Detection of point sources

Transcription

1 A&A manuscript no. (will be inserted by hand later) Your thesaurus codes are: 23( ; ; ; ) ASTRONOMY AND ASTROPHYSICS Detection of point sources R. C. Snel, P. Linde and S. Spännare Lund Observatory, Box 43, S Lund, Sweden, January 15, 1998 Abstract. Photometry of individual sources requires an initial phase of detection. The detection process can be separated into two phases: (1) preprocessing of the image, and (2) actual detection, yielding a list of discrete sources. Given a distribution of background intensities, the probabilityof intensity deviations for light sources as a result of the background can be calculated. Setting a maximum acceptable probability defines a unique intensity, closely related to the standard deviation of the background intensity and the signal to noise (S/N) ratio of a pixel. Expressions are derived for the S/N ratio of pixels containing stars, both for isolated stars and crowded fields. A general expression for completeness of detection of sources in crowded stellar fields is derived, using the detection threshold, the distribution of intensity measurements for an initial estimated intensity, the number of pixels occupied by a single stellar image, and the stellar luminosity function obtained with a groundbased telescope. Three different kinds of preprocessing operations were applied on an image of a globular cluster core, testing the effect on source detection. Preprocessing was done through matched filtering with an estimate of the point spread function (PSF), fitting of the PSF to each point in the image, and deconvolution. As a reference for positions and intensities, an HST WFPC2 image of the same field was used. Efficiency of detection was expressed using measured completeness of detection, mapping the effect of crowding by dividing the image into six zones with varying stellar density. PSF fitting and matched filtering yielded comparable results, while deconvolution performed equally well only for the most crowded regions, but somewhat worse for the less crowded regions. Validity of the derived expressions was evaluated with the results obtained from the test images. Key words: Methods: data analysis Methods: observational Techniques: image processing Stars: imaging 1. Introduction Accurate photometry is together with spectrometry the foundation for most observational astrophysical knowledge. Spectrometry allows more detailed investigation, but is generally difficult to apply in situations in which many and weak sources have to be studied. Photometry of crowded fields, using well selected passbands, permits far-reaching conclusions of distant stellar systems. While several powerful photometry tools are generally available (DAOPHOT, Stetson 1987; ROMAFOT, Buonanno et al. 1983; CAPELLA, Debray et al. 1994; SPS, Janes and Heasley 1993; DoPHOT, Schechter et al. 1993), there are still challenging data analysis problems for which improvement is both necessary and possible. One such problem is optimal photometry in severely crowded stellar fields. Photometry of individual sources requires an initial phase of detection. In fact, reliable detection is fundamental for acquiring high photometric accuracy in crowded fields. This is true also for measurements of brighter sources if undetected objects are nearby. In this paper, we will investigate some theoretical aspects related to detection and present results using several different detection techniques What defines a detection? A common criterion for discerning a source from the background is its signal to noise (S/N) ratio: if the S/N ratio calculated from the observation exceeds a preset limit, we feel it safe to state that a source has been detected. Another possibility is to set a maximum acceptable probability for an intensity deviation to be the result of background and noise in the observation. Deviations above this limit are considered to be real sources. Errors in the detection can occur in two ways: a true source evades detection, or a source is detected when no true source is present. The latter we will call artifact. The number of artifacts obviously increases when lowering the detection threshold. The detection threshold is the intensity corresponding to the S/N ratio or probability which defines a detection. Setting the detection threshold too high causes fewer true sources to be detected. In most cases, the goal is to detect as many true sources as possible, while detecting as few artifacts as possible. Thus, artifacts must be separated from true sources, using appropriate criteria, such as the S/N ratio of the source, the probability of the source being an artifact of the noise, and the shape of the source image. Discrimination can be done either before or af-

