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Markov Random Fields (MRF)

A graphical model for describing spatial consistency in images

Suppose you want to label image pixels with some labels {l1,…,lk} , e.g., segmentation, stereo disparity, foreground-background, etc.

Ref:

1. S. Z. Li. Markov Random Field Modeling in Image Analysis.

Springer-Verlag, 1991

2. S. Geman and D. Geman. Stochastic relaxation, gibbs distribution

and bayesian restoration of images. PAMI, 6(6):721–741, 1984.

CS 534 – Stereo Imaging - 1

From Slides by S. Seitz - University of Washington

Definition

MRF Components:

A set of sites: P={1,…,m} : each pixel is a site.

Neighborhood for each pixel N={Np | p P}

A set of random variables (random field), one for each site F={Fp | p P} Denotes the label at each pixel.

Each random variable takes a value fp from the set of labels L={l1,…,lk}

We have a joint event {F1=f1,…, Fm=fm} , or a configuration, abbreviated as F=f

The joint prob. Of such configuration: Pr(F=f) or Pr(f)

CS 534 – Stereo Imaging - 2

From Slides by S. Seitz - University of Washington

Definition

MRF Components:

Pr(fi) > 0 for all variables fi.

Markov Property: Each Random variable depends on other RVs only through its neighbors. Pr(fp | fS-{p})=Pr (fp|fNp), p

So, we need to define a neighborhood system: Np (neighbors for site p).

No strict rules for neighborhood definition.

Cliques for this neighborhood

CS 534 – Stereo Imaging - 3

From Slides by S. Seitz - University of Washington

Definition

MRF Components:

The joint prob. of such configuration:

Pr(F=f) or Pr(f)

Markov Property: Each Random variable depends on other RVs only through its neighbors. Pr(fp | fS-{p})=Pr (fp|fNp), p

So, we need to define a neighborhood system: Np (neighbors for site p)

Hammersley-Clifford Theorem:Pr(f) exp(-C VC(f))

Sum over all cliques in the neighborhood system

VCis clique potential

We may decide

1. NOT to include all cliques in a neighborhood; or

2. Use different Vc for different cliques in the same neighborhood

CS 534 – Stereo Imaging - 4

From Slides by S. Seitz - University of Washington

Optimal Configuration

MRF Components:

Hammersley-Clifford Theorem:

Pr(f) exp(-C VC(f))

Consider MRF’s with arbitrary cliques among neighboring pixels

Sum over all cliques in the neighborhood system

VCis clique potential: prior probability that elements of the clique C have certain values

Typical potential: Potts model:

CS 534 – Stereo Imaging - 5

From Slides by S. Seitz - University of Washington

Optimal Configuration

MRF Components:

Hammersley-Clifford Theorem:

Pr(f) exp(-C VC(f))

Consider MRF’s with clique potentials of pairs of neighboring pixels

Most commonly used….very popular in vision.

Energy function:

There are two constraints to satisfy:

- Data Constraint: Labeling should reflect the observation.

- Smoothness constraint: Labeling should reflect spatial consistency (pixels close to each other are most likely to have similar labels).

CS 534 – Stereo Imaging - 6

Probabilistic interpretation

The problem is we are not observing the labels but we observe something else that depends on these labels with some noise (eg intensity or disparity)

At each site we have an observation ip

The observed value at each site depends on its label: the prob. of certain observed value given certain label at site p : g(ip,fp)=Pr(ip|Fp=fp)

The overall observation prob. Given the labels: Pr(O|f)

We need to infer about the labels

given the observation Pr(f|O) Pr(O|f) Pr(f)

CS 534 – Stereo Imaging - 7

Using MRFs

How to model different problems?

Given observations y, and the parameters of the MRF, how to infer the hidden variables, x?

How to learn the parameters of the MRF?

