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Statistics of natural images

May 30, 2010Ofer BartalAlon Faktor

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Outline

• Motivation• Classical statistical models• New MRF model approach• Learning the models• Applications and results

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Motivation

• Big variance in appearance • Can we even dream of modeling this?

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Motivation

• Main questions:– Do all natural images obey some common

“rules”?– How can one find these “rules”?– How to use “rules” for computer vision

tasks?

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Motivation

• Why bother to model at all?

• “Noise”, uncertainty

• Model helps choose the “best” possible answer

• Lets see some examples

Natural image model

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Noise-blur removal

• Consider the classical De-convolution problem

• Can be formulated as linear set of equations:

?Y h X N X

cs cs csy Hx n

H Y+X

N

7

X

=

YNh

X̂

?

Noise-blur removal

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Inpainting

Y AX n

Y

?X

1 0 0 ... 0

0 1 0 0 ... 0

0 0 0 0 0 1 0 0 ... 0

0 0 0 0 0 0 1 0 ... 0

0 0 .... 0 1

A

Missing lines of identity matrix = missing pixels (under-determined system)

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Motivation

• Problems: – Unknown noise– H may be singular (Deconvolution)– H may be under-determined (Inpainting)

• So there can be many solutions. • How can we find the “right” one?

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Motivation

• Goal: Estimate x– Assume:

• Prior model of natural image:• Prior model of noise:

– Use MAP estimator to find x:

* arg max ( | ) arg max ( | ) ( )x x

x P x y P y x P x

( )xP x

* arg max ( ) ( )n xx

x P y Hx P x

( )nP n

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Energy Minimization problem

• The MAP problem can be reformulated as:

data term( | )+prior term( )

ˆ arg min x

E y x x

x E

E

x

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Classical models

• Smoothness prior (model of image gradients) – Gaussian prior (LS problem)– L1 Prior and sparse prior (IRLS problem)

Image gradient

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Gaussian Priors

• Assume:

– Gaussian priors on gradients of x:

– Gaussian noise:• Using this assumption:

2

221( )

2

x

p x e

* arg min 2T T

xx x Tx x b

* arg max ( ) ( )n xx

x P y Hx P x

2~ (0, )n N

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Non-Gaussian Priors

• Empirical results: image gradients have a Non-Gaussian heavy tailed distribution

• We assume L1 or sparse prior• We solve it by IRLS –iterative re-weighted LS

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De-convolution Results

Gaussian prior Sparse priorBlurred image

Good results on simple images

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De-noising Results

De-noising resultNoisy image

Poor results on real natural images

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Classical models – Pro’s and Con’s

• Advantages:– Simple and easy to implement

• Disadvantages:– Too Heuristic– Only one property - Smoothness– Bias towards totally smooth images:

P P

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Going Beyond Classical Models

0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

number of similar patches (in log10 scale)

prob

abilt

y

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

number of similar patches (in log10 scale)

prob

abilt

y

0 1 2 3 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

number of similar patches (in log10 scale)

prob

abilt

y

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Modern Approach

• Model is based on image properties• Choose properties using image dataset

• Questions:1. What types of properties?

Responses to linear filters.2. How to find good properties?

Either pre-determined bank or learn from data.3. How should combine properties to one distribution?

We will see how.

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Mathematical framework

• Want: A model p(I) of real distribution f(I).• Computationally hard:

– A 100x100 pixel image has 10,000 variables• Can explicitly model only a few dimensions at a time

Arrow = viewpoint of few dimensions

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Mathematical framework

• A viewpoint is a response to a linear filter• A distribution over these responses is a

marginal of real distribution f(I)• (Marginal = Distribution over a subset of variables)

Arrow = marginal of f(I)

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Mathematical framework

• If p(I) and f(I) have the same marginal distributions of linear filters then p(I)=f(I) (proposition by Zhu and Mumford)

• “Hope”: If we will choose K “good” filters then p(I) and f(I) will be “close”.

How do we measure “close?”