2 2 R. C. Snel, P. Linde and S. Spännare: Detection of point sources ter the actual detection of the source. Appropriate filtering of the signal can enhance structures before detection, while an accurate analysis of the shape of the source image can typically be done after detection. We thus identify two stages of the detection process, (1) preprocessing of the image, to separate potential sources from background and noise, and (2) actual detection, which in its simplest form is the truncation of selection parameters at appropriate values. Preprocessing is an image enhancement operation, while the actual detection converts the image to a list of discrete objects. 2. Theoretical considerations 2.1. Probability of intensity deviations The S/N ratio describes the accuracy with which the signal can be measured, but it is not the best selection criterion for detection of very faint signals. For identification of a source, it is not necessary to measure its intensity with a certain accuracy. It suffices to say with a certain confidence that the source is not a spurious peak in the noise, irrespective of intensity. With this detection criterion, the distribution function of non-stellar intensities, primarily background and noise, can be used to define a detection threshold. E.g., for a Poisson distribution of photon counts, the probability P I of detecting an intensity equal to or higher than I, resulting from a source-free background b is P I = 1X m=i e?b bm m! ; (1) or for a Gaussian distribution with standard deviation and mean value b Z 1 1 P I = p 2 2 I e? (m?b) dm: (2) When P I is chosen sufficiently small, any detected intensity deviation above the level defined by P I is with high probability due to a source. A fixed probability defines a unique intensity, which, expressed in standard deviations above the background, is equal to the S/N ratio of the pixel, if the background noise is dominating Signal to noise ratio The S/N ratio is an estimate of the accuracy with which the intensity of a source can be determined. This estimate can either be calculated from known noise properties and an assumed intensity, or from the actually measured intensity and intensity fluctuations. For a given observation, the value of the S/N ratio depends on the way it is calculated. E.g., the S/N ratio can be defined for a single detection element (pixel), or for the total flux of the source, integrated over a large number of pixels. For a CCD image, the S/N ratio for a single pixel can be calculated from the expected signal in the pixel, divided by the total expected noise in the pixel. Newberry (1991) gives an excellent discussion of the contributions from various kinds of noise. Consider a CCD with gain g electrons per analog-to-digital unit (ADU), read-out noise R electrons, and a pixel on the CCD with contributions I sky t from the sky background (and possible dark-current) and I source t from a source, with the flux I in electrons per second and the exposure time t in seconds. The signal S is equal to I source t, and the noisep N is the sum of the read-out noise R, truncation noise T = (g 2? 1)=12, processing noise P, and Poisson noise in the intensities of the sky and source. Truncation noise is caused by limited intensity resolution of the detector. Processing noise results from correction for calibrated effects, such as bias, dark, and preflash images, which are not free of noise. I source t has to be calculated from the total observed intensity I source t + I sky t, by estimating I sky t from p pixels in some other part of the CCD containingonly sky contribution. This introduces an extra contribution to the noise, and leads to the following expression for S/N ratio in a single pixel (see also Newberry 1991): S=N = p Isource t q 1 + ( Isky I + R2 +T 2 +P 2 source I sourcet )(1 + 1 p ) (3) We will call p R 2 + T 2 + P 2 the base noise B (the minimum noise for a single pixel), and will assume p large. Depending on which terms in the denominator can be neglected, we identify three simplifications: 1. The denominator is equal to 1, which implies that the source intensity is large compared to the sky intensity and base noise p p S=N = I source t = S (4) 2. The sky intensity is dominating S=N = I p s source t p I source pisky = S (5) I sky 3. The base noise is dominating S=N = I sourcet = S B B Flat-field effects, being multiplicative, add a relative contribution to the noise term. Low S/N ratio values, with high noise relative to the signal, are not significantly affected by noise in the flat-field. High S/N ratio values, with low relative noise, are limited by the S/N ratio of the flat-field. Equation 3 is illustrated in Fig. 2 as dashed lines. Note how the S/N ratio increases linearly with intensity for low intensity, but with the square root of the intensity for high intensity Effects of sampling The approximation of a single pixel containing the entire signal from a source is only valid for a severely under-sampled point spread function (PSF). If the PSF covers more than one pixel, the S/N ratio for a source must be calculated from the sum of the signal and the noise, respectively, over a number of pixels. The contribution to the signal increases asymptotically with the number of pixels, approaching the total signal. The contribution to the noise increases for each additional pixel (6)

