Total Variation Blind Deconvolution - University of Oxford
Transcript of Total Variation Blind Deconvolution - University of Oxford
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Engineering Science — University of Oxford
Total Variation Blind Deconvolution: The Devil
is in the Details*Paolo Favaro
Computer Vision Group University of Bern
*Joint work with Daniele Perrone
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Blur in pictures• When we take a picture we expose the sensor of
our camera to the incoming light through the lens
• The lens needs to be placed at the right distance between the scene and the sensor, otherwise…
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Out of focus blur
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Engineering Science — University of Oxford
Blur in pictures• When we take a picture we expose the sensor of
our camera to the incoming light through the lens
• The camera or the scene should not move during the exposure otherwise…
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Motion Blur
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A blur model• When the captured image is blurry then we have no
choice but to try and remove the degradation computationally
• The first step is to model blur degradation
f = k ⇤ u+ n
= * +
blurry image kernel sharp image noise
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Deblurring• When the kernel k is known then we are essentially
inverting a linear system
• Deblurring can be posed as a convex optimization problem
minu
�kukBV +1
2kf � k ⇤ uk22
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Kernel k is known: Deblurring
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Kernel k is known: Deblurring
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Blind deconvolution• Neither the kernel nor the sharp image are known
• We need to recover both the blur and the sharp image
• The problem is non convex
minu,k
�kukBV +1
2kf � k ⇤ uk22
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Blind deconvolution• Neither the kernel nor the sharp image are known
• We need to recover both the blur and the sharp image
• The problem is non convex
minu,k
�kukBV +1
2kf � k ⇤ uk22
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Prior Work
• Before 1996-1998 the general belief was that blind deconvolution was not just impossible, but that it was hopelessly impossible
• How can we extract more data than we observe?
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Ambiguities• The main difficulty in solving blind deconvolution is that
the problem is ill-posed
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Ambiguities• The main difficulty in solving blind deconvolution is that
the problem is ill-posed
• For example, if (u,k) is a solution, then also (au,k/a) and (u(x+d),k(x-d)) for any d and for any a>0 are solutions
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Engineering Science — University of Oxford
Ambiguities• The main difficulty in solving blind deconvolution is that
the problem is ill-posed
• For example, if (u,k) is a solution, then also (au,k/a) and (u(x+d),k(x-d)) for any d and for any a>0 are solutions
• Consider the Fourier transform: F = KU, where F, K and U are Fourier transforms of f, k, and u respectively
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Engineering Science — University of Oxford
Ambiguities• The main difficulty in solving blind deconvolution is that
the problem is ill-posed
• For example, if (u,k) is a solution, then also (au,k/a) and (u(x+d),k(x-d)) for any d and for any a>0 are solutions
• Consider the Fourier transform: F = KU, where F, K and U are Fourier transforms of f, k, and u respectively
• Then, for any K that is not 0 at any frequency there exists always a U such that F = KU (simply let U = F/K)
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The role of the image prior• To reduce the set of ambiguities to a unique sensible answer one
can use a regularization term
• One of the first regularization terms proposed in blind deconvolution was the H1 prior (You and Kaveh 1996)
• Total Variation (strongly related to sparse gradient and natural image prior) was also proposed at the same time (You and Kaveh 1996) kruk2
kruk22
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Chan and Wong (1998)• Total Variation Blind Deconvolution (similar work
appeared earlier in You and Kaveh, 1996)
• Solve
• Use an alternating minimization algorithm (fix the blur and compute the sharp image, then fix the sharp image and compute the blur)
minu,k
�kukBV +1
2kf � k ⇤ uk22
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Chan and Wong (1998)sharp image
restored image
restored image
out of focus
Gaussian blur
it works!
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Fergus et al (2006)• Alternating minimization does not work (MAPu,k)
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Fergus et al (2006)• Alternating minimization does not work (MAPu,k)
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Fergus et al (2006)• Alternating minimization does not work (MAPu,k)
• Use instead a MAPk approach (based on Miskin and McKay 2000)
• Marginalize wrt a distribution of the sharp images
• Compute k by maximizing the marginalized dist.
• Compute u by solving deblurring given k
• Technical details: Use a variational bayesian approach (Jordan et al 1999) and a Gaussian mixture model
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Fergus et al (2006)motion blurred
motion blurred restored
restored
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Shan et al (2008)• Impose that noise is iid
• Use alternating minimization (MAPu,k) but on the image gradients
• Impose that sharp image and blurry image should coincide where the blurry image is very smooth
• Then estimate sharp image given kernel k
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Shan et al (2008)
motion blurred iteration 1 iteration 6 iteration 10
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Cho and Lee (2009)• Success of prior work is: Sharp edge restoration and noise
suppression in smooth regions
• Blur can be estimated reliably at edges
• Try and predict edges with a shock filter
• Use a modified alternating minimization (MAPu,k)
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Cho and Lee (2009)motion blurred
restored
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Xu et al (2013)
• Use a saturated L1 prior (they call it unnatural L0)
• Use alternating minimization (MAPu,k)
• Technical details: Many intermediate steps
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Xu et al (2013)
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Levin et al (2011)
• Stop using MAPu,k! It should not work! Use MAPk
• Compare the following true solution (u,k) with the no-blur solution (f, )
• Then, solution is based only on the image prior; however, the prior favors the no-blur solution!