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Distance between distributions

• Kullback-Leibler divergence:

• Problem - f(I) unknown• Proposition - use instead:

• Measures fit of model to observations

( ), ( ; , ) log ( ) log ( ; , )f fKL f I p I S E f I E p I S

~

( ), ( ; , ) log ( ; , ) log ( ; , )P X

KL f I p I S p I S p I S

( ; , )p I S X

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Illustration

log ( ; , )p I S

log ( ; , )p I S

~

KL

~

( ), ( ; , ) log ( ; , ) log ( ; , )P X

KL f I p I S p I S p I S

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Getting synthesized images

• Get synthesized images by sampling the learned model

• Sample using Markov Chain Monte Carlo (MCMC).

• Drawback: Learning process is slow

xp

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Our model P(I) – A MRF

• MRF = Markov Random Field• A MRF is based on a graph G=(V,E).

V – pixels E – between pixels that affect each other

• Our distribution is the MRF:

( )

1( ) exp ( )c c

c Cliques

p I U IZ

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Simple grid MRF

• Here, cliques are edges• Every pixel belongs to 4 cliques

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MRF

• We limit ourselves to:

– Cliques of fixed size (over-lapping patches)

– Same for all cliques

• We get:

( ) ( )( ) ( )

1

( ) ( )K

Tc cU I F I

( ) ( )( )

1 1

1( ) exp ( )

C KT

cc

p I F IZ

U

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MRF simulation

( ) ( )( )

1 1

1( ) exp ( )

C KT

cc

p I F IZ

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Histogram simulation

( ) ( )obsnH I

Histogram of a marginal

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MRF

• In terms of convolutions:

• Denote: Set of potential functions:

• Denote: Set of filters:

( )

1...K

( )

1...KS F

( ) ( )

1 ( , )

,1( ; , )

K

x y

F I x y

p I S eZ

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MRF - A simple example

• Cliques of size 1• Pixels are i.i.d and distributed by grayscale histogram

grayscale histogram

Drawback: cliques are too small

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MRF - Another simple example

• Clique = whole image• Result: Uniform distribution on images in dataset

Px

Drawback: cliques are too big

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Revisiting classical models

• Actually, the classical model is a pairwise MRF:

• Has cliques of size 2:

• Has only 2 linear filters => 2 marginals

• No guarantee that p(I) will be close to f(I)

( , )( ( , )) ( ( , ))1

( )x yx yI x y I x y

p I eZ

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Zhu and Mumford’s approach (1997)

• We want to find K “good” filters• Strategy:

– Start off with a bank B of possible filters– Choose subset that minimizes the

distance between p(I) and f(I)– For computational reasons, choose filters one by

one using a greedy method

, | |S B S K

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Choosing the next filter

• AIG = the difference between the model p(I) and the data from the viewpoint of marginal

• AIF = the difference in between different images in dataset from the viewpoint of marginal

( ) ( ) ( )IC AIG AIF

( ) ( )( ; , )

1

( ) ( )

1

1( ) ( ) ( )

2

1( ) ( )

2obs

Mobsn P I S

n

Mobsn

n

AIG H I E H IM

AIF H IM

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Algorithm – Filter selection

Bank of filters

IC

IC

IC arg max

max

Model ( )learn

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Learning the potentials

( ) Model

( )IC

Calculate update

Init

(Using maximum entropy on P)

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The bank of filters

• Filter types: – Intensity filter (1X1)– Isotropic filters - Laplacian of Gaussian (LG, )– Directional filters - Gabor (Gcos, Gsin)

• Computation in different scales - image pyramid

Laplacian of Gaussian Gabor

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Running example of algorithmExperiment I

Use only small filters

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Results

All learned potentials have a diffusive nature

( ) ( )( )

1 1

1( ) exp ( )

C KT

cc

p I F IZ

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Running example of algorithmExperiment II

• Only gradient filters, in different scales• Small filters -> diffusive potential (as expected)• Surprisingly: Large filters -> reactive potentials

Diffusive Reactive

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Examples of the synthesized images

Experiment I Experiment II

This image is more “natural” because it has some regions with sharp boundaries

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Outline

• We have seen:– MRF models – Selection of filters from a bank – Learning potentials

• Now:– Data-driven filters – Analytic results for simple potentials– Making sense in results– Applications

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Roth and Black’s approach

filters potentials

Chosen from bank Learn a-parametricallyX XLearn from data Learn parametrically

Learn together

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Motivation – model of natural patches

• Why learn filters from data?• Inspiration from models of natural patches:

– Sparse coding– Component analysis– Product of experts

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Motivation – Sparse Coding of patches

• Goal: find a set s.t.