3 R. C. Snel, P. Linde and S. Spännare: Detection of point sources 3 with at least the noise due to the sky intensity and the base noise. Selecting an appropriate number of pixels, or weighting with the PSF, yields an optimal S/N ratio for each source. Howell (1989) describes the problem of aperture photometry in detail, and illustrates how the S/N ratio depends on the aperture radius. Weighted aperture photometry, as described by e.g. Cool and King (1995), eliminates the problem of decreasing S/N ratio at large aperture radii. However, fixed aperture photometry yields nearly identical S/N ratio as weighted aperture photometry, if the optimal fixed aperture radius is chosen. This radius is a function of source intensity, and can be chosen to optimise detection of faint sources Effects of crowding For a single pixel, it is impossible to distinguish between background intensity from the sky and that from nearby sources. If the images of sources overlap, the net effect is to raise the local background, resulting in a lower S/N ratio for all sources involved. This means that faint sources that would have been detected if they were isolated, can evade detection in the presence of a bright source. If the presence of extra sources is unknown, this can introduce systematic errors in the estimate of intensity and position of the detected source. If an estimate for the intensity of the object (source or artifact) is available, the S/N ratio can be defined as S=N = I source t q P psfiisourcet+iskyt+b 2 psf 2 i ; (7) I sky being the contribution from all other stars as well as the diffuse sky brightness, and psf i the PSF normalised to unity integral. I sky can be calculated from the measurable intensities in the image and the known I source and psf i, and B can be estimated from an area of the image devoid of stars, or calculated from detector characteristics. Around each source, there is a zone where no other sources below a minimum intensity, I min t, can be detected. In the absence of other sources, this minimum intensity decreases with distance from the source. Specifically, for a normalised PSF with radial profile psf(r), with r the distance from the centre of the source, the S/N ratio for sources in the wings of a bright source, with flux I bright electrons per second, can be approximated by S=N = p I source t pibright : (8) psf(r) For a PSF profile psf(r) and neglecting the contributions from base noise and sky intensity, the minimum intensity I min t a source in the wings of a bright source must have to be detected with S=N Q can be derived from Eq. (3) and is given by q I min t = Q2 2 ( I bright t psf(r)=q 2 ): (9) This assumes that all the flux I min falls withina single pixel, and that the positionand intensity of the bright source is known. For small psf(r), the intensity converges to Q 2, for large I bright t psf(r), I min t = Q p I bright t psf(r) Completeness of detection The probability of detecting a star at a given magnitude is not always unity, and depends on the detection threshold and the degree of crowding. If the number of stars in a magnitude interval is large, the fraction of detected stars is a good approximation for the probability of detection. When determining the luminosity function (LF) of a field of stars, incompleteness of detection has to be taken into account. Normally this is done through artificial star experiments; a small number of stars is added to the image, after which an attempt is made to detect and measure these stars. This procedure is repeated many times with different positions and intensities of the artificial stars. Completeness of detection (C) is calculated from the number of detected and added artificial stars. The probability of detection can also be calculated from the LF, PSF, detection threshold, and distribution of measured magnitudes given an estimated magnitude Detection threshold In the absence of crowding effects, completeness of detection is determined by the detection threshold and accuracy of intensity measurement. For a hypothetical perfect measuring algorithm, the probability of detection, and thus C, is unity for intensities above the detection threshold, and zero for intensities below. Most detection and measuring algorithms truncate detection at a threshold using the estimated intensity, after which a more accurate measurement of the intensity is made. The slope of C around the detection threshold is determined by the distribution of measured intensities, given an estimated intensity. A reasonable assumption for this distribution is the one resulting from Poisson and read-out noise. Expressed as a function of magnitude, C will be steeper for higher detection thresholds, since higher intensity implies higher S/N ratio, and thus a narrower distributionof measured magnitudes. C can be calculated from the measurement accuracy for a given magnitude C 1 (m 0 ) = Z mth?1 f(m 0 ; m)dm; (10) with m 0 the measured magnitude, m the estimated magnitude, m th the magnitude corresponding to the detection threshold, and f(m 0 ; m) the function describing the distribution of measured magnitudes given an estimated magnitude, normalised to unity integral Crowding When two stars lie too close together in an image, they can not be separated and will be detected as a single star. This effect causes completeness of detection to be lower than unity, even for stars that are bright enough to be detected had they been isolated. Considering a star with a given magnitude, and random

