Blind galaxy survey images Deconvolution with Shape Constraint · Laplacian Graph P = D - J, where D is the degree matrix and J is the adjacent matrix of the graph P tP = V V t Degree

Jean-Luc Starck http://jstarck.cosmostat.org

Collaborators:

Blind galaxy survey images Deconvolution with Shape Constraint

Morgan Schmitz Fred Nole Fadi Nammour

http://www.cosmostat.org

- Part I: Introduction to the Blind Deconvolution Problem

- Part II: Point Spread Fonction (PSF) Field Recovery

- Part III: Shape Measurements & Deconvolution

Weak Gravitational Lensing & Blind Deconvolution

CEA - Irfu

Weak Lensing

Observer Gravitational lensBackground galaxies

Euclid Mission • Medium-class mission in ESA’s Cosmic Vision program • 6 year survey, launch in Q4 2022

Optimized for weak lensing (and galaxy clustering) — 15,000 deg2 survey area — over 1.5 billion galaxies — redshifts out to z = 2

Euclid ESA Space Mission

2022

!probe the properties and nature of dark energy, dark matter, gravity and distinguish their effects decisively

Gains in space: Stable data: homogeneous data set over the whole sky !Systematics are small, understood and controlled !Homogeneity : Selection function perfectly controlled

Galaxies

Detection + Classification stars/galaxies

Galaxies Stars

Motivation for spatial observations

Convolution Operator + Sampling

Space Variant PSF

Euclid PSF Wavelength Dependency

• PSF Modeling ➡Undersampling ➡Space dependency ➡Wavelength dependency ➡Time dependency

Shape Measurements

Galaxies are convolved by an asymetric PSF+ Images are undersampled

Shape measurements must be deconvolved

Methods: Moments (KSB), Shapelets, Forward-Fitting, Bayesian estimation, etc

State of the art: PSFextractor [E. Bertin, 2011]

where h is Lanczos interpolation kernel

Regularization

1. Forward Modelling approach (FM) leaded by Lance Miller• Model the exit pupil using pre-defined Zemax modes.• Fit to all stars.

➡PRO: No other existing method can achieve the very strong requirement on the PSF.➡CON: But hard to validate since i) the simulations are done with the same model, and ii)

the model may change (vibrations at launch).

2. Need a data driven approach • Validate the FM solution on real data.• All surveys ((KIDS, CFHTLenS, DES) have used the data driven approach.• HST: TinyTim HST modelling software (Krist 1995) for the Hubble Space Telescope not

as good as a data driven approach (Jee et al, PASP, 2007; Hoffmann and Anderson, Instrument Science Report ACS 2017-8, 2017).

• What does not exist can be invented.• Combination of both approaches could lead to the optimal solution.

State of the art: PSF Modeling

– Introduction to the Blind Deconvolution Problem

– Euclid PSF Field MeasurementMonochromatic PSF: Matrix Factorization + Laplacian Graph + Sparsity

Polychromatic PSF and Optimal Transport

– Shape Measurements & Deconvolution

Blind Deconvolution

• F. M. Mboula, J.-L. Starck, S. Ronayette, K. Okumura, and J. Amiaux, Super-resolution method using sparse regularization for point-spread function recovery. A&A, 575, id.A86, 2015.

• F. Ngole, J.-L Starck, et al, “Constraint matrix factorization for space variant PSFs field restoration”, Inverse Problems, 2016.• M.A. Schmitz, J.-L. Starck and F.M. Ngolè, "Euclid Point Spread Function field recovery through interpolation on a Graph

Laplacian", submitted, 2018.

• F.M. Ngolè Mboula, and J.-L. Starck, "PSFs field learning based on Optimal Transport distances", SIAM Journal on Imaging Sciences, 10, 3, pp. 979-1004, 2017. DOI: 10.1137/16M1093677.

