Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.adass.org/adass/proceedings/adass03/reprints/P6-6.pdf Дата изменения: Tue Aug 31 00:19:54 2004 Дата индексирования: Tue Oct 2 10:06:21 2012 Кодировка: Поисковые слова: arp 220

Astronomical Data Analysis Software and Systems XIII ASP Conference Series, Vol. 314, 2004 F. Ochsenbein, M. Al len, and D. Egret, eds.

Image Centering with a Maximum Likeliho o d Estimator: Application to Infrared Astronomical imaging
Damien Gratadour
1 2 1,2

, Laurent M. Mugnier1 , Daniel Rouan

2

atil DOTA / ONERA, BP 72, 92322 Ch^ lon cedex, France LESIA / Observatoire de Paris-Meudon, 92195 Meudon, France Abstract. We present a new, Maximum Likeliho o d (ML) based, metho d for the estimation of the shift b etween two images. It notably outp erforms the classical cross-correlation metho d esp ecially in the case of low photon levels. Moreover, it is arbitrarily subpixel, without any resampling of the image, through the maximisation of a criterion. The metho d was tested with simulations and was applied to the case of infrared astronomical imaging where the signal is usually very weak. We have also extended our metho d to the joint estimation of the shifts in a sequence of N images, and preliminary results are presented in last section.

1.

Intro duction

An accurate centering of a sequence of images is mandatory in thermal IR astrophysics to increase SNR while preserving the resolution of an instrument. A classical metho d to estimate the translation parameters is the linear crosscorrelation of noisy images with a reference (Vanderlugt 1964, Kumar et al. 1992). Downie and Walkup (1994) showed that taking into account the noise statistic can greatly improve the accuracy. Carfantan & Rougґ have studied the e case of the subpixel estimation of the maximum of the intercorrelation of two images with various interp olation for a stationnary gaussian noise (Carfantan and Rougґ 2001). Finally Guillaume et al. have studied the pixel accurate shift e estimation problem in the case of p oissonian noise at low photon level (Guillaume et al. 1998). We have developp ed a new, Maximum Likeliho o d (ML) based, metho d for the estimation of the shift b etween two images. It is arbitrarily subpixel, without any resampling of the image, through the maximisation of a criterion. We describ e in section 2 the theoretical basis of the metho d. Then in section 3, we present the results obtained with simulated images for different typ es of noise (pure gaussian additive or mixture of stationnary gaussian and p oissonian noise). Results on real data are presented in section 4. We have also extended our metho d to the joint estimation of the shifts in a sequence of N images, and preliminary results are presented in last section. 558 c Copyright 2004 Astronomical So ciety of the Pacific. All rights reserved.


Image Centering with a ML Estimator 2. Description of the metho d

559

If we assume a reference r , the intensity at pixel (k , l) of the observed translated image i1 can b e written as: i1 (k , l) = [r (x, y ) (x - x1 , y - y1 )]x (k , l) + b1 (k , l) (1)

where (x1 , y1 ) are the translation parameters, b is an additive noise, and x is the sampling op erateur. If the image is Nyquist sampled, one can reconstruct, via the Fourier domain, a shifted version of the image for any subpixel shift. If we approximate the noise in the image, i.e. a mixture of gaussian (detector) and p oissonian noise, as a non-stationnary gaussian noise, then the anti loglikeliho o d of observing an intensity i 1 (k , l) for the reference intensity r (x, y ) and for the hyp othesis µ = (x1 , y1 ) is given by: J (x1 , y1 ) =
k ,l

1 |i1 (k , l) - [r (x, y ) (x - x1 , y - y1 )]x (k , l)| 2 (k , l)
2 1

2

(2)

2 where 1 is the noise variance which can b e directly estimated on the image. It is easy to show that, following the two hyp othesis of stationnarity of the noise and of p erio dicity of the reference, the ML estimate of the translation b etween the two images is the maximum of the linear cross-correlation of the images. When the reference is not known, one has to consider a noisy frame as a reference.

i1 (k , l) = [i0 (x, y ) (x - x1 , y - y1 )]x + b(k , l)

(3)

Where b includes b oth the noise in the image used as a reference and the noise in the image to b e recentered. Then the anti log-likeliho o d to b e minimize has 2 the same expression as in equation 2 changing r (x, y ) into i 0 (x, y ) and 1 (k , l) 2 (k , l ) = 2 (k , l ) + 2 (x, y ) (x - x , y - y ) = 2 2 (k , l ) : into 1 1 1 0 1 J (x1 , y1 ) =
k ,l

1 |i1 (x, y ) - [i0 (x - x1 , y - y1 )]x | 4 (k , l)
2 1

2

(4)

To find the minimum of this criterion, we used a gradient typ e adaptive step minimization algorithm, issued from a collab oration of our team with the Group e des Problemes Inverses at Lab oratoire des Signaux et Systemes (GPI 1997). However, one has to notice that the criterion, in the case of unknown reference and considering the real noise variance contains a lot of lo cal minima. This make the minimization difficult and so should decrease the p erformance of the metho d in this case. 3. Results with simulation

The metho d has b een tested with simulations in the case of a mixture of gaussian (detector) and p oissonian noise. The gaussian noise variance is constant (10) and the photon level in the images ranges from 1 to 10 6 . The cross-correlation of the two images is interp olated around its maximum to provide a sub-pixel estimation. In the case of the known reference, our metho d, considering a constant


560

Gratadour et al.

