Transcript gghgh
Unnatural L0 Representation
for Natural Image Deblurring
Speaker: Wei-Sheng Lai
Date: 2013/04/26
Outline
1.
2.
3.
4.
Introduction
Related work
L0 Deblurring
Conclusion
2
1. Introduction
β’ Form of image blur :
1. Object motion
2. Camera Shake
3. Out of focus (defocus)
β’
Blur model:
π΅ =πΏβπΎ+π
B: blurred(observed) image
L: latent(sharp) image
K: blur kernel
N: noise
β¨: convolution
Point Spread Function (PSF)
3
1. Introduction
β’ Ill-posed problem:
observation (B) < unknown variables (L + K)
4
1. Introduction
β’ Early method:
1. RichardsonβLucy deconvolution (RL) [1][2]
πΏπ‘+1
=
πΏπ‘
π΅
.β πΎ β π‘
πΏ βπΎ
πΎ: flipped blur
kernel
2. Wiener filter [3]
πΏ(πΉ) = π΅(πΉ) .β
ο
πΎ β (πΉ)
2
πΎ πΉ 2+π π΄
π : noise ratio
A : constant
Both are known to be sensitive to noise.
[1] Richardson, William Hadley. "Bayesian-based iterative method of image restoration." JOSA 62.1 (1972): 55-59.
[2] Lucy, L. B. "An iterative technique for the rectification of observed distributions."The astronomical journal 79 (1974): 745.
[3] Wiener, Norbert. Extrapolation, interpolation, and smoothing of stationary time series: with engineering applications. Technology
5
Press of the Massachusetts Institute of Technology, 1950.
1. Introduction
β’ Recent framework: Maximum-a-Posteriori (MAP)
πΏβ , πΎ β = πππ min π΅ β πΏ β πΎ
πΏ,πΎ
2
2 + ΟπΏ
πΏ + ΟπΎ πΎ
β ΟπΏ πΏ : prior of latent image
β ΟπΎ πΎ : prior of kernel
β’ Non-linear problem, iterative optimization :
πΏβ = πππ min π΅ β πΏ β πΎ
πΏ
πΎ β = πππ min π΅ β πΏ β πΎ
πΎ
2
2 + ΟπΏ
2
2 + ΟπΎ
πΏ
πΎ
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2. Related work
β’ Fergus et al. Siggraph 2006 [4]
β Heavy tails distribution of nature image gradient
β Assume kernel prior as Gamma distribution
π₯ π π βπ₯/π
π π₯ π, π =
π! π π+1
[4] R. Fergus et al, βRemoving camera shake from a single photograph,β Siggraph 2006
7
2. Related work
β’ Prior (regularization) :
β Gaussian prior (L2 regularization) [5]:
πΈ(πΏ) = π΅ β πΏ β πΎ 22 + π π»πΏ
β TV-L1 prior [6]:
πΈ(πΏ) = π΅ β πΏ β πΎ
β Sparse prior [7]:
πΈ(πΏ) = π΅ β πΏ β πΎ
2
2
+ π π»πΏ
2
2
+ π π»πΏ
2
2
1
πΌ
,πΌ
[5] S Cho et al, βFast motion deblur,β Siggraph 2009
[6] Xu, Li, and Jiaya Jia. "Two-phase kernel estimation for robust motion deblurring." ECCV 2010.
[7] Levin, Anat, et al. "Image and depth from a conventional camera with a coded aperture." ACM TOG 2007
β€1
8
2. Related work
β’ Q.Suan et al. Siggraph 2008 [8]
β N and ππ should follow the zero-mean Gaussian
distribution
πβπ΅ β πβπΏ β πΎ
E L =
2
2
+ π1 π ππ₯ πΏ + π ππ¦ πΏ
πβ
+ π2
ππ₯ πΏ β ππ₯ π΅
2
2
+ ππ¦ πΏ β ππ¦ π΅
2
2
+ πΎ
[8] Q. Shan et al, βHigh quality motion deblurring from a single image,β Siggraph 2008
1
1
9
2. Related work
β’ Cho et al. Siggraph 2009 [5]
β Accelerate the deblurring procedure by first estimating a
predicted image and using L2 regularization
β’ Kernel estimation :
πβπ΅ β πβπ β πΎ
πΈ πΎ =
2
2
+π½ πΎ
2
2
+ π π»πΏ
2
2
πβ
β’ Image deconvolution:
πβπ΅ β πβπΏ β πΎ
πΈ πΏ =
2
2
πβ
[5] S Cho et al, βFast motion deblur,β Siggraph 2009
10
2. Related work
β’ Anat Levin et al. CVPR 2009 [9] :
β MAP x,k approach will favor blur image with delta kernel.
