The Photometry of Undersampled Point‐Spread Functions

The method of PSF reconstruction advocated here stacks the dithered images in the Fourier domain. The Fourier transform of a discretely sampled data set is periodic, repeating to ±∞ in both x and y. When an image is undersampled, or aliased, the higher order "satellites" in the Fourier domain overlap with and contaminate the fundamental transform. This contamination cannot be eliminated in any single image, but as the object being imaged is shifted with respect to the pixel grid, the phases within the Fourier satellites vary. With a sufficient number of dithered images, each having different phases, the aliasing can be eliminated algebraically. The sampling frequency in the final superimage is determined by the spatial scale at which P^' no longer has significant power. This approach is nicely summarized by Bracewell (1978) for the case of one‐dimensional data; I present a tutorial on its extension to images, with particular application to HST WFPC2 images, in Lauer (1999). The attractive features of this method are that the superimage is well‐sampled, there are no arbitrary parameters controlling its construction, and there is no blurring at the Nyquist scale. The method requires a minimum of N² images with nondegenerate dithers to construct a superimage with N × N subsampling; when more images are available the superimage is overdetermined and becomes the best fit to the dither set.

The HST WFPC2 and NICMOS imagers provide contrasting test cases for exploring the effects of intrapixel sensitivity variations. Intrapixel effects are subtle in the WFC chips of WFPC2 but severe in the NIC3 camera. Figures 1– 4 show PSFs reconstructed with 3 × 3 subsampling for the V (F555W) and I (F814W) filters in the WFC camera of WFPC2 and the J (F110W) and H (F160W) filters for NIC3. The WFPC2 PSFs were constructed from dithered observations of ω Cen (STScI program 4819) actually obtained by the WFPC2 Investigation Definition Team for the purpose of understanding variations in the WFPC2 PSF as a function of pixel location. The image set consists of 20 images in each of two filters: F555W and F814W. The dither pattern consists of 0025 steps in the row and column directions, the 20 images mapping out a 0075 × 0 1 rectangle with a square grid.

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The NIC3 PSFs were constructed from images obtained for the Hubble Deep Field South (HDFS) program. The image set comprised 146 exposures, of which 98 were used (the star in the discarded images was either out of the field or too close to its border). The PSFs shown are from the brightest star in the HDFS field. The HDFS dither pattern consisted of both small and large angular offsets, largely dictated by the needs of the other cameras used in the HDFS program. While this data set was fine for the present purposes, in many ways the dither pattern was far from optimal, an issue that I will discuss in further detail below.

It is simple to calibrate the effects of intrapixel sensitivity variations and undersampling on a PSF, once P^' has been constructed. Since P^' is well‐sampled, its centroid can be shifted to any desired fractional pixel location, without loss of resolution or information. Once shifted, coarse samples can be drawn from P^' to simulate a PSF, P₀(δx,δy), as would be observed at that location. P₀ is thus

where the III function refers to the spacing of the detector (rather than subsampled) pixels. The photometric error, (δx,δy), at the given offset is

One can thus systematically map over the entire domain of fractional centroid offsets at any desired resolution; the map essentially consists of a lookup table of photometric offsets to be applied to a reduced photometric data set. It is critical to use a method of interpolation for P^' that does not degrade the resolution; I do this with sinc‐function interpolation, which is the theoretically appropriate sampling kernel for well‐sampled data. Last, I emphasize that no "integration over a pixel" is included in equation (5), nor should be. Remember, P^' already reflects convolution of the optical PSF with the detector pixel response; thus this integration has already implicitly taken place.

