Image Deconvolution

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Image Deconvolution

• Applied to CCD astronomical images
• with astrometric purposes

Maria Teresa Merino Espasa Universidad de
Barcelona (Spain)
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Summary

• Introduction to Image Restoration
• Adaptative deconvolution with wavelets
• Applications of our deconvolution algorithm with
  astrometric purposes
• Increase of faint detections in Baker-Nunn images
  for faint asteroid discovering
• Quest aplication to increase astrometrical
  resolution to detect gravitational lenses
  candidates
• Work in progress to increse faint detection
  increase astrometrical resolution
• Conclusions

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1. Introduction to image Restoration

• Objective
• Presentation of our algorithm for MLE
  deconvolution in linear space-invariant PSF
  systems with complete basic CCD images statistics
  (Poisson Gaussian)

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Image restoration
Our detected data images are not exactly the
radiation from sources of interest. Its
important to consider that may be affected by

• atmosphere effects
• optical distorsions and aberrations
• different pixel sensitivity
• additional noise (in emision, travelling and/or
  detection)

Image Restoration is the removal or reduction of
those degradations that were incurred while the
digital image was being obtained to bring the
image toward what it would have been if it had
been recorded without degradation.
In astronomy, the most usual objectives are

• deblurring
• removal atmospheric seeing radiation
• image sharpening
• fusion of data from diferent instruments
• increase S/N of sources
• astrometric resolution improvement

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Linear image formation system with additive noise
p(x,y) projection (measured) data g(x,y)
blurred image n(x,y) additive noise h(x,y)
blurring effects d(x,y) sourcebackground
radiation
When space-invariant PSF
In Fourier Space
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Deconvolution ill-posed problem
Deconvolution Image restoration based on some
knowledge of the degradation processes effects
and/or the expected source distribution using
models of the statistical and blurring effects of
the image formation system.
...?
Why not..

• at those points the division operation is
  undefined or results in meaningless values !!!!
• for points having very small H(u,v), although
  the division can be done, the noise will be
  amplified to an intorelable extent !!!!

Because
H(u,v) has zero o near zero values in most of
(u,v) range
Conclusion
Image deconvolution is an inverse solution
ill-posed problem !!!
All the algorithms which try to solve it should
be concerned about obtaining a solution with the
three following properties

• existence
• uniqueness
• stability

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Deconvolution algorithms

• Image formation, degradation and recovering model
  basic hypotesis
• Classical approach
• Direct inverse filter in Fourier space
• Linear Regularized Filters (Wiener)
• Minimum root mean squares
• CLEAN
• Bayessian approach
• Fixed Model MLE, MAP
• Variable Model Pixons
• Regularization hypotesis (positiveness, energy
  conservation, frequency bounders, )
• Convergence iterative algorithm (gradient
  methods, EM, successive substitutions,)
• Iteration Stopping protocol (a posteriori,
  feasibility tests, crossvalidation maximum,)

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The Bayesian Approach (fixed models)
P(ap) a posterior probability, radiation a
given the recorded data p P(a) a prior
probability of the source of radiation P(pa)
likelihood P(p) Normalizing constant
Bayes Hypotesis The most probable image a(x,y),
given measured data p(x,y) is obtained by
maximizing P(ap) with a suposed P(pa) image
formation model (and maybe also a P(a) a priory
knowledge).
The most usual deconvolution algorithms division
regarding how they maximize P(ap) are

• MAP (Maximum A Posteriori), maximizes P(pa)P(a)
• MLE (Maximum Likelihood Estimator), maximizes
  P(pa) with P(a)cte
• MEM (Maximum Entropy Method) is MAP with p(a)
  given by maximum entropy hypotesis (minimum data
  uncertainty)

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GP_MLE deconvolution

• sucessive sustitutions convergence method
• energy conservation
• Poisson photon gatheringGauss readout noise

Algorithm basics
Iterative and nonlinear algorithm
NOTE This resulting algorithm, without flat and
background, is similar to MLE Poisson (Richardson
Lucy) computation structure if pp and K1
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2. Adaptative Deconvolution with Wavelets

• Objective
• Presentation of our multiresolution adaptative
  deconvolution algorithm based on wavelets

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Wavelets descomposition
The wavelet transform replaces the Fourier
transforms sinusoidal waves by a family
generated by translations and dilations of a
function called Mother Wavelet.
- space and scale -
base functions

• Discrete wavelet transform (DWT) algorithm
  specific properties
• DWT is computed with à trous algorithm.
• Mother wavelet function ? is derived from a B3
  cubic spline scaling function.
• This is a dyadic decomposition (scale ? 2i ,
  i1,...,n).
• No subsampling is applied ? wavelet planes have
  same size as original image.

• General properties of wavelets descomposition
• Spatial and frequential content is almost
  decoupled.
• Better noise vs.signal discrimination than
  Fourier transform.
• Good processing flexibility.

