NON-BLIND IMAGE RESTORATION WITH SYMMETRIC GENERALIZED PARETO PRIORS

Xing Mei1,2 Bao-Gang Hu2 Siwei Lyu1 1Computer Science Department, University at Albany, State University of New York

2National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences

Introduction

• Non-Blind Image Restoration

= ⊗ +y x k nDegraded 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 yx

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(| | )

p x xx

ω

ω

ωγω γγ += ∈+

, 0ω γ >

• 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

GaussianLaplacianhyper-LaplacianSGP

- 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

1min ( ) log(| | )

2 i j ii j

λ γ=

⊗ + ⊗ +

∑ ∑x

y - x k x f

Data likelihood

SGP-based regularizers

- pixel index, - band-pass (gradient) filter typei 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

2min ( ) ( ) log(| z | )2z

g z z vβ γ= − + +

Note that when '( ) 0 ( - )( ) 1 0g z z v zβ γ= ⇒ + + =0,z >Quadratic equations A closed-form solution

1 2,z z

1 2

2 22 2

, 1 1min ( ) ( ) log(| | )

2 2i j j i j ii j j

λ β γ= =

⊗ + ⊗ − + +

∑ ∑ ∑x,z z

y - x k x f z z

Quadratic penalty terms,

β→∞

1 2,z zx

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