Transcript mf_eccv12_vwt

Filter-based Mean-Field Inference for
Random Fields with Higher-Order
Terms and Product Label-Spaces
Vibhav Vineet*, Jonathan Warrell*,
Philip H.S. Torr
http://cms.brookes.ac.uk/research/visiongroup/
*Joint first authors
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Labelling problems
Many vision problems can be expressed
as dense image labelling problems
Object segmentation
Stereo
Optical flow
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Overview
• Graph cuts so far have proved the method of choice for CRFs
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Overview
• Graph cuts so far have proved the method of choice for CRFs
• Recently message passing methods have started to achieve equal
performance with much faster run times
• But only for pairwise CRFs
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Overview
• Graph cuts so far have proved the method of choice for CRFs
• Recently message passing methods have started to achieve equal
performance with much faster run times
• But only for pairwise CRFs
• Some problems require higher order information
• Co-occurrences terms
• Product label spaces
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Overview
• Graph cuts so far have proved the method of choice for CRFs
• Recently message passing methods have started to achieve equal
performance with much faster run times
• But only for pairwise CRFs
• Some problems require higher order information
• Co-occurrences terms
• Product label spaces
• Our contribution is to develop fast message passing based methods
for certain classes of higher order information
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Importance of co-occurrence terms
Context is an important cue for global scene understanding
Can you identify this object?
Slide courtesy A Torralba
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Importance of co-occurrence terms
We can identify it as keyboard through scene context
Slide courtesy A Torralba
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Importance of co-occurrence terms
The keyboard, table and monitor often co-occur together
Shown to improve accuracy recently in Ladický et al (ECCV ’10)
Slide courtesy A Torralba
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Importance of PN Potts terms
• PN Potts enforce region consistency
• Detector-based PN potentials are
formed by applying grab-cut to
bounding box to create a clique
• Improves over pairwise terms only
Result without
detections
Slide courtesy L Ladicky
Set of detections
Final Result
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Importance of higher order terms
We use higher order information to improve object class
segmentation …
Image
Object labels
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Importance of higher order terms
… and also to improve joint object and stereo labelling using
product label spaces
Image
Object labels
Disparity labels
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CRF formulation
Standard CRF energy formulation
Pairwise
CRF
Data term
Inference
Smoothness term
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CRF formulation
Standard CRF energy formulation
Higher
Order
CRF
Data term
Inference
Smoothness term
Higher order terms
Co-occurrence
term
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Inference
Standard CRF energy
Data term
Smoothness term
Higher order term
Co-occurrence
term
Can be solved using graph-cuts based method
But with co-occurrence ~10 times slower than pairwise only
Relatively fast but still computationally expensive!
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Our inference
Standard CRF energy
Data term
Smoothness term
Higher order term
Co-occurrence
term
We use filter-based mean-field inference approach
Our method achieves almost 10-40 times speed up
compared to graph cuts based methods
Much faster due to efficient filtering
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Efficient inference in pairwise CRF
• Krähenbühl et al (NIPS ’11) propose an efficient method for
inference in pairwise CRF under two assumptions:
• Mean-field approximation to CRF
• Pairwise weights take a linear combination of Gaussian
kernels
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Efficient inference in pairwise CRF
• Krähenbühl et al (NIPS ’11) propose an efficient method for
inference in pairwise CRF under two assumptions:
• Mean-field approximation to CRF
• Pairwise weights take a linear combination of Gaussian
kernels
• They achieve almost 5 times speed up over graph cuts + also
allow dense connectivity
Fully
connected
(dense)
pairwise CRF
Slide courtesy P Krahenbuhl
Inference
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Mean-field based inference
• Mean-field approximation
approximate intractable P with Q from a tractable family
P
• Minimize the KL-divergence between Q and P
Slide courtesy S Nowozin
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Mean-field based inference
• Mean-field update for pairwise terms:
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Mean-field based inference
• Mean-field update for pairwise terms:
• This can be evaluated using Gaussian convolutions
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Mean-field based inference
• Mean-field update for pairwise terms:
• This can be evaluated using Gaussian convolutions
• We evaluate two approaches for Gaussian convolution
• **Permutohedral lattice based filtering
• ***Domain transform based filtering
**Adams
et.al. Fast high-dimensional filtering using the permutohedral lattice. CG-10
***Gasta et.al. Domain transform for edge-aware image and video processing. TOG-11
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Q distribution
Q distribution for different classes across
different iterations
Iteration 0
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Q distribution
Q distribution for different classes across
different iterations
Iteration 1
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0.2
0.3
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0.5
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0.8
0.9
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Q distribution
Q distribution for different classes across
different iterations
Iteration 2
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0.2
0.3
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0.5
0.6
0.7
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0.9
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Q distribution
Q distribution for different classes across
different iterations
Iteration 10
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Higher order mean-field update
• Marginal update in mean-field
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Higher order mean-field update
• Marginal update in mean-field
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Labels:
