Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.stsci.edu/stsci/meetings/irw/proceedings/ratnatungak.ps.gz Äàòà èçìåíåíèÿ: Mon May 23 18:43:59 1994 Äàòà èíäåêñèðîâàíèÿ: Sun Dec 23 06:55:02 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

The Restoration of HST Images and Spectra II
Space Telescope Science Institute, 1994
R. J. Hanisch and R. L. White, eds.
Maximum Likelihood Estimation of Galaxy Morphology: Faint HST Wide
Field Camera Images
Kavan U. Ratnatunga, Richard E. Grif®ths, and Stefano Casertano
Center for Astrophysical Sciences, Johns Hopkins University, Baltimore, MD 21218
Abstract. A modeling approach based on the maximum likelihood method has been
developed to extract quantitative morphological and structural parameter estimates for faint
galaxy images obtained with the Hubble Space Telescope (HST) Wide Field Planetary
Camera (WFPC). We model both the galaxy image and the instrumental characteristics of
the WFPC, including the complex Point Spread Function (PSF), the error in the Analog­to­
Digital Converter (ADC), and the positive noise bias due to faint cosmic rays and undetected
warm pixels. Because convolved galaxy images are compared directly with the observations,
we avoid the need for deconvolution, which is dif®cult and potentially unstable for faint
images.
1. Introduction
The HST Medium Deep Survey (MDS) Key Project (Grif®ths et al. 1992, 1994) includes obser­
vations of a large number of random WFC ®elds. They are taken in parallel mode, when another
detector is being used on a primary target at a single pointing extending over two or more orbits.
Most of the objects detected are faint galaxy images. For statistical analysis of the large scale struc­
ture of the universe, the survey is generating a catalog of galaxy magnitudes, colors, half light radii,
axis ratios and morphology. In order for this catalog to be well­de®ned despite the relatively long
span over which the observations have been obtained, and to ensure uniformity in the processing,
we need reliable error estimates for all evaluated parameters to combine information from different
®elds, with different integrated exposure times.
In the traditional approach to image analysis, CCD observations are ®rst calibrated to subtract
the bias and dark current and eliminate variations in the detector sensitivity. Bright cosmic rays
are then removed, by stacking multiple observations if available, otherwise by using one of several
algorithms that identify cosmic ray events in the image. Defective pixels may be removed by
interpolation. For HST, the brighter images are then deconvolved to restore the images to the high
resolution of the core of the PSF.
This approach can be very successful when dealing with bright (high signal­to­noise) images.
The result is a cosmetically clean image on which morphological classi®cation and other measure­
ments can be carried out in a non­parametric, model­independent way. However, this procedure
does have some drawbacks when applied to a large sample of galaxies. Some quantities, such
as magnitudes, are very dif®cult to measure accurately and without bias on deconvolved images;
even for those quantities that can be measured, the error distribution is distorted and probably non­
Gaussian. Intercomparison of galaxy properties does require, implicitly or explicitly, the adoption
of a model. The primary disadvantage is that deconvolution of faint images is likely to be unstable
(Schade and Elson 1993).
In the alternative approach described below we assume a simple parametrized model for the
galaxy image and transform it to the observed domain. Parameters used in the model de®nition
are then estimated by maximizing the likelihood function, which is the probability of obtaining
each observed image for any set of galaxy parameters. Tests of the procedure on simulated galaxy
333

334 Ratnatunga, Grif®ths, & Casertano
images show that derived model parameters and their error estimates are unbiased. The likelihood
ratio between ®ts to different types of models can be used for a coarse morphological classi®cation.
2. Model Fitting
The properties of an ensemble of faint galaxies can be de®ned by the distribution of the parameters
of the models that best ®t each individual object. This is only practical for faint images if the
number of adjustable parameters to be ®tted is kept to a minimum; a possible initial set would
include total magnitude, half­light radius, ellipticity, orientation and centroid position. For brighter
images, where more signal is available, we can ®t additional shape parameters, such as bulge­to­
disk ratio. The model also includes instrumental parameters, such as the sky background (estimated
locally) and amplitude of the noise. The PSF and the distribution of faint cosmic rays are measured
separately. It should be pointed out that the modelling approach for extended objects is relatively
insensitive to small­scale errors in the description of the PSF.
For any set of model parameters, a smooth galaxy image is generated and convolved with the
expected PSF. The expected sky brightness is added and the image is multiplied by the detector
sensitivity (#at ®eld). The probability of the observed value is then computed for each pixel using
the adopted error distribution. Since the parameters are estimated as the maximum likelihood
model that ®t the observations, we can ignore defective pixels without any need to interpolate
them. The likelihood function is de®ned as the sum of the natural logarithm of the probability
for individual pixels, including both galaxy pixels and neighboring sky pixels (in fact, pixels do
not have to be separated into ‘galaxy' and ‘sky'). For a Gaussian error distribution, the likelihood
function is proportional to the ü 2 of the distribution; however, the method allows for a different error
distribution can be used, to take into account both the uneven probability distribution of different
data numbers due to the ADC error, and the wide positive tail due to faint cosmic rays and warm
pixels.
The choice of models is based on standard galaxy models and include an exponential disk and an
r 1=4 law. Because the method involves a non­linear optimization, the choice of initial guesses for the
parameters is quite important; we generally use initial values estimated from moments of the image,
as well as the results of the deconvolution if available. Optimization uses the Numerical Algorithm
for Maximum Likelihood Estimation (NAMaLiE) developed by Ratnatunga and Casertano (1991).
The information on errors of the estimated parameters and their correlation can be obtained from
the covariance matrix, which is the inverse of the Hessian (matrix of second order derivatives) at the
peak of the likelihood function. Extensive numerical simulations have shown that both parameter
estimates and their errors are unbiased.
This approach also requires an error map for the calibrated image, which is not generated by
the current HST WFPC pipeline calibration. We use a simple noise model to estimate the rms noise
in each pixel. Special consideration must be given to noise in the calibration images (bias, dark and
#at ®eld), which, unlike image noise, does not decrease as multiple exposures are stacked together,
unless the exposures are shifted with respect to one another. In some cases involving multiple
exposures, the noise could be estimated from the internal scatter between individual exposures, plus
the contribution of the calibration ®les.
The contribution of faint cosmic rays to the noise is found to be described adequately by a
Weibull distribution with power law index of 0.25; convolved with a Gaussian distribution for shot
and read noise, this is a good representation for the total ‘noise' distribution over the full range
in data numbers. The two additional parameters required to describe the distribution of cosmic
rays are derived from a single ®t for the whole CCD, and therefore do not increase the number of
parameters to be ®tted for each image. Empirically, the scatter between different exposures is found
to be small.

