28.5 Dithering

The recovery of high frequency spatial information is fundamentally limited by the pixel response function (PRF). The PRF of an ideal CCD with square pixels is simply a square boxcar function the size of the pixel. In practice, the PRF is a function not only of the physical size of the pixels, but also the degree to which photons and electrons are scattered into adjacent pixels, as well as smearing introduced by telescopic position wandering. The image recorded by the CCD is the "true" image (that which would be captured by an ideal detector at the focal plane) convolved with this PRF. Thus, at best, the image will be no sharper than that allowed by an ideal square pixel. In the case of WFPC2, in which at least 20% of the light falling on a given pixel is detected in adjacent pixels, the image is even less sharp.

The PRF of an ideal square pixel, that is a boxcar function, severely suppresses power on scales comparable to the size of the pixel. According to the Shannon-Nyquist theorem of information theory the sampling interval required to capture nearly all of the information passed by square pixels is 1/2 the size l of a pixel. This corresponds to dithering the CCD from its starting position of (0,0) to three other positions, (0,1/2 l), (1/2 l, 0) and (1/2 l, 1/2 l); however, in practice, much of the information can be regained by a single dither to (1/2 l , 1/2 l).

The process of retrieving high-spatial resolution information from dithered images can be thought of as having two stages. The first, reconstruction, removes the effect of sampling and restores the image to that produced by the convolution of the PSF and PRF of the telescope and detector. The more demanding stage, deconvolution (sometimes called restoration), attempts to remove much of the blurring produced by the optics and detector. In effect, deconvolution boosts the relative strength of the high-frequency components of the Fourier spectrum to undo the suppression produced by the PSF and PRF.

If your observations were taken with either of the two dither patterns discussed above, and if the positioning of the telescope was accurate to about a tenth of a pixel (this is usually but not always the case), then you can reconstruct the image merely by interlacing the pixels of the offset images. In the case of a two-fold dither-that is images offset by a vector (n + 1/2, n + 1/2) pixels, where n is an integer-the interlaced images can be put on a square grid rotated 45° from the original orientation of the CCD (see Figure 28.2, top). In the case of a four-fold dither, the images are interlaced on a grid twice as fine as the original CCD and coaligned with it (see Figure 28.2, bottom).

Figure 28.2: Interlacing Pixels of Offset Images

The drizzle algorithm was implemented as the STSDAS task drizzle, as part of the dither package, which helps users combine dithered images. The dither package is included in STSDAS release v2.0.1 and later, and includes the following tasks:

• precor: Determines regions of the image containing astrophysical objects and nulls the remainder of the image, substantially reducing the effect of cosmic rays and chip defects on the offset measurement. The ouput from precor is only used for offset determination and not final image creation.
• offset: Cross-correlates all four images in a WFPC image, creating output cross-correlation images with names that can be appropriately grouped by later tasks. Uses the task crossdriz to perform the cross-correlation.
• crossdriz: Cross-correlates two images, after preprocessing which includes trimming, and, if requested, drizzling to remove geometric distortion or rotation. crossdriz will also perform a loop over a range of test rotation angles.
• shiftfind: Locates the peak in a cross-correlation image and fits for sub-pixel shift information. The search region and details of the fitting can be adjusted by the user.
• rotfind: Fits for the rotation angle between two images. rotfind is called when crossdriz has been used to loop over a range of test rotation angles between two images.
• avshift: Determines the shifts between two WFPC2 images by averaging the results obtained on each of the groups after adjusting for the rotation angles between the four groups. avshift can also be used to estimate the rotation angle between two different WFPC2 images, when the rotation angle is a small fraction of a degree.
• blot: Maps a drizzled image back onto an input image. This is an essential part of the tasks we are developing for removing cosmic rays from singly-dithered images. Additional information on these tasks is available in Fruchter et al. (1997) and Mutchler and Fruchter (1997), as well as the on-line help files.

  Although reconstruction largely removes the effects of sampling on the image, it does not restore the information lost to the smearing of the PSF and PRF. Deconvolution of the images, however, does hold out the possibility of recapturing much of this information. Figure 28.3, supplied by Richard Hook of the ST-ECF, shows the result of applying the Richardson-Lucy deconvolution scheme to HST data, used extensively in the analysis of WF/PC-1 data. The upper-left image shows one of four input images. The upper-right image shows a deconvolution of all of the data, and the lower two images show deconvolutions of independent subsets of the data. A dramatic gain in resolution is evident.

  A version of the Richardson-Lucy (RL) deconvolution scheme capable of handling dithered WFPC2 data is already available to STSDAS users. It is the task acoadd in the package stsdas.contrib. In order to use acoadd, users will need to supply the program both with a PSF (which in practice should be the convolution of the PRF with the optical PSF) and with the offsets in position between the various images. The position offset between the two images can be obtained using the task crossdriz in the dither package.

  In principle, image deconvolution requires an accurate knowledge of both the instrument PSF and PRF. At present, our best models of the WFPC2 PSF come from the publicly available TinyTim software (Krist, 1995). The quality of the TinyTim model can be improved substantially by taking into account the exact position of the source within the pixel. Remy et al. (1997) discuss how this can be accomplished by generating multiple TinyTim images at various focus and jitter values, oversampled with respect to the camera pixels. At present, this is very labor-intensive, and the results cannot be easily integrated into the existing deconvolution software. Another limitation of the existing software is that it cannot incorporate the significant variation of the PSF across the field of view. As a result, the Richardson-Lucy approach can only be applied to limited regions of a chip at a time. Nonetheless, tests done on WFPC2 images suggest that RL deconvolution can give the WFPC2 user a substantial gain in resolution even in the presence of typical PSF and PRF errors. Users interested in more information on dithering, reconstruction, and deconvolution should consult the February and September 1995 issues of the ST-ECF Newsletter, where these issues are discussed in detail.