Slides - Uwe Schmidt
A Generative Perspective on MRFs in Low-Level Vision
Uwe Schmidt
Qi Gao
Stefan Roth
Department of Computer Science TU Darmstadt Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010
Low-Level Vision SuperResolution
Discriminative ! performance Generative " versatility ! versatility " learning
Image Restoration Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 2
Stereo
Optical Flow
Low-Level Vision SuperResolution
Generative
MRF
Stereo
Priors
Image Restoration Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 2
Optical Flow
Common MRF Evaluation MRF prior
p(x)
Application likelihood
p(y|x)
y generate/ measure
Posterior
p(x|y)
MAP estimation
Indirect model evaluation
(Gradient methods, Graph cuts, ...)
^ x
Compare # Measure PSNR, ...
Ground truth
MAP estimate
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 3
Desirable MRF Evaluation ■ Purpose of MRF priors • Model statistical properties of
Difficult!
natural images and scenes
$ Evaluate generative properties [Zhu & Mumford ’97]
MRF prior Draw samples (MCMC)
• e.g. derivative statistics of the model • neglected ever since
Compare statistical properties
Data
MRF samples
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 4
Agenda 1. Evaluate generative properties of common image priors • Pairwise & high-order MRFs • Based on a flexible MRF framework with an efficient sampler 2. Learn improved generative models 3. Find that in the context of MAP estimation our models do not perform as well as expected for image denoising 4. Address this problem (and others) by changing the estimator Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 5
Flexible MRF Model ■ Fields-of-Experts (FoE) framework [Roth & Black ’05, ’09] • Subsumes popular pairwise & high-order MRFs N YY 2 1 ⇤x⇤ /2
p(x;
)=
Z( )
Image
e
Expert function
T Ji x(c) ; ↵i
c⇥C i=1
Linear filter Parameters
e.g.
= {Ji , ↵i } i = 1, . . . , N
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 6
Vector of nodes in clique c
Flexible MRF Model ■ Fields-of-Experts (FoE) framework [Roth & Black ’05, ’09] • Subsumes popular pairwise & high-order MRFs N YY 2 1 ⇤x⇤ /2
p(x; −1
10
)=
Z( )
e
Mixture Weights
T Ji x(c) ; ↵i
c⇥C i=1
GSM example
Gaussian Scale Mixture (GSM) Distribution
−3
10
[Wainwright & Simoncelli ’99, Weiss & Freeman ’07]
−5
10
−200
−100
⇤(JT i x(c) ; ↵i ) =
XJ
0
j=1
100
ij
200
2 · N (JT x ; 0, ⇥ i (c) i /sj )
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 7
Sampling from the MRF ■ Obtain joint distribution:
Mixture Weights
• Product of GSMs = GSM • Augment MRF with auxiliary variables z for the mixture components and do not marginalize them out
X z
Gaussian
p(z) p(x|z) | {z } p(x, z)
■ Gibbs sampling from the joint distribution p(x, z; ) [Geman & Yang ’95; Welling et al. ’02] p(x|z) ) and p(z|x; p(z|x) ) • Alternate block sampling from p(x|z; • The z can be discarded in the end • Least-squares method for sampling p(x|z; ) [Weiss ’05, Levi ’09] Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 8
MRF Sampling – Example Pairwise MRF
High-order MRF with 3 3 cliques
Subsequent iterations of the Gibbs sampler
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 9
Generative Properties of Pairwise MRFs 30 25 20 15
■ Consider simplest pairwise MRFs
−90 −95
MRF
−100 −105
−1
10
Potential function Fit to the Generalized marginals Laplacian
−3
−1
10
Derivative marginals KLD=1.57 Natural KLD=1.37 images Marginal KL-divergence
−3
10
10
−5
−5
10
10 −200 −100
0
100
200
−200 −100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 10
100
200
Generative Properties of High-order MRFs [Roth & Black ’09] 24 5 5 filters Student-t experts
■ Common FoE models • Evaluate filter statistics of model filters Ji
■ Apparent contradiction: " Poor generative properties
−1
10
Natural Images
−3
10
−5
10
KLD=2.10
−1
Why?
