NON-BLIND IMAGE RESTORATION WITH SYMMETRIC GENERALIZED PARETO PRIORS Xing Mei 1,2 Bao-Gang Hu 2 Siwei Lyu 1 1 Computer Science Department, University at Albany, State University of New York 2 National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences Introduction • Non-Blind Image Restoration = ⊗ + y x k n Degraded Image Gaussian Noise Blurring Kernel Clean Image - Assuming known & noise level, estimate from - An ill-posed problem, requiring priors on k • Challenges - Find good image priors - Develop efficient numerical solutions x y x Our Proposal • A New Parametric Image Prior • A Fast & Effective Image Restoration Method - Closed-form numerical solutions - State-of-the-art image restoration quality & processing speed - Captures heavy-tailed statistics of gradient distributions - Handles other band-pass filter responses - Fitting ability comparable to hyper-Laplacian The SGP Prior • Symmetric Generalized Pareto (SGP) - Symmetrizes generalized pareto to the whole real line - Tail heaviness is controlled by 1 ( | , ) , 2(| | ) px x x ω ω ωγ ωγ γ + = ∈ + , 0 ωγ > • A Fitting Example Collect Gradients MLE Fitting - Tails are better captured by the SGP model • Quantitative Evaluation 0 0.5 1 1.5 2 2.5 3 3.5 Gradient X Gradient Y Sub-band 1 Sub-band 2 Sub-band 3 Sub-band 4 Gaussian Laplacian hyper-Laplacian SGP - 100 images from van Hateren’s data set - 6 band-pass filter responses - Average likelihood scores - SGP comparable to hyper-Laplacian Restoration with SGP • The Maximum-a-posterior (MAP) Formulation 2 2 1 min ( ) log(| | ) 2 i j i i j λ γ = ⊗ + ⊗ + ∑ ∑ x y-x k x f Data likelihood SGP-based regularizers - pixel index, - band-pass (gradient) filter type i j - Difficult to solve due to the non-differentiable regularizers • Half-Quadratic Splitting Solution - Decouples from SGP regularizers using auxiliary variables and quadratic penalty terms - Solves two sub-problems using block coordinate descent 1 2 , ⊗ ⊗ x f x f • Fixed , solve with 2D FFTs and IFFTs 1 2 , z z • Fixed , solve in a common 1D form independently on each pixel x 2 min () ( ) log(| z | ) 2 z gz z v β γ = − + + Note that when '( ) 0 ( - )( ) 1 0 g z z v z β γ = ⇒ + + = 0, z > Quadratic equations A closed-form solution 1 2 , z z 1 2 2 2 2 2 , 1 1 min ( ) ( ) log(| | ) 2 2 i j j i j i i j j λ β γ = = ⊗ + ⊗ − + + ∑ ∑ ∑ x,z z y-x k x f z z Quadratic penalty terms, β →∞ 1 2 , z z x Experimental Results • Experimental Settings - 12 grayscale images, 10 blurring kernels, 3 noise levels - Comparison to L1 [Wang et al., SIAM JIS 2008], LUT [Krishnan and Fergus, NIPS 2009], GISA [Zuo et al., ICCV 2013] • Quantitative Results - Average PSNR with 4 Gaussian kernels and 3 noise levels L1 SGP LUT GISA > > ≈ - Average PSNR with 6 motion kernels and 3 noise levels SGP LUT GISA>L1 > ≈ • Visual Comparison - The ‘Barbara’ example: 27x27 motion kernel + 10% noise Original Image Corrupted Image L1 20.74dB LUT 21.96dB GISA 21.96dB SGP 22.06dB • Running Time (in Seconds) - The z-step: L1, SGP - 0.014 , LUT - 0.048, GISA - 0.023 Acknowledgement This work is supported by the National Science Foundation under Grant. Nos. IIS-0953373 and CCF-1319800, and National Institute of Justice Grant No 2013-IJ-CX-K010. - The x-step: 0.576