we study the problem of compressing a signal buried in additive white Gaussian noise. The approach we propose is to design an asymptotically optimal quantizer in the the wavelet domain. The undecimated discrete wavelet transform is shift-invariant and redundant, thus it is robust compared with the decimated and orthogonal discrete wavelet transform (DWT). Compared with the original wavelet shrinkage method of Donoho and Johnstone, our method signiicantly improves the noise reduction performance. Our method of noise reduction is in essence equivalent to applying the original approach to all the possible circular shifts of the data and average the denoised results, and we implement it in an eecient way. The basis functions of the undecimated DWT form a frame of the signal space. Quantization of the frame coeecients is less harmful compared with quantization of orthogonal transform coeecients. Noise reduction facilitates data compression, while data compression introduces additional noise. We study the noise reduction and data compression as integrated parts of one problem, in which the undecimated DWT plays an important role.