Transformations

For spectra a transformation from wavelength to frequency is used to
identify spectral regions which follow a power law. This
transformation is given by

where is the frequency and is wavelength. The intensities

In the classification of galaxies, ellipticals can be distinguished on
their radial surface brightness profile which can be approximated
by
.
This gives the transformation formula

Since the intensity profile only should be resampled, the flux correction is not applied in this case. A logarithmic scale is also used here to achieve a linear relation. An example is given in Figure 2.9.

Whereas the transforms mentioned above only perform a non-linear
rebinning of the data, the Fourier transform converts a spatial image
into the frequency domain. This transform has two main applications
namely analysis of periodic phenomena and convolution due to its
special properties. The transform can be given as

where (

Special numeric techniques, called Fast Fourier Transforms or FFT, can significantly decrease the time needed to compute these transforms. They are optimized for images with a size equal to a power of 2 (see e.g. Hunt 1969) but can also be used in other cases.

To analysis the periodic occurrence of features in time series,
spectra, or images the amplitude of the Fourier transform or the power
spectrum is used. The power spectrum
of the function *F* is
given by

Peaks in indicate the presence of specific frequencies while the continuum originates from a combination of objects and noise. Since the Fourier transform assumes the image to occur periodically, discontinuities at the edges of the image will produce artificial contributions to the power spectrum. Thus, care should be taken to remove systematic background variations to avoid this happening. As an example of using Fourier transforms to extract information from a frame, an azimuthal intensity profile of the spiral galaxy A0526-16 is shown in Figure 2.10.

The radius was chosen so that the profile intersects the spiral pattern in the inner parts of the galaxy. In the amplitude plot of the Fourier transform, it can be seen that the spiral pattern mainly contains even frequency components. The 4

It is possible to use the transformation for convolutions because this
operation corresponds to a multiplication in the frequency domain :

where and denote the Fourier transform and convolution operators, respectively. Especially when the convolution matrix is large it is more efficient to perform the operation in frequency than in spatial domain.