Image Deconvolution Image Deconvolution

Applied to CCD astronomical Applied to CCD astronomical imagesimages

with astrometric purposeswith astrometric purposes

Maria Teresa Merino EspasaUniversidad de Barcelona (Spain)

SummarySummary

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

1. Introduction to image 1. Introduction to image RestorationRestoration

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

Image restorationImage restoration

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

detection) …

Our detected data images are not exactly the radiation from sources of interest. It’s important to consider that may be affected by:

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 radiationimage sharpeningfusion of data from diferent instrumentsincrease S/N of sourcesastrometric resolution improvement…

Linear image formation system with additive Linear image formation system with additive noisenoise

),(),(),( yxnyxgyxp p(x,y): projection (measured) data

g(x,y): blurred image

n(x,y): additive noise

h(x,y): blurring effects

d(x,y): source+background radiation111111 ),(·),;,(),( dydxyxdyxyxhyxg

When When space-invariant PSFspace-invariant PSF::

),(),(),(·),(),( 111111 yxdyxhdydxyxdyyxxhyxg

),(),(),(),( yxnyxdyxhyxp

In Fourier In Fourier Space:Space: ),(),()·,(),( vuNvuDvuHvuP

Deconvolution ill-posed problemDeconvolution ill-posed problem

Why not…..

),(),(

),(),(

),(vuHvuN

vuHvuP

vuD ……...

?

Because:H(u,v) has zero o near zero values in most of (u,v) range

1. at those points the division operation is undefined or results in meaningless values !!!!

2. for points having very small |H(u,v)|, although the division can be done, the noise will be amplified to an intorelable extent !!!!

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

Deconvolution: Image restoration based on some knowledge of the Image restoration based on some knowledge of the degradation processes effects and/or the expected source distribution degradation processes effects and/or the expected source distribution using models of the statistical and blurring effects of the image using models of the statistical and blurring effects of the image formation system.formation system.

Deconvolution algorithmsDeconvolution algorithms

1. 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

2. Regularization hypotesis (positiveness, energy conservation, frequency bounders, …)

3. Convergence iterative algorithm (gradient methods, EM, successive substitutions,…)

4. Iteration Stopping protocol (a posteriori, feasibility tests, crossvalidation maximum,…)

The Bayesian Approach The Bayesian Approach (fixed (fixed models)models)

)(

)()|()|(

p

aappa

P(a|p): “a posterior probability”, radiation a given the recorded data p

P(a): “a prior probability” of the source of radiation

P(p|a): likelihood

P(p): Normalizing constant

• MAP (Maximum A Posteriori), maximizes P(p|a)·P(a)

• MLE (Maximum Likelihood Estimator), maximizes P(p|a) with P(a)=cte

• MEM (Maximum Entropy Method) is MAP with p(a) given by maximum entropy hypotesis (=minimum data uncertainty)

Bayes Hypotesis:Bayes Hypotesis: The most probable image a(x,y), given The most probable image a(x,y), given measured data p(x,y) is obtained by maximizing P(a|p) with a measured data p(x,y) is obtained by maximizing P(a|p) with a suposed P(p|a) image formation model (and maybe also a suposed P(p|a) image formation model (and maybe also a P(a) “a priory” knowledge).P(a) “a priory” knowledge).

The most usual deconvolution algorithms division regarding how they maximize P(a|p) are:

G+P_MLE deconvolutionG+P_MLE deconvolution

• sucessive sustitutions convergence method

• energy conservation:

• Poisson photon gathering+Gauss readout noise:

0

2

)(

0

2

)(

!

)(

!

)(

'2

2

2

2

k

kj

pk

k

kj

pk

j

k

he

k

hke

pj

j

!

j

k

jhj

hP k h e

k

NOTE: This resulting algorithm, without flat and background, is similar to MLE Poisson (Richardson Lucy) computation structure if p’=p and K=1

Iterative and nonlinear algorithm

2

2

2

)(

2

1)|(

kp

j

j

ekpP

B

iji

j

ijj ba

C

fh

1

·

m

D

jjj

B

l

sljl

jji

i

si

si

bCaf

pf

qaa

1

1

)(

)()1( '1

B

i

m

D

jjj

B

l

sljl

jjisi

D

jjj

s

bCaf

pfa

bp

1 1

1

)(

)(

1)(

'

)(

AlgorithAlgorithmmbasics:basics:

D

jj

D

jj

B

i

B

i

D

j

ijiii bp

Cj

afaq

111 1 1

2. Adaptative 2. Adaptative Deconvolution with Deconvolution with

WaveletsWavelets

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

Wavelets descompositionWavelets descompositionThe wavelet transform replaces the Fourier transform’s sinusoidal waves The wavelet transform replaces the Fourier transform’s sinusoidal waves

by a family generated by by a family generated by translationstranslations and and dilationsdilations of a function called of a function called

Mother WaveletMother Wavelet. .

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

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 , i=1,...,n). No subsampling is applied wavelet planes have same size as original image.