Modeling image pixel labels as MRF

MRF-based segmentation

real image

1

label image

Slides by R. Huang – Rutgers University

Modeling image pixel labels as MRF

MRF-based segmentation

real image

1

label image

Slides by R. Huang – Rutgers University

MRF-based segmentation

Classifying image pixels into different regions under the constraint of both local observations and spatial relationships

Probabilistic interpretation:

region labels

model param.

image pixels

Slides by R. Huang – Rutgers University

Model joint probability

region labels

model param.

image pixels

How did we factorize?

image-label

compatibility

Function

enforcing

Data

Constraint

label-label

compatibility

Function

enforcing Smoothness constraint

label

image

local

Observations

neighboring

label nodes

Slides by R. Huang – Rutgers University

Probabilistic interpretation

We need to infer about the labels given the observation

Pr( f | O ) Pr(O|f ) Pr(f)

MAP estimate of f should minimize the posterior energy

Data (observation) term:

Data Constraint

Neighborhood term: Smoothness Constraint

CS 534 – Stereo Imaging - 14

MRF-based segmentation

EM algorithm

E-Step: (Inference)

M-Step: (learning)

Applying and learning MRFMethods to be described.

Pseduo-likelihood method.

Slides by R. Huang – Rutgers University

Applying and learning MRF: Example

Slides by R. Huang – Rutgers University

Inference in MRFs

Inference in MRFs

Classical:

Gibbs sampling, simulated annealing Self study

Iterated condtional modes (ICM) Also Self study

State of the Art

Graph cuts

Belief propagation

Linear Programming (not covered in this lecture)

Tree-reweighted message passing (not covered in this lecture)

Slides by R. Huang – Rutgers University

Gibbs sampling and simulated annealing

Gibbs sampling:

A way to generate random samples from a (potentially very complicated) probability distribution

Simulated annealing:

A schedule for modifying the probability distribution so that, at “zero temperature”, you draw samples only from the MAP solution.

Simulated Annealing algorithm:

x := x0; e := E(x) // Initial state, energy.

k := 0 // Energy evaluation count.

while k < kmax and e > emax // While time remains & not good enough:

xn := neighbour(x) // Pick some neighbour.

en := E(xn) // Compute its energy.

if P(e, en, temp(k/kmax)) > random() then // Should we move to it?

x := xn; e := en // Yes, change state.

k := k + 1 // One more evaluation done

return x // Return current solution

Slides by R. Huang – Rutgers University

Gibbs sampling and simulated annealing cont.

Simulated annealing as you gradually lower the “temperature” of the probability distribution ultimately giving zero probability to all but the MAP estimate.

finds global MAP solution.

takes forever. (Gibbs sampling is in the inner loop…)

Slides by R. Huang – Rutgers University

Iterated conditional modes

Start with an estimate of labeling x

For each node xi:

Condition on all the neighbors

Find the label decreasing the energy function the most

Repeat till convergence

Fast

Heavily depend on initialization, local minimum

Described in: Winkler, 1995. Introduced by Besag in 1986.

Slides by R. Huang – Rutgers University

Solving Energy Minimization with Graph Cuts

- Many classes of Energy Minimization problems in Computer Vision can be reduced to Graph Cuts
- Solve multiple-labels problems with binary decisions

Yevgeny Doctor IP Seminar 2008, IDC

Approximate Energy Minimization

- “Fast Approximate Energy Minimization via Graph Cuts.” Yuri Boykov, Olga Veksler, Ramin Zabih, 1999
- For two classes of interaction potentials V (Esmooth):
- V is semi-metric on a label space L if for every :
- V is metric on L if in addition, triangle inequality holds:
- For example, truncated L2 distance and Potts Interaction Penalty are both metric.

Yevgeny Doctor IP Seminar 2008, IDC

Solution for Semi-metric Class

- Swap-Move algorithm:
- 1. Start with an arbitrary labeling f
- 2. Set success := 0
- 3. For each pair of labels
- 3.1. Find f* = argmin E(f') among f' within one a-b swap of f
- 3.2. If E(f*) < E(f), set f := f* and success := 1
- 4. If success = 1 goto 2
- 5. Return f
- a-b swap:
- In the new labeling f’, some pixels that were labeled a in f are now labeled b, and vice versa.

