Dithering and the OmegaCAM focal plane arrangement.

Konrad Kuijken, 28 Oct 1999

The following discussion addresses the issue of the impact on the dithering patterns to be used with OmegaCAM, for different arrangements of the science CCDs in the focal plane. In particular, the effect of a large cross-shaped gap through the center of the array, necessary if the filters are segmented, is addressed.

For this discussion, the following is assumed:

Considering a cross-shaped dead area in the center of the array of width W, it is clear that a given part of the sky sould at worst lie in two such crosses. Some puzzling then shows that the most compact arrangement of the centers of the dithers is something like (where each cell has size WxW): anything more compact would wipe out parts of the sky three or more times.
X
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I have investigated some dither patterns using the full array, opened up with an 11mm central cross, to see the effects on the other, narrower gaps of the CCD array. The array looks as follows:
The outer yellow square indicates the area to be filled by dithering; it falls almost completely within the VST fully-corrected field, except for the very outer corners. It consists of a total of 319.4 million pixels, of which 268.4 million (84%) are covered in a single exposure. The inner square delineates a square degree, and covers 16384x16384 pixels (as many as the full array). Of these 268.4 million pixels, 221.9 million (83%) are covered by a single exposure.

Dither 5: First the exact pattern above, with dither steps of W=11mm (736 pixels): where black, white and red are areas covered 0,1 or 2 times, and green, dark blue and light blue 3,4, and 5 times. The centers of the pointings are indicated by triangles. In this arrangement, the number of pixels with 2 or fewer exposures is rather low, as are the green (3) areas. This latter point is important as bad columns, cosmic rays etc will turn green into red. Statistics of the number of times pixels inside the yellow square are covered by this dither pattern:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 69.6 69.6
Pixels covered 4 times (Million) 133.6 133.1
Pixels covered 3 times (Million) 89.3 57.7
Pixels covered <3 times (Million) 26.9 8.0



Dither 5a: Opening up this dither pattern by a factor 1.3 (W=960 pixels) results in the following pattern. The red areas have been reduced greatly. The pattern has been fragmented a bit more, but this looks better than Dither5. The size of the square area which is covered at least 3 times has only been reduced very slightly at the corners. Statistics:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 47.7 47.7
Pixels covered 4 times (Million) 137.4 137.4
Pixels covered 3 times (Million) 105.6 76.5
Pixels covered <3 times (Million) 28.7 6.8



Dither 5b:  Conversely, here's a more compact dither pattern (steps scaled to half of those of dither5). This arrangement leaves much larger areas of homogeneous coverage (i.e., the context map is much less fragmented than in dithers 5 and 5a show above). However, it must be borne in mind that many bad columns will end up on top of the large green areas. A further disadvantage is that the center of the image contains some pixels which are covered only once. Statistics:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 108.8 108.3
Pixels covered 4 times (Million) 114.2 97.7
Pixels covered 3 times (Million) 78.3 52.5
Pixels covered <3 times (Million) 18.1 9.9



Given the large fragmentation of these images, I also looked into more compact dither configurations. These will necessarily lead to areas with poor coverage in the center of the resulting image, but the trade-off against homogeneity, dithering software effort, or effects of image distortion, may be worth it. I tried a simple cross-shaped pattern

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Dither 3: Dither steps of 11mm, exactly equal to the width of the dead cross. Such an arrangement causes some regions with only double coverage, but leaves much larger areas of homogeneous coverage (i.e., the context map is much less fragmented than in the 5-point dithers show above). However, there are many more pixels covered 3 or 5 times than 4 times, leading to sharp jumps in sensitivity. Statistics:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 142.5 141.9
Pixels covered 4 times (Million) 37.7 37.7
Pixels covered 3 times (Million) 106.2 71.1
Pixels covered <3 times (Million) 33.0 17.8



Dither 3a:is a hybrid of the above. Dither steps of 11mm in x, and 13mm in y, then repeated rotated through 90, 180, 270 degrees. This resembles the 5-point dither pattern above, but somewhat more compact and at a different angle. There is little advantage of this pattern over dither3: red areas are hardly reduced, fragmentation is greatly increased.
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 104.0 104.0
Pixels covered 4 times (Million) 99.7 99.2
Pixels covered 3 times (Million) 87.7 53.0
Pixels covered <3 times (Million) 28.0 12.3



Summary of pixel coverage of the various schemes:

Pixel coverage of the central square degree (million pixels covered):
Scheme 5 5a 5b 3 3a
covered 5 times 69.6 47.7 108.3 141.9 104.0
covered 4 times 133.1 137.4 97.7 37.7 99.2
covered 3 times 57.7 76.5 52.5 71.1 53.0
covered <3 times 8.0 6.8 9.9 17.8 12.3

What if there is no central cross?

Finally, some investigations of the dithering if the central cross is replaced by just a column or line, 11mm wide. Because of its success above, I will only look at the dither5a pattern, with the offsets in x and y scaled in proportion to the maximum gap in that direction. This possibility is relevant in case it proves possible to split the filters of OmegaCAM in half, rather than into quadrants.

With only a horizontal (perpendicular to the readout-port edges of the CCDs) central gap:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 78.9 78.9
Pixels covered 4 times (Million) 139.0 139.0
Pixels covered 3 times (Million) 73.4 47.5
Pixels covered <3 times (Million) 17.3 3.0

And with only a vertical central gap:
Statistics of this arrangement:
Outer
Square		 Inner
Square
Pixels covered 5 times (Million) 80.5 79.2
Pixels covered 4 times (Million) 153.2 148.6
Pixels covered 3 times (Million) 66.7 39.4
Pixels covered <3 times (Million) 8.1 1.2

Conclusions?

The cross-shaped gap that results when the filters are implemented as butted quadrants can be filled with dithering, but this requires taking dither steps of order at least 15mm, or 1000 pixels (200arcsec). At least about 2% of pixels inside a square degree area will be covered fewer than 3 times in such an arrangement. Most of these lie near the outer edges of the field.

5-point dither schemes with smaller steps result in less fragmented context maps of the final combined image, but at the price of a serious ghost of the central cross. A good compromise appears to be the dither5a scheme.

If the filters can be manufactured in halves, the covered fraction within a square degree can be increased substantially by adopting the same dither5a scheme, with the x- and y-offsets rescaled in proportion to the size of the maximum gap in that direction. In this case aligning the CCD array so that the central gap is parallel to the sides of the CCDs with readout ports appears to give best results.

To Be Done:

Further optimize the pattern. There may be a better compromise than 5a, by expanding/rotating the pattern a different amount. A criterion for optimizing this should be developed. (e.g. minimize f(<3) over the central square degree). Also the statistical effect of bad columns will have to be assessed; while it is unlikely that a given pixel of the co-added image will be hit by two different bad columns, any area covered 3 times in the analysis here will be vulnerable to bad columns. Preliminary analysis suggests that, since OmegaCAM will be sampling seeing well, it is unlikely that isolated bad columns will be a grave problem.