4 4 R. C. Snel, P. Linde and S. Spännare: Detection of point sources distribution of stars, the probability that this star evades detection due to its proximity to a brighter star is proportional to the fraction of the image covered by brighter stars. This fraction can be calculated from the LF of the image, yielding the probability P e that a star evades detection: P e (m) = Z b m (1? P e (m 0 ))LF(m 0 )dm 0 ; (11) N pix?1 where b is the number of pixels occupied by a single star, N pix the number of pixels in the image, and LF(m) the luminosity function for the image. Note that P e occurs both on the left and right hand side of the equation. Iterative numerical determination of P e yields a correct solution. Completeness of detection for a crowded field, ignoring detection threshold, is given by C 2 (m) = 1? P e (12) Completeness of detection for the general case with crowding and finite accuracy of measurement is given by C(m 0 ) = Z mth?1 C 2 (m)f(m 0 ; m)dm: (13) In Fig. 5 an illustrationof this relation is given for a real image. 3. Experiments To investigate and verify some of the theoretical considerations given above, a set of experiments were performed, using both simulated and real images. Several techniques for image enhancement and subsequent detection have been applied Test images A set of test images was created, simulating various strong crowding conditions. In Linde et al. (1998) a description is available. Since some image properties are not easily included in simulations, an exposure of the globular cluster M15 was also added to the test sample. The exposure was taken with the Nordic Optical Telescope (NOT) and has a resolution of approximately 0.5 seconds of arc. An advantage of using a globular cluster as a test object is the strong gradient in stellar density across the image. Crowding increases as one approaches the cluster core, and the efficiency of the detection algorithm can readily be ascertained. In Fig. 1 is shown an excerpt of the M15 exposure, with six zones marked, centred on the approximate core of the globular cluster. In order to obtain true positions of all observable stars in the image obtained with the NOT, exposures taken with the Wide Field and Planetary Camera 2 were retrieved from the Hubble Space Telescope science archive. They have a resolution of approximately 0.1 seconds of arc Preprocessing The efficiency of detection depends strongly on preprocessing of the image. Preprocessing should transform the image in such a way that the stars are isolated, bright and sharp with a unique local maximum, and enhance the contrast between stars and background. Three methods have been used in this investigation: matched filtering, deconvolution and PSF fitting Matched filtering Matched filtering with a PSF-like function is a fast and easily implemented preprocessing step capable of removing sloping backgrounds in the image, sharpening and separating blended stellar images, and discriminating against cosmic ray effects, image defects and extended sources. Sharpening of images and removal of the background is achieved only if the matched filter contribution approaches zero at a finite radius, and the integral over the matched filter vanishes. The amount of sharpening is determined by the radius at which the filter contribution vanishes. A side effect of the filtering is that rings with negative values may appear around bright objects, hiding possible fainter stars from subsequent detection. Matched filtering of our test images was performed with the FIND algorithm in DAOPHOT (Stetson 1987), which uses an estimate of the full width at half maximum (FWHM) of the PSF to create a matched filter consisting of a lowered truncated Gaussian, with a radius of 1.5 and zero integral Deconvolution An observed image can be regarded as the convolution of the true image and the PSF, with additional noise. In the Fourier domain the PSF suppresses high spatial frequencies of the true image, while the noise (if white) is present at all frequencies. Deconvolution is used to regain some of the high-frequency information, enforcing positive values of intensities on the deconvolved image. The result is sharpening of the image, thus separating blended stars and increasing contrast against the diffuse background. The deconvolution algorithm used in the present work is based on the Lucy-Richardson method, with modifications by Murtagh, Starck and Bijaoui (1995) PSF fitting Linde et al. (1998) have shown that fitting the PSF to each pixel (or sub-pixel position) in the image helps in subsequent detection of stars, without detecting too many artifacts. Spännare (1998) has developed a code, FINDSTARS, that performs detection and photometry in crowded stellar fields. The algorithm operates in two modes. In the first mode, a PSF is systematically fitted to selected positions in order to scan for star-like objects. The square sum of the residuals between fitted PSF and pixel data is calculated for each position. If predetermined threshold conditions on minimum intensity and