KLD=5.26
10
! Good application performance
[Weiss & Freeman ’07] 25 15 15 filters fixed GSM experts
MRF Samples
−3
10
−5
10
−150 −75
0
75
150
−150 −75
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 11
0
75
150
Learning Better Generative MRFs ■ Learn shapes of flexible GSM experts and linear filters Jii Ji (for high-order model) • Use efficient sampler • Otherwise training similar to [Roth & Black ’09]
■ Learned models: 1. Pairwise MRF with single GSM potential (fixed first-derivative filters)
2. FoE with 3 3 cliques and 8 GSM experts (including filters) Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 12
Generative Properties of Our Pairwise MRF ■ Our pairwise MRF compared to previously shown
200
−100
250
100
−200
200
0
−300
150
30 25 20 15
−400
−90 −95
MRF
−100 −105
−1
10
Potential function Our learned GSM
−1
10
−3
Derivative marginals KLD=0.006 Natural images
−3
10
10
−5
−5
10
10 −200 −100
0
100
200
−200 −100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 13
100
200
15 10 50 0 −5
Our Learned FoE in Comparison Our learned 3 3 FoE
GSM −150 −75
0
75
150
[Roth & Black ’09]
[Weiss & Freeman ’07]
Student-t
GSM
−150 −75
0
75
150
−150 −75
Learned linear filters
Much more peaked!
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 14
0
75
150
Generative Properties of our FoE ■ Filter statistics of our learned 3 3 FoE • Much better than previous models • Room for improvement −1
10
Natural Images
KLD=0.08
MRF Samples
−3
10
−5
10
−150 −75
0
75
150
−150 −75
0
75
150
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 15
Image Denoising ■ Image denoising assuming i.i.d. Gaussian noise with known standard deviation σ Gaussian Likelihood
p(x|y;
)
N y; x, λ=1
2
Our 3 3 FoE
I · p(x;
)
Regularization weight
optimal λ
[Roth & Black ’09] MAP, optimal λ
PSNR=29.18dB
PSNR=30.06dB
MAP
PSNR=22.18dB
PSNR=26.64dB
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 16
Image Denoising – MAP ■ Recent works point to deficiencies of MAP [Nikolova ’07, Woodford et al. ’09] ■ We find only modest correlation between: • Image quality of the MAP estimate • Generative quality of the MRF Better generative properties
!
Better application performance
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 17
Image Denoising – MMSE ■ We propose to use Bayesian minimum mean squared error estimation (MMSE) Z 2 ˆ = arg min ||˜ x x x|| p(x|y; ) dx = E[x|y] ˜ x
Samples
average
■ [Levi ’09] extended sampler to the posterior • Only used a single sample in applications
■ We approximate the MMSE estimate • Average samples from the posterior
■ We find high correlation between: • Image quality of the MMSE estimate • Generative quality of the MRF Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 18
MMSE
Image Denoising – Results ■ Compared the MMSE estimate for our learned models with other popular methods Average PSNR (dB) for 68 test images (σ = 25) 5 5 FoE [Roth & Black ’09] – MAP w/λ 27.44 Non-local means [Buades et al. ’05] pairwise (ours) – MMSE
27.50 27.54 27.86
5 5 FoE [Samuel & Tappen ’09] – MAP
27.95
3 3 FoE (ours) – MMSE
28.02
BLS-GSM [Portilla et al. ’03] – (MMSE) 26.5
27.0
27.5
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 19
28.0
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics MAP
MMSE
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics Derivative marginals
• No piecewise constant regions −1
10 MAP
MMSE
−3
10
−100
0
100
Original images (noise-free) Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics Derivative marginals
• No piecewise constant regions
MAP
KLD=1.64
−1
10 MAP
MMSE
−3
10
−100
0
100
[Woodford et al. ’09] Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics • No piecewise constant regions • Works with standard MRFs MAP
MMSE
Derivative marginals −1
10
MAP MMSE
KLD=1.64 KLD=0.03
−3
10
−100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
100
Summary ■ Evaluated MRFs through their generative properties • Based on a flexible framework with an efficient sampler
■ Common image priors are surprisingly poor generative models ■ Learned better generative MRFs (pairwise & high-order) • Potentials more peaked
■ Sampling makes MMSE estimation practical • Several advantages over MAP • Excellent results from generative, application-neutral models
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 21
Thanks! Acknowledgments: Yair Weiss, Arjan Kuijper, Michael Goesele, Kegan Samuel, Marshall Tappen
Please come to our poster! Code and models available soon at http://bit.ly/mmse-mrf
Questions? Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 22
More Generative Properties KLD
Random filters 10
−1
9 9 7 7 5 5 3 3
0.13 0.09
−3
10
0
−5
0
75
150
Natural images
1
−150 −75
0
75
150
Our pairwise MRF
−150 −75
0
75
10
150
Our 3 3 FoE
KLD
−1
KLD
0.09 0.04 0
−3
10
2
0
10
−150 −75
Multiscale derivative filters
KLD
0
−5
10
4
−150 −75
0
75
150
Natural images
−150 −75
0
75
150
Our pairwise MRF
−150 −75
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 26
0
75
150
Our 3 3 FoE
Uwe Schmidt
Qi Gao
Stefan Roth
Department of Computer Science TU Darmstadt Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010
Low-Level Vision SuperResolution
Discriminative ! performance Generative " versatility ! versatility " learning
Image Restoration Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 2
Stereo
Optical Flow
Low-Level Vision SuperResolution
Generative
MRF
Stereo
Priors
Image Restoration Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 2
Optical Flow
Common MRF Evaluation MRF prior
p(x)
Application likelihood
p(y|x)
y generate/ measure
Posterior
p(x|y)
MAP estimation
Indirect model evaluation
(Gradient methods, Graph cuts, ...)