Figures 5– 8 show the error maps for the WFPC2 and NIC3 PSFs as a function of the fractional pixel location of the PSF centroid. The square area of the maps corresponds to the domain - 1 / 2<δx<1 / 2, - 1 / 2<δy<1 / 2, in steps of 0.05 pixels; (δx,δy) = (0,0) is at the center of the maps. An important caveat is that in practice finite limits of integration must be used in equation (6); thus the absolute size of the errors will vary somewhat with aperture. In the present case I measure the flux in a 15 × 1 5 box for the WFC PSFs and a 30 × 3 0 box for NIC3. Overexposed stellar images show that significant scattered light falls outside these limits, but in the case of WFPC2 at least, the aperture includes nearly all the pixels above the background for stellar images not saturated in the core.

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The WFC error maps show that the photometric effects of undersampling are subtle but are still significant for bright sources with sufficient signal. The error map for the WFC V‐band PSF has a peak‐to‐peak range of 0.030 mag and an rms dispersion of 0.008 mag. The effects of undersampling are slightly reduced in I band, as might be expected given its larger PSF width; the peak‐to‐peak error range is 0.023 mag, with a 0.006 mag dispersion. The random color error is thus limited to 0.01 mag. Intriguingly, however, the V and I maps qualitatively resemble each other; thus errors in V and I may correlated depending on how the telescope was pointed for the two images. Both filters show that the error maps are anisotropic in the CCD row (y) and column (x), with maximal flux (>0) actually corresponding to when the PSF falls between two CCD rows; however, the error maps cannot be simply described as separable x‐ and y‐functions. These results are in excellent agreement with the simple measurements presented in Holtzman et al. (1995), who found little dependence of the photometry on fractional x‐location, but a few percent variation dependent on fractional y‐location, again with more light detected for stars centered between rows. Note that this implies that the intrapixel response itself for CCDs is more complex than a simple picture that might have fairly uniform pixels separated by less sensitive "cracks." Jorden et al. (1994) emphasized, for example, that at some wavelengths the CCD column stops corresponded to regions of enhanced sensitivity. At the same time, they saw this pattern reverse in sign at other wavelengths—the reader is cautioned that the present results are valid only for the F555W and F814W filters.

If undersampling effects are subtle in the WFPC2 CCDs, they completely dominate the photometric errors of stellar photometry done with the NIC3 camera. The peak‐to‐peak error range in the NIC3 J band is 0.39 mag; the dispersion is 0.10 mag. The H‐band PSF is broader, given the longer wavelength of the bandpass, thus reducing the undersampling effects—still the errors remain large, with a 0.22 mag peak‐to‐peak range and a 0.06 mag dispersion. The greatest sensitivity indeed occurs for PSFs centered on a pixel, unlike the case for the WFPC2 CCDs; given the architecture of the NICMOS arrays, the picture of losing light in the cracks between the pixels may be more valid for these devices. In passing, I emphasize that the range of photometric variation implied from the reconstructed PSF in general will exceed that seen in the individual images in the dither set, unless some of the PSF observations fortuitously fell on the locations of both maximal and minimal detected flux. A naive analysis of the flux variations observed within a dither set will always underestimate the effects of undersampling; the reconstructed PSF cannot be understood as a simple interpolated average of the individual observations.

A critical issue is understanding if the present calibrations can be used to correct for undersampling effects in WFPC2 or NIC3 images. Unfortunately, in the case of WFPC2, it appears that the form of the PSF core is the dominant contributor to undersampled structure. Maps made with different stars within the same image set have error maps that differ significantly from each other. The WFC PSFs shown above were taken from a bright star near the center of the W2 CCD of WFPC2. A map generated from another star only 147 pixels away in W2 is shown in Figure 9. Both maps show that the row position dominates the error term, with maximal flux detected for centroids falling between rows, and have about same dynamic range. The detailed structure of the first map is different enough from the second map, however, such that it would provide little help for correcting the photometry of a star imaged at the second location on W2.

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The situation is better in for NIC3, where the pixel response appears to dominate. Figure 10 shows the map for another star in the same H‐band data set as the star shown in Figure 4. The peak‐to‐peak amplitude of the map is within a few hundredths of a magnitude of the map shown in Figure 8, and its morphology is similar enough that corrections derived from the former star would work well for the latter.