=1
=2
=3
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Higher order mean-field update
• Marginal update in mean-field
• High time complexity for general higher order terms: O(L|C|)
We show how these can be solved for PN Potts
and co-occurrence terms efficiently
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PN Potts example
PN Potts enforces region consistent labellings
Label set consists of 3 labels
Potts
patterns
Clique of 6
variables
Example: Detector potentials
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Expectation update
Sum across possible states of the clique
Clique takes label l
Clique does not
taking label l
By rearranging the expectation as above, we reduce
the time complexity from O(LN) to O(NL)
Can be extended to pattern-based potentials (Komodakis et al CVPR ’09)
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Global co-occurrence terms
Co-occurrence models which objects belong together
Λ(x)={ aeroplane, tree, flower,
building, boat, grass, sky }


Λ(x)={ building, tree, grass, sky }
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Global co-occurrence terms
Associates a cost with each possible label subset
={
,
,
}
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Global co-occurrence terms
Associates a cost with each possible label subset
={
,
,
}
We use a second order assumption to cost function
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Our model
We define a cost over a set of latent variables: Y{1…L}
Each latent variable represents a label
Y:
Costs include unary and pairwise cost
Each latent variable node is connected
to each image variable node
K
Latent variable binary states:
:on
:off
X:
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Global co-occurrence constraints
Constraint on the model
Constraint violation
Y:
K
K
X:
Pay cost K for each violation
If latent variable is off, no image
variable should take that label
Overall complexity: O(NL+L2)
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Product label space
Assign an object and disparity label to each pixel
Joint energy function defined over product label space:
data term
Left Camera Image
Right Camera Image
smoothness term
Inference in
product label
space
higher order term
Object Class Segmentation
Dense Stereo Reconstruction 37
PascalVOC-10 dataset - qualitative
Image
Ground truth
Fully connected
pairwise CRF*
alpha-expansion**
Ours
Observe an improvement over alternative methods
*Krahenbuhl
et al. Efficient Inference in Fully Connected CRFs with Gaussian Edge Potentials, NIPS 11
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**Ladicky L. et.al. Graph cut based inference with co-occurrence statistics, ECCV-10
PascalVOC - quantitative
Algorithm
Time (s)
Overall
Av. Recall
Av. I/U
AHCRF+Cooc
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81.43
38.01
30.9
Dense pairwise 0.67
71.43
34.53
28.40
Dense pairwise 4.35
+ Potts
79.87
40.71
30.18
Dense pairwise 4.4
+ Potts + Cooc
80.44
43.08
33.2
Observe an improvement of 2.3% in I/U score over Ladicky et.al*
*Ladicky L. et.al. Graph cut based inference with co-occurrence statistics, ECCV-10
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PascalVOC - quantitative
Algorithm
Time (s)
Overall
Av. Recall
Av. I/U
AHCRF+Cooc
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81.43
38.01
30.9
Dense pairwise 0.67
71.43
34.53
28.40
Dense pairwise 4.35
+ Potts
79.87
40.71
30.18
Dense pairwise 4.4
+ Potts + Cooc
80.44
43.08
33.2
Observe an improvement of 2.3% in I/U score over Ladicky et.al*
Achieve 8-9x speed up compared to alpha-expansion based
method of Ladicky et.al*
*Ladicky L. et.al. Graph cut based inference with co-occurrence statistics, ECCV-10
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Leuven dataset - qualitative
Left image
Ground truth
Ours
Right image
Ground truth
Ours
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Leuven dataset - quantitative
Algorithm
Time (s)
Object
(% correct)
Stereo
(% correct)
GC + Range (1)
24.6
95.94
76.97
GC + Range (2)
49.9
95.94
77.31
GC + Range (3)*
74.4
95.94
77.46
Extended CostVol
4.2
95.20
77.18
Dense + HO (PLBF)
3.1
95.24
78.89
Dense + HO (DTBF)
2.1
95.06
78.21
Dense + HO + CostVol +DTBF
6.3
94.98
79.00
Achieve 12-35x speed up compared to alpha-expansion based method of
Ladicky et.al*
*Ladicky L. et.al. Joint optimisation for object class segmentation an dense stereo
reconstruction. BMVC-2010
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Conclusion
• We provide efficient ways of incorporating higher-order terms
into fully connected pairwise CRF models
• Demonstrate improved efficiency compared to previous models
with higher-order terms
• Also demonstrate improved accuracy over previous approaches
• Similar methods applicable to a broad range of vision problems
• Code is available for download:
http://cms.brookes.ac.uk/staff/VibhavVineet/
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EXTRA …
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Joint object-stereo model
Introduce two different set of variables
Xi: object variable
Yi: disparity variable
Z_i: [ x_i y_i ]
Messages exchanged between object and stereo variables
Joint energy function:
Unary
Pairwise
Higher order
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Marginal update for object variables
Message from disparity variables to object variables
Filtering is done using permutohedral lattice based filtering*
strategy
*Adams A. et.al. Fast high-dimensional filtering using the permutohedral lattice.
Computer Graphics Forum-2010
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Marginal update for disparity variables
Message from object variables to disparity variables
Filtering is done using domain transform based filtering* strategy
*Gasta E.S.L. et.al. Domain transform for edge-aware image and video processing.
ACM Trans. Graph.-2011
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Mean-field Vs. Graph-cuts
• Measure I/U score on PascalVOC-10 segmentation
• Increase standard deviation for mean-field
• Increase window size for graph-cuts method
• Both achieve almost similar accuracy
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Mean-field Vs. Graph-cuts
• Measure I/U score on PascalVOC-10 segmentation
• Increase standard deviation for mean-field
• Increase window size for graph-cuts method
•Time complexity very high, making infeasible to work with large neighbourhood system
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Window sizes
• Comparison on matched energy
Algorithm
Model
Time (s)
Av. I/U
Alpha-exp (n=10)
Pairwise
326.17
28.59
Mean-field
pairwise
0.67
28.64
Alpha-exp (n=3)
Pairwise + Potts
56.8
29.6
Mean-field
Pairwise + Potts
4.35
30.11
Alpha-exp (n=1)
Pairwise + Potts
+ Cooc
103.94
30.45
Mean-field
Pairwise + Potts
+ Cooc
4.4
32.17
Impact of adding more complex costs and increasing window size50