• M. A. Schmitz et a; "Wasserstein Dictionary Learning:Optimal Transport-based unsupervised representation learning", SIAM Journal on Imaging Sciences, 11, 2018.

http://arxiv.org/abs/1410.7679

http://arxiv.org/abs/1410.7679

https://arxiv.org/abs/1703.06066

http://epubs.siam.org/loi/sjisbi






DecimationRandomshifting

Noise

Datapoints

Pixels

0 K...

0

N

matrix

Degradation

Pixels

0 K...

0

N/4

matrix

Flatto1-Dvector

ObservedPSF

Datapoints

...... ... ...

...

...

“True”PSF(punctualstarconvolvedwithtruePSF)

Datapoint

Datapoint

The PSF Field Recovery Problem

:truePSFsatdifferentpositionsoftheFOV.

:decimationandshifting.

:observedPSFs.

X � RN�K+

minX

� Y �MX �2

X

matrix M

The PSF Super-resolution Challenge

Y = HX +N

Inverse Problems & Regularization

Physical Knowledge on X (ex: Gaussian Random Field, etc). ==> Gaussian smoothing, Wiener reconstruction, etc

==> Log normal distribution prior

Knowledge on the histogram of X in pixel space or in another one. ==> Positivity constraint, sparsity constraint, etc.

X properties are understood through a representative data set. ==> Machine Learning

minX

||Y �HX||2 C(X)+

ai,k = coefficient corresponding to contribution of the i-th vector to the k-th PSF.

PSF(k) = xk =rX

i=1

ai,ksisi = ith vector (2D image)

Eigen PSF

Joint estimations of super-resolved PSFs at stars positions ➡ Low rank constraint: Constraint the PSFs to be a

linear combination of the eigenvectors PSFs:

Constraints

a1,i

a2,i

aK,i

ak,i

a.,i

a.,i

ui � Ustars

f(�uk1

� uk2�2)

➡ Proximity constraint on the ai,k coefficients: the closer are the stars, the more the coefficients of the linear combination are similar.

X

r

k �si k1<latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit>

➡ Smoothness constraint on each si

➡ Positivity constraint (xk > 0)

PSF Laplacian Graph Laplacian Graph P = D - J, where D is the degree matrix and J is the adjacent matrix of the graph

P tP = V �V t

Degree matrix= diagonal matrix with information about the degree of each vertex—that is, the number of edges attached to each vertex.

Adjacent matrix A: element Aij is one when there is an edge from vertex i to vertex j, and zero when there is no edge.

si are ”eigen PSF”

Matrix Factorization

PSF(k) = xk =rX

i=1

ai,ksi

S = [s1, ..., sp]

http://www.cosmostat.org/software/rca/

Each eigen PSF is sparse in a wavelet dictionaryX

r

k �si k1<latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit><latexit sha1_base64="BGy0wU3NROsLcxBQ3kBsivhF+Lc=">AAACEHicbVBNS8NAEJ3Ur1q/oh69LBbRU0lE0GPRi8cKthWaEDbbbbt0dxN2N0IJ/Qle/CtePCji1aM3/43bNqC2Phh4vDfDzLw45Uwbz/tySkvLK6tr5fXKxubW9o67u9fSSaYIbZKEJ+ouxppyJmnTMMPpXaooFjGn7Xh4NfHb91RplshbM0ppKHBfsh4j2Fgpco8DnYlIoSDFCnNOOQoaA4Z0xH6kyEcocqtezZsCLRK/IFUo0Ijcz6CbkExQaQjHWnd8LzVhjpVhhNNxJcg0TTEZ4j7tWCqxoDrMpw+N0ZFVuqiXKFvSoKn6eyLHQuuRiG2nwGag572J+J/XyUzvIsyZTDNDJZkt6mUcmQRN0kFdpigxfGQJJorZWxEZ2BiIsRlWbAj+/MuLpHVa872af3NWrV8WcZThAA7hBHw4hzpcQwOaQOABnuAFXp1H59l5c95nrSWnmNmHP3A+vgFBk5wN</latexit>