πΏβ , πΎ β = πππ min π΅ β πΏ β πΎ
πΏ,πΎ
2
2 +π
π»πΏ
2
2
β Estimate kernel K first, then use non-blind deconvolution to
solve the latent image.
[9] Levin, Anat, et al. "Understanding and evaluating blind deconvolution algorithms." CVPR 2009.
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Unnatural L0 Sparse Representation
for Natural Image Deblurring
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3. L0 Deblurring
β’ Li Xu et al. CVPR 2013 [10]
β Predict image with L0 optimization
β’ L0-norm:
π π₯ = π₯
0
=
0,
1,
π₯ =0
π₯ β 0
β’ Approximate L0 sparsity function:
1
2
π»
πΌ
,
β
π π»β πΌ = π 2
1,
ππ π»β πΌ β€ π
ππ‘βπππ€ππ π
[10] Xu, Li, Shicheng Zheng, and Jiaya Jia. "Unnatural L0 Sparse Representation for Natural Image Deblurring.β CVPR 2013
13
3. L0 Deblurring
β’ Main objective function:
πππ
πΎβπΌβπ΅
2
2
+π
π0 πβ πΌ + πΎ πΎ
2
2
ββ{π₯,π¦}
where π0 πβ πΌ =
π π(πβ πΌπ ),
1
π2
π π»β πΌ =
π»β πΌ 2 , ππ π»β πΌ β€ π
1,
ππ‘βπππ€ππ π
β’ Iteratively solve:
πΌ (π‘+1)
= πππ min
πΌ
πΎ (π‘+1) = πππ min
πΎ
πΎ (π‘)
βπΌβπ΅
2
2
+π
πΎ β πΌ (π‘+1) β π΅
π0 πβ πΌ
ββ{π₯,π¦}
2
+πΎ πΎ
2
2
2
[10] Xu, Li, Shicheng Zheng, and Jiaya Jia. "Unnatural L0 Sparse Representation for Natural Image Deblurring.β CVPR 2013
14
3. L0 Deblurring
β’ Solving
πΌ (π‘+1) = πππ min
πΌ
where π0 πβ πΌ =
πΎ (π‘) β πΌ β π΅
π π(ππ₯ πΌπ )
,
2
2
+π
π πβ πΌ =
π0 πβ πΌ
ββ{π₯,π¦}
1
π2
πβ πΌ 2 , ππ πβ πΌ β€ π
1,
ππ‘βπππ€ππ π
β’ Equivalent to solving
πΌ (π‘+1) = πππ min
πΌ,π€
πΎ (π‘) β πΌ β π΅
2
+π
2
0,
π€βπ =
πβ πΌ,
π€βπ
ββ{π₯,π¦} π
ππ πβ πΌ β€ π
ππ‘βπππ€ππ π
0
+
1
(π‘) β π€
π
πΌ
β
βπ
π2
2
π β {1, 2β1 , 4β1 , 8β1 }
[10] Xu, Li, Shicheng Zheng, and Jiaya Jia. "Unnatural L0 Sparse Representation for Natural Image Deblurring.β CVPR 2013
15
3. L0 Deblurring
πΉ π₯ =
πΉ π΅π .β πΉ π + π/π 2 πΉ ππ₯ .β πΉ π€π₯ + πΉ ππ¦ .β πΉ π€π¦
πΉ π΅π .β πΉ π΅π + π/π 2 πΉπ·2
πΎ
(π‘+1)
= πππ min
πΎ
πΎβπΌ
(π‘+1)
(π‘+1)
π (π‘+1)
=
πΉ β1
πΉ(π΄π
πΉ
π (π‘+1) = π (π‘)
βπ΅
2
+πΎ πΎ
2
2
2
) .β πΉ(π¦)
(π‘+1)
π΄π
2
+πΎ
πΌ
π΄ππ
π¦
(π΄ππ
π΄π
+ πΎ)π
π
[10] Xu, Li, Shicheng Zheng, and Jiaya Jia. "Unnatural L0 Sparse Representation for Natural Image Deblurring.β CVPR 2013
16
3. L0 Deblurring
Unnatural
Fast Hyper-Laplacian
deconvolution (πΏ0.5 norm) [11]
Representation
Input image
Deblurring
result
L0 optimization
Predict map
kernel
[11] Krishnan, Dilip, and Rob Fergus. "Fast image deconvolution using hyper-Laplacian priors." ANIPS 2009
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3. L0 Deblurring
β’ Other results
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3. L0 Deblurring
β’ Advantage of L0 deblurring:
β Fast convergence
β High quality
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4. Conclusion
β’ A naïve MAP x,k estimation will fail.
πΏβ , πΎ β = πππ min π΅ β πΏ β πΎ
πΏ,πΎ
2
2 +π
π»πΏ
2
2
β’ How to estimate correct kernel is important.
β’ It is not as simple as what I have shown, there are
many implementation details.
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Thanks for Attention !
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