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High-quality Motion Deblurringfrom a Single Image
High-quality Motion Deblurringfrom a Single Image
QiQi ShanShanLeo Leo JiayaJiaya JiaJiaAseemAseem AgarwalaAgarwala
The Chinese University of Hong KongThe Chinese University of Hong KongAdobe Systems, Inc.
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2
The Problem
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3
Blur Model
Blurred image I Sharp image L
Camera Noise n
Blur kernel f
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4
Previous Work (1)
Hardware solutions:
[Raskar et al. 2006]
[Ben-Ezra and Nayar 2004]
[Levin et al. 2008]
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5
Previous Work (2)
Multi-frame solutions:
[Petschnigg et al. 2004]
[Jia et al. 2004] [Rav-Acha and Peleg 2005]
[Yuan et al. 2007]
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6
Previous Work (3)
Single image solutions:
[Jia 2007][Fergus et al. 2006]
[Levin et al. 2007] [Yuan et al. 2008]
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7
The Problems
Non-blind deconvolution
Blind deconvolution
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An Example
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9
Challenges (1)
Assuming no noise
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10
Challenges (2)
With noise
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An Example
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An Example
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An Example
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An Example
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15
n
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16
n
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17
Noise constraint
n
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18
0( | 0, )iiN n ζ∏
1( | 0, )iiN n ζ∇∏
0 1( | 0, ) ( | 0, )i ii iN n N nζ ζ∇∏ ∏
Noise constraint
n
Possible noise models:
(1)
(2)
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19
0 1
2
( | 0, ) ( | 0, )
( | 0, )i ii i
ii
N n N n
N n
ζ ζ
ζ
∇
∇∇
∏ ∏∏
Noise constraint
n
0( | 0, )iiN n ζ∏
1( | 0, )iiN n ζ∇∏
Possible noise models:
(1)
(2)
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20
0 1
2
( | 0, ) ( | 0, )
( ( )
(
| 0, )
) i ii i
ii
P n N n N n
N n
ζ ζ
ζ
= ∇
∇ ∇
∏ ∏∏
A random variable following an independent Gaussian distribution also has its any order derivative following it. [Simon 2002]
n
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21
n
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22
n
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23
n
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24
exponentially distributed
, ( 0) jfj
i
ep f fτ−= ≥∏
Kernel Statistics
f
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25
L
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26
L
Prior p(L) trabajando en dos escalas:Global: para ayudar a mitigar la
ill-possedness del problema (a nivel estadístico)
Local: para corregir los ringingartifacts
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27
Image Global Statistics
L
…
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28
Image Global Statistics
L
…
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29
Image Global Statistics
…
L
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30
Image Global Statistics
L
21
| |( ( )
( ))
k L x ca L
P Lb x c
− ∇ ≤⎧= ⎨− ∇ + >
∇⎩
))(log( 1 LP ∇
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31
Image Global Statistics
L
21
| |( ( )
( ))
k L x ca L
P Lb x c
− ∇ ≤⎧= ⎨− ∇ + >
∇⎩
))(log( 1 LP ∇
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32
Image Local Constraint
L
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33
Image Local Constraint
L
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34
LI
Image Local Constraint
L
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35
LI
Image Local Constraint
L
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36
LI
Image Local Constraint
L
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37
LI
Image Local Constraint
12 ( | 0,) )(i white
p N L IL σ∈
= ∇ −∇∏
L
∏∈
∇−∇whitei
ii ILN ),0|( 1σ
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38
LI
Image Local Constraint
12 ( | 0,) )(i white
p N L IL σ∈
= ∇ −∇∏
L
∏∈
∇−∇whitei
ii ILN ),0|( 1σ
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39
Combining All constraints
1 2min ( , ) min log[ ( ) ( ) ( ) ( )]E L f p n p L p L p f= ∇
L f n
Two-step iterative optimization• Optimize L• Optimize f
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40
1 2( ) log[ ( ) ( ) ( )]E L p n p L p L= ∇
**
* * 22( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
Convolution is time-consuming
log ( )p n
1log ( )p L∇ 2log ( )p L1 1 1|| log ( ) ||p Lλ ∇ 2
2 2( || || )i ii white
L Iλ∈
+ ∇ −∇∑
Optimize L
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41
Idea: separate convolution
**
* * 22( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
log ( )p n
Optimize L
1log ( )p L∇ 2log ( )p L1 1 1|| log ( ) ||p Lλ ∇ 2
2 2( || || )i ii white
L Iλ∈
+ ∇ −∇∑
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42
Idea: separate convolution
**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
log ( )p n
Optimize L
1log ( )p L∇ 2log ( )p L1 1 1|| log ( ) ||pλ Ψ 2
2 2( || || )i ii white
L Iλ∈
+ ∇ −∇∑
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43
Idea: separate convolution
**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
log ( )p n
