Pixel sampling

An important consideration in photometric and astrometric measurements made with a CCD is how well the PSF is sampled on the two-dimensional array. PSFs that are well sampled by a CCD observation will lead directly to the result that the center and shape of the PSF will be known to higher precision, and thus one will obtain a final answer that will be of higher accuracy. We can define a sampling parameter, r, as follows (Howell et al., 1996; Buonanno & Iannicola, 1989):

FWHM

where FWHM is the full-width half-maximum value of the source PSF and P is the pixel size, both values given in the same units. For r less than about 1.5, digital data are considered undersampled. As can be seen from the above expression, r will be small for the case of a CCD with large pixel sizes compared with the total areal coverage of the imaged PSF. The other possible case of small r values is if the CCD image contains very tight PSFs such as those that might be obtained at observing sites with very good seeing, if using adaptive optics systems, or for CCD images obtained outside the Earth's atmosphere (i.e., space-based telescopes). Real life examples of cases that will produce undersampled images (i.e., small r values) are a typical wide-field telescope outfitted with a large-format CCD, such as a Schmidt telescope or a camera lens, or a space-based telescope such as the Hubble Space Telescope wide-field planetary camera (WFPC) (Holtzman, 1990; Howell et al., 1996, and Section 7.1).

Anytime the value of r approaches the limiting case of undersampling, standard software methods and techniques of astrometric and photometric data analysis will begin to produce increasingly larger errors and poorer fits as r decreases further (see Section 7.1). Photometric and astrometric errors obtained from CCD observations are related in that the analysis techniques for each type of measurement are very similar. Both photometry and astrometry require intimate knowledge of the centroid position of the source PSF. However, it has been shown (King, 1983; Stone, 1989; Howell & Merline, 1991) that for undersampled data the photometric error is least for source PSFs

4000 r

3000

3000

Fig. 5.12. The effects of pixel sampling are shown. The top PSF is a wellsampled star image with a S/N of near 230. The bottom panel shows the same PSF but now severely undersampled and centered at the middle of a pixel and (next page) at the corner of four pixels respectively. Note that the undersampled profiles are not well represented by a Gaussian function. From Howell et al. (1996).

Fig. 5.12. The effects of pixel sampling are shown. The top PSF is a wellsampled star image with a S/N of near 230. The bottom panel shows the same PSF but now severely undersampled and centered at the middle of a pixel and (next page) at the corner of four pixels respectively. Note that the undersampled profiles are not well represented by a Gaussian function. From Howell et al. (1996).

that are centered on the edge of a pixel or exactly at its center, whereas for astrometric data, the resulting error is least when the source PSF is centered midway between a pixel's edge and its center.

The rule of thumb for pixel sampling on a CCD follows directly from the statistical result of Nyquist sampling. That is, sampling of the PSF of an astronomical source will be optimal in terms of S/N, error rejection, data analysis, and so on for a source PSF that has its FWHM value sampled over about two pixels (i.e., FWHM ~2 ■ pixel size). For example, if the average seeing at an observing site produces source PSFs with FWHM values of 2 arcsec, then an ideal (optimal) CCD pixel size to use would be one for which each pixel within the array has a projected image size of 1 arcsec across.1 A rigorous mathematical definition of undersampling, based on the Nyquist theorem, identifies critical sampling as the sampling interval that is equal to the width (i.e., standard deviation) of the PSF. For a Gaussian PSF this corresponds to a FHWM equal to 2.355 pixels. Of course an ideal image scale is hard to meet in reality as seeing and telescope focus change with time, source PSFs generally do not fall onto the CCD pixel grid exactly on a pixel boundary, and one generally has only a limited number of available CCD cameras with fixed pixel sizes.

1 The determination and measurement of CCD pixel size or plate scale was discussed in Sec tion 4.1.

Since CCD detectors do indeed sample astronomical sources in a quantized manner, pixel sampling within the array will cause even identical PSFs to change their detailed appearance slightly, even within the same observation. The effects of such sampling differences become worse as the sampling parameter (r) becomes smaller (Howell & Merline, 1991; Merline & Howell, 1995). Figure 5.12 illustrates some examples of various values of r caused by CCD pixel size. The top panel shows a well-sampled source PSF that appears to be more or less Gaussian in shape. The remaining two panels in Figure 5.12 show the same model PSF but now imaged as a poorly sampled (r = 1) source. The undersampled cases are for a source PSF with a pixel-centered centroid and a corner-centered centroid respectively. Notice in Figure 5.12 that the undersampled PSFs are not well represented by a Gaussian function (Buonanno & Iannicola, 1989; Holtzman, 1990; Howell & Merline, 1991; Howell etal., 1996).

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