1 Active Random Fields Adrian Barbu. FSU 2 The MAP Estimation Problem Estimation problem: Given input data y, solve Example: Image denoising Given noisy.

1

Active Random Fields

Adrian Barbu

FSU2

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

FSU3

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

FSU4

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

FSU5

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) 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)

FSU6

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

FSU7

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

Model m

Trainining PhaseNatural Images

MCMC samplesfrom current

model

Sufficientstatistics

Update m

FSU8

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

FSU9

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:

FSU10

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

Model m

Algorithm a

Trainining PhaseInput images

Current results

Desired results

BenchmarkMeasure

Update mand a

FSU11

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

FSU12

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

FSU13

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

FSU14

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

Trained Active

FOE filters, niter=1

FSU15

Training 1-Iteration ARF, =25Follow Marginal Space Learning Consider a sequence of subspaces

Represent marginals by propagating particles between subspaces

Propagate only one particle (mode)1. Start with one filter, size 3x3

Train until no improvement We found the particle in this subspace

2. Add another filter initialized with zeros Retrain to find the new mode

3. Repeat step 2 until there are 5 filters4. Increase filters to 5x5

Retrain to find new mode5. Repeat step 2 until there are 13 filters

PSNR training (blue), testing (red)while training the 1-iteration ARF, =25 filters

5 Filters3x3

5 Filters5x5

6 Filters5x5

13 Filters5x5

2 Filters3x3

1 Filter3x3

0 1 2 3 4 5

x 104

21

22

23

24

25

26

27

28

Steps x10000

PS

NR

Training DataTesting Data (unseen)

FSU16

Training other ARFs Other levels initialized as

Start with one iteration, =25 Each arrow takes about one day on a 8-core machine 3-iteration ARFs can also perform 4 iterations

n iter =1=15

n iter =2=15

n iter =1=10

n iter =2=10

n iter =1=20

n iter =2=20

n iter =1=25

n iter =2=25

n iter =1=50

n iter =2=50

n iter =3=10

n iter =4=10

n iter =4=15

n iter =3=15

n iter =3=20

n iter =4=20

n iter =4=25

n iter =3=25

n iter =3=50

n iter =4=50

FSU17

Concerns 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

Long training time Easily parallelizable 1 -3 days on a 8 core PC Good news: CPU power increases exponentially (Moore’s law)

FSU18

Results

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

Original Image

FSU19

Standard Test Images

Lena Barbara Boats

House Peppers

FSU20

Evaluation, Standard Test Images

noise=25

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

FSU21

Evaluation, Berkeley Dataset68 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.

FSU22

Evaluation, Berkeley Dataset

Average PSNR on 68 images from the Berkeley dataset, not used for training.

1: Wiener Filter2: Nonlinear diffusion3: 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)1 2 3 4 5 6 7 8 9 10 11 12

21

22

23

24

25

26

27

28

29

30

31

Algorithm, sigma = 50 (PSNR=14.15)

PS

NR

1 2 3 4 5 6 7 8 9 10 11 1221

22

23

24

25

26

27

28

29

30

31

Algorithm, sigma = 25 (PSNR=20.17)

PS

NR

1 2 3 4 5 6 7 8 9 10 11 12

25

26

27

28

29

30

31

32

33

34

35

36

Algorithm, sigma=20 (PSNR=22.11)

PS

NR

FSU23

Speed-Performance Comparison

noise=25

0 1 2 3 4 5 6 7 8 9 1026

26.5

27

27.5

28

28.5

29

Frames per second

PS

NR

ARFBM3DKSVDDCTWaveletFOENonlocalNonLinDiffWiener

FSU24

Performance on Different Levels of Noise

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

10 15 20 25 30 35 40 45 5018

20

22

24

26

28

30

32

34

Level of noise sigma

PS

NR

3000-iter. FOE1-iter. ARF, sigma=101-iter. ARF, sigma=151-iter. ARF, sigma=201-iter. ARF, sigma=251-iter. ARF, sigma=50

FSU25

Adding a Data Term Active FOE

1-iteration version has no data term

Modification with data term

Equivalent

FSU26

Performance with Data Term

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

noises

10 15 20 25 30 35 40 45 50

16

18

20

22

24

26

28

30

32

Level of noise sigma

PS

NR

3000-iteration FOE1-iteration FOE1-iter FOE,retrained coeffsARF, no data term, train 15-25ARF w/ data term, train 15-25ARF w/ data term, train 15-40

FSU27

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

FSU28

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

FSU29

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.