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Example of Wavelets descomposition
w2
c0
w1
w4
c4
w3
Original image
Residual plane at scale 4
Wavelet planes
co
w1
w2
w3
w4
c4





High frequency
Low frequency
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Adaptative deconvolution with wavelets
To deconvolve the image, this is decomposed in
wavelet planes at every iteration of the method.
The signal detection masks only applies
deconvolution to those pixels with significant
signal.
Objectives of using wavelets Multiescale
deconvolution Objectives of adaptative
deconvolution - Do not amplify noise. - To
detect and deconvolute only those features with
signal.
Signal detection mask
if
with
noise std. desv. at wavelet plane n sorrounding
pixel j
if
GP_AWMLE
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3. Applications of our deconvolution algorithm
with astrometric purposes

• Objective
• Show astrometric objectives, aplications and
  deconvolution artifacts and effects

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Deconvolution of Baker-Nunn (RAO) images
Data overview

• 50cm - f/1 Baker-Nunn instrument!!!
• very large FOV 4.2ºx4.2º,
• moderately undersampled data 3.9/pixel,
• significant object blending,
• bright stars cause overblooming,
• PSF is low correlated with seeing and dominated
  by systematic distortions across FOV, which can
  be modelled
• due to large pixel size, background level is
  dominated by Poisson noise.

0.8degx0.8deg subframe note object blending and
blooming around bright stars
Deconvolution objective
Increase limit magnitude to detect faint NEOs
with minimum false detections
RAO Rothney Astrophysical Observatory University
of Calgary (Canada)
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Protocol for Baker-Nunn images
1) CCD image calibration bias, dark and
flatfield correction
iraf.ccdproc
2) Compute astrometric plate transformation with
reference catalogue
wcstools.sua2, sextractor,iraf.ccmap
3) Perform aperture photometry of selected stars
(bright but not saturated)
iraf.phot, iraf.pstselect
4) PSF fitting with list created in 3)
iraf.psf, iraf.seepsf
5) Image deconvolution
6) sextractor object detection and matching with
USNOA2.0 catalogue
7) Compute limiting magnitude gain and rest of
analysis
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Deconvolution of Baker-Nunn images
Benefits of image deconvolution
1) Increase of SNR
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Limiting magnitude gain
2) Increase of resolution
?
Object deblending
Deconvoluted image 60 iterations yellow and red
are new matched detections
Original image in blue matched detections
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Deconvolution of Baker-Nunn images
Matched vs. non-matched detections
Adaptative wavelet deconvolution introduces small
number of non-matched detections. In contrast to
classical Richardson-Lucy algorithms, false
detections increase linearly, not exponentially.
Evolution of raw and matched detections with the
number of iterations. Asymptotic converge
translated into stable number of detections for
large number of iterations.
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Deconvolution of Baker-Nunn images
Where do the non-matched detections come from?
Left original image with explanation of the
origin of non-matched detections displayed in red
in the right frame, which corresponds to a 60
iteration deconvoluted image.
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Deconvolution of Baker-Nunn images
Limiting magnitude gain
But deriving magnitude gain directly from ratio
of the number of detections is not allways
reliable, due to intrinsic magnitude distribution
of studied stellar field.
From estimating the completeness of magnitude
histograms with respect to USNOA2.0 distribution
USNOA2.0 R magnitude distribution for the
catalogue itself (top), deconvoluted image 180
iterations (middle) and original image (bottom)
?mR ? 0.6
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Deconvolution of QUEST images
QUEST QUasar Ecuatorial Survey Team
Data overview
QUEST images to deconvolve
WIYN images to evaluate results

• V filter, frames from 2 CCDs
• 1/pixel,
• 2 diferent fields in 2 single CCDs
• Limiting magnitude (S/Ngt10) aprox. 19.2

• 2 MiniMosaic WIYN
• 0.141/pixel
• Covers aprox. the same field
• 638 QSO candidates detected by varability
  criteria in those fields

Deconvolution objective

• Detect a subsample of QSO candidates likely to be
  lensed by
• improving the resolution (deblend close
  companions)
• improving the limiting magnitude

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Deconvolution of QUEST images
Aplication of image deconvolution to 6 QSO
candidates in the field. Each panel includes
original QUEST image, deconvolved QUEST (400
iterations) and high resolution WIYN image,
repectively. Those in green correspond to objecte
present in all three images, in black those only
resolved by WIYN and in red those resolved in
both deconvolved QUEST and WIYN.
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Algorithm properties

• Powerful technique for increasing the number of
  useful science objects from the faint part of
  magnitude distribution.

• Enhances spatial resolution which translates into
  better object deblending discrimination.

• Generalizable for whatever observing facility
  employing a CCD.

• Cheap only requires fast computers. The
  algorithm is highly parallelizable to be run in
  distributed computers.

• Low speed convergence and heavy CPU and RAM usage.

• Systematic usage best suited for astrometric
  surveys with moderate data throughput.

• Selective usage is suited for all kind of images,
  either high or low resolution.

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Work in progress

• Use multi frame reduction for removing cosmics
  and hot pixels.

• Improve PSF fitting to decrease false detections
  around bright stars.

• Compare internal astrometric accuracy of original
  and deconvoluted images.

• Accurate determination of CCD flats, gain and
  readout noise for better noise distribution
  modelling when deconvoluting.

• Define strategy to assess space variant PSF
  across the FOV

• Further evaluation of astrometric improvements
  and photometric effects.

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To sum up

• In this presentation we have presented our
  selected algorithm
• Gausian and Poison Adaptative Wavelets Maximum
  Likehood Estimator

• This algorithm has been proved with two
  diferent astrometrical type of data and diferent
  scientific goals

• Those examples have shown how our algorithm
  can produce in the particular case of astrometry
  of point sources
• recover faint objects
• improve resolution

Final Conclusions
Deconvolution can be a powerful technique in
astrometry projects.

• But its crucial to select the proper algorithm
  carefully considering
• your data
• your scientifical objectives