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 which 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 harmonics 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 to study the correlation between images as a function of scale. The cross-correlation coefficient strongly depends on the scale. The classical cross-correlation coefficient can be misleading if a bright, extended central region or an extended disk exists in the galactic images. An analysis of the scaling and cross-correlation characteristics of 9 optical and radio maps of the nearby spiral galaxy NGC6946 is presented. The wavelet spectra allow to separate structures on different scales like spiral arms and diffuse extended emission. Only the images of thermal radio emission and Hα emission give indications of 3-dimensional Kolmogorov-type turbulence on the smallest resolved scales (160–800 pc). The cross-correlations between the images of NGC6946 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 Hα emission are best correlated on a scale of about 1 ≃ 1.6 kpc, the typical width of a spiral arm. On a similar scale, the images of polarised radio and Hα emission are anticorrelated, which remains undetected with classical cross-correlation analysis.