Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors

• Published in: IEEE transactions on image processing Vol. 15; no. 4; pp. 937 - 951
• Main Author: Bioucas-Dias, J.M.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.04.2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Bayes Theorem; Bayesian; Bayesian analysis; Bayesian methods; Computational complexity; Computer Simulation; Deconvolution; Density; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; expectation maximization (EM); Gaussian; Gaussian scale mixtures (GSM); generalized expectation maximization (GEM); GSM; heavy-tailed priors; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image processing; Information Storage and Retrieval - methods; Information, signal and communications theory; Iterative algorithms; Mathematical analysis; Mathematical models; Maximization; Maximum a posteriori estimation; Miscellaneous; Models, Statistical; Noise reduction; Numerical Analysis, Computer-Assisted; Reproducibility of Results; Sensitivity and Specificity; Signal and communications theory; Signal processing; Signal Processing, Computer-Assisted; Signal representation. Spectral analysis; Signal, noise; Studies; Sufficient conditions; Telecommunications and information theory; wavelet; Wavelet coefficients; Wavelet domain; Performance evaluation; Second order; State of the art; Image processing; Necessary and sufficient condition; Threshold detection; Noise reduction; Wireless telecommunication; Iterative method; expectation maximization (EM); A posteriori estimation; Gaussian scale mixtures (GSM); Bayes estimation; Mobile radiocommunication; GSM system; Bayesian; heavy-tailed priors; Scale factor; Computational complexity; Wavelets; deconvolution; Gaussian process; Wavelet transformation; generalized expectation maximization (GEM); Signal processing; wavelet; EM algorithm; Signal analysis
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2005.863972

Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys , and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of O(NlogN) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.