MODEL:


Matrix Factorization


Sparsity on the eigen PSF: the PSF should have a sparse representation in an appropriate basis

Positivity ConstraintSmoothness of the PSF field constraint : the smaller the difference between two PSFs’ positions ui , uj , the smaller the difference between their estimated representations

Data fidelity term

minS,�

12�Y �M(S�V �)�22 +

r�

i=1

�wi � �si�1 + �+(S�V �) + ��(�)


The PSF Interpolation Challengeui � Ustars

1� uk1 � uk2 �

ek2

Numerical Experiments

With undersampling (upsampling factor of 2)

Center

Local

Corner

Obs Ref RCAPSFEX

Data: 500 Euclid-like PSFs (Zemax), field observed with different SNRsTheese PSFs account for mirrors polishing imperfections, manufacturing and alignments errors and thermal stability of the telescope.


Stars SNR

Multiplicative Bias• M.A. Schmitz, J.-L. Starck and F.M. Ngolè, "Euclid Point Spread Function field recovery through

interpolation on a Graph Laplacian", submitted, 2018.

20% Improvement

PSF Wavelength Dependency

G. Monge, “Mémoire sur la théorie des déblais et des remblais”, 1781

OptimalTransportTheMonge-Kantorovichproblem

Barycenter

OT

Linear interpolation

Plausible average distribution in presence of shifts and changing of shape

Optimal Transport & PSF

Euclideanbarycenter

,

Optimal Transport IfinsteadwereplacetheEuclideandistancebytheWassersteindistance..

Wassersteinbarycenter

,

Optimal Transport

0

1

Wassersteinbarycenter

,

ATTRACTIVETOOL,BUTCOSTLYTOCOMPUTE:==>fixedusingaregularisation(Sinkhorniterations)

• M. A. Schmitz et a; "Wasserstein Dictionary Learning:Optimal Transport-based unsupervised representation learning", SIAM Journal on Imaging Sciences, 11, 2018.




Optimal Transport

BacktothePSFestimationproblem:

… … … … … … …

Howtobreakthedegeneracy?

0 1.0

…

Hypothesis:monochromaticPSFsofaintermediarywavelengthareWassersteinbarycentersoftheextremes.

Optimal Transport

Hypothesis:monochromaticPSFsofaintermediarywavelengthareWassersteinbarycentersoftheextremes.

Experimentresults:

SimulatedEuclid-likePSFs(groundtruth)

OTModeling

linearprojectionofwavelengthsinto[0,1].

