METHODS FOR IMAGE RESTORATION

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METHODS FOR IMAGE RESTORATION

• Michele Piana
• Dipartimento di Informatica
• Universita di Verona

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• Inverse problems and ill-posedness

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INVERSE PROBLEMS
gh Kf h
two known quantities (data and model)
one equation
two unknowns (source function and noise)
solving IPs consists in thinking backward
thinking backward is difficult
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ILL-POSEDNESS
Hadamard (1901)

• ill-posed problems do not have a solution, or
  their solution
• is not unique, or their solution does not
  depend continuously
• on the data

• ill-posed problems are dépourvu de signification
  physique

Examples

• X-ray CT (and all kinds of tomography)

• Electro- and Magneto-encephalograpy (EEG and MEG)

• Pattern recognition

• Image deblurring

All inverse problems are ill-posed problems!!
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A MODEL FOR IMAGE FORMATION
Given the object, the computation of the image
through the imaging equation defines the direct
problem
The inverse problem of image restoration given
the noisy image g determine the unknown object f
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DEBLURRING
Helpful approximations
Then, through 2D Fourier transform
The inverse problem of image restoration is
often a deconvolution problem
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NON-UNIQUENESS
If
Then
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NON-EXISTENCE
g(?) is an irregular function (due to the noise
affecting the measurements process)
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INSTABILITY
If
h(x) has a compact support,
Then
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SOLUTION METHODS
Statistical approach

• inverse problems are reformulated as problems of
  
• statistical inference by means of Bayesian
  statistics

• all quantities are modelled as random variables

• the goal is to determine the posterior
  probability density
• function

• many estimators error on the solution for free

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SOLUTION METHODS
Deterministic approach

• functional spaces provide the mathematical
  framework

• stability is obtained by constraining the
  solutions to
• belong to some particular subspace

• many algorithms historically well-established
• Tikhonov method (1963) is the first
  regularization method
• for linear inverse problems

• difficulty in determining the uncertainty on the
  solution

• extension to non-linear problems not established

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• Statistical approach

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STATISTICAL GLOSSARY

• A random variable X is a function associating a
  (real)
• number x to the outcome of an experiment

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STATISTICAL GLOSSARY
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STATISTICAL GLOSSARY

• Random variable X the unknown

• Random variable Y the measurement

• Random variable E the noise

• Functional relation Yf(X,E) the physical model

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STATISTICAL INVERSE PROBLEMS
Probability densities
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BAYESIAN APPROACH
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BAYES THEOREM
Then
Remark 1 only conditional probabilities, no
joint probability
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BAYES THEOREM
Thanks to Bayes formula, an inverse problem can
be solved by three steps

• based on all prior knowledges on the unknown,
• find a prior density which judiciously reflects
  these
• apriori information on the solution

• find the likelihood function describing the
  interrelation
• between measurement and unknown (i.e.,
  formulate
• an accurate mathematical model of the problem)

• formulate effective computational techniques to
• assess the posterior density

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PRIORS
The problem is the one to transform qualitative
information into a quantitative form to be
included in the prior density
Examples

• Subsurface electromagnetic sounding of the
  earth
• how to probabilistically describe
  layered/non-layered
• structures and cracks?

• Anatomical medical imaging how to
  probabilistically
• describe the location and surface structures of
  a tumor?

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PRIORS

• easily tractable

• approximations to non-Gaussian distributions
• (thanks to the central limit theorem)

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LIKELIHOODS
The likelihood function contains

• the forward model

• information on the noise

(Easy) example
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ESTIMATORS
Given the posterior density, the solution of the
inverse problems may be estimated in different
statistically-based fashions
Examples