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Regardless of the utility of the present error maps for correcting undersampled stellar photometry directly, they do show what errors are likely to be encountered and how well various dither patterns sample the error pattern. The present analysis emphasizes constructing a well‐sampled image from a dither set to counter undersampling present in any single image. Clearly, a simpler approach of averaging the photometry from a star observed at different positions in general may reduce the photometric scatter due to undersampling by √N, where N is the number of dither steps available. A caveat is that neither the error maps nor dither pattern is necessarily random; thus this simple scheme may produce less noise reduction than expected or still include biases among stars at differing positions, particularly if the number of dithers is small. The large range of the NIC3 error maps further implies that an extremely large data set may be required to obtain 1% photometry by the simple combination of random dithers.

In this context, I have been surprised by how often dithered image sets fail to sample the full range of fractional pixel space adequately. Figure 11, for example, shows the fractional dithers realized in the NIC3 HDFS image set. Despite the availability of nearly 100 images, the pattern misses covering the fractional pixel space as well as a simple regular 3 × 3 pattern would; note that no or very few dithers fall within three of the corners of the figure. To be fair, the dithers in the HDFS program were optimized for the other cameras on board HST, rather than NIC3, but clearly the assumption that such a large data set would randomly sample the full fractional space is not justified. In the present case, failure to include many dithers landing near the pixel corners in the case of NIC3 clearly produces a strong bias, since these are the regions in which the flux deficit due to undersampling is most severe; furthermore, other stars in the same image set will have their dithers phased differently; thus biases due to incomplete dither coverage in this particular image set would be presented as large scatter among stars at different locations. Note that these biases will not be removed by image reconstruction algorithms, such as Drizzle, that simply redistribute the image flux; in the end one is effectively still just averaging the stellar images. The Fourier reconstruction methods that I discuss in Lauer (1999), in contrast, can reconstruct an unbiased PSF from even nonoptimal dither patterns, such as that in Figure 11. The trick is that the complete set of Fourier components that describe the PSF may still be represented in the dither set and isolated algebraically, even if it is not optimally encoded in the data.

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While the error maps encode the photometric error as a function of the fractional location of the PSF centroid, an important caveat is that measurement of the centroid itself will be affected by undersampling. This issue is central to the concerns of Anderson & King (2000), who discuss methods to obtain high‐precision relative astrometry on WFPC2 images for the goal of measuring the relative proper motions of stars within globular clusters. Thus, I will not dwell extensively on this issue myself. Nevertheless, being able to obtain accurate centroids of PSFs is critical to constructing the effective PSF in the first place and then evaluating the photometric error of stars at any position in the image set.

Figure 12 gives an error map of the total differences between the centroids of undersampled NIC3 H‐band PSFs (drawn from the effective PSF presented in Fig. 4) as a function of the true offset. In this case, the centroids were computed as the simple center of weight of a 5 × 5 box centered on the brightest pixel of the extracted PSF. The size of the error varied smoothly over the fractional shift domain, ranging from 0.12 to 0.27 pixels; if the computed centroids were used to look up the corresponding H‐band photometric error in Figure 8, clearly the implied photometric correction could be substantially in error. Now it is true that this simple method of calculating centroids perhaps would never be the algorithm of choice for undersampled data, but this is the point—methods that do work well for centroiding well‐sampled PSFs may work poorly for undersampled data, motivating the use of more sophisticated approaches.

