^{2} Space Telescope-European Coordinating Facility

Fortunately, much of the information lost in sampling can be recovered. In the lower right we display the image recovered using one of the family of techniques we refer to as "linear reconstruction." The most commonly used of these techniques are shift-and-add and interlacing. The image in the lower right corner has been restored by interlacing a 3x3 array of dithered images. However, due to poor placement of the sampling grid or the effects of geometric distortion, true interlacing of images is often infeasible. On the other hand, the other standard technique, shift-and-add, convolves the image yet again with the orginal pixel, adding to the blurring of the image and the correlation of the noise. (The deterioration in image quality between the upper and lower right hand images is due entirely to convolution with the WF pixel). Here we present a method which has the versatility of shift-and-add yet largely maintains the resolution and independent noise statistics of interlacing.

The new shrunken pixels, or "drops", rain down upon the subsampled output image.

In the case of the HDF, the drops had linear dimensions one-half that of the input pixel --- slightly larger than the dimensions of the output subsampled pixels. The flux in each drop is divided up among the overlapping output pixels in proportion to the areas of overlap. Note that if the drop size is sufficiently small not all output pixels have data added to them from each input image. One must therefore choose a drop size that is small enough to avoid degrading the image, but large enough that the after all images are "dripped" the coverage is fairly uniform.

In averaging the input image with the output image, the size of
the drop is further adjusted to reflect the geometric distortion of the
camera before the overlap of the drop with pixels
in the ouput image is determined. When a drop with value
i_{xy} and user defined weight w_{xy} is added
to an image with pixel value I_{xy}, weight W_{xy},
and fractional pixel overlap 0 < a_{xy} < 1, the resulting
value of the image I'_{xy} and weight W'_{xy}
is

In order to test the effect of drizzling on stellar photometry, we created a
19x19 grid of artificial stars on a 4 times oversampled grid
using the Tiny Tim WFPC PSF modelling code and
then convolved these stars with a narrow Gaussian which approximates the smearing
caused by the cross-talk between neighboring pixels in the WFPC.
These images were then multiplied by the Jacobian of the geometric distortion
of the WF3 camera, to represent the effect of the geometric distortion on
point source photometry.
This image was then shifted and sampled on a 2x2 grid and the results combined using
the **drizzle** algorithm with an output pixel size one-half that of the original WF pixels,
and
a drop size with linear dimensions 0.65 that of the WF pixel. The geometric distortion
of the chip was removed during drizzling using the polynomial model of Trauger * et al. *
The amount of data
and dithering pattern, therefore,
resemble ones that a typical user might produce. (In contrast, the HDF contained 11 different
pointings.)

To determine the effectiveness of drizzling in correcting the photometric effects of geometric distortion, we then obtained aperture photometry on the stars in one of the original input images and on the the stars in the output drizzled image. In the image to the lower left we display the results obtained from the input image. The photometric measurements of the 19x19 stars are represented by a 19x19 pixel image. The effect of the distortion on the photometry is obvious -- the stars in the corners are up to ~4% brighter than those in the center of the chip. The image on the right similarly displays the results of aperture photometry on the 19x19 grid of stars after drizzling. The effect of geometric distortion on the photometry is dramatically reduced: the r.m.s.~photometric variation in the drizzled image is 0.004 mags.

Below are shown two PSFs. The upper PSF is taken directly from the HDF F450W drizzled image. The PSF shows substantial variations about the best fit Gaussian due to the effects of non-uniform sampling (note, however, that these variations do not noticeably affect aperture photometry performed with a radius greater than or equal to 5 output pixels -- that is 2 WF pixels). The lower PSF is bright star taken from a deep image with a nearly perfect four-point dither. The uniform sampling produces a smooth PSF. The difference in the apparent widths of the PSFs is due to the use of larger output pixels in the second image than in the HDF (0."05 vs. 0."04).

Although the removal of cosmic rays using **drizzle**
is still very much a work
in progress, we have developed the following procedure which appears quite
promising.

- Drizzle each image onto a separate sub-sampled output image, but preserve the initial pixel size (thus, for instance, an initial pixel might cover the area four output pixels, but because of the effects of the shifting and geometric distortion, the output image contains sub-pixels which are a weighted average of adjacent pixels, rather than just a block replication of the input).
- Take the median of the output drizzled images.
- Map the median image back to the input plane of each of the individual images, including the
image shifts and geometric distortion. The is done using a program named
**blot**(blotting is, in effect, the inverse of drizzling). - Take the spatial derivative of each of the blotted output images.
- Compare each original image with the corresponding blotted image. Where the difference is larger than can be explained by noise statistics, or the flattening effect of taking the median or perhaps an error in the shift (the magnitudes of the latter two effects are estimated using the image derivative), the suspect pixel is masked.
- Repeat the previous step on pixels adjacent to pixels already masked, using a more stringent comparison.
- Finally, drizzle the input images onto a single output image using the pixel masks created in the previous steps.

Most of the information displayed on this page can be found in a paper, written by Richard and myself, for an SPIE conference. This version is easily printable, and can be retrieved from astro-ph , an "e-print" archive of astronomical papers.

This page accessed times since September 15, 1996.