-RCA �<latexit sha1_base64="1QWkvc23bZ9uQzoxNspzdWFe2Tk=">AAAB8XicdVDLSgMxFL1TX7W+qi7dBFvBVZkp+NoV3bisYB/YDiWTybShSWZIMkIZ+hduXCji1r9x59+YtiOo6IHA4Zxzyb0nSDjTxnU/nMLS8srqWnG9tLG5tb1T3t1r6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMr2Z+554qzWJ5ayYJ9QUeShYxgo2V7qp9brMhrqJBueLW3DnQN3LiehenHvJypQI5moPyez+MSSqoNIRjrXuemxg/w8owwum01E81TTAZ4yHtWSqxoNrP5htP0ZFVQhTFyj5p0Fz9PpFhofVEBDYpsBnp395M/MvrpSY69zMmk9RQSRYfRSlHJkaz81HIFCWGTyzBRDG7KyIjrDAxtqSSLeHrUvQ/addrnlvzbuqVxmVeRxEO4BCOwYMzaMA1NKEFBCQ8wBM8O9p5dF6c10W04OQz+/ADztsnbpqQFw==</latexit><latexit sha1_base64="1QWkvc23bZ9uQzoxNspzdWFe2Tk=">AAAB8XicdVDLSgMxFL1TX7W+qi7dBFvBVZkp+NoV3bisYB/YDiWTybShSWZIMkIZ+hduXCji1r9x59+YtiOo6IHA4Zxzyb0nSDjTxnU/nMLS8srqWnG9tLG5tb1T3t1r6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMr2Z+554qzWJ5ayYJ9QUeShYxgo2V7qp9brMhrqJBueLW3DnQN3LiehenHvJypQI5moPyez+MSSqoNIRjrXuemxg/w8owwum01E81TTAZ4yHtWSqxoNrP5htP0ZFVQhTFyj5p0Fz9PpFhofVEBDYpsBnp395M/MvrpSY69zMmk9RQSRYfRSlHJkaz81HIFCWGTyzBRDG7KyIjrDAxtqSSLeHrUvQ/addrnlvzbuqVxmVeRxEO4BCOwYMzaMA1NKEFBCQ8wBM8O9p5dF6c10W04OQz+/ADztsnbpqQFw==</latexit><latexit sha1_base64="1QWkvc23bZ9uQzoxNspzdWFe2Tk=">AAAB8XicdVDLSgMxFL1TX7W+qi7dBFvBVZkp+NoV3bisYB/YDiWTybShSWZIMkIZ+hduXCji1r9x59+YtiOo6IHA4Zxzyb0nSDjTxnU/nMLS8srqWnG9tLG5tb1T3t1r6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMr2Z+554qzWJ5ayYJ9QUeShYxgo2V7qp9brMhrqJBueLW3DnQN3LiehenHvJypQI5moPyez+MSSqoNIRjrXuemxg/w8owwum01E81TTAZ4yHtWSqxoNrP5htP0ZFVQhTFyj5p0Fz9PpFhofVEBDYpsBnp395M/MvrpSY69zMmk9RQSRYfRSlHJkaz81HIFCWGTyzBRDG7KyIjrDAxtqSSLeHrUvQ/addrnlvzbuqVxmVeRxEO4BCOwYMzaMA1NKEFBCQ8wBM8O9p5dF6c10W04OQz+/ADztsnbpqQFw==</latexit><latexit sha1_base64="1QWkvc23bZ9uQzoxNspzdWFe2Tk=">AAAB8XicdVDLSgMxFL1TX7W+qi7dBFvBVZkp+NoV3bisYB/YDiWTybShSWZIMkIZ+hduXCji1r9x59+YtiOo6IHA4Zxzyb0nSDjTxnU/nMLS8srqWnG9tLG5tb1T3t1r6zhVhLZIzGPVDbCmnEnaMsxw2k0UxSLgtBOMr2Z+554qzWJ5ayYJ9QUeShYxgo2V7qp9brMhrqJBueLW3DnQN3LiehenHvJypQI5moPyez+MSSqoNIRjrXuemxg/w8owwum01E81TTAZ4yHtWSqxoNrP5htP0ZFVQhTFyj5p0Fz9PpFhofVEBDYpsBnp395M/MvrpSY69zMmk9RQSRYfRSlHJkaz81HIFCWGTyzBRDG7KyIjrDAxtqSSLeHrUvQ/addrnlvzbuqVxmVeRxEO4BCOwYMzaMA1NKEFBCQ8wBM8O9p5dF6c10W04OQz+/ADztsnbpqQFw==</latexit>

Simpleconstraintondictionary:probabilitydistribution.

