Scaling and correlation analysis of galactic images

Different scaling and autocorrelation characteristics and their application to astronomical images are discussed: the structure function, the autocorrelation function, Fourier spectra and wavelet spectra. The choice of the mathematical tool is of great importance for the scaling analysis of images. The structure function, for example, cannot resolve scales that are close to the dominating large-scale structures, and can lead to the wrong interpretation that a continuous range of scales with a power law exists. The traditional Fourier technique, applied to real data, gives very spiky spectra, in which the separation of real maxima and high harmonies can be difficult. We recommend as the optimal tool the wavelet spectrum with a suitable choice of the analysing wavelet. We introduce the wavelet cross-correlation function, which enables us to study the correlation between images as a function of scale. The crosscorrelation coefficient strongly depends on the scale. The classical cross-correlation coefficient can be misleading if a bright, extended central region or an extended disc exists in the galactic images. An analysis of the scaling and cross-correlation characteristics of nine optical and radio maps of the nearby spiral galaxy NGC 6946 is presented. The wavelet spectra allow us to separate structures on different scales like spiral arms and diffuse extended emission. Only the images of thermal radio emission and H alpha emission give indications of three-dimensional Kolmogorov-type turbulence on the smallest resolved scales (160-800pc). The crosscorrelations between the images of NGC 6946 show strong similarities between the images of total radio emission, red Light and mid-infrared dust emission on all scales. The best correlation is found between total radio emission and dust emission. Thermal radio continuum and Ha emission are best correlated on a scale of about 1 arcmin similar or equal to 1.6 kpc, the typical width of a spiral arm. On a similar scale, the images of polarized radio and H alpha emission are anticorrelated, a fact that remains undetected with classical cross-correlation analysis.