5 R. C. Snel, P. Linde and S. Spännare: Detection of point sources 5 Fig. 1. The ground-based M15 test image. Six different zones, used to investigate varying degree of crowding, are marked by circles. Two sub-fields, shown in Fig. 3 in greater detail, are marked by rectangles. residual square sum are met, the fit parameters are stored for further processing. These intermediate results may be organised in the form of a new, enhanced, image. See Fig. 3d for an example. In the second mode, the enhanced image is analysed, using a sorting procedure to record the detected object positions. The sorting begins by finding the highest intensity in the intermediate image resulting from phase one. Subsequently, from all data points inside a preset radius around the highest intensity, the one with smallest residual is selected, while all others are rejected. In this way, positional accuracy is improved and multiple identifications are avoided. Usually the radius preset is chosen slightly smaller than the FWHM of the PSF in crowded fields which means that no faint stars are detected closer to a brighter star than given by the FWHM. The procedure is repeated for the remaining data sets in decreasing intensity order. The final output contains the position, intensity, magnitude, background and residual square sum of every detected star in the image Detection of local maxima The actual detection of sources converts an enhanced twodimensional image into a list of sources, each with two spatial coordinates and possibly other parameters, like estimates of intensity, residuals, and probability. In the cases of matched filtering and deconvolved images, a straightforward detection algorithm has been used. It checks each pixel in the image against its eight closest neighbours. If the pixel value is above a detection threshold and above the intensity of its neighbours, it is regarded as a source, with coordinates equal to the pixel coordinates. The disadvantage of this detection algorithm is that multiple sources, with small enough separation to have a single maximum, will be detected as a single source, despite the fact that the appearance of the source may strongly suggest multiple sources. A more refined algorithm may use the shape of the intensity distribution around the maxima to decide upon the presence of possible, probable or definite source multiplicity. However, such an algorithm could be considered as part of the preprocessing, and thus be included as such Source or artifact? Before the efficiency of a detection algorithm can be evaluated, the problem of identification of a source must be considered. In very crowded fields this is not a trivial task, due to the high spatial density of objects. Each detected object was checked against a list of true sources. An object was considered successfully detected if the distance between true and detected position differed less than a preset threshold, which was determined from study of the distribution of the positional differences. If an estimate for the intensity of the source was available, an additional constraint using a maximum magnitude difference was used. If a source was detected without any correspondence in the reference list, it was considered an artifact. The reference list for each zone was truncated at a magnitude resulting in an approximately constant source density for all zones, resulting in a reference list source density considerably above the density of detected sources in the preprocessed images. This was done in order to prevent confusion, especially at the cluster core, with the numerous faint stars in the list Thresholding The number of sources is very sensitive to the threshold value used to distinguish between sources and background. The threshold value should balance artifacts from the background and faint stars near the detection limit. As Saha (1995) states, specifying a number of spurious detections (Eqs. 1 and 2), fixes the completeness for any given S/N ratio (see also Section 4.3 below). An objective way to set the threshold value involves use of those parts of the image where no stars are expected to be present. The detection algorithm is applied to the image, and the

6 6 R. C. Snel, P. Linde and S. Spännare: Detection of point sources threshold value is adjusted until the number of artifacts detected in the area devoid of stars is judged to be of adequate order. In the cases investigated here, crowding is so high that no part of the image is devoid of stars. The detection threshold can then be set to a low value. Since every part of the image contains stars, artifacts of the diffuse background can not exist. However, artifacts resulting from inaccurate measurement of positions and intensities are still possible. 4. Results 4.1. S/N ratio in crowded fields Equations (4) (6) show two regimes for a fixed exposure time: for bright stars the S/N ratio increases with the square root of the flux of the star, for stars in fields dominated by the sky background, the S/N ratio increases linearly with the flux of the star. Near bright stars the S/N ratio is additionally subject to effects described by Eq. (8). In crowded fields, the effects given by Eq. (8) will dominate over those presented in Eq. (5). The sky background is composed of unresolved faint stars. This is illustrated in Fig. 2. Two simulated crowded field images were used (see Linde et al. 1998), constructed using a Wielen (1983) LF and with FWHM of the PSF of 2 and 6 pixels, respectively. Each dot represents an S/N ratio computation (Eq. 7) for a star. Local variations in image crowding, effectively causing a locally higher background, cause stars with identical intensities to have differing S/N ratios. Figure 3 shows examples from application of the preprocessing algorithms to the