Yevgeny Doctor IP Seminar 2008, IDC

Solve a-b swap step with Graph Cut

- Graph:

Fast Approximate Energy Minimization via Graph Cuts

Yuri Boykov, Olga Veksler, Ramin Zabih, 1999

Yevgeny Doctor IP Seminar 2008, IDC

Solve a-b swap step with Graph Cut

- Cut and Labeling:
- Weights:

Fast Approximate Energy Minimization via Graph Cuts

Yuri Boykov, Olga Veksler, Ramin Zabih, 1999

Yevgeny Doctor IP Seminar 2008, IDC

Computing a multiway cut

With two labels: classical min-cut problem

Solvable by standard network flow algorithms

polynomial time in theory, nearly linear in practice

More than 2 labels: NP-hard

But efficient approximation algorithms exist

Within a factor of 2 of optimal

Computes local minimum in a strong sense

even very large moves will not improve the energy

Yuri Boykov, Olga Veksler and Ramin Zabih, Fast Approximate Energy Minimization via Graph Cuts, International Conference on Computer Vision, September 1999.

Basic idea

reduce to a series of 2-way-cut sub-problems, using one of:

swap move: pixels with label l1 can change to l2, and vice-versa

expansion move: any pixel can change it’s label to l1

Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 26

Belief propagation

Message Passing (Original: Weiss & Freeman ‘01, faster: Felzenswalb & Huttenlocher ‘04)

Send messages between neighbors.

Messages estimate the cost (or Energy) of a configuration of a clique given all other cliques.

s3

q

q

p

=

s2

s1

Messages are initialized to zero

Belief propagation

Gathering belief

After time T, the messages are combined to compute a belief.

p3

p4

p2

q

p1

Label with largest belief wins.

Inference in MRFs

Loopy BP

tractable, good approximate in network with loops

Not guaranteed to converge, may oscillate infinitely.

Stereo as energy minimization

Matching Cost Formulated as Energy:

At pixel p = (x , y)

“data” term penalizing bad matches

“neighborhood term” encouraging spatial smoothness

(truncated)

Norm of the difference between labels at neighboring x, y.

(also, truncated)

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 30

Stereo as a Graph cut

Terminals (possible disparity labels)

From Slides by Yuri Boykov, Olga Veksler, Ramin Zabih “Markov Random Fields with Efficient Approximations” – CVPR 98

CS 534 – Stereo Imaging - 31

Stereo as a graph problem [Boykov, 1999]

Pixels

edge weight

d3

d2

d1

edge weight

Labels

(disparities)

CS 534 – Stereo Imaging - 32

From Slides by S. Seitz - University of Washington

Graph definition

Initial state

Each pixel connected to it’s immediate neighbors

Each disparity label connected to all of the pixels

d3

d2

d1

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 33

Stereo matching by graph cuts

Graph Cut

Delete enough edges so that

each pixel is (transitively) connected to exactly one label node

Cost of a cut: sum of deleted edge weights

Finding min cost cut equivalent to finding global minimum of the energy function

d3

d2

d1

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 34

Motion estimation as energy minimization

Matching Cost Formulated as Energy:

At pixel p = (x , y)

“data” term penalizing bad matches

“neighborhood term” encouraging spatial smoothness

(truncated)

Norm of the difference between labels at neighboring x, y.

(also, truncated)

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 35

Results with window search

Window-based matching

(best window size)

Ground truth

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 36

Better methods exist...

- State of the art method
- Boykov et al., Fast Approximate Energy Minimization via Graph Cuts,
- International Conference on Computer Vision, September 1999.

Ground truth

From Slides by S. Seitz - University of Washington

CS 534 – Stereo Imaging - 37

GrabCut

GrabCut

Rother et al 2004

Magic Wand(198?)

Intelligent ScissorsMortensen and Barrett (1995)

User Input

Result

Regions

Regions & Boundary

Boundary

Slides C Rother et al., Microsoft Research, Cambridge

Data Term

R

Foreground &Background

Gaussian Mixture Model

(typically 5-8 components)

G

Background

D() is log-likelihood given the mixture model \Theta

Slides C Rother et al., Microsoft Research, Cambridge

Smoothness term

An object is a coherent set of pixels:

Probability of a configuration:

Iterate until convergence:

1. Compute a configuration given the mixture model. (E-Step)

2. Compute the model parameters given the configuration. (M-Step)

Slides C Rother et al., Microsoft Research, Cambridge

Moderately simple examples

… GrabCut completes automatically

Slides C Rother et al., Microsoft Research, Cambridge

Difficult Examples

Camouflage &

Low Contrast

Fine structure

No telepathy

Initial Rectangle

InitialResult

Slides C Rother et al., Microsoft Research, Cambridge

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