Transcript Cluster Algorithms and Applications in Computer Vision

Active Random Fields
Adrian Barbu
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The MAP Estimation Problem
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Estimation problem:
Given input data y, solve
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Example: Image denoising
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Given noisy image y, find denoised image x
Issues
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Modeling: How to approximate
Computing: How to find x fast?
Noisy image y
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Denoised image x
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MAP Estimation Issues
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Popular approach:
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Problems with this approach
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Find a very accurate model
Find best optimum x of that model
Hard to obtain good
Desired solution needs to be at global maximum
For many models
, the global maximum cannot be
obtained in any reasonable time.
Using suboptimal algorithms to find the maximum leads to
suboptimal solutions
E.g. Markov Random Fields
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Markov Random Fields
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Bayesian Models:
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Markov Random Field (MRF) prior
E.g. Image Denoising model
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Gaussian Likelihood
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Fields of Experts MRF prior
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Differential Lorentzian
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Image filters Ji
Image Filters Ji
Roth and Black, 2005
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MAP Estimation (Inference) in MRF
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Exact inference is too hard
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For the Potts model, one of the simplest MRFs
it is already NP hard (Boykov et al, 2001)
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Approximate inference is suboptimal
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Gradient descent
Iterated Conditional Modes (Besag 1986)
Belief Propagation (Yedidia et al, 2001)
Graph Cuts (Boykov et al, 2001)
Tree-Reweighted Message Passing (Wainwright et al, 2003)
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Gradient Descent for Fields of Experts
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Energy function:
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Analytic gradient (Roth & Black, 2005)
FOE filters
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Gradient descent iterations
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3000 iterations with small 
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Takes more than 30 min per image on a modern computer
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Training the MRF
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Gradient update in model parameters
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Minimize KL divergence between learned prior and true probability
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Gradient ascent in log-likelihood
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Need to know Normalization Constant Z
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EX from training data
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Z and Ep obtained by MCMC
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Slow to train
Suf ficient
statistics
Update m
Training the FOE prior
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Contrastive divergence (Hinton)
Model m
MCMC samples
f rom current
model
Natural Images
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An approximate ML technique
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Initialize at data points and run a fixed number of iterations
Trainining Phase
Takes about two days
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Going to Real-Time Performance
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Wainwright (2006)
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In computation-limited settings, MAP estimation is not the best
choice
Some biased models could compensate for the fast inference
algorithm
How much can we gain from biased models?
Proposed denoising approach:
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1-4 gradient descent iterations (not 3000)
Takes less than a second per image
1000-3000 times speedup vs MAP estimation
Better accuracy than FOE model
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Active Random Field
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Active Random Field = A pair (M,A) of
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a MRF model M, with parameters M
a fast and suboptimal inference algorithm A with parameters A
They cannot be separated since they are trained together
E.g. Active FOE for image denoising
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Fields of Experts model
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Algorithm: 1-4 iterations of gradient descent
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Parameters:
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Training the Active Random Field
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Discriminative training
Training examples = pairs
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inputs yi+ desired outputs ti
Training=optimization
Benchmark
Measure
Update m
and a
Desired results
Model m
Algorithm a
Current results
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Loss function L
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Input images
Trainining Phase
Aka benchmark measure
Evaluates accuracy on training set
End-to-end training:
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covers entire process from input image to final result
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Related Work
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Energy Based Models (LeCun & Huang, 2005)
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Train a MRF energy model to have minima
close to desired locations
Assumes exact inference (slow)
Shape Regression Machine (Zhou & Comaniciu, 2007)
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Train a regressor to find an object
Uses a classifier to clean up result
Aimed for object detection, not MRFs
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Related Work
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Training model-algorithm combinations
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CRF based on pairwise potential trained for object classification
Torralba et al, 2004
AutoContext: Sequence of CRF-like boosted classifiers for object
segmentation, Tu 2008
Both minimize a loss function and report results on another
loss function (suboptimal)
Both train iterative classifiers that are more and more complex
at each iteration – speed degrades quickly for improving
accuracy
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Related Work
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Training model-algorithm combinations and reporting results
on the same loss function for image denoising
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Tappen, & Orlando, 2007 - Use same type of training for obtaining
a stronger MAP optimum in image denoising
Gaussian Conditional Random Fields: Tappen et al, 2007 – exact
MAP but hundreds of times slower. Results comparable with 2iteration ARF
Common theme: trying to obtain a strong MAP optimum
This work: fast and suboptimal estimator balanced by a
complex model and appropriate training
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Training Active Fields of Experts
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Training set
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40 images from the Berkeley dataset (Martin 2001)
Same as Roth and Black 2005
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Separate training for each noise level 
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Loss function L = PSNR
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Same measure used for reporting results
Trained Active
FOE filters, niter=1
is the standard deviation of
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Training 1-Iteration ARF, =25
Follow Marginal Space Learning
 Consider a sequence of subspaces
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Represent marginals by propagating
particles between subspaces
Propagate only one particle (mode)
1. Start with one filter, size 3x3
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2.