Figure 1. Adaptive optics image of ARP 220 in the L-band with NACO at VLT. Left image, frames registred with a classical crosscorrelation metho d and averaged, right, frames registred with our algorithm and averaged.. noise variance, outp erforms the cross-correlation at very low photon level (i.e. numb er of photons smaller or equal to variance of the detector noise). When we consider the real noise variance, our metho d gives more accurate results and slightly outp erforms the cross-correlation at high photon levels. In the case of an unknown reference, the p erformance of our metho d is quite identical considering or not the real noise distribution since the criterion contains a lot of lo cal minima in the first case. It notably outp erforms the cross-correlation at low photon level and allows subpixel accuracy as so on as the numb er of photon p er pixel is greater than the variance of the detector noise as the accuracy of the interp olated cross correlation is worst than the pixel. 4. Results with real data

The metho d has also b een tested and used with a set of raw images of Arp 220 from NAOS-CONICA (NACO) at VLT. Arp 220 is a typical Ultra Luminous Infrared Galaxy, caracterised by a very p owerfull emission in infrared bands but very faint counterpart at the visible wavelengths. NACO is the only adaptive optics system that allows to servo infrared source and so achieve diffraction limited images at a large telescop e of such galaxies. A series of 85 images of this galaxy have b een aquired in the L-band in March 2003. The background dynamics of each image is around 80000 photons p er pixel and the source dynamic at the maximum is around 200 photons p er pixel. This is the case where the classical correlation of images is inefficient. Our metho d allows to recenter each frame with a subpixel accuracy, and so to obtain the image displayed on Fig. 1. The resolution of this image on the sky is ab out 0.1 ", i.e. diffraction limited for a 8-m telescop e in the L-band. This allows to compare this image to the one obtain with the space telescop e in other bands giving insightfull astophysical interpretations (see Gratadour et al. 2003).


Image Centering with a ML Estimator 5.

561

Joint estimation of the reference and the translation parameters

If we consider now a series of images {i j (k , l)} randomly shifted, and if we try to find simultaneously the shift parameters {µ j } = {(xj , yj )} and the reference image r (x, y ), then the anti log-likeliho o d can b e written as: J ({ij (k , l)}; r (x, y ), {µj }) =
m k ,l

1 |im (k , l) - 2 (k , l)
2 1 2

[r (x - xm , y - ym )]x (k , l)|

One can show that minimizing J ({ij (k , l)}; r (x, y ), {µj }) on r (x, y ) and {µj } is equivalent to minimize: J ({ij (k , l)}; r (x, y ) = rM L (k , l), {µj }) on {µj }, with:

r

ML

(k , l) =
m

im (k , l) (x + xm , y + ym )

(5)

It can additionnaly b e shown (Blanc et al. 2003) that this joint ML solution on r (x, y ) and {µj } is identical to the ML solution on the sole {µ j } assuming a gaussian prior probability on r (x, y ). The preliminary results show that, as in the previous metho d (estimation of the shift b etween two images) the criterion which considers the real noise variance contains a lot of lo cal minima. This induce low p erformance of the metho d in this case. But, if we consider a constant noise variance, and we use the shift estimated with the previous metho d as guess for the minimization of this joint criterion, the p erformance are b etter in the low photon level domain (10 to 100). References Blanc, A. & Mugnier, L. M. & Idier, J. 2003, JOSA A, 20, 6 Carfantan, H. & Rougґ B. 2001, GRETSI XVI I I, 849 e, Downie & Walkup 1994, JOSA A, 11, 1599 Gratadour, D. & Rouan, D. & Clenet, Y. & Gendron, E. & Lacomb e, F. 2003 IAU Symp. 221, 265 Group e des Probl` emes Inverses, "GPAV une grande o euvre collective" internal rep ort, LSS, 1997 Guillaume, M. & Melon, P. & Refregier, P. & Llebaria, A. 1998, JOSA A, 15, 2841 Slo cumb, B. & Snyder, D. 1990, Acqu. Track. and Point. IV, 1304, 165 Vander Lugt, A. 1964, IEEE Transactions on information theory, 10, 139 Vijaya Kumar, B. & Dickey F. & DeLaurentis J. 1992, JOSA A, 9, 678