Eigen-PSFextremerightEigen-PSFextremeleft

ReconstructedPSFs

Groun

d-truth

Very Preliminary Results

• M. A. Schmitz et al, in preparation, 2018


- Part I: Weak Lensing and Euclid

- Part II: Point Spread Fonction (PSF) Field Recovery

- Part III: Shape Measurements & Deconvolution

Weak Gravitational Lensing: The prospects of Euclid

Astronomical Image Deconvolution

y = Hx+ n

ObservedImagePSFConvolutionTrueImageNoise

Sandard deconvolution framework:

H is huge !!!

argminX

1

2kY �HXk22 + k�tXkp s.t. X � 0

Sandard deconvolution framework:

argminX

1

2kY �H(X)k22 + �k�tXkp s.t. X � 0

Big Astronomical Image Deconvolution

Y = H(X) +N

[y0,y1,…,yn]

[H0x0,H1x1,…,Hnxn]

[n0,n1,…,nn]

Object Oriented Deconvolution

For each galaxy, we use the PSF related to its center pixel:

min↵

kY �H�↵k2 + �k↵kpp

➡ Wavelet Denoising introduces a bias.➡ Fitting a galaxy model directly on significant wavelet coefficients

introduces a bias.

➡ Need to find a way to preserve the ellipticity during the restoration process

Inverse Problems: Sparse Recovery

Measuring Galaxies shapes

a

b

θ

: GalaxyX<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

e(X) =

1��ba

�2

1 +

�ba

�2 exp(2i✓) = e1(X) + ie2(X)

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

The complex ellipticity of a galaxy image uses quadrupole moments

e = � + eg < e >� �and

X 2 Mn(R)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

µs,t(X) =X

i,j

X[i, j](i� ic)s(j � jc)

t


a

b

θ a

b

θ

From Ellipticity to Moments

is an unbiased estimator of “ellipticity” (Schneidez & Seitz 1994)

�1(X) =µ2,0(X)� µ0,2(X)µ2,0(X) + µ0,2(X)

�2(X) =2µ1,1(X)

µ2,0(X) + µ0,2(X)

� = �1 + i�2 =µ2,0 � µ0,2 + 2µ1,1

µ2,0 + µ0,2

�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

=<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Observed galaxy Matched Gaussian window

Windowed galaxy

Y<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

W<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

W � Y<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Moments WindowingDoes not work in presense of noise! Need to apply a window function.

• “HSM” (or adaptive moments, Hirata & Seljak 2003; Mandelbaum et al 2004): match the gaussian window to the object

• Handling the PSF: Kaiser, Squires & Broadhurst 1995 (KSB).

High Accuracy Requirements�̃ = (1 + m)� + c� = �1 + i�2 = |�|ei2�

m = m1 + im2 = |m|ei2�m

c = c1 + ic2 = |c|ei2�c

�m < 2� 10�3

�c < 5� 10�4

Requirements for the ESA Euclid Space Telescope

Additive bias Multiplicative bias

Criteria for a good method

➡ Fast Calculation.➡ Level of bias (multiplicative and additive bias).➡ Capacity to measure the bias (number of simulations, depth, resolution, etc).➡ Robustness to calibration errors (PSF measurement errors, detectors effects,

background subtraction, etc).

X = image properties

e1(X) =µ2,0(X)� µ0,2(X)

µ2,0(X) + µ0,2(X), e2(X) =

2µ1,1

µ2,0(X) + µ0,2(X)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

X 2 Mn(R)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

µs,t(X) =X

i,j

X[i, j](i� ic)s(j � jc)

t


Moments using Inner Products

e1(X) =hX,U5ihX,U3i � hX,U1i2 + hX,U2i2

hX,U5ihX,U3i � hX,U1i2 � hX,U2i2, e2(X) =

2(hX,U6ihX,U3i � hX,U1ihX,U2i)hX,U5ihX,U3i � hX,U1i2 � hX,U2i2






U1 =

0

BBB@

1 1 . . . 12 2 . . . 2...

.... . .

...n n . . . n

1

CCCA


U2 =

0

BBB@

1 2 . . . n1 2 . . . n...

.... . .

...1 2 . . . n

1

CCCA


U3 =

0

BBB@

1 1 . . . 11 1 . . . 1...

.... . .

...1 1 . . . 1

1

CCCA


U4 =

0

BBB@

12 + 12 12 + 22 . . . 12 + n2

22 + 12 22 + 22 . . . 22 + n2

......