�
f ⌘ � ⇤ f ⌘ k ⇤ u
krfk2 kruk2
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Levin et al (2011)
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MAPk• After marginalization Levin et al. 2011 obtain
• which is an alternating minimization
• weights are updated sequentially
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A conundrum• On the one hand many MAPu,k implementations
and (heuristic) variants work very well, and on the other hand they are not supposed to work at all
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A conundrum• On the one hand many MAPu,k implementations
and (heuristic) variants work very well, and on the other hand they are not supposed to work at all
• Rather than developing yet another blind deconvolution algorithm, should we not try to understand what is going on first?
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Engineering Science — University of Oxford
A conundrum• On the one hand many MAPu,k implementations
and (heuristic) variants work very well, and on the other hand they are not supposed to work at all
• Rather than developing yet another blind deconvolution algorithm, should we not try to understand what is going on first?
• Could MAPk be just another recipe for MAPu,k?
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Recent analysis• Wipf and Zhang arxiv 2013: MAPk equivalent to a MAPu,k
• See also Babacan et al. 2012 and Krishnan et al. 2014
MAPk
MAPu,k
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Recent analysis
• So, current conclusion is that it is not about MAPk vs MAPu,k, but rather about the choice of priors
• Still, this does not explain why current so-called MAPu,k approaches (that use TV-like priors) work
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Removing the bells and whistles
• We start by applying the golden rule in analysis: Remove all the unnecessary
• Result: Total Variation Blind Deconvolution (1996!)
min
u,k�J(u) + kk ⇤ u� fk22
subject to k < 0, kkk1 = 1
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Attempt #1: Exact solution
• The alternating minimization (AM) algorithm
• Actually, it does not work!
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AM does not work
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A toy example in 1D• Let us consider a 1D signal (a hat function) and a 1D
blur of 3 pixels
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A toy example in 1D• Let us consider a 1D signal (a hat function) and a 1D
blur of 3 pixels
• Because the blur components add to 1, we only have 2 free parameters
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A toy example in 1D• Let us consider a 1D signal (a hat function) and a 1D
blur of 3 pixels
• Because the blur components add to 1, we only have 2 free parameters
• For each possible combination of these parameters we minimize the TV problem wrt the sharp image (a deblurring problem)
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A toy example in 1D• Let us consider a 1D signal (a hat function) and a 1D
blur of 3 pixels
• Because the blur components add to 1, we only have 2 free parameters
• For each possible combination of these parameters we minimize the TV problem wrt the sharp image (a deblurring problem)
• We show the energy at the minimum wrt the sharp image for each possible blur
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A toy example in 1Dk[2]
k[1]
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
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A toy example in 1Dk[2]
k[1]
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
true minimum
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A toy example in 1Dk[2]
k[1]
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
true minimum
initial solution
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A toy example in 1Dk[2]
k[1]
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
true minimum
initial solution
value of energy at no-blur solutions is lower than at the true minimum
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Attempt #2: Approximate solution• Projected alternating minimization (PAM) —
implementation of Chang and Wong (1998)
• It works!
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Where’s Wally?
• What is the difference between AM and PAM that makes PAM work?
• Why does it make it work?
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Comparison of AM and PAM• The first step (image deblurring) is identical
• The second step separates the normalization and the positivity constraints from the minimization step
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k[2]
k[1]
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
A gradient descent??
� = 0.01final solution
initial solution
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Normalization is the keyk[2]
k[1]
1 2
1
2
k[2]
k[1]
1 2
1
2
k[2]
k[1]
1 2
1
2� = 0.1
kkk1 = 1 kkk1 = 1.5 kkk1 = 2.5
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k[2]
k[1]
1 2
1
2� = 1.5
k[2]
k[1]
1 2
1
2
k[2]
k[1]
1 2
1
2
Normalization is the key
kkk1 = 1 kkk1 = 1.5 kkk1 = 2.5
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k[2]
k[1]
1 2
1
2
k[2]
k[1]
1 2
1
2
k[2]
k[1]
1 2
1
2
Normalization is the key
kkk1 = 1 kkk1 = 1.5 kkk1 = 2.5
� = 2.5
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AM on a step function
−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
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Engineering Science — University of Oxford
AM on a step function
−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
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AM on a step function
−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
AM on a step function
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
AM on a step function
no-blur error
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
AM on a step function
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV SignalBlurred TV Signal
AM on a step function
additional true-blur error
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalScaled TV Signal
PAM on a step function
−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalTV Signal
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalScaled TV Signal
PAM on a step function
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Engineering Science — University of Oxford−10 −5 0 5 10
−0.5
0
0.5
x
f[x]
Blurred signalSharp SignalScaled TV Signal
PAM on a step function
Detailed proofs of convergence of PAM in CVPR 2014
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Technical details
• As in most current implementations we used a pyramid scheme
• Adaptation of the regularization parameter is needed
• Boundary conditions: None as we use the exact blur model
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The PAM algorithm
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Engineering Science — University of Oxford
Experiments
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ourLevinChoFergus
error ratio
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Blurry image
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Cho and Lee (2009)
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Fergus et al (2006)
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Hirsch et al (2011)
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Shan et al (2008)
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Whyte et al (2011)
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Xu and Jia (2010)
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Our (PAM)
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Blurred
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Xu and Jia (2010)
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Our (PAM)
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One more exampleblurry Cho and Lee (2009) Goldstein and
Fattal (2012)
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our (PAM)One more example
Zhong et al (2013) Levin et al (2011)
be wary of the results of others!
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our (PAM)One more example
Zhong et al (2013) Levin et al (2011)
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Engineering Science — University of Oxford
Conclusions• We have shown (with theory and experiments) why
many alternating minimization algorithms work
• The reason lies in the normalization of blur (scaling) + regularization parameter
• This 1998 algorithm competes very well with recent more sophisticated algorithms
• Perhaps we should rethink our formulation of blind deconvolution?