^ x
Compare # Measure PSNR, ...
Ground truth
MAP estimate
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 3
Desirable MRF Evaluation ■ Purpose of MRF priors • Model statistical properties of
Difficult!
natural images and scenes
$ Evaluate generative properties [Zhu & Mumford ’97]
MRF prior Draw samples (MCMC)
• e.g. derivative statistics of the model • neglected ever since
Compare statistical properties
Data
MRF samples
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 4
Agenda 1. Evaluate generative properties of common image priors • Pairwise & high-order MRFs • Based on a flexible MRF framework with an efficient sampler 2. Learn improved generative models 3. Find that in the context of MAP estimation our models do not perform as well as expected for image denoising 4. Address this problem (and others) by changing the estimator Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 5
Flexible MRF Model ■ Fields-of-Experts (FoE) framework [Roth & Black ’05, ’09] • Subsumes popular pairwise & high-order MRFs N YY 2 1 ⇤x⇤ /2
p(x;
)=
Z( )
Image
e
Expert function
T Ji x(c) ; ↵i
c⇥C i=1
Linear filter Parameters
e.g.
= {Ji , ↵i } i = 1, . . . , N
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 6
Vector of nodes in clique c
Flexible MRF Model ■ Fields-of-Experts (FoE) framework [Roth & Black ’05, ’09] • Subsumes popular pairwise & high-order MRFs N YY 2 1 ⇤x⇤ /2
p(x; −1
10
)=
Z( )
e
Mixture Weights
T Ji x(c) ; ↵i
c⇥C i=1
GSM example
Gaussian Scale Mixture (GSM) Distribution
−3
10
[Wainwright & Simoncelli ’99, Weiss & Freeman ’07]
−5
10
−200
−100
⇤(JT i x(c) ; ↵i ) =
XJ
0
j=1
100
ij
200
2 · N (JT x ; 0, ⇥ i (c) i /sj )
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 7
Sampling from the MRF ■ Obtain joint distribution:
Mixture Weights
• Product of GSMs = GSM • Augment MRF with auxiliary variables z for the mixture components and do not marginalize them out
X z
Gaussian
p(z) p(x|z) | {z } p(x, z)
■ Gibbs sampling from the joint distribution p(x, z; ) [Geman & Yang ’95; Welling et al. ’02] p(x|z) ) and p(z|x; p(z|x) ) • Alternate block sampling from p(x|z; • The z can be discarded in the end • Least-squares method for sampling p(x|z; ) [Weiss ’05, Levi ’09] Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 8
MRF Sampling – Example Pairwise MRF
High-order MRF with 3 3 cliques
Subsequent iterations of the Gibbs sampler
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 9
Generative Properties of Pairwise MRFs 30 25 20 15
■ Consider simplest pairwise MRFs
−90 −95
MRF
−100 −105
−1
10
Potential function Fit to the Generalized marginals Laplacian
−3
−1
10
Derivative marginals KLD=1.57 Natural KLD=1.37 images Marginal KL-divergence
−3
10
10
−5
−5
10
10 −200 −100
0
100
200
−200 −100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 10
100
200
Generative Properties of High-order MRFs [Roth & Black ’09] 24 5 5 filters Student-t experts
■ Common FoE models • Evaluate filter statistics of model filters Ji
■ Apparent contradiction: " Poor generative properties
−1
10
Natural Images
−3
10
−5
10
KLD=2.10
−1
Why?
KLD=5.26
10
! Good application performance
[Weiss & Freeman ’07] 25 15 15 filters fixed GSM experts
MRF Samples
−3
10
−5
10
−150 −75
0
75
150
−150 −75
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 11
0
75
150
Learning Better Generative MRFs ■ Learn shapes of flexible GSM experts and linear filters Jii Ji (for high-order model) • Use efficient sampler • Otherwise training similar to [Roth & Black ’09]
■ Learned models: 1. Pairwise MRF with single GSM potential (fixed first-derivative filters)
2. FoE with 3 3 cliques and 8 GSM experts (including filters) Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 12
Generative Properties of Our Pairwise MRF ■ Our pairwise MRF compared to previously shown
200
−100
250
100
−200
200
0
−300
150
30 25 20 15
−400
−90 −95
MRF
−100 −105
−1
10
Potential function Our learned GSM
−1
10
−3
Derivative marginals KLD=0.006 Natural images
−3
10
10
−5
−5
10
10 −200 −100
0
100
200
−200 −100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 13
100
200
15 10 50 0 −5
Our Learned FoE in Comparison Our learned 3 3 FoE
GSM −150 −75
0
75
150
[Roth & Black ’09]
[Weiss & Freeman ’07]
Student-t
GSM
−150 −75
0
75
150
−150 −75
Learned linear filters
Much more peaked!