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The method that I use is to cross‐correlate the undersampled PSFs with a well‐sampled PSF. This, of course, is fine if one has already generated a PSF from other stars in the image set or can fold in knowledge of the pixel response with the construction of a theoretical PSF (see § 2.4). In practice, however, I have found that the Fourier reconstruction method is fairly tolerant of centroid errors, and for WFPC2 and NIC3 data, an initial ad hoc effective PSF can be constructed from simple centroiding algorithms as discussed in the previous paragraph. The penalty is some blurring in the PSF core, but the ad hoc effective PSF can then be used to derive more accurate centroids from the cross‐correlation method—indeed, this can lead to an iterative loop where one is continually refining the centroids and the effective PSF in successive stages. If a preexisting effective PSF is available, however, a critical step is to normalize it as closely as possible to the expected flux of the new PSF being constructed. With undersampled PSFs, particularly when much of the flux in contained in a single bright pixel, positional information is lost and there can be strong covariance between intensity scaling and centroid measurement, a point emphasized by Anderson & King (2000). Again in practice, however, I have found with WFPC2 and NIC3 data that one can readily construct an initial ad hoc PSF with a rough initial normalization. Last, of course, good information on the dither steps may already be available from external information or measurements conducted from an ensemble of other sources in the image set.

Equation (4) shows that the effective PSF, P^', is the convolution of the intrinsic optical PSF with the pixel response, R. If knowledge of the optics‐only PSF is available, or if it can be calculated by an algorithm such as Tiny Tim (Krist & Hook 1997), then it may be possible to isolate the pixel response by deconvolution. If R is largely constant over the array, as Jorden et al. (1994) suggest is true for CCDs, then is may be possible to use it in conjunction with theoretical spatially variable PSFs to construct improved subsampled PSFs at any point within the field.

Figure 13 shows an attempt to isolate R for the F555W filter and the W2 CCD of WFPC2 by deconvolving the effective PSF in Figure 1 with a theoretical PSF constructed with the Tiny Tim package. The particular star selected has V−I = 1.09 (in the WFPC2 filters), which corresponds very closely to spectral type K0. The theoretical PSF was constructed for the star's location on W2 and with 3 × 3 subsampling. Deconvolution was done with 160 iterations of Lucy‐Richardson deconvolution (Richardson 1972; Lucy 1974); however, convergence occurred well in advance of this many iterations.

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The F555W R kernel is in excellent agreement with previous, but full‐pixel rather than subpixel, estimates of the WFPC2 pixel response. Holtzman et al. (1995) noted that some diffusion of light across pixel boundaries appeared to be occurring in the WFPC2 CCDs. Krist & Hook (1997) suggest a kernel that has 75% of its integral in a central pixel, with 5% flanking pixels in the row and column of the central pixel. The present kernel is given below with 3 × 3 subsampling, with the pixel values given as percentages of the total integral:

Pixels outside the 5 × 5 kernel listed are essentially zero; the diffusion out of the central (full‐sized) pixel is actually limited to only a thin margin a single subpixel in width.

Figure 14 shows a similar attempt to isolate R for the F110W filter in the NIC3 camera. As expected given the more severe undersampling effects in NIC3, the R kernel is more compact and sharply peaked than the WF2 pixel response. The NIC3 pixel kernel is given below with 3 × 3 subsampling, with the pixel values given as percentages of the total integral:

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A one‐sentence summary of this paper is that the precision of stellar photometry may be significantly limited by undersampling. The common assumption that a CCD consists of an array of contiguous and uniform pixels is an excellent initial approximation, but this is not correct in detail. One may be tempted to adopt a refined picture in which the array consists of uniform pixels, but surrounded by dead "moats"; however, this still is likely to be an oversimplification. In truth, the sensitivity pattern within a pixel is likely to be complex and highly dependent on the specifics of the detector architecture—indeed, once one allows for possible diffusion of photons or photoelectrons within the detector, the total spatial response function of a single pixel may be more complex than can be described by pure sensitivity variations alone. The import of the pixel response depends directly on the severity of the undersampling and the structural content of the astronomical source being imaged. It should also be understood that the pixel response may be even more important than the core of the optics PSF in setting the final resolution of an image.