. . ....

n2 + 12 n2 + 22 . . . n2 + n2

1

CCCA


U5 =

0

BBB@

12 � 12 12 � 22 . . . 12 � n2

22 � 12 22 � 22 . . . 22 � n2

......

. . ....

n2 � 12 n2 � 22 . . . n2 � n2

1

CCCA


Matrices of indices combination

U6 =

0

BBB@

1 2 . . . n2 4 . . . 2n...

.... . .

...n 2n . . . n2

1

CCCA

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Idea : Conserving the inner products is equivalent to conserving the ellipticity parameters.

Advantage : These inner products are linear functions of the image we want to restore so it is mathematically easier to work with.

Question : How to formulate a constraint using these inner products? What would be the additional contribution of this constraint regarding the data fidelity term?

Ellipticities as inner products

The Shape Contraint :

The constraint looks just like a data fidelity term expressed in the moments space.

Idea : • Remove noise as much as possible inside the constraint in a inverse pb.

M(X) =6X

i=1

µi hX ⇤H � Y, Uii2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

The Moment Shape Constraint

M(X) =6X

i=1

µi hW � (X ⇤H � Y ), Uii2<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

=<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Observed galaxy Matched Gaussian window

Windowed galaxy

Y<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

W<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

W � Y<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Formulation with a Gaussian window :

Constraint with a Gaussian Window

Multi-Window ConstraintsRather than trying to fit a Gaussian window to the data, why not trying all windows in all directions and all scales ?

This can be done easily using a Curvelet like Decomposition.

Contrast Enhancement

Formulation with shearlets :

Aleternative form :

• : adjoint of the shearlet transform operator ⇤j


• : Curvelet/Shearlet transform (Kutyniok & Labate, 2012) j<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

M(X) =1

6

6X

i=1

1

N

NX

j=1

µij

⌦X ⇤H � Y, ⇤

j (Ui)↵2


M(X) =1

6

6X

i=1

1

N

NX

j=1

µij h j(X ⇤H � Y ), Uii2


Advantage : is a constant. ⇤j (Ui)


Measuring the Multi-Scale Moments

• Number of bands

• : galaxy observationY 2 Mn(R)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

• : noise standard deviation in �<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> Y


• : trade-off parameter between the data fidelity term and the moments constraint�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

• : weighting tensor��MAD<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

• : starlet transform operator�<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

• : moments constraintM(.)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

L(X) =1

2�2kX ⇤H � Y k2F +

�

2�2M(X) + k��MAD � �Xk0


• : Point Spread FunctionH 2 Mn(R)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Functional to Minimise

Denoising Experiment: Gaussian window v/s shearlets300 GALSIM galaxies

Gaussian window v/s shearlets

Experiment: Deconvolution

100 GALSIM galaxies + Optical GALSIM PSF

Experiment: Deconvolution

Deconvolution : GALSIM galaxies + MeerKat PSF

1024 GALSIM galaxies + Radio PSF (MeerKat, CASA)

" Weak Lensing and Shape Measurements➡ Serious mathematical challenges to well control systematics.➡ Need the best methods.

"Euclid PSF: RCA is new method which uses all available PSFs to derive the PSF field. ➡ Use Optimal Transport, Graph Laplacian and Sparsity.

" Moments Constraint: ➡ A new approach to preserve the ellipticity information during the restoration process.➡ Shape constraints using shearlets outperform the standard Gaussian windowing.

" Perspectives: ➡ Radio: Is this shape constraint reconstruction method competitive with galaxy model fitting in

Fourier space ?➡ Optical: Compare the constraint denoising method to the Gaussian windowing.

Conclusions

Documents

Blind galaxy survey images Deconvolution with Shape Constraint · Laplacian Graph P = D - J, where D is the degree matrix and J is the adjacent matrix of the graph P tP = V V t Degree