A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations

• Published in: IEEE transactions on image processing Vol. 18; no. 2; pp. 310 - 321
• Main Authors: Dupe, F.-X., Fadili, J.M., Starck, J.-L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive noise; Algorithms; Applications; Applied sciences; Computer Science; Data Interpretation, Statistical; Deconvolution; Degradation; Dictionaries; Engineering Sciences; Exact sciences and technology; forward-backward splitting; Gaussian noise; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image processing; Image restoration; Information, signal and communications theory; Iterative algorithms; iterative thresholding; Mathematical models; Mathematics; Minimization; Miscellaneous; Models, Statistical; Noise; Nonlinear equations; Optimization and Control; Poisson Distribution; Poisson equations; Poisson noise; proximal iteration; Regularization; Representations; Reproducibility of Results; Sensitivity and Specificity; Signal and communications theory; Signal and Image Processing; Signal processing; Signal representation. Spectral analysis; Signal, noise; sparse representations; Statistics; Statistics Theory; Studies; Telecommunications and information theory; Wavelet transforms; Dictionaries; Image processing; Threshold detection; forward-backward splitting; sparse representations; Iterative method; Poisson process; Poisson noise; Degradation; Non linear equation; Gaussian noise; Image representation; Sparse representation; Additive noise; Model selection; iterative thresholding; proximal iteration; Image restoration; Signal representation; Algorithm; Noisy image; Deconvolution; Microscopy; Wavelet transformation; Signal processing; Iterative thresholding; Sparse representations; Proximal iteration
• ISSN: 1057-7149; 1941-0042
• DOI: 10.1109/TIP.2008.2008223

We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are as follows. First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the minimization of a convex functional with a data-fidelity term reflecting the noise properties, and a nonsmooth sparsity-promoting penalty over the image representation coefficients (e.g., lscr 1 -norm). An additional term is also included in the functional to ensure positivity of the restored image. Third, a fast iterative forward-backward splitting algorithm is proposed to solve the minimization problem. We derive existence and uniqueness conditions of the solution, and establish convergence of the iterative algorithm. Finally, a GCV-based model selection procedure is proposed to objectively select the regularization parameter. Experimental results are carried out to show the striking benefits gained from taking into account the Poisson statistics of the noise. These results also suggest that using sparse-domain regularization may be tractable in many deconvolution applications with Poisson noise such as astronomy and microscopy.