Sub-pixel Sampling and Dithering

The major advantage to space based observations, such as those obtained with

HST, is the removal of seeing a↵ects due to turbulence in the Earth’s atmosphere.

This significantly improves the resolution of the point spread function (PSF) of the resulting image, transitioning from a seeing dominated case, to one in which space-

bourne instruments may, in principle, reach the di↵raction limited regime. Whilst

this may sound like a significant improvement upon the quality of the incoming data, there are additional resolution restrictions enforced by the detectors used by space based instruments. Frequently the pixel scale of the primary science instrument’s CCD under samples the width of the PSF.

One solution to this conundrum lies in the ability to accurately o↵set the

telescope spatially by small increments. Dithering is the practice of shifting the

detector such that sky coordinates within the field of view land on di↵erent detector

co-ordinates, and is typically used to negate the e↵ects of bad pixels and CCD de-

fects. When these detector shifts are much smaller than the relative scale size of the detector pixels with non-integer deviations of the telescope pointing (i.e. dithering

at a sub-pixel level), the target is moved through a number of di↵erent locations

on the detector chip. This allows for the recovery of some of the information lost through undersampling the PSF, as each separate sub-dithered imaged now samples it. The dither pattern employed directs the level to which the PSF is sampled, with 4 dither pointings recovering the majority of the information contained within the image. Unfortunately, the final combination of these dithered images to produce a better sampled final PSF is a non-trivial task.

Drizzling

To some, this may merely suggest typical British weather conditions3_{, however,}

to the astrophysical community, drizzling is a method of reconstructing sub-pixel dithered images.

Drizzling [Fruchter and et al., 2009] combines two di↵erent image recon-

struction methods; “interlacing” and the “shift-and-add” method. Interlacing, as it’s name suggests, alternates the placing of pixels from two or more input images within the CCD grid, such that the final output is a uniformly woven “mesh” of

these input images. Whilst this technique naturally mitigates the a↵ects of bad

3_{Or perhaps the practice of moistening sponge-based confectionary with slightly sweetened citrus}

or hot pixels within the CCD chip, any small errors within the positioning of the telescope or any geometric distortion of the image by the telescope optics become apparent when combining images this way. The “shift-and-add” method on the other hand takes each input pixel from the dithered images and moves it into lo- cation within a finer sub-sampled grid on the final output image, simply summing the sub-pixel inputs to determine the value of each final output pixel. Whilst the absolute positions of the dither locations are less consequential using this method, the resulting image is still convolved to the scale of the original detector pixel.

Variable-pixel linear reconstruction, or more colloquially, “drizzling”, at- tempts to alleviate the issues inherent to the previous two techniques. As with the “shift-and-add” technique, pixels from the original dithered images are mapped onto a subsampled output grid, in which the user may specify the final size of the output pixels. However, drizzle allows the user to shrink the size of the input pixels

relative to the final output pixels. This parameter, known as the pixfrac, runs

between 0 and 1, permitting the user to dictate the degree to which pixels are con- volved with the PSF of the detector (although it should be noted that setting the

pixfrac too small will result in some output pixels receiving no data from input pixels, [Gonzaga and et al., 2012]).

These shrunken pixels are then drizzled onto the output grid of pixels, ac- counting for any shifts or rotations in the input frames, and the optical distortion of the camera. The flux contained within each of the input pixels is proportionally divided between the output pixels depending upon the fractional overlap of the in- put pixel. Consequently, this introduces correlated noise to the final uncertainties of each pixel values, which is discussed later in this section.

Drizzling may be easily performed for HST images using AstroDrizzle soft-

ware [Fruchter and Hook, 2002] within PyRAF, which I do for all of the HST im-

ages used within this thesis. Given the 4 dither positions employed for all of the

WFC3/UVIS and ACS images used within this thesis, I drizzle these UV images

to a final pixel scale of 0.0250 0 pix 1_{, whilst for the nIR images used I retain the}

native 0.130 0 pix 1 scale due to the lack of dithering.

Within the UV data set, images are subject to greater Charge Transfer Effi-

ciency (CTE) losses, which arise due to inefficient transfer of charge between pixels

during CCD readout, a consequence of cumulative radiation damage in a low Earth orbit environment [Bourque et al., 2013]. To mitigate against this, all early images taken under programme GO-13025 utilised a pre-flash to fill charge traps. In the latter observations of this program, and indeed for all of the UV observations taken within programs GO-13022 and GO-13026, sources were additionally relocated to

the corners of the UVIS chip to minimise the number of transfers employed before readout. The final individual images were then cleaned for CTE tails using the method of Anderson and Bedin [2010] prior to drizzling.

The UV images were also re-drizzled again to match the plate scale of the

nIR imaging (0.130 0 pix 1). Though this lowers the resolution of the image, the

technique allows for easier detection of low surface brightness features, and for a direct comparison between the nIR and UV imaging used within both Chapters 3 and 4.

Correlated Noise

A consequence of combining dithered images through drizzle is the correlation of noise between adjacent pixels due to the way in which the flux is spread over multiple output pixels. This not only produces an underestimate of the measurement of the noise in an object within the final drizzled image on the output pixel scale, but also creates uncertainty on the measurement on an individual pixel. As demonstrated

within Figure 2.1, for any given input pixel, covering areas a,b,c,d etc within the

final output pixel plane, the noise contained within the entirety of the input pixel is greater than the sum of that contained within the fractional areas in the output pixel, as the cross terms caused by the division of flux with neighbouring pixels

cannot be incorporated in the error measurement of any one pixel (i.e. (a2_{+ b}2₊

c2+d2)✏2<✏2).

The noise correlation ratio,R, depends upon the drizzle parameters chosen

and the geometry and orientation and geometry of the dithered input images. The full derivation of this parameter is beyond the scope of this work, however it is useful to note that it may be simplified to the following expression:

R= r

1 ₃1_r forr 1

= r

1 r₃ forr 1

(2.1)

wherer is the ratio of the pixfrac value used within drizzling and the pixel

scale of the output image.