If countering the effects of undersampling on stellar photometry is important, then I argue that the best solution is to dither the images in a regular pattern that permits easy reconstruction of a well‐sampled superimage. The information content of the superimage is as complete as can be allowed for the particular properties of the camera's detector and optics.

The optimal dither pattern is a regular N × N grid of 1/N subpixel steps. In practice, one may want to add full integral steps to the fractional steps as a way of stepping over hot pixels, bad columns, traps, or any other compact detector defects; however, if there are significant scale variations over the detectors field, then it is best to keep the total spatial extent of the dither pattern as compact as possible. In an ideal case, one would also obtain two or more exposures at each dither step, so as to eliminate cosmic‐ray events or any other variable noise feature. If large angular steps are desirable as well to counter any large‐scale variations in the detector response, then I suggest that the best way to proceed is to obtain the full data set as subsets of complete compact dither sequences separated by the larger offsets. Each subset will make a well‐sampled superimage; combining the superimages into a final image is then simple.

If a regular dither pattern can be executed exactly, then construction of a superimage requires nothing fancier than simple interlacing of the individual images. If the dither positions fall somewhat away from their optimal locations, but the fractional pixel domain still has good coverage or the image set is overdetermined, then I suggest the Fourier method used in this paper as a possible reconstruction algorithm. However, even if no formal reconstruction is attempted, a regular dither pattern will optimize the information content of the image set. Last, I emphasize that in general this is a fully two‐dimensional problem. The WFCP2 pixel response, for example, is more important in the column direction, but it cannot be cleanly separated into separate x‐ and y‐functions. For cameras such as NIC3, two orthogonal one‐dimensional patterns will fall well short of mapping the fractional pixel domain.

The choice of a pixel scale for an astronomical camera often requires a compromise between having as a large a field as possible versus obtaining well‐sampled images. Since even rather poorly sampled CCD cameras, such as the WFC channel of WFPC2, produce excellent stellar photometry for most problems, it is difficult to argue against tipping their design toward the largest field that the optics can accommodate. However, for many of the near‐ and mid‐IR cameras contemplated for space missions now in the early design phases, one must recognize that IR arrays may be less forgiving of undersampled PSFs and may limit the photometric accuracy to unacceptable levels if the undersampling is too extreme.

If designing a Nyquist‐sampled camera causes unacceptable limitations on the field, however, then I suggest that one may want to include a dither capability directly in the camera itself. HST has demonstrated the value of dithering undersampled images, but dithering HST images is both awkward and prone to error or nonoptimal patterns since the full spacecraft must be moved. An in‐camera dither capability, in contrast, can likely be made simple, highly accurate, and easy to invoke; ideally, the dither capability would be accurate enough to allow for direct interlace reconstruction of the superimage. The caveats are that the dithering must be conducted on timescales shorter than those on which significant variation in image structure will occur; when readout noise or overhead becomes significant, the minimum timescale between dithers may make dithering difficult. Last, I emphasize that even fine dithering is unlikely to lift the effects of severe undersampling. As a camera becomes increasingly poorly sampled, not only does the size of the image set required rise as the square of the inverse dither step (and with it the attendant difficulty of maintaining a stable image over the length of a dither sequence), but the precision to which small errors in the pattern can be detected and corrected for declines, making accurate image reconstruction all the more difficult. In practice, pushing beyond N = 3 subsampling for either WFPC2 or NIC3 appears to be highly cumbersome and, thus, may imply final upper limits to the pixel scales of other cameras in general if accurate photometry is both important and limited by undersampling.

I thank Ken Mighell, Todd Boroson, and Dave Monet for useful conversations. I thank Jay Anderson and Ivan King for an advance look at their manuscript on astrometry from undersampled PSFs and many e‐mail exchanges, which helped inform this discussion. Harry Ferguson kindly provided the reduced NICMOS images used in the analysis. Alex Storrs kindly provided the Tiny Tim NICMOS PSFs.