M15 test image. Two sub-fields are shown enlarged, to the left an area 1.5 minutes of arc from the centre, to the right an area partly covering the centre. The sub-fields are shown at very different contrast scales (cf. Fig. 1). The top pair shows original data obtained with the ground-based NOT, and the bottom show the corresponding fields as observed with the HST. The three pairs in between show preprocessed versions of the NOT images resulting from matched filtering, deconvolution and PSF fitting, respectively. For the cases of matched filtering and deconvolution, the preprocessed images were subjected to the local maximum detection algorithm discussed in Sect Intensity estimations were scaled to approximately match the integrated intensities in the HST image. Subsequently, the positions and intensities of the detected objects were matched against the reference list, derived from the HST images Matched filtering Figure 3b shows the effect of matched filtering. Comparison with Fig. 3e shows that most stars are enhanced and that some blended stars become separated. The gradient in the background disappears. The brightest stars show clearly how filtering causes negative (black in the figure) values around sharp objects. Fainter stars located in these rings, with a maximum value below the detection threshold, will not be detected, even if they do have a local maximum. Saturated stars show holes, which affect detection, but the effects of which can be taken into account during final measurement of position and intensity. While the matched filtering of the image was accomplished with DAOPHOT s FIND algorithm, the actual detection of maxima differed slightly from the one used by DAOPHOT. Where DAOPHOT checks for nearby stars and estimates a subpixel precision position, the local maximum detection algorithm used here does not Deconvolution Fig. 2. Signal to noise ratio as a function of intensity for stars in crowded fields. The lines are the calculated S/N ratio (Eq. 3), valid for a single pixel. The dots show the calculated S/N ratio (Eq. 7) for individual stars in a simulated crowded field image. Figure 3c shows the effect of deconvolution on the original image. Noise suppression effectively prevents amplification of noise in the deconvolved image. The familiar ring patterns around bright stars are still present, causing similar effects as the rings around bright stars in the case of matched filtering; fainter stars in these rings evade detection. Noise suppression causes some unwanted side-effects for detection. As is evident from the area around the globular cluster core, the deconvolved noise-suppressed image shows structure suggesting the presence of stars, but no local maximum is present. This prevents the local maximum algorithm from detecting some stars, which are detected in the image resulting from matched filtering PSF fitting 4.2. Preprocessing In Fig. 3d PSF fitting has been applied (see Sect ) to the NOT image. A very high amplification of star-like objects is noted and separation of the objects is favourable (although the high contrasts prevent a good representation of this in fig. 3d).

7 R. C. Snel, P. Linde and S. Spännare: Detection of point sources 7 (a) Original NOT (b) Matched filtering (c) Deconvolution (d) PSF fitting (e) Original HST Fig. 3. Inner field (right) and outer field (left, approx. 1.5 minutes of arc from the core) of the globular cluster M15, observed with the Nordic Optical Telescope (a) and with Hubble Space Telescope (e). Panels (b) show the result after applying matched filtering on the images in (a); (c) and (d) show the corresponding images applying Lucy-Richardson deconvolution with noise suppression and PSF fitting, respectively.

8 8 R. C. Snel, P. Linde and S. Spännare: Detection of point sources The fitting algorithm does not allow negative intensities, so no negative valued rings are present around the stars. Nevertheless, rings devoid of fainter stars are present around bright stars Completeness of detection Completeness of detection for each zone shown in Fig. 1 was calculated from the detected and true numbers of stars, using 0.5 magnitude intervals for stars fainter than V=17, and 1 magnitude intervals for those brighter than V=17. Since the number of stars per magnitude bin was small for brighter stars, smoothed interpolated curves were used to represent the results. Figure 4 shows completeness of detection (C) for the case of PSF fitting. On the left, C is expressed as a function of magnitude, on the right as a function of S/N ratio. The large gradient in crowding is clearly visible as a change in limiting magnitude (defined here as the magnitude at which C = 0:5) by as much as 4.5 magnitudes. Expressed in S/N ratio, the difference is only 1.2 dex, which can be compared directly with the magnitudes after multiplication with a factor?2.5 (see discussion). Comparison with Fig. 5 shows the validity of Eq. (13), especially for the upper part of the figure. The number of detected sources per zone is given in Table 1. Table 1. Detected number of stars and artifacts for the three preprocessing methods. Zones are numbered from the core outwards, see Fig. 1. zone PSF-fitting deconvolution matched filtering stars artifacts stars artifacts stars artifacts total Theoretical completeness Figure 5 shows the expected completeness of detection, using Eq. 13, and the LF measured in the HST images, assuming a PSF area of 40 pixels, and Gaussian distribution in intensities i for f(i 0 ; i). Read-out noise and intensity scaling correspond to the properties of the NOT image. 