Maximum Likelihood Estimation of Galaxy Morphology 335
3. Simulations
The galaxy models that have been extensively tested include an exponential law, meant to represent
a disk galaxy, and an r 1=4 law, representing ellipticals. Both are fully described by six adjustable
parameters: total magnitude, half­light radius, axis ratio, position angle, and the coordinates of the
center. When both ®ts are performed on a simulated image of either kind, the value of the likelihood
function is very effective in choosing the #right# model down to typical magnitudes of V = 23 or
I = 22 with a single­orbit WFPC exposure, and about one magnitude fainter with the expected
parameters for WFPC 2 observations (see Table 1). (Six orbits are required to reach one magnitude
fainter with either camera.)
Fits can be successfully performed for fainter galaxies, but at one magnitude fainter than the
value in Table 1, the ability to discriminate between models is substantially impaired, and therefore
the ®tted half­light radius may be biased by the model adopted. Not surprisingly, the bias is smaller
than the error in the estimated parameters.
Table 1. Magnitude limits for classi®cation of faint galaxies (single HST orbit).
Detector Filter Exp. Time Sky Mag. Lim. Mag.
WFC F555W (V) 2400 23.00 23.00
WFC F785LP (I) 2400 21.75 22.00
WFPC2 F555W (V) 2400 23.00 24.00
WFPC2 F814W (I) 2400 21.75 23.50
In order to ®t more complex models successfully, better signal­to­noise ratio is required.
Typically, we estimate that the image must be one magnitude brighter for each additional parameter
to be ®tted. An additional complication arises for combined models consisting of an exponential
(disk) and an r 1=4 law (bulge) together. We ®nd that the magnitude difference between the disk and
the bulge correlates strongly with the ratio of their sizes, even for bright simulated images. This
indicates that an image ®t has poor leverage to determine the parameters of the two components
independently.
We note also that stellar images can be easily ®tted about two magnitudes fainter than the limits
in the table, thanks to the smaller number of free parameters. However, we have not investigated
the effects of errors in the PSF, which may have a greater impact on stellar than on galaxy images.
Such issues have been discussed extensively for stellar photometry packages that use the ®tting
approach, such as DAOPHOT.
4. Discussion
While extensive experience has been garnered in the application of maximum likelihood to stellar
photometry with HST (see for example Stetson 1994), comparatively little has been done for galaxy
images (see Schade and Elson 1994). What we present here is only a ®rst step, albeit a successful
one. The real test of the method comes from its application to actual data.
In the application to real HST observations, we generally do achieve convergence on galaxy
parameters to the magnitude limits given in Table 1. However, signi®cant ambiguity remains in
the interpretation of the results. Real galaxy images do not follow the simple models used here:
spiral arms, bars, double nuclei, dust lanes, and bright knots of star formation all add complications
that cannot be properly modeled with the limited information available in typical images. The only
recourse in our exploratory runs has been to estimate a best ®t using a simple galaxy pro®le and
to inspect the residual distribution for additional visual clari®cation. More detailed results will be
reported in Grif®ths et al. (1994) and Ratnatunga, Grif®ths, and Casertano (1994).

336 Ratnatunga, Grif®ths, & Casertano
We should also point out that a key ingredient in achieving our limited success on actual data
has been the improved calibration of WFC images, optimized for the quantitative analysis of faint
images. The details are reported in Ratnatunga et al. (1994).
Acknowledgments. This work is based on observations taken with the NASA/ESA Hubble
Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the
Associations of Universities for Research in Astronomy, Inc., under NASA contract NAS5±26555.
Coordination and analysis of data for the Medium­Deep Survey is funded by STScI grants GO
2684.0X.87A and GO 3917.0X.91A. We acknowledge the helpful comments from the full HST
Medium Deep Survey team.
References
Grif®ths, R. E. et al. 1992, in Science with the Hubble Space Telescope, P. Benevenuti & E. Schreier,
eds., European Southern Observatory, Garching, 13
Grif®ths, R. E. et al. 1994, ApJ, submitted
Ratnatunga, K. U.,& Casertano, S. 1991, AJ, 101, 1075
Ratnatunga, K. U., Grif®ths, R. E., Casertano, S., Neuschaefer, L. W., & Wyckoff, E. W. 1994, in
HST Calibration Workshop Proceedings, C. Blades, ed., Space Telescope Science Institute,
Baltimore
Ratnatunga, K. U., Grif®ths, R. E.,& Casertano, S. 1994, in preparation
Schade, D. J., & Elson, R. A. W. 1993, AJ, 105, 1581
Schade, D. J., & Elson, R. A. W. 1994, in preparation
Stetson, P. B. 1994, this volume