Fourier filtering

For all three observing runs at the NOT the pixel sampling used was sufficient to record Fourier spatial frequencies higher than the telescope diffraction limit of $\frac{d}{\lambda}$ in certain orientations with respect to the pixel array. As the photon shot noise and multiplication register noise are stochastic they contribute equally at all spatial frequencies. The spatial frequencies beyond the diffraction limit of the telescope will contain no signal from the astronomical source but as much photon shot noise and multiplication register noise as are found in lower spatial frequencies. By suppressing spatial frequencies beyond $\frac{d}{\lambda}$ in the data it is thus possible to obtain an improvement in the overall signal-to-noise ratio of the images. This will lead directly to an improvement in the limiting magnitude of reference star which can be used, and in the limiting magnitude of faint source which can be detected in the field.

The readout noise from the analogue camera electronics is distributed at a range of spatial frequencies within the images. A significant fraction of this noise appears at spatial frequencies beyond the telescope diffraction limit, and this component of the readout noise is also reduced if high spatial frequencies are suppressed.

There are two clear applications for Fourier filtering in my approach to the data reduction, as listed in Table 5.6.

Table 5.6: The two applications of Fourier filtering in the revised data-reduction scheme.

Application	Description	Filter used
1	The measurement of the Strehl ratio and position of the brightest speckle in each of the short exposures	Diffraction-limited modulation transfer function
2	For the production of a high signal-to-noise image from the re-centred selected exposures	Modified Hanning window

In the first application we want to find bright speckles in the noisy short exposure images. A very effective approach is to search for the peak cross-correlation between the short exposure image and a diffraction-limited telescope PSF. This cross-correlation process is equivalent to multiplying the images by the modulation transfer function of the diffraction-limited telescope in the Fourier domain (effectively convolving the image with a diffraction-limited telescope PSF). In order to minimise the additional computation required, this was implemented during the sinc-resampling process, which is also performed in the Fourier domain. The model for the modulation transfer function which I used was calculated from the autocorrelation of the simple model of the NOT aperture described by Figure 3.4, and I have included a graphical description of the transfer function in Figure 5.6. I used the same geometrical approach for calculating this function as was used for the autocorrelation of two circles in Appendix A (see Equation A.15).

Figure 5.6: The modulation transfer function for a diffraction-limited telescope (this modulation transfer function corresponds to the amplitude of the Fourier transform of the diffraction-limited PSF shown in Figures 3.8b--d). On the left is a greyscale plot of the modulation transfer function for the simple model of the NOT aperture shown in Figure 3.4, with white corresponding to unity and black corresponding to zero in the function. The red circle has a radius of $\frac{d}{\lambda}$ corresponding to the diffraction-limit of the telescope aperture at a wavelength of $\lambda =0.86$ $\mu m$. The green box corresponds to the Nyquist limit of the CCD pixel array for $40\times 40$ $mas$ pixels - no Fourier components were recorded beyond this limit. The origin of the spatial frequency domain is at the centre. A horizontal cross-section along the blue line is shown on the right.

After the filtering and resampling have been completed, the peak in the resampled image corresponds to the most likely location of the brightest speckle. The height of the peak provides a measure of the flux in the brightest speckle, and hence the Strehl ratio. The measured Strehl ratio and position would then be used to select and re-centre the sinc-resampled (but unfiltered) exposures. In the high signal-to-noise regime the peak flux in the filtered short exposures was related to the peak flux in the original exposure by a non-linear, monotonically increasing function. The non-linearity of this function does not introduce complications, however, as the measurements made on the filtered images are simply used to sort the exposures according to their quality, and then to re-centre the selected exposures. Strehl ratios quoted in the text are based on measurements of the brightest pixel in the final Lucky Exposures image.

In the second application of filtering described in Table 5.6, only spatial Fourier components beyond $\frac{d}{\lambda}$ can be suppressed without blurring the astronomical image. However, the dynamic range of sinc-resampled images is limited by Gibb's phenomena, which can be suppressed only if the spatial Fourier components of the image are smoothly reduced to zero below the Nyquist cutoff spatial frequency of the CCD pixel array. A two dimensional Fourier filter was developed based on the Hanning window which attempted to suppress both the noise and Gibb's phenomena whilst minimising the blurring of the image. The filter function was flat-topped but dropped smoothly to zero before reaching the Nyquist frequency in all orientations, and also dropped smoothly to zero at spatial frequencies beyond $\frac{d}{\lambda}$ in orientations where $\frac{d}{\lambda}$ was lower than the Nyquist frequency ($\lambda$ was taken as the centroid of the observing band). A greyscale representation of the filter and cross-section are shown in Figure 5.7. The value of this filter function $F_{N}$ can be defined in polar coordinates as follows:

$$F_{N}\left ( r,\theta \right )= \left\{ \begin{array}{ll} ... ...{if $r\ge r_{c}\left ( \theta \right )+w$} \end{array}\right.$$ (5.1)

where $r_{c}\left ( \theta \right )$ was the spatial frequency beyond which Fourier components should be suppressed, $w$ was the width of region over which the filter function dropped smoothly to zero and $\theta$ and $r$ describe a polar coordinate system with the origin at zero spatial frequency. The cut-off spatial frequency $r_{c}\left ( \theta \right )$ is defined in terms of $\theta$ because the filter must drop to zero sooner in some orientations due to the Nyquist sampling of the detector (following the rules described earlier and as shown in Figure 5.7). A width $w$ corresponding to one-fifth of the Nyquist sampling frequency was used for the analyses presented here. For observing runs using CCDs with pixels which were not square, the Nyquist sampling frequency of the CCD was different in the horizontal and vertical directions, and the filter function used was modified appropriately obeying the same rules.

Figure 5.7: Filter function used to suppress noise. The function is based on a Hanning window but has a flat top. A linear greyscale plot is shown on the left with white corresponding to unity and black corresponding to zero. The function dropped smoothly to zero before reaching the Nyquist frequency for the CCD pixel array in all orientations. The Nyquist frequency is indicated by the green box. The function also dropped smoothly to zero at spatial frequencies beyond the diffraction limit of $\frac{d}{\lambda}$ in orientations where this was lower than the Nyquist frequency. Spatial frequencies of $\frac{d}{\lambda}$ are indicated in the plot by the red circle. A horizontal cross-section along the blue line is shown on the right.

The two applications of spatial filtering mentioned above were incorporated into my approach to the data reduction relatively straightforwardly. In both applications the filtering was implemented in the Fourier domain at the same time as the sinc-resampling of the relevant images. A modified version of the data reduction flow chart of Figure 3.13 incorporating the filtering is shown in Figure 5.8.

Figure 5.8: Flow chart describing the data reduction method.