On the Statistical Analysis of Dirty Pictures

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On the Statistical Analysis of Dirty Pictures

• Julian Besag

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Image Processing

• Required in a very wide range of practical
  problems
• Computer vision
• Computer tomography
• Agriculture
• Many more
• Picture acquisition techniques are noisy

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Problem Statement

• Given a noisy picture
• And 2 source of information (assumptions)
• A multivariate record for each pixel
• Pixels close together tend to be alike
• Reconstruct the true scene

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Notation

• S 2D region, partitioned into pixels numbered
  1n
• x (x1, x2, , xn) a coloring of S
• x (realization of X) true coloring of S
• y (y1, y2, , yn) (realization of Y) observed
  pixel color

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Assumption 1

• Given a scene x, the random variables Y1, Y2, ,
  Yn are conditionally independent and each Yi has
  the same known conditional density function
  f(yixi), dependent only on xi.
• Probability of correct acquisition

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Assumption 2

• The true coloring x is a realization of a
  locally dependant Markov random field with
  specified distribution p(x)

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Locally Dependent M.r.f.s

• Generally, the conditional distribution of pixel
  i depends on all other pixels, S\i
• We are only concerned with local dependencies

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Previous Methodology

• Maximum Probability Estimation
• Classification by Maximum Marginal Probabilities

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Maximum Probability Estimation

• Chose an estimate x such that it will have the
  maximum probability given a record vector y.
• In Bayesian framework x is MAP estimate
• In decision theory 0-1 loss function

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Maximum Probability Estimation

• Iterate over each pixel
• Chose color xi at pixel i from probability
• Slowly decreasing T will guarantee convergence

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Classification by Maximum Marginal Probabilities

• Maximize the proportion of correctly classified
  pixels
• Note that P(xi y) depends on all records
• Another proposal use a small neighborhood for
  maximization
• Still computationally hard because P is not
  available in closed form

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Problems

• Large scale effects
• Favors scenes of single color
• Computationally expensive

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Estimation by Iterated Conditional Modes

• The previously discussed methods have enormous
  computational demands, and undesirable
  large-scale properties.
• We want a faster method with good large-scale
  properties.

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Iterated Conditional Modes

• When applied to each pixel in turn, this
  procedure defines a single cycle of an iterative
  algorithm for estimating x

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Examples of ICM

• Each example involves
• c unordered colors
• Neighborhood is 8 surrounding pixels
• A known scene x
• At each pixel i, a record yi is generated from a
  Gaussian distribution with mean and
  variance ?.

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The hillclimbing update step
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Extremes of ß

• ß 0 gives the maximum likelihood classifier,
  with which ICM is initialized
• ß 8, xi is determined by a majority vote of its
  neighbors, with yi records only used to break
  ties.

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Example 1

• 6 cycles of ICM were applied, with ß 1.5

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Example 2

• Hand-drawn to display a wide array of features
• yi records were generated by superimposing
  independent Gaussian noise, v? .6
• 8 cycles, ß increasing from .5 to 1.5 over the
  1st 6

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Models for the true scene

• Most of the material here is speculative, a topic
  for future research
• There are many kinds of images possessing special
  structures in the true scene.
• What we have seen so far in the examples are
  discrete ordered colors.

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Examples of special types of images

• Unordered colors
• These are generally codes for some other
  attribute, such as crop identities
• Excluded adjacencies
• It may be known that certain colors cannot appear
  on neighboring pixels in the true scene.

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More special cases

• Grey-level scenes
• Colors may have a natural ordering, such as
  intensity. The authors did not have the computing
  equipment to process, display, and experiment
  with 256 grey levels.
• Continuous intensities
• p(x) is a Gaussian M.r.f. with zero mean

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More special cases

• Special features, such as thin lines
• Author had some success reproducing hedges and
  roads in radar images.
• Pixel overlap

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Parameter Estimation

• This may be computationally expensive
• This is often unnecessary
• We may need to estimate ? in l(yx ?)
• Learn how records result from true scenes.
• And we may need to estimate F in p(xF)
• Learn probabilities of true scenes.

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Parameter Estimation, cont.

• Estimation from training data
• Estimation during ICM

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Example of Parameter Estimation

• Records produced with Gaussian noise, ? .36
• Correct value of ? , gradually increasing ß gives
  1.2 error
• Estimating ß 1.83 and ? .366 gives 1.2 error
• ? known but ß 1.8 estimated gives 1.1 error

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Block reconstruction

• Suppose the Bs form 2x2 blocks of four, with
  overlap between blocks
• At each stage, the block in question must be
  assigned one of c4 colorings, based on 4 records,
  and 26 direct and diagonal adjacencies

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Block reconstruction example

• Univariate Gaussian records with ? .9105
• Basic ICM with ß 1.5 gives 9 error rate
• ICM with ß 8 estimated gives 5.7 error

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Conclusion

• We began by adopting a strict probabilistic
  formulation with regard to the true scene and
  generated records.
• We then abandoned these in favor of ICM, on
  grounds of computation and to avoid unwelcome
  large-scale effects.
• There is a vast number of problems in image
  processing and pattern recognition to which
  statisticians might usefully contribute.