%
\documentclass[11pt,twoside]{article} % Leave intact
\usepackage{adassconf}
%\usepackage[latin1]{inputenc}
%\usepackage[T1]{fontenc}
\newfont{\cyr}{wncyr10 scaled 1200}
\newcommand{\Psixvishah}{\mbox{\cyr sh}}
\newcommand{\PsixviShah}{\mbox{\cyr Sh}}
\newcommand{\Psixvidiscretize}[1]{{\left[#1\right]}_{\Psixvishah}}
% If you have the old LaTeX 2.09, and not the current LaTeX2e, comment
% out the \documentclass and \usepackage lines above and uncomment
% the following:
%\documentstyle[11pt,twoside,adassconf]{article}
\begin{document} % Leave intact
\paperID{P6-6}
%%%% ID=P6-6
\title{Image Centering with a Maximum Likelihood Estimator: Application to Infrared Astronomical imaging}
\titlemark{Image Centering with a ML Estimator}
\iffalse %%%% ORIGINAL !!!
\author{Damien Gratadour}
\affil{DOTA / ONERA, BP 72, 92322 Ch\^atillon cedex, France.}
\affil{LESIA / Observatoire de Paris-Meudon, 92195 Meudon, France}
\author{Laurent M. Mugnier}
\affil{DOTA / ONERA, BP 72, 92322 Ch\^atillon cedex, France.}
\author{Daniel Rouan}
\affil{LESIA / Observatoire de Paris-Meudon, 92195 Meudon, France}
%\altaffiltext{1}{DOTA-ONERA}
%\altaffiltext{2}{LESIA}
\else %%%% Changed FO to gain some space
\author{Damien Gratadour\altaffilmark{1,2},
Laurent M. Mugnier\altaffilmark{1},
Daniel Rouan\altaffilmark{2}}
\vspace*{2ex}
\altaffilmark{1}\affil{DOTA / ONERA, BP 72, 92322 Ch\^atillon cedex, France}
\altaffilmark{2}\affil{LESIA / Observatoire de Paris-Meudon, 92195 Meudon, France}
\fi
\contact{Damien Gratadour}
\email{firstname.surname@obspm.Fr}
\paindex{Gratadour, D.}
\aindex{Mugnier, L. M.}
\aindex{Rouan, D.}
% Remove this line if there is only one author
%-----------------------------------------------------------------------
% Author list for page header
%-----------------------------------------------------------------------
% Please supply a list of author last names for the page header. in
% one of these formats:
%
% EXAMPLES:
% \authormark{Lastname}
% \authormark{Lastname1 \& Lastname2}
% \authormark{Lastname1, Lastname2, ... \& LastnameN}
% \authormark{Lastname et al.}
%
% Use the "et al." form in the case of seven or more authors, or if
% the preferred form is too long to fit in the header.
\authormark{Gratadour et al.}
%-----------------------------------------------------------------------
% Subject Index keywords
%-----------------------------------------------------------------------
% Enter a comma separated list of up to 6 keywords describing your
% paper. These will NOT be printed as part of your paper; however,
% they will be used to generate the subject index for the proceedings.
% There is no standard list; however, you can consult the indices
% for past proceedings (http://adass.org/adass/proceedings/).
%
% EXAMPLE: \keywords{visualization, astronomy: radio, parallel
% computing, AIPS++, Galactic Center}
%
% In this example, the author noticed that "radio astronomy" appeared
% in the ADASS VII Index as "astronomy" being the major keyword and
% "radio" as the minor keyword. The colon is used to introduce another
% level into the index.
\keywords{ML}
%-----------------------------------------------------------------------
% Abstract
%-----------------------------------------------------------------------
% Type abstract in the space below. Consult the User Guide and Latex
% Information file for a list of supported macros (e.g. for typesetting
% special symbols). Do not leave a blank line between \begin{abstract}
% and the start of your text.
\begin{abstract} % Leave intact
% Place the text of your abstract here - NO BLANK LINES
We present a new, Maximum Likelihood (ML) based,
method for the estimation of the shift between two images. It notably outperforms the classical cross-correlation method especially 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 method 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 method to the joint estimation of the shifts in a sequence of N images, and preliminary results are presented in last section.
\end{abstract}
%-----------------------------------------------------------------------
% Main Body
%-----------------------------------------------------------------------
% Place the text for the main body of the paper here. You should use
% the \section command to label the various sections; use of
% \subsection is optional. Significant words in section titles should
% be capitalized. Sections and subsections will be numbered
% automatically.
%
% EXAMPLE: \section{Introduction}
% ...
% \subsection{Our View of the World}
% ...
% \section{A New Approach}
%
% It is recommended that you look at the sample papers, sample1.tex
% and sample2.tex, for examples for formatting references, footnotes,
% figures, equations, html links, lists, and other special features.
\section{Introduction}
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 method to estimate the translation parameters is the linear
cross-correlation 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 \& Roug\'e have studied
the case of the subpixel estimation of the maximum of the intercorrelation of
two images with various interpolation for a stationnary gaussian noise (Carfantan and Roug\'e 2001). Finally Guillaume et al. have studied the pixel accurate shift estimation problem in the case of poissonian noise at low photon level (Guillaume et al. 1998).