Optimize L
1log ( )p L∇ 2log ( )p L1 1 1|| log ( ) ||pλ Ψ 2
2 2( || || )ii white
i Iλ∈
+ −∇Ψ∑
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**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
log ( )p n
Idea: separate convolution
22(|| || )Lγ+ Ψ −∇
Optimize L
1log ( )p L∇ 2log ( )p L1 1 1|| log ( ) ||pλ Ψ 2
2 2( || || )ii white
i Iλ∈
+ Ψ −∇∑
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45
**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
1 1 1|| log ( ) ||pλ Ψ 22(|| || )Lγ+ Ψ −∇2
2 2( || || )i ii white
Iλ∈
+ Ψ −∇∑
Iteratively optimize L:Update LUpdate Ψ
**
* * 22|| ||w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇⎜ ⎟⎝ ⎠∑ 2
2(|| || )Lγ Ψ −∇+
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46
**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑ 2
2(|| || )Lγ+ Ψ −∇
Plancherel’s theorem implies:the sum of the square of a function equals the sum of the square of its Fourier transform
Iteratively optimize L:Update LUpdate Ψ
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47
**
* * 22'( ) || ||E L w L f I
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
Iteratively optimize L:Update LUpdate Ψ
22(|| || )Lγ+ Ψ −∇
We minimize E’(F(L)) instead, with a closed-formsolution.
**
* * 22'( ( )) || ( ) ( ) ( ) ||E F L w F L F f F I
∇∇
⎛ ⎞= ∇ − ∇ +⎜⎝
• ⎟⎠
∑22(|| ( ) ( ) || )F F Lγ Ψ − ∇
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48
Iteratively optimize L:Update LUpdate Ψ
**
* * 22|| ||w L f I
∇∇
⎛ ⎞∇ ⊗ −∇ +⎜ ⎟
⎝ ⎠∑
1 1 1|| log ( ) ||pλ Ψ 22(|| || )Lγ+ Ψ −∇2
2 2( || || )i ii white
Iλ∈
+ Ψ −∇∑
'( )E Ψ =
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49
Iteratively optimize L:Update LUpdate Ψ
1 1 1|| log ( ) ||pλ Ψ 22(|| || )Lγ+ Ψ −∇2
2 2( || || )i ii white
Iλ∈
+ Ψ −∇∑'( )E Ψ =
'i
iE Ψ=∑
each only contains a single variable Ψi'i
E Ψ
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50
Iteratively optimize L:Update LUpdate Ψ
Iteration 0Iteration 2Iteration 1Iteration 4Iteration 3Iteration 7Iteration 5Iteration 6Iteration 8 (converge)
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51
Time: about 30 seconds for an 800x600 image
Iteration 8 (converge)
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52
A comparison
RL deconvolution
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53
A comparison
Our deconvolution
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54
Two-step iterative optimization• Optimize L• Optimize f
**
* * 22 1( ) || || || ||E f w L f I f
∇∇
⎛ ⎞= ∇ ⊗ −∇ +⎜ ⎟⎝ ⎠∑
1 2min ( , ) min log[ ( ) ( ) ( ) ( )]E L f p n p L p L p f= ∇
A form of L1-norm regularized problem and is solved using an interior point method
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55Iteration 0
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56Iteration 1
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57Iteration 6
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58Iteration 10
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59
Iteration 1
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60
Iteration 2
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61
Iteration 4
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62
Iteration 6
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Iteration 8
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64
Convergence
Time: about 350 seconds for an 800x600 image
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65
Results
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66
Results
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67
Results
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68
Results
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69
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70
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71
More results
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72
More results
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73
More results
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74
More results
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75
More results
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76
More results
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77
More results
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78
More results
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Running Time• Non-blind deconvolution (an 800 x 600 image)
• Blind deconvolution– Approximately 10 minutes for an 800 x 600 image
Approximately 30 seconds in our Matlab implementation
Less than 10 seconds in our C++ implementation
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80
Program Available
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81
Conclusion
• A well-constrained single-image deblurring model
• Advanced optimization
• Improvement is possible for– Spatially-variant blur– Saturated images
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General Blur Images
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General Blur Images
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85
Initialization
• User specified initialization
• Multi-Scale approach– Initialize the clear image in each scale by the one
recovered in a upper layer
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86
Color channel
• For a blur images with a large PSF, we take all the three channels (RGB) into computation
• For a blur image with a small PSF, only the luminance channel are taken into computation