•

• Learn from database of natural patches

• Only few filters should fire on a given patch

1

, are sparseN

i i ii

patch a F a

iF

,i ia patch F

iF

1 2 3 4 5

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Motivation – Component analysis

• Learn by component analysis:– PCA– ICA

• Results in “filters like” components– PCA – first components look like contrast filters– ICA - components look like Gabor filters

iF

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PCA results

high

low

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ICA results

• Independent filters • Can derive model for patches:

1

( ) ( )n

Ti i

i

P x p F x

TiF x

ip

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Motivation – Product of experts

• More sophisticated model for natural patches:

• Training of MLE => “intuitive” filters:

2

1

1( ; ) ( ; ), , ( ) 1

( 2, )

i

i i i

KT

POE i sti ii

F Fz

zZ F

p X X

texturecontrast

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• extension of POE to FOE:

Field of experts (FOE)

( )1 1

1( ; ) exp ( ; ) ,,

( , )

C KT

FOE i iii

i ii

cc

F Fp I IFZ

log( )st

( )1 1

( ; )iC K

TFOE i c

c i

E F I

Roth S., Black M. J., Fields of experts IJCV, 2009

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The experts

• Student-t experts2

( ) 12

i

i

zz

( )st z

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Meaning of

• Higher means:– Punishes high responses more severely – A filter with higher weight

( )st z

1

2

( )

1

1( ; , ) exp

( , )

g log 12

K

FOE i i i iii i

TCi c

ic

p I F gZ F

F I

Learning the model

log ( ; , )p I S

log ( ; , )p I S

~

KL

Model

1

2K

MCMCinit

random

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Results of learning FOE

Filters aren’t “intuitive”

F

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So far…

filters potentials

Chosen from bank Learned a-parametrically

diffusive reactive

Small filters Large filters

non-intuitive?

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So far…

filters potentials

Learned from database Learned parametrically

non-intuitive?

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What now?

• Revisiting POE and FOE with Gaussian potentials

• Relation to non-Gaussian potentials• Making sense of previous results

Weiss Y., Freeman W. T. What makes a good model of natural images?. CVPR, 2007

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Gaussian POE

2

2

1

2

1

* 2

1

1; exp ( )

( )

ln ; ( ) ln ( )

arg min ( ) ln ( )i

z

KT

GPOE i iii

KT

GPOE i i ii

KT

i i iML F i

e

p x F F xZ F

p x F F x Z F

F F x Z F

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• Claim: Z is constant for any set of K orthonormal vectors

•

• This has an analytic solution – the K minor components of the data

Gaussian POE

* 2

1

arg min ( )i

KT

i iF orthonormal i

F F x

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• Non-intuitive high-frequency filters• Reminder - PCA

ResultsExample of learned filters

high

low

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Gaussian FOE

2

2( )

1 1

2 2( )

1 1 1 ,

2 2 2

1

2

1

1( ;{ }) exp ( )

({ })

( ) ( * ) ( , )

{ }( ) { }( ) ( ) { }( )

ln ({ }) ln { }( ) ln ( )

K CT

GFOE i i ci ci

K C KTi c i

i c i x y

K

ii

K

i ii

z

p I F F IZ F

F I F I x y

F I G I

Z F F G

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Gaussian FOE

* 2

( )1 1

2*

( )

*2

arg min ( ) ln ( )

( ) arg min ( ) { }( ) ln ( )

1( )

{ }( )

i

K CT

i i c iML F i c

MLG

ML

F F I Z F

G G I G

GI

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Gaussian FOE

• satisfies:

=> Optimal filters have high frequencies

2*2

1

1{ }( )

{ }( )

K

ii

FI

2*

1

{ }( )K

ii

F

2{ }( )I

*iF

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• Non-Gaussian potentials -> modeled by GSM

• Properties of GFOE hold for GSM

Gaussian Scale Mixture (GSM)

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Revisiting FOE

• Student t expert – fit GSM• Filters have the property of

Natural image Roth and Black filters

2

21

1{ }( )

{ }( )

K

ii

FI

high-frequency filters

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Learning FOE with fixed filters

Algorithm prefers high-frequency filters

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Conclusion

• For Gaussian potentials and GSM’s:learning => High frequency filters

• Experimental evidence to this phenomena • Maybe there is a “logic” behind this non-intuitive

result?

