Abstract
We study weighted $l^2$ fidelity in variational models for Poisson noise related image restoration problems. Gaussian approximation to Poisson noise statistic is adopted to deduce weighted $l^2$ fidelity. Different from the traditional weighted $l^2$ approximation, we propose a reweighted $l^2$ fidelity with sparse regularization by wavelet frame. Based on the split Bregman algorithm introduced in [21], the proposed numerical scheme is composed of three easy subproblems that involve quadratic minimization, soft shrinkage and matrix vector multiplications. Unlike usual least square approximation of Poisson noise, we dynamically update the underlying noise variance from previous estimate. The solution of the proposed algorithm is shown to be the same as the one obtained by minimizing Kullback-Leibler divergence fidelity with the same regularization. This reweighted $l^2$ formulation can be easily extended to mixed Poisson-Gaussian noise case. Finally, the efficiency and quality of the proposed algorithm compared to other Poisson noise removal methods are demonstrated through denoising and deblurring examples. Moreover, mixed Poisson-Gaussian noise tests are performed on both simulated and real digital images for further illustration of the performance of the proposed method.
Highlights
We consider an imaging system whose output data is a vector f ∈ (R+)M and the true underlying image is u =Ni=1 ∈ (R+)N
We focus on variational image restoration from observation f contaminated by Poisson or mixed Poisson-Gaussian noise
In [25], a Stein’s unbiased risk estimator (SURE) based on Poisson-Gaussian statistics is constructed in wavelet transform for mixed PoissonGaussian noise denoising, but the construction of estimators is complicated that it cannot be extended for more general image restoration problem such as deblurring, where the Poisson noise is added to the signal f = Au and the difficulty comes from the additional linear operator A
Summary
In [25], a Stein’s unbiased risk estimator (SURE) based on Poisson-Gaussian statistics is constructed in wavelet transform for mixed PoissonGaussian noise denoising, but the construction of estimators is complicated that it cannot be extended for more general image restoration problem such as deblurring, where the Poisson noise is added to the signal f = Au and the difficulty comes from the additional linear operator A. A reweighted 2 method for image restoration with Poisson and mixed Poisson-Gaussian noise 3 representation and total variation has been revealed in [10], and theoretically total variation can be interpreted as a special form of 1 framelets regularization. The main contribution of this paper is introducing a reweighted least square fidelity, that well approximates KL-divergence, and developping an efficient iterative algorithm solving the optimization problem with framelets regularization. The last part is dedicated to numerical simulation where other fidelity terms and models are compared numerically to ours for Poisson image denoising/deblurring and the mixed Poisson-Gaussian denoising/deblurring cases
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