Image deconvolution under poisson noise using sparse representations and proximal thresholding iteration

• Published in: 2008 IEEE International Conference on Acoustics, Speech and Signal Processing pp. 761 - 764
• Main Authors: Dupe, F.-X., Fadili, M.J., Starck, J.L.
• Format: Conference Proceeding
• Language: English
• Published: IEEE 01.03.2008
• Subjects: Additive noise; Deconvolution; Degradation; Dictionaries; forward-backward splitting; Gaussian noise; Image restoration; Iterative algorithms; Iterative thresholding; Nonlinear equations; Poisson noise; Proximal iteration; Sparse representations; Technological innovation; Wavelet transforms
• ISBN: 9781424414833; 1424414830
• ISSN: 1520-6149; 2379-190X
• DOI: 10.1109/ICASSP.2008.4517721

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 transform. Our key innovations are: First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a non-linear 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 non-smooth sparsity-promoting penalties over the image representation coefficients (e.g. l 1 -norm). Third, a fast iterative backward-forward 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. 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, e.g. astronomy or microscopy.