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Retrain to find the new mode
Repeat step 2 until there are 5 filters
Increase filters to 5x5
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5.
Train until no improvement
We found the particle in this subspace
Add another filter initialized with zeros
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3.
PSNR
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Training Data
Testing Data (unseen)
0
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2
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Steps x10000
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5
4
x 10
PSNR training (blue), testing (red)
while training the 1-iteration ARF, =25 filters
Retrain to find new mode
Repeat step 2 until there are 13 filters
1 Filter
3x3
2 Filters
3x3
5 Filters
3x3
5 Filters
5x5
6 Filters
5x5
13 Filters
5x5
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Training other ARFs
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Other levels initialized as
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Start with one iteration, =25
Each arrow takes about one day on a 8-core machine
3-iteration ARFs can also perform 4 iterations
niter=1
=10
niter=1
=15
niter=1
=20
niter=1
=25
niter=1
=50
niter=2
=10
niter=2
=15
niter=2
=20
niter=2
=25
niter=2
=50
niter=3
=10
niter=3
=15
niter=3
=20
niter=3
=25
niter=3
=50
niter=4
=10
niter=4
=15
niter=4
=20
niter=4
=25
niter=4
=50
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Concerns about Active FOE
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Avoid overfitting
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Use large patches to avoid boundary effect
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Totally 6 million nodes
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Full size images instead of smaller patches
Lots of training data
Use a validation set to detect overfitting
Long training time
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Easily parallelizable
1 -3 days on a 8 core PC
Good news: CPU power increases exponentially (Moore’s law)
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Results
Original Image
Corrupted with Gaussian
noise, =25, PSNR=20.17
4-iteration ARF,
PSNR=28.94, t=0.6s
3000-iteration FOE,
PSNR=28.67, t=2250s
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Standard Test Images
Lena
House
Barbara
Peppers
Boats
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Evaluation, Standard Test Images
Lena
Barbara
Boats
House
Peppers
Average
FOE (Roth & Black, 2005)
30.82
27.04
28.72
31.11
29.20
29.38
Active FOE, 1 iteration
30.15
27.10
28.66
30.14
28.90
28.99
Active FOE, 2 iterations
30.66
27.49
28.99
30.80
29.31
29.45
Active FOE, 3 iterations
30.76
27.57
29.08
31.04
29.45
29.58
Active FOE, 4 iterations
30.86
27.59
29.14
31.18
29.51
29.66
Wavelet Denoising (Portilla et al, 2003)
31.69
29.13
29.37
31.40
29.21
30.16
Overcomplete DCT (Elad et al, 2006)
30.89
28.65
28.78
31.03
29.01
29.67
Globally Trained Dictionary (Elad et al, 2006)
31.20
27.57
29.17
31.82
29.84
29.92
KSVD (Elad et al, 2006)
31.32
29.60
29.28
32.15
29.73
30.42
BM3D (Dabov et al, 2007)
32.08
30.72
29.91
32.86
30.16
31.15
noise=25
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Evaluation, Berkeley Dataset
68 images from the Berkeley dataset
 Not used for training, not overfitted by other methods.
 Roth & Black ‘05 also evaluated on them.
 A more realistic evaluation than on 5 images.
