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### Active Random Fields

Adrian Barbu

The MAP Estimation Problem

- Estimation problem:
Given input data y, solve

- Example: Image denoising
- Given noisy image y, find denoised image x

- Issues
- Modeling: How to approximate ?
- Computing: How to find x fast?

Noisy image y

Denoised image x

MAP Estimation Issues

- Popular approach:
- Find a very accurate model
- Find best optimum x of that model

- Problems with this approach
- 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

Markov Random Fields

- Bayesian Models:
- Markov Random Field (MRF) prior

- E.g. Image Denoising model
- Gaussian Likelihood
- Fields of Experts MRF prior
- Differential Lorentzian
- Image filters Ji

Image Filters Ji

Roth and Black, 2005

MAP Estimation (Inference) in MRF

- Exact inference is too hard
- For the Potts model, one of the simplest MRFs
it is already NP hard (Boykov et al, 2001)

- For the Potts model, one of the simplest MRFs
- Approximate inference is suboptimal
- 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)

Gradient Descent for Fields of Experts

- Energy function:
- Analytic gradient (Roth & Black, 2005)
- Gradient descent iterations
- 3000 iterations with small
- Takes more than 30 min per image on a modern computer

FOE filters

Training the MRF

- Gradient update in model parameters
- Minimize KL divergence between learned prior and true probability
- Gradient ascent in log-likelihood
- Need to know Normalization Constant Z
- EX from training data
- Z and Ep obtained by MCMC
- Slow to train

- Training the FOE prior
- Contrastive divergence (Hinton)
- An approximate ML technique
- Initialize at data points and run a fixed number of iterations

- Takes about two days

- Contrastive divergence (Hinton)

Going to Real-Time Performance

- Wainwright (2006)
- 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:
- 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

Active Random Field

- Active Random Field = A pair (M,A) of
- 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
- Fields of Experts model
- Algorithm: 1-4 iterations of gradient descent
- Parameters:

Training the Active Random Field

- Discriminative training
- Training examples = pairs
- inputs yi+ desired outputs ti

- Training=optimization
- Loss function L
- Aka benchmark measure
- Evaluates accuracy on training set
- End-to-end training:
- covers entire process from input image to final result

Related Work

- Energy Based Models (LeCun & Huang, 2005)
- Train a MRF energy model to have minima close to desired locations
- Assumes exact inference (slow)

- Shape Regression Machine (Zhou & Comaniciu, 2007)
- Train a regressor to find an object
- Uses a classifier to clean up result
- Aimed for object detection, not MRFs

Related Work

- Training model-algorithm combinations
- 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

Related Work

- Training model-algorithm combinations and reporting results on the same loss function for image denoising
- 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 2-iteration ARF

- Common theme: trying to obtain a strong MAP optimum
- This work: fast and suboptimal estimator balanced by a complex model and appropriate training

Training Active Fields of Experts

- Training set
- 40 images from the Berkeley dataset (Martin 2001)
- Same as Roth and Black 2005

- Separate training for each noise level
- Loss function L = PSNR
- Same measure used for reporting results
is the standard deviation of

- Same measure used for reporting results

Trained Active

FOE filters, niter=1

Follow Marginal Space Learning

Consider a sequence of subspaces

Represent marginals by propagating particles between subspaces

Propagate only one particle (mode)

Start with one filter, size 3x3

Train until no improvement

We found the particle in this subspace

Add another filter initialized with zeros

Retrain to find the new mode

Repeat step 2 until there are 5 filters

Increase filters to 5x5

Retrain to find new mode

Repeat step 2 until there are 13 filters

Training 1-Iteration ARF, =25PSNR training (blue), testing (red)

while training the 1-iteration ARF, =25 filters

Start with one iteration, =25

Each arrow takes about one day on a 8-core machine

3-iteration ARFs can also perform 4 iterations

Training other ARFsConcerns about Active FOE

- Avoid overfitting
- Use large patches to avoid boundary effect
- Full size images instead of smaller patches

- Totally 6 million nodes
- Lots of training data

- Use a validation set to detect overfitting

- Use large patches to avoid boundary effect
- Long training time
- Easily parallelizable
- 1 -3 days on a 8 core PC
- Good news: CPU power increases exponentially (Moore’s law)

Results

Original Image

4-iteration ARF,

PSNR=28.94, t=0.6s

Corrupted with Gaussian

noise, =25, PSNR=20.17

3000-iteration FOE,

PSNR=28.67, t=2250s

Evaluation, Standard Test Images

noise=25

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.

Evaluation, Berkeley Dataset

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)

12: BM3D (Dabov et al, 2007)

Speed-Performance Comparison

noise=25

Performance on Different Levels of Noise

- Trained for a specific noise level
- No data term
- Band-pass behavior

Adding a Data Term

- Active FOE
- 1-iteration version has no data term
- Modification with data term
- Equivalent

Performance with Data Term

- Data term removes band-pass behavior
- 1-iteration ARF as good as 3000-iteration FOE for a range of noises

Conclusion

- An Active Random Field is a pair of
- A Markov Random Field based model
- A fast, approximate inference algorithm (estimator)

- Training = optimization of the MRF and algorithm parameters using
- A benchmark measure on which the results will be reported
- Training data as pairs of input and desired output

- Pros
- Great speed and accuracy
- Good control of overfitting using a validation set

- Cons
- Slow to train

Future Work

- Extending image denoising
- Learning filters over multiple channels
- Learning the robust function
- Learn filters for image sequences using temporal coherence

- Other applications
- Computer Vision:
- Edge and Road detection, Image segmentation
- Stereo matching, motion, tracking etc

- Medical Imaging
- Learning a Discriminative Anatomical Network of Organ and Landmark Detectors

- Computer Vision:

References

- 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 10-thInternational 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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