METHODS FOR IMAGE RESTORATION

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METHODS FOR IMAGE RESTORATION

• Michele Piana
• Dipartimento di Informatica
• Universita di Verona

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REGULARIZATION ALGORITHM
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REGULARIZATION ALGORITHMS
A regularization algorith satisfies the
semiconvergence property
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HEURISTICS

• There are plenty of choice criterions there is
  no reliable
• choice criterion

• Some choice recipe is valid for all
  regularization methods
• but there are recipes specifically built up for
  a specific
• regularization method

• Simulation is the basis for the adoption of a
  selection
• criterion semiconvergence allows the
  determination of
• the typical size of the optimal regularization
  parameter
• for a specific problem

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SELECTION CRITERIONS

• Discrepancy principle all regularization
  methods (and
• even non-linear problems) generalization to
  the case
• of noisy models general trend oversmoothing

• Generalized Cross Validation (GCV) typically
  for
• Tikhonov method generalization to problems
  different
• than inverse problems general trend
  undersmoothing

• L-curve typically for Tikhonov method weak
  theoretical
• basis

• Semi-heuristic approach analysis of the
  cumulative residuals

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DISCREPANCY PRINCIPLE
Remarks
generalization to non-linear problems is
straightforward
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DISCREPANCY PRINCIPLE
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GENERALIZED CROSS VALIDATION (GCV)
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GENERALIZED CROSS VALIDATION
Some linear algebra some rotations
GCV find the minimizer of
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L-CURVE
Warning use the log-log scale!!
Choice criterion the optimal regularization
parameter is the value for which the L-curve has
its corner
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L-CURVE
Remarks
this is a very heuristic recipe there are cases
where the L-curve is not L-shaped
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WIENER PROCESSES
is the discretization of a Wiener process
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REGULARIZATION
Tikhonov regularization and Discrepancy Principle

• find the one-parameter family of solutions of
  the minimum problem

Heuristic remark the Discrepancy Principle is
oversmoothing
Cumulative residuals
Optimality criterion
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• Tikhonov method for the analysis of satellite data

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MICROSCOPIC VIEW
Bremsstrahlung

• Accelerated electrons collide against heavy ions
  in the
• solar chromosphere.

• The scattered electrons decrease their velocity.

• A big amount of electromagnetic radiation is
  emitted in
• the hard X-ray range

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A BIT OF PHYSICS - I
Collisional bremsstrahlung
The relation between the measured photon spectrum
g(e) and the mean electron flux spectrum F(E) is
described by a Volterra integral equation of the
first kind
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A BIT OF PHYSICS - II
Advantages

• Electrons carry more physics than photons

• Electron flux maps reach energies higher than
  photon maps
• (thanks to bremsstrahlung)

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A BIT OF PHYSICS - III
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A BIT OF PHYSICS - IV
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THE ALGORITHM - I

• For each detector and each (u,v) point

• construct the count visibility spectrum
• (count visibility vs count energy)

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THE ALGORITHM - II
(Detector 9 u0.00105 arcsec-1, v0.00252
arcsec-1)
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THE ALGORITHM - III

• For each detector and each (u,v) point

• construct the count visibility spectrum
• (count visibility vs count energy)

• apply regularized inversion to obtain an
  electron
• visibility spectrum (electron visibility vs
  count energy)

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THE ALGORITHM - IV

• The regularized solution is computed by means of
  the SVD of A

• The regularization parameter is fixed by
  analyzing the cumulative
• residuals

• The error on the regularized solution is
  assessed by means of
• the confidence strip

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THE ALGORITHM - V

• For each detector and each (u,v) point

• construct the count visibility spectrum
• (count visibility vs count energy)

• apply regularized inversion to obtain an
  electron
• visibility spectrum (electron visibility vs
  count energy)

• For each detector and each electron energy

• construct the electron visibilities

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THE ALGORITHM - VI
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THE ALGORITHM - VII

• For each detector and each (u,v) point

• construct the count visibility spectrum
• (count visibility vs count energy)

• apply regularized inversion to obtain an
  electron
• visibility spectrum (electron visibility vs
  count energy)

• For each detector and each electron energy

• construct the electron visibilities

• For each electron energy

• apply some Fourier-based imaging method
• to construct the electron flux map

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10-14 keV
14-18 keV
18-22 keV
22-26 keV
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26-30 keV
30-34 keV
34-38 keV
38-42 keV
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42-46 keV
46-50 keV
50-54 keV
54-58 keV
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58-62 keV
62-66 keV
66-70 keV
70-74 keV
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74-78 keV
78-82 keV
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POWER OF THE METHOD

• Spatially integrated electron spectroscopy

• Spatially resolved electron spectroscopy