Image Analysis and Markov Random Fields (MRFs)

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Image Analysis and Markov Random Fields (MRFs)

• Quanren Xiong

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Statistical models

• Some image structures are not deterministic, are
  best characterized by their statistical
  properties.
• For example, textures can be represented by their
  first and second statistics.
• Images are often distorted by statistical noise.
  To restore the true image, images are often
  treated as realizations of a random process.

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Uses of Markov Random Fields

• MRFs are a kind of statistical model.
• They can be used to model spatial constrains.
• smoothness of image regions
• spatial regularity of textures in a small region
• depth continuity in stereo construction

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What are MRFs

• Neighbors and cliquesLet S be a set of
  locations, here for simplicity, assume S a
  grid. S (i, j) i, j are integers
  .Neighbors of s S are defined as ?((i, j))
  (k, l) 0lt(k - i)2 (l - j)2 ltconstant
  A subset C of S is a clique if any two
  different elements of C are neighbors. The set
  of all cliques of S is denoted by O.

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Examples of neighborhood

• 4-neighborhood

cliques
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Examples of neighborhood

• 8-neighborhood

cliques
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Random fields

• The random vector on S is
  called a random field and assumed to have density
  p(x).
• Images as Random fieldsIf vector X represents
  intensity values of an image, then its component
  Xs is the intensity value at location s(i, j).

S
X

640x480
640x480
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Markov Random Fields

• If p(x) of a random field fulfills the so called
  Markov condition with respect to a neighborhood
  system, it is called a Markov Random Field.

I.E, the value Xs at location S is only depend on
its neighbors.
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MRFs versus Markov Chains

• MRFs replace temporal dependency of Markov
  chains with spatial dependency.

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Markov Random fields

• p(x) can also be factorize over cliques due to
  its Markov properties. i.e.?C is a
  function of of X determined by clique C.

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Markov Random Fields

• MRFs are equivalent to Gibbs Fields and p(x) has
  the following form
• H(x) is called energy function.
• The summation in the denominator is over all
  possible configurations on S. In our case are
  over all possible images. For 256 grey values
  and 640x480 grid, it will have 256640x480 terms.
  Z is impractical to evaluate.
• so p(x) is only known up to a constant.

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Local Characteristics of MRFs

• For every , we have,S\I
  means complement of I
• If I is a small set, since X only changes over
  I, ZI can be evaluated in reasonable time.
• So p(yIxS\I) is known.

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Using MRFs in Image Analysis

• In image analysis, p(x) is often the posterior
  probability of Bayesian inference. That is, p(x)
  p(xy0).
• For example y0 may be the observed image with
  noise, and we want to compute the estimate x0 of
  the true image x0 based on p(x) p(xy0).

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Using MRFs in Image Analysis
MRF Model (either learned or known)
Sampling
X0
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Difficulties in computing X0

• A way to compute the estimate X0 is to
  let,
• But p(xy0) is only known up to a constant Z, How
  to do above integration?

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Monte Carlo integration

• One solution is to construct a Markov chain
  having p(x) as its limiting distribution.
• If the Markov chain starting at state X0, and
  going through states X1, X2, X3, , Xt,, then
  E(X)p(x) can be approximated by
• m is a sufficiently long burn-in time. Xm1,
  Xm2,...... can be considered as samples drawn
  from p(x).

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Gibbs Sampler

• Because X is a high dimension vector. (For a
  640x480 image its dimension is 640x480).
• It is not practical to update all components of
  Xt to Xt1 in one iteration.
• One version of Metropolis-Hastings algorithm,
  called Gibbs Sampler, builds the Markov chain and
  updates only a single component of Xt in one
  iteration.

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Gibbs Sampler Algorithm

• Let the vector X has k components,
  X(X0,X1,X2,,Xk). and presently it is in state
  Xt (x0,x1,x2,,xk).
• An index that is equally likely to be any of
  1,,k is chosen. say index i.
• A random variable w with density P wx P
  Xix Xj xj, j ? i is generated.
• If wx, the updated Xt is Xt1
  (x0,x1,x2,xi-1,x,xi1,,xk).

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2 aspects of using MRFs

• Find an appropriate model class, the general form
  of H(x).
• Identify suitable parameters in H(x) from
  observed samples of X.
• This is the most difficult part in applying MRFs.

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An Example

• Suppose we want to restore a binary (1/-1) image
  with pepper-and-salt noise added.
• The Ising model is chosen.

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Open Issues / Discussion

• Code Development
• What should our MRF library look like?
• Challenges Build MRF model from image samples
  and then generate new images using Gibbs sampler
• Need to a way to determine the parameters in H(x)
  based on image samples.

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Reference

1. Ross Kindermann and J. Laurie Snell, Markov
   Random Fields and Their Applications,
   http//www.ams.org/online_bks/conm1/, 1980.
2. Gerhard Winkler, Image Analysis, Random Fields
   and Markov Chain Monte Carlo methods. Springer,
   2003
3. W. R. Gilks, Markov Chain Monte Carlo in
   Practice, Chapman Hall/CRC, 1998.
4. Sheldon M. Ross, Introduction to Probability
   Models, Academic Press, 2003.