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Making Sense of results

• Criterion for “good” filters for patches – Rarely fire on natural images and fire frequently on all other images

Patches from Natural images

Histogram of filter responses

White noise

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Making Sense of results

• An image was modeled by what you don’t expect to find in it

• This is satisfied by the classical prior of smooth gradients

• But why limit ourselves to intuitive filters?• Maybe non-intuitive filters can do better…

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reactivediffusive

White noise

Patches from Natural images

Revisiting diffusive and reactive potentials

White noise

Patches from Natural images

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Inference

• We learned a model• We can use it for inference problems

– Corrupted information– Missing information

• Exact inference – Loopy BP • Approximate inference - gradient based

optimization

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Belief Propagation

• Observed data is incorporated to model byiy i

ix

iy

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Belief Propagation

Message passing Algorithm

• Exact only on tree MRFs • Efficient only on pairwise MRFs

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Alternative by Roth and Black

• Reminder:

• Approximate inference by gradient-based optimization :

• Advantage: Low computational cost

• Drawback: only local minimum if not convex

= argmin ( ( | )) ( ( ))MAPI

I Log P I I Log P I

Uncertainty \Noise model Learned model

( 1) ( ) , ( , )t tII I I I E I I

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Partition function

=> No need to estimate partition function

• We get:

( , ) 1

arg min ( ( | )) * ( , );n

i iI x y i

Log P I I F I x y

( , ) 1

argmin ( ( | )) log( ( , )) * ( , );n

i i i iI x y i

Log P I I Z F F I x y

X

(Doesn’t depend on )

I

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The gradient step

( ) ( )

( , ) 1 1

* ( , ); * '( * ; )n N

i iI i i i

x y i i

F I x y F F I

( )iF( )iF

• How to derivate the second term?• By a mathematical “trick” we get:

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• Assume Gaussian noise

• So the Gradient step is:

De-noising

2

2

( | ) ( )

1( ( | )) ( )

2

nP I I P I I

Log P I I I I

( ) ( )2

1

1( ) * '( * ; )

Ni i

ii

I I I F F I

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Results

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Results

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Results

Original Noisy(20.29dB)

FOE(28.72dB)

Poritilla (Wavelets)(28.9dB)

Non-local means

(28.21dB)

StandardNon-Linear diffusion (27.18dB)

State of the art

Generalprior

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Results on Berkeley databaseWiener filter

Non-Linear diffusion FOE

Poritilla1Poritilla2

Out

put P

SNR

Low noise

High noise

Input PSNRLow noise

High noise

Input PSNR

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How many 3x3 filters to take?

Number of filters

Size of filter – 3X3Performance start saturating when we reach 8 filters

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Dependence on size and shape of clique

What is the best filter?

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Inpainting - Reminder

Y AX n

Y X

Problem: pixels outside mask can change

Solution: constraint them

Inpainting

• Assume pixels outside mask M don’t change

• So the gradient step is: ( ) ( )

1

* '( * ; )N

i ii

i

I M F F I

Advanced Topics In Computer

Vision CourseSpring 2010

Advanced Topics In Computer

Vision CourseSpring 2010

0-1 Mask Image we want to inpaint98

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Results

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Results

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ResultsFOE Bertalmio

FOE Bertalmio

PSNR 29.06dB 27.56dB

SSIM 0.9371 0.9167

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Pro’s and Con’s

• Perform well on narrow straws or small holes (even if they cover most of the image)

• Isn’t able to fill large holes• Isn’t designed to handle textures

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Thank you for Listening…