Weighting exposures

Instead of selecting or rejecting individual exposures, measurements of the image quality could be used to weight the exposures in the final re-centred and summed image. In this section I will compare the performance of one formula for exposure weighting with the exposure selection method. In order to determine the best approach for the exposure weighting, it is necessary to define a more quantitive measure of the signal to noise for high resolution imaging. In order to allow rapid measurements on large datasets I chose a rather simple three step approach:

1. The Fourier power at high spatial frequencies in the core of the PSF was summed, to represent a measure of the signal at high spatial frequencies;
2. The Fourier power at high spatial frequencies in a region of the wings of the PSF was summed, to represent a measure of the noise at high spatial frequencies; and
3. My estimate of the signal-to-noise $R$ was taken simply as a ratio of the two numbers calculated in steps 1 and 2.

In order to ensure that the measurements of the signal were not significantly contaminated by noise, these signal-to-noise measurement were applied to images obtained after shifting and co-adding a number of exposures.

The short exposure images of $\zeta$ Boötis in the first run were first sorted in order of the Strehl ratio measured on the brighter component. The exposures were then binned into groups of exposures with similar Strehl ratios, each group containing $1$% of the total exposures. The exposures in each group were then shifted and co-added, reducing the dataset to $100$ images, each representing one of the groups.

The signal-to-noise measurement described above was applied to these $100$ images using the b component of $\zeta$ Boötis (the a component had been used as the reference star). The region around the b component used for the ``signal'' measurements was limited to the circle around the star shown in Figure 3.29a. A circular flat-topped window which dropped smoothly to zero at the edges (similar to a Hanning window) was used to extract a finite region of the image data without introducing high frequency noise components at the boundaries of the circle. A similar section of the image away from the stars was used for noise measurements (shown by the upper circle in the image). The range of spatial frequencies used to represent high resolution in the image was initially chosen (rather arbitrarily) as those ranging between $6.25$ $cycles/as$ and $12.5$ $cycles/as$, and the image power spectrum was summed in two dimensions over these spatial frequencies. The effect of varying this range of spatial frequencies will be discussed later.

Figure 3.29: a) shows the Lucky Exposures image generated using the best 41% of exposures with circles overlaid to indicate regions used for signal and noise measurements at high spatial frequencies. b) shows an image where each exposure is weighted by Strehl ratio to the power of $2.416$. The left-hand star was used as the reference, and the Strehl ratio for the companion star in both these images is $0.11$. For simplicity only data from the first run on $\zeta$ Boötis was used for this analysis.

The signal-to-noise ratio $R$ for high spatial frequencies calculated in this way is plotted against the Strehl ratio $\mathcal{S}$ for the reference star in the images in Figure 3.30a. Also shown is the best fitting function of the form:

$$R=A\mathcal{S}^{b}$$ (3.4)

where the values of $A$ and $b$ were determined by least-squares fit. For the best fit line shown in Figure 3.30a, $b=2.4$. Figure 3.30b shows the same data plotted on logarithmic scales.

Figure 3.30: Panel a) shows an estimator for the signal-to-noise of the high spatial frequency components in the image of binary component b when component a is used as the reference star, as described in the text. Panel b) shows the same data on a logarithmic scale. The first run of $\zeta$ Boötis was used for this analysis.

Now that we have a relationship between the Strehl ratio of the short exposures and $R$, our measure of the signal-to-noise ratio, we can make a concerted effort to produce the image with the maximum signal-to-noise ratio using the data on $\zeta$ Boötis. If the individual exposures are treated as independent, uncorrelated measurements, then the signal-to-noise ratio should be maximised if all the exposures are selected, but the individual exposures are weighted according to their Strehl ratios.

The data from the first run on $\zeta$ Boötis were processed in this way, with the individual exposures weighted by a value $W$ proportional to our signal-to-noise estimate:

$$W=A\mathcal{S}^{b}$$ (3.5)

The exposures were then re-centred and co-added to give the image in Figure 3.29b. This image has a Strehl ratio of $0.11$ and a signal-to-noise ratio of $R=936$. Alongside this in Figure 3.29a is the image generated from simple exposure selection (without weighting of the exposures) which has the same Strehl ratio. The fraction of exposures required to give this Strehl ratio was $41\%$ (determined by trial and error in a semi-automated procedure). The signal-to-noise ratio $R$ for the image in Figure 3.29a is $935$, essentially identical to that provided by the weighted exposures approach in Figure 3.29b. The conventional shift-and-add approach performs less favourably, with a signal-to-noise ratio of $479$.

If the signal-to-noise criteria $R$ used to determine the signal-to-noise ratio for high resolution imaging is modified, and the same analysis is followed through, then the Strehl ratio of the final image from the weighted exposures approach will be different. A number of different measures of signal-to-noise were tested, either utilising different ranges of spatial frequencies, or weighted proportionately with the image Strehl ratio. For all the weighting models tested, I also generated images with similar Strehl ratios using the simple exposure selection method without weighting. The images generated using exposure selection always gave similar signal-to-noise ratios to the images generated using exposure weighting. With faint reference stars the accuracy of the Strehl ratio measurements is dependent on the Strehl ratio itself, and the choice of optimum weighting function becomes very complex. The complexity of the various weighting models, their dependence on the numerous aspects of the observations which affect the accuracy of Strehl ratio measurements, and the increased computational requirements make this approach less favourable than simple exposure selection. The analyses in the remainder of this thesis will be restricted to exposure selection without weighting of the exposures.