Power Spectral Analysis

The first step in PSA is to Fourier transform the image and calculate the square modulus of the FT to generate the power spectrum, . So,

Here FT denotes the Fourier transform. The power spectrum generally has a very wide dynamic range so we rescale it for display with the following operations. First, to reduce the level of the DC component we reset it to equal the value of an adjacent frequency:

Next, we rearrange the power spectrum quadrant by quadrant so that the zero frequency is in the center of the array, the 1st quadrant is in the upper right, the second in the upper left, etc. Then we rescale the array to have values from 1.0 to 10.0 and take the base 10 logarithm, so the final array used for display, has the form

where and are the minimum and maximum of the arrays, respectively.

Display of this rescaled array with a false color display program such as SAOIMAGE or IDL allows observation of a range of sometimes subtle effects which relate to high resolution detail in the image. Working in frequency space has the advantage that the smallest scale features in the image are the largest scale in the power spectrum, so close binaries or image elongation may be immediately obvious in the display. Conversely, the large scale features in the image, the nonuniform background and the extensive wings in the point spread function affect mostly the low frequency region of the power spectrum, so they can be ignored in the inspection process.

An important application of PSA is to detect and characterize binary images. The power spectrum for a binary is just a cosine function whose frequency is proportional to the binary separation, whose modulation is related to the relative intensity of the two components, and whose fringes are perpendicular to the position angle of the binary. Since close binaries give widely spaced fringes, it is easy to fit the fringes to obtain these three parameters. After determining the position angle, one can average the power spectrum in a direction perpendicular to the direction of the fringes, increasing the SNR for the fit and reducing the fit to a one-dimensional operation. This allows accurate measurement even for low SNR data.

For resolved objects, one can fit other simple models to extract diameters. For example, the power spectrum of a disk just has the form and this may be modified to account for a limb-darkening function. If just the diameter of the object is of interest, one can azimuthally average the power spectrum into an array of increasing radius annuli, increasing the SNR for the fit. For asymmetric objects, one can fit an elliptical function to the power spectrum, giving two more extracted parameters (the ratio of minor to major axes of the ellipse, and its position angle).

In all cases, one can restrict the range of frequencies to which the fit is applied. Specifically, one excludes the very lowest frequencies where the large scale nonuniformities in the image have the greatest effect, and one limits the fit to a frequency range where the SNR is sufficiently high. In all cases, calibration of the fits are required by fitting to power spectra of unresolved stars with roughly the same SNR and from a relatively close field position. One can also use PSFs from a program such as Tiny TIM for this calibration, though using both is probably the safest option. Again, because one is doing a global fit, the detailed two-dimensional structure in the PSF, which is still poorly known, is not nearly as important as it is for image deconvolution.

Another operation that proves to be useful for many objects is to apply a spatial filter to the image before generating the power spectrum. One can perform a range of operations including simple extraction of regions of interest, subtraction of background, removal of cosmic rays, etc. These operations need to be performed very carefully to minimize their effect on measurements made in the power spectra.

Another interesting feature of PSA is that one can use it to detect resolved structure on scales small compared to the telescope diffraction limit and the camera sampling limit. The telescope diffraction limit is the frequency at which the telescope aperture transfer function drops to zero. However, objects which are somewhat smaller than the diffraction limit, or binaries with separations less than the diffraction limit, show significant drops in power in the high frequency region of their power spectra as compared to truly unresolved objects. Therefore, while the HST diffraction limit in the visible is only 40 mas (and the PC sampling limit is only 86 mas) stellar diameters and binary separations to a few mas could be estimated from high enough SNR data.