5. Discussion We have emphasized the detection aspects of crowded field photometry, realising that detection of sources in crowded fields can not be completely disconnected from measurement of the sources. Incomplete detection of sources affects measurement, while inaccurate measurement may hide undetected sources from later identification. Especially in severely crowded fields, confusion of sources limits photometric accuracy. More complete identification of sources will increase the accuracy of subsequent measurements. In moderately crowded fields, iterative detection and subtraction of stars yields good results, albeit with the possibility of strong correlation between the intensities of overlapping stars. For severely crowded fields, iterative methods do generally not give an adequate solution for the fainter stars. The results of the three preprocessing algorithms are presented in Table 1. The resulting difference between the three algorithms is limited. Considering the successfully detected number of stars, matched filtering performs best. The deconvolution algorithm used in the present work yields conservative results, with somewhat fewer detected stars but also fewer artifacts. The result that PSF fitting and matched filtering with the PSF yielded nearly equivalent completeness of detection is not surprising, since the methods are, in principle, mathematically equivalent (Cool and King 1995). Differences between the three resulting data sets are relatively small, especially in the crowded regions of the image. The number of detected artifacts is, especially for background-limitedfields, a function of the detection threshold. This threshold was set to a low value, detecting each local maximum in the preprocessed image. For the outer parts of the globular cluster, crowding is low enough to observe diffuse sky background in some areas of the image. Here, as expected, the number of artifacts is higher than with proper choice of detection threshold. The majority of artifacts is faint. For the inner parts of the globular cluster, crowding is so high that every pixel contains significant light from at least one star. Artifacts are the result of incorrect measurement of the positions or intensities of stars. Artifacts resulting from the background (Eqs. 1 and 2) are not present. The distribution of measured intensities for the artifacts confirms this. Magnitude measurement errors larger than one magnitude cause artifacts. A likely scenario for this to occur is that one or more bright stars dominate the intensity distribution in the image, but a much fainter star is located at the measured position. The measured intensity, composed of the sum of the bright and faint stars, is matched against that of the faint star, after which the object is identified as an artifact due to too large a magnitude error. It should be noted that several similar exposures often exist in realistic observational data sets. Obviously, since artifacts of the background, if present, would generally be randomly distributed, a comparison between detection in such exposures provides a further means of artifact discrimination. Comparing the completeness of detection measured in the image with the theoretical expression (13), an estimate of the number of pixels b per resolution element is needed for each preprocessing algorithm. As can be seen with the help of Eq. (9), a zone exists around each star where no fainter stars can be detected. The area of this zone is equal to b. Equation (9) defines the zone through the S/N ratio, which makes b a function of magnitude, and assumes that only the S/N ratio is the limiting factor for detection. A typical value for b is 4 2, where

9 R. C. Snel, P. Linde and S. Spännare: Detection of point sources 9 Fig. 4. Completeness of detection for the sections shown in Fig. 1. To the left expressed as a function of magnitude, to the right as a function of log(s/n). Fig. 5. Theoretical completeness of detection (Eq. (13) for the sections in Fig. 1. is the standard deviation of the Gaussian function describing the PSF, which equals 28 pixels for the PSF in the image. In practice, the preprocessing algorithm too imposes zones of non-detection on the image. This can be clearly seen in the preprocessed images of Fig. 3. Irrespective of the type of preprocessing performed, zones exist around all bright stars where no faint stars are detected. Only stars of comparable brightness are detected in close proximity to bright stars. The effective area of each resolution element, and through Eq. (13) the completeness of detection, is an objective way of comparing different preprocessing algorithms. The value for b of 40 pixels, used to generate Fig. 5, corresponds well with the radii of the zones of nondetection of typically 4 pixels. The average number of pixels per detected star is 54, calculated from Table 1 and the number of pixels in the image. For the individual zones this value varies between 60 and 47. Correspondence between