We have developped a new, Maximum Likelihood (ML) based,
method for the estimation of the shift between two images. It is arbitrarily subpixel, without any resampling of the image, through the maximisation of a criterion.
We describe in section 2 the theoretical basis of the method. Then in section 3, we present the results obtained with simulated images for different types of noise (pure gaussian additive or mixture of stationnary gaussian and poissonian noise). Results on real data are presented in section 4. We have also extended our method to the joint estimation of the shifts in a sequence of N images, and preliminary results are presented in last section.
\section{Description of the method}
If we assume a reference $r$, the intensity at pixel $(k,l)$ of the
observed translated image $i_1$ can be written as:
\begin{equation}\label{P6-6:eqn:first}
i_1(k,l)=\Psixvidiscretize{r(x,y) \ast \delta(x-x_1,y-y_1)} (k,l)+b_1(k,l)
\end{equation}
where $(x_1,y_1)$ are the translation parameters, $b$ is an
additive noise, and $ \Psixvishah $ is the sampling operateur. 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 poissonian noise, as a non-stationnary gaussian noise, then the anti log-likelihood of observing an intensity $i_1(k,l)$ for the
reference intensity $r(x,y)$ and for the hypothesis
$\mu=(x_{1},y_{1})$ is given by:
\begin{equation}\label{P6-6:eqn:second}
\displaystyle \mathcal{J} (x_1,y_1) =\sum_{k,l}
\frac{1}{2\sigma_1^2(k,l)} \big|i_1(k,l)- \Psixvidiscretize{r(x,y) \ast \delta(x-x_1,y-y_1)} (k,l)\big|^2
\end{equation}
where $\sigma_1^{2}$ is the noise variance which can be directly
estimated on the image. It is easy to show that, following the two hypothesis of stationnarity of the noise and of periodicity of the reference, the ML estimate of the translation between 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.
\begin{equation}\label{P6-6:eqn:third}
i_1(k,l)=\Psixvidiscretize{i_0(x,y)\ast \delta(x-x_1,y-y_1)}+b(k,l)
\end{equation}
Where $b$ includes both the noise in the image used as a reference and the noise in the image to be recentered. Then the anti log-likelihood to be minimize has the same
expression as in equation 2 changing $r(x,y)$ into $i_0(x,y)$ and
$\sigma_1^2(k,l)$ into
$\sigma^2(k,l)=\sigma_1^2(k,l)+\sigma_0^2(x,y) \ast
\delta(x-x_1,y-y_1)=2\sigma^2_1(k,l)$~:
\begin{equation}\label{P6-6:eqn:fourth}
\displaystyle \mathcal{J} (x_1,y_1) =\sum_{k,l}
\frac{1}{4\sigma_1^2(k,l)} |i_1(x,y)-\Psixvidiscretize{i_0(x-x_1,y-y_1)}|^2
\end{equation}
To find the minimum of this criterion, we used a gradient type
adaptive step minimization algorithm, issued from a collaboration
of our team with the Groupe des Problemes Inverses at Laboratoire
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 local minima. This make the minimization difficult
and so should decrease the performance of the method in this case.
\section{Results with simulation}
The method has been tested with
simulations in the case of
a mixture of gaussian (detector) and poissonian
noise.
%Results are presented in Figure 1, where the mean square
%error (bias$^2$+variance)$^{1/2}$ is plotted
%as a function of the SNR.
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 interpolated around its maximum to provide a sub-pixel estimation.
In the case of the known reference, our method, considering a constant noise variance, outperforms the cross-correlation at very low photon level (i.e. number of photons smaller or equal to variance of the detector noise). When we consider the real noise variance, our method gives more accurate results and slightly outperforms the cross-correlation at high photon levels.
In the case of an unknown reference, the performance of our method is quite identical considering or not the real noise distribution since the criterion contains a lot of local minima in the first case. It notably outperforms the cross-correlation at low photon level and allows subpixel accuracy as soon as the number of photon per pixel is greater than the variance of the detector noise as the accuracy of the interpolated cross correlation is worst than the pixel.
\section{Results with real data}
The method has also been 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 powerfull 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 telescope of such galaxies. A series of 85 images of this galaxy have been aquired in the L-band in March 2003. The background dynamics of each image is around 80000 photons per pixel and the source dynamic at the maximum is around 200 photons per pixel. This is the case where the classical correlation of images is inefficient. Our method 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 about 0.1 ", i.e. diffraction limited for a 8-m telescope in the L-band.
\begin{figure}
\epsscale{.80}
\plotone{P6-6_f1.eps}
\caption{Adaptive optics image of ARP 220 in the L-band with NACO at VLT. Left image, frames registred with a classical cross-correlation method and averaged, right, frames registred with our algorithm and averaged..} \label{P6-6:Grat-fig-1}
\end{figure}
This allows to compare this image to the one obtain with the space telescope in other bands giving insightfull astophysical interpretations (see Gratadour et al. 2003).