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 14
0
75
150
Generative Properties of our FoE ■ Filter statistics of our learned 3 3 FoE • Much better than previous models • Room for improvement −1
10
Natural Images
KLD=0.08
MRF Samples
−3
10
−5
10
−150 −75
0
75
150
−150 −75
0
75
150
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 15
Image Denoising ■ Image denoising assuming i.i.d. Gaussian noise with known standard deviation σ Gaussian Likelihood
p(x|y;
)
N y; x, λ=1
2
Our 3 3 FoE
I · p(x;
)
Regularization weight
optimal λ
[Roth & Black ’09] MAP, optimal λ
PSNR=29.18dB
PSNR=30.06dB
MAP
PSNR=22.18dB
PSNR=26.64dB
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 16
Image Denoising – MAP ■ Recent works point to deficiencies of MAP [Nikolova ’07, Woodford et al. ’09] ■ We find only modest correlation between: • Image quality of the MAP estimate • Generative quality of the MRF Better generative properties
!
Better application performance
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 17
Image Denoising – MMSE ■ We propose to use Bayesian minimum mean squared error estimation (MMSE) Z 2 ˆ = arg min ||˜ x x x|| p(x|y; ) dx = E[x|y] ˜ x
Samples
average
■ [Levi ’09] extended sampler to the posterior • Only used a single sample in applications
■ We approximate the MMSE estimate • Average samples from the posterior
■ We find high correlation between: • Image quality of the MMSE estimate • Generative quality of the MRF Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 18
MMSE
Image Denoising – Results ■ Compared the MMSE estimate for our learned models with other popular methods Average PSNR (dB) for 68 test images (σ = 25) 5 5 FoE [Roth & Black ’09] – MAP w/λ 27.44 Non-local means [Buades et al. ’05] pairwise (ours) – MMSE
27.50 27.54 27.86
5 5 FoE [Samuel & Tappen ’09] – MAP
27.95
3 3 FoE (ours) – MMSE
28.02
BLS-GSM [Portilla et al. ’03] – (MMSE) 26.5
27.0
27.5
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 19
28.0
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics MAP
MMSE
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics Derivative marginals
• No piecewise constant regions −1
10 MAP
MMSE
−3
10
−100
0
100
Original images (noise-free) Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics Derivative marginals
• No piecewise constant regions
MAP
KLD=1.64
−1
10 MAP
MMSE
−3
10
−100
0
100
[Woodford et al. ’09] Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
Advantages of the MMSE ■ Denoising performance highly correlated with the generative quality of the model ■ No regularization weight λ required to perform well ■ Denoised image does not exhibit incorrect statistics • No piecewise constant regions • Works with standard MRFs MAP
MMSE
Derivative marginals −1
10
MAP MMSE
KLD=1.64 KLD=0.03
−3
10
−100
0
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 20
100
Summary ■ Evaluated MRFs through their generative properties • Based on a flexible framework with an efficient sampler
■ Common image priors are surprisingly poor generative models ■ Learned better generative MRFs (pairwise & high-order) • Potentials more peaked
■ Sampling makes MMSE estimation practical • Several advantages over MAP • Excellent results from generative, application-neutral models
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 21
Thanks! Acknowledgments: Yair Weiss, Arjan Kuijper, Michael Goesele, Kegan Samuel, Marshall Tappen
Please come to our poster! Code and models available soon at http://bit.ly/mmse-mrf
Questions? Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 22
More Generative Properties KLD
Random filters 10
−1
9 9 7 7 5 5 3 3
0.13 0.09
−3
10
0
−5
0
75
150
Natural images
1
−150 −75
0
75
150
Our pairwise MRF
−150 −75
0
75
10
150
Our 3 3 FoE
KLD
−1
KLD
0.09 0.04 0
−3
10
2
0
10
−150 −75
Multiscale derivative filters
KLD
0
−5
10
4
−150 −75
0
75
150
Natural images
−150 −75
0
75
150
Our pairwise MRF
−150 −75
Uwe Schmidt, Qi Gao, Stefan Roth: A Generative Perspective on MRFs in Low-Level Vision | CVPR 2010 | 26
0
75
150
Our 3 3 FoE