the observed and theoretical completeness is satisfactory. A single parameter was adjusted to match all six curves to the observed completeness. The difference between the observed and theoretical completeness has a number of explanations. Theoretical completeness is calculated using assumptions concerning the distribution of measured magnitudes given a true magnitude. After matching against the reference table, magnitudes of successfully detected objects are replaced with their true magnitudes, which are allowed to differ from the measured data by one magnitude. Additionally, Eq. (13) assumes that crowding is homogeneous, while in fact there is a gradient in the stellar density. Faulty assumptions concerning f(m 0 ; m) influence the tail of the the curves describing completeness of detection (C-curves), while normalisation of f(m 0 ; m) ensures that the integrated effect, at high completeness, is negligible. Magnitude measurement errors resulting from crowding are not modelled in the expression for f(m 0 ; m) used to create Fig. 5. The simplification of neglecting the magnitude-dependence of b may cause systematic errors as well, which could become significant at different slopes of the LF. Equation (9) assumes that the exact flux and position of the bright star is known. In practice, these parameters can not be known with infinite accuracy. A way to properly approach the problem is to use the probability that a detected distribution of photons is not the result of a single source. Lucy (1992) describes how one can resolve two sources of equal intensity using the fourth moment of the distribution of the photons positions;

10 10 R. C. Snel, P. Linde and S. Spännare: Detection of point sources for a perfect, noise-free and linear detector, the required number of photons to resolve the two sources scales with r?8. Figure 4 shows completeness of detection both as a function of magnitude and of S/N ratio. The spread between the curves resulting from use of the magnitude is 50 % larger than when expressing C using the S/N ratio. The effects of crowding on the S/N ratio can be clearly seen, shifting the bright stars to lower S/N ratio values. Figure 2 shows that the spread in S/N ratio for a single intensity can be large, which affects detection of fainter stars, especially due to bright source wings as detailed through Eq. (8). Therefore, it is not only the magnitude of the star itself that is determining for its detection, but also the magnitudes of its nearest neighbours, and thus crowding. Since the S/N ratio is also sensitive to crowding, while the estimated magnitude of the star is not, the spread in the C-curves is smaller when expressed as a function of S/N. Thus it can be concluded that image processing algorithms which increase the S/N ratio in a single pixel, will serve well as preprocessing algorithms for detection. PSF fitting and matched filtering replace each pixel with the PSF-weighted average of its surroundings, which is exactly what is done in weighted aperture photometry. The result is a signal estimate for a potential star that is centred on the pixel for which the estimate is made. The noise in the same pixel is the weighted average of the noise of its surroundings, which means that stochastic noise will be reduced. The result is increased S/N ratio for the pixel in question, but also correlated noise and signal between the pixels in the processed image. For those pixels that actually have a star approximately centred on them, the estimates for the signal and noise will be realistic. Thus, the expression (3) for the S/N ratio, which is valid for a single pixel, can be applied, but now with a different denominator, representing the weighted noise in the surroundingsof the pixel. This is why the curves in Fig. 2 match the upper envelope of the points relatively well, despite the fact that the simplification was made that all light falls in a single pixel. The two curves differ only in a single parameter value, the effective base noise in the image, which was adjusted to make the low-intensity end of the curves and the points match. With effective base noise we denote the noise that replaces the combined contributionsfrom PSF-weighted sky and base noise. Newberry M.V., 1991, PASP 103, 122 Saha P., 1995, AJ 110, 916 Schechter P., Mateo M., Saha A., 1993, PASP 105, 1342 Spännare S., in preparation Stetson P.B., 1987, PASP 99, 191 Wielen, R., Jahreiss, H., Krüger, R., In Philip A.G.D., Upgren A.R. (eds.) The Nearby Stars and the Stellar Luminosity Function, IAU Coll. 76, 163 Acknowledgements. We thank Jean-Luc Starck for deconvolving our test image. References Buonanno R., Buscema G., Corsi C. E., Ferraro I., Iannicola G., 1983, A&A 126, 278 Cool A.M., King I.R., In: Koratkar A.P., Leitherer C. (eds.) Calibrating Hubble Space Telescope: Post Servicing Mission, Baltimore: Space Telescope Science Institute Debray B., Llebaria A., Dubout-Crillon R., Petit M., 1994, A&A 281, 613 Howell S.B., 1989, PASP 101, 616 Janes K., Heasley J., 1993, PASP 105, 527 Linde P., Snel R., Spännare S., Vistas in Astronomy, 1998, in press Lucy L.B., 1992, AJ 104, 1260 Murtagh F., Starck J.-L., Bijaoui A., 1995, A&AS 112, 179 This article was processed by the author using Springer-Verlag L A TEX A&A style file L-AA version 3.