\section{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 $\{\mu_j\}=\{(x_j,y_j)\}$ and the reference image $r(x,y)$, then the anti log-likelihood can be written as:
\begin{eqnarray*}
\displaystyle \mathcal{J}\left(\{i_j(k,l)\};r(x,y),\{\mu_j\}\right)
=\sum_{m}\sum_{k,l}\frac{1}{2\sigma_1^2(k,l)} \big|i_m(k,l)- \\
\big[r\ast \delta(x-x_m,y-y_m)\big]_{\Psixvishah}(k,l) \big|^2
\end{eqnarray*}
One can show that minimizing $\displaystyle
\mathcal{J}\left(\{i_j(k,l)\};r(x,y),\{\mu_j\}\right)$ on $r(x,y)$ and
$\{\mu_j\}$ is equivalent to minimize:
$\displaystyle \mathcal{J}\left(\{i_j(k,l)\};r(x,y)=r_{ML}(k,l),\{\mu_j\}\right)$ on $\{\mu_j\}$, with:
\begin{center}
\begin{equation}
r_{ML}(k,l)=\sum_{m}i_m(k,l)*\delta(x+x_m,y+y_m)
\end{equation}
\end{center}
It can additionnaly be shown (Blanc et al. 2003) that this joint ML solution on $r(x,y)$ and $\{\mu_j\}$ is identical to the ML solution on the sole $\{\mu_j\}$ assuming a gaussian prior probability on $r(x,y)$.
The preliminary results show that, as in the previous method (estimation of
the shift between two images) the criterion which considers the real noise
variance contains a lot of local minima. This induce low performance of the
method in this case. But, if we consider a constant noise variance, and we use
the shift estimated with the previous method as guess for the minimization of
this joint criterion, the performance are better in the low photon level
domain (10 to 100).
%-----------------------------------------------------------------------
% References
%-----------------------------------------------------------------------
% List your references below within the reference environment
% (i.e. between the \begin{references} and \end{references} tags).
% Each new reference should begin with a \reference command which sets
% up the proper indentation. Observe the following order when listing
% bibliographical information for each reference: author name(s),
% publication year, journal name, volume, and page number for
% articles. Note that many journal names are available as macros; see
% the User Guide listing "macro-ized" journals.
%
% EXAMPLE: \reference Hagiwara, K., \& Zeppenfeld, D.\ 1986,
% Nucl.Phys., 274, 1
% \reference H\'enon, M.\ 1961, Ann.d'Ap., 24, 369
% \reference King, I.\ R.\ 1966, \aj, 71, 276
% \reference King, I.\ R.\ 1975, in Dynamics of Stellar
% Systems, ed.\ A.\ Hayli (Dordrecht: Reidel), 99
% \reference Tody, D.\ 1998, \adassvii, 146
% \reference Zacharias, N.\ \& Zacharias, M.\ 2003,
% \adassxii, \paperref{P7.6}
%
% Note the following tricks used in the example above:
%
% o \& is used to format an ampersand symbol (&).
% o \'e puts an accent agu over the letter e. See the User Guide
% and the sample files for details on formatting special
% characters.
% o "\ " after a period prevents LaTeX from interpreting the period
% as an end of a sentence.
% o \aj is a macro that expands to "Astron. J." See the User Guide
% for a full list of journal macros
% o \adassvii is a macro that expands to the full title, editor,
% and publishing information for the ADASS VII conference
% proceedings. Such macros are defined for ADASS conferences I
% through XI.
% o When referencing a paper in the current volume, use the
% \adassxii and \paperref macros. The argument to \paperref is
% the paper ID code for the paper you are referencing. See the
% note in the "Paper ID Code" section above for details on how to
% determine the paper ID code for the paper you reference.
%
\bibliography{Acronymes,EnglishAcronyms,Livres,Articles,laurent,damien}
\bibliographystyle{pasp}
\begin{references}
\reference Blanc, A. \& Mugnier, L. M. \& Idier, J. \ 2003, JOSA A, 20, 6
\reference Carfantan, H. \& Roug\'e, B. \ 2001, GRETSI XVIII, 849
\reference Downie \& Walkup \ 1994, JOSA A, 11, 1599
\reference Gratadour, D. \& Rouan, D. \& Clenet, Y. \& Gendron, E. \&
Lacombe, F. \ 2003 IAU Symp. 221, 265
\reference Groupe des Probl\`emes Inverses, ``GPAV une grande oeuvre collective'' internal report, LSS, 1997
\reference Guillaume, M. \& Melon, P. \& Refregier, P. \& Llebaria, A. \ 1998, JOSA A, 15, 2841
\reference Slocumb, B. \& Snyder, D. \ 1990, Acqu. Track. and Point. IV, 1304, 165
\reference Vander Lugt, A. \ 1964, IEEE Transactions on information theory, 10, 139
\reference Vijaya Kumar, B. \& Dickey F. \& DeLaurentis J. \ 1992, JOSA A, 9, 678
\end{references}
% Do not place any material after the references section
\end{document} % Leave intact