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Evaluation, Berkeley Dataset
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PSNR
PSNR
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Algorithm, sigma=20 (PSNR=22.11)
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PSNR
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Algorithm, sigma = 50 (PSNR=14.15)
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Algorithm, sigma = 25 (PSNR=20.17)
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Average PSNR on 68 images from the Berkeley dataset,
not used for training.
1: Wiener Filter
2: Nonlinear diffusion
3: Non-local means (Buades et al, 2005)
4: FOE model, 3000 iterations,
5,6,7,8: Our algorithm with 1,2,3 and 4 iterations
9: Wavelet based denoising (Portilla et al, 2003)
10: Overcomplete DCT (Elad et al, 2006)
11: KSVD (Elad et al, 2006)
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12: BM3D (Dabov et al, 2007)
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Speed-Performance Comparison
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ARF
BM3D
KSVD
DCT
Wavelet
FOE
Nonlocal
NonLinDiff
Wiener
28.5
PSNR
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27.5
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26.5
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noise=25
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Frames per second
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Performance on Different Levels of Noise
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3000-iter. FOE
1-iter. ARF, sigma=10
1-iter. ARF, sigma=15
1-iter. ARF, sigma=20
1-iter. ARF, sigma=25
1-iter. ARF, sigma=50
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PSNR
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Level of noise sigma
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Trained for a specific noise level
No data term
Band-pass behavior
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Adding a Data Term
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Active FOE
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1-iteration version has no data term
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Modification with data term
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Equivalent
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Performance with Data Term
3000-iteration FOE
1-iteration FOE
1-iter FOE,retrained coeffs
ARF, no data term, train 15-25
ARF w/ data term, train 15-25
ARF w/ data term, train 15-40
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PSNR
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Level of noise sigma
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Data term removes band-pass behavior
1-iteration ARF as good as 3000-iteration FOE for a range of
noises
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Conclusion
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An Active Random Field is a pair of
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Training = optimization of the MRF and algorithm parameters using
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A benchmark measure on which the results will be reported
Training data as pairs of input and desired output
Pros
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A Markov Random Field based model
A fast, approximate inference algorithm (estimator)
Great speed and accuracy
Good control of overfitting using a validation set
Cons
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Slow to train
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Future Work
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Extending image denoising
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Learning filters over multiple channels
Learning the robust function
Learn filters for image sequences using temporal coherence
Other applications
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Computer Vision:
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Edge and Road detection, Image segmentation
Stereo matching, motion, tracking etc
Medical Imaging
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Learning a Discriminative Anatomical Network
of Organ and Landmark Detectors
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References
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A. Barbu. Training an Active Random Field for Real-Time Image Denoising. IEEE Trans. Image Processing,
18, November 2009.
Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. Pattern Analysis
and Machine Intelligence, IEEE Transactions on, 23(11):1222–1239, 2001.
A. Buades, B. Coll, and J.M. Morel. A Non-Local Algorithm for Image Denoising. Computer Vision and Pattern
Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, 2, 2005.
K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian. Image Denoising by Sparse 3-D Transform-Domain
Collaborative Filtering. Image Processing, IEEE Transactions on, 16(8):2080–2095, 2007.
M. Elad and M. Aharon. Image denoising via sparse and redundant representations over learned dictionaries.
IEEE Trans. Image Process, 15(12):3736–3745, 2006.
G.E. Hinton. Training Products of Experts by Minimizing Contrastive Divergence. Neural Computation,
14(8):1771–1800, 2002.
Y. LeCun and F.J. Huang. Loss functions for discriminative training of energy-based models. Proc. of the 10thInternational Workshop on Artificial Intelligence and Statistics (AIStats 05), 3, 2005.
D. Martin, C. Fowlkes, D. Tal, and J. Malik. A Database of Human Segmented Natural Images and its
Application to Evaluating Segmentation Algorithms. Proc. of ICCV01, 2:416–425.
J. Portilla, V. Strela, MJ Wainwright, and EP Simoncelli. Image denoising using scale mixtures of Gaussians
in the wavelet domain. Image Processing, IEEE Transactions on, 12(11):1338–1351, 2003.
S. Roth and M.J. Black. Fields of Experts. International Journal of Computer Vision, 82(2):205–229, 2009.
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