- space and scale

- base

functions

)(, xbab a

dxxfabafWT abx )()()),(( 2

1

Example of Wavelets descompositionExample of Wavelets descomposition

co 1 2 3 4 c4 = + + + + Wavelet planes

High frequency Low frequency

Original image

Residual plane at scale 4

cc00 ww11

cc44ww33ww44

ww22

Adaptative deconvolution with Adaptative deconvolution with waveletswavelets

Signal detection mask:

with

Objectives of using wavelets: Multiescale deconvolutionObjectives of adaptative deconvolution:

- Do not amplify noise. - To detect and deconvolute only those features with signal.

To deconvolve the image, this is decomposed in wavelet planes at every To deconvolve the image, this is decomposed in wavelet planes at every iteration of the method. The signal detection masks only applies iteration of the method. The signal detection masks only applies deconvolution to those pixels with significant signal.deconvolution to those pixels with significant signal.

m

D

jjj

B

l

sljl

N

v

shjv

pjvjv

shjvji

i

si

si

bCaf

mf

qaa

wj

1

1

)(

)(,

',,

)(,

)()1( 1

0

)(2

2

3

exp12

,

2

,,

,r

jv

rjvjv

jvm

f

n

t

shjv

pjv

jv n

f

2)(,

',

,

rjv,

noise std. desv. at wavelet plane sorrounding pixel j

if 0,, rjvjv

if 0,, rjvjv

G+P_AWMLE G+P_AWMLE ::

3. Applications of our 3. Applications of our deconvolution algorithm with deconvolution algorithm with

astrometric purposes:astrometric purposes:

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

-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.

Data overview

0.8degx0.8deg subframe: note object blending and blooming around bright starsDeconvolution objective:

Increase limit magnitude to detect faint NEOs with minimum false detections

RAO: Rothney Astrophysical ObservatoryUniversity of Calgary (Canada)

Deconvolution of Baker-Nunn (RAO) Deconvolution of Baker-Nunn (RAO) imagesimages

Protocol for Baker-Nunn imagesProtocol for Baker-Nunn images

7) Compute limiting magnitude gain and rest of

analysis

1) CCD image calibration: bias, dark and flatfield correction

2) Compute astrometric plate transformation with reference catalogue

3) Perform aperture photometry of selected stars (bright but not saturated)

4) PSF fitting with list created in 3)

6) sextractor object detection and matching with USNOA2.0 catalogue

iraf.ccdproc

wcstools.sua2, sextractor,iraf.ccmap

iraf.phot, iraf.pstselect

iraf.psf, iraf.seepsf

5) Image deconvolution

Deconvolution of Baker-Nunn imagesDeconvolution of Baker-Nunn images

Benefits of image deconvolution

1) Increase of SNR

Limiting magnitude gain

2) Increase of resolution

Object deblending

Original image: in blue matched detections

Deconvoluted image 60 iterations: yellow and red are new matched

detections

Matched vs. non-matched detections

Evolution of raw and matched detections with the number of iterations.Asymptotic converge translated into stable number of detections for large number of iterations.

Adaptative wavelet deconvolution introduces small number of non-matched detections. In contrast to classical Richardson-Lucy algorithms, false detections increase linearly, not exponentially.

Deconvolution of Baker-Nunn imagesDeconvolution 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.

Deconvolution of Baker-Nunn imagesDeconvolution of Baker-Nunn images

Limiting magnitude gain

USNOA2.0 R magnitude distribution for the catalogue itself (top), deconvoluted image 180 iterations (middle) and original

image (bottom)

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:

mR 0.6

Image# matched detections

Original 1559

Deconvoluted 180 iterations

2365

Deconvolution of Baker-Nunn imagesDeconvolution of Baker-Nunn images

Deconvolution of QUEST imagesDeconvolution of QUEST images

V filter, frames from 2 CCDs 1”/pixel, 2 diferent fields in 2 single CCDs Limiting magnitude (S/N>10)

aprox. 19.2

Data overview

Deconvolution objective:

Detect a subsample of QSO candidates likely to be lensed by:

1. improving the resolution (deblend close companions)

2. improving the limiting magnitude

2 MiniMosaic WIYN 0.141”/pixel Covers aprox. the same field 6+38 QSO candidates detected by

varability criteria in those fields

QUEST= QUasar Ecuatorial Survey Team

QUEST images to deconvolve

WIYN images to evaluate results

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.

Deconvolution of QUEST imagesDeconvolution of QUEST images

Algorithm propertiesAlgorithm properties Powerful technique for increasing the number of useful

science objects from the faint part of magnitude distribution.

Low speed convergence and heavy CPU and RAM usage.

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

Generalizable for whatever observing facility employing a CCD.

Enhances spatial resolution which translates into better object deblending discrimination.

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.

Work in progressWork 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.

To sum up:To sum up:

Deconvolution can be a powerful technique in astrometry projects.

Those examples have shown how our algorithm can produce in the particular case of astrometry of point sources:

• recover faint objects• improve resolution

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

But it’s crucial to select the proper algorithm carefully considering:

1. your data2. your scientifical objectives

Final ConclusionsFinal Conclusions