Filter banks and the EM algorithm

The authors present a wavelet based modification of the ML-EM algorithm for reconstructing positron emission tomography images. By using the filter bank implementation of the wavelet transform, this algorithm has the flexibility to incorporate a priori information, while maintaining the same computational complexity as the standard ML-EM algorithm. Thus, it has a significant computational advantage over usual Bayesian methods. It differs from recent wavelet-based Bayesian methods as it achieves “regularization” by an adaptive, wavelet-based method of thresholding which minimizes Stein´s Unbiased Estimate of Risk. The basic method consists of applying Donoho and Johnstone´s (1993) SureShrink wavelet denoising of the Poisson data, and then applying the standard ML-EM algorithm to the denoised data. A more elaborate method is discussed in which a wavelet denoising step is inserted after each EM iteration. This technique differs from previous smoothing techniques applied to the ML-EM algorithm since it is able to recover edges in discontinuous images