Statistical invariances in artificial, natural, and urban images

To answer the question about the way our visual system processes images it has to work with every day, it is necessary to investigate the statistical structure of these pictures. For this purpose we investigated several ensem bles of artificial and real-world greyscale images to find different invari ance properties: translation invariance by determining an average pair-corr elation function, scale invariance by investigating the power spectrum and the coarse graining of the images, and a new hierarchical invariance recent ly proposed [D. L. Ruderman, Network 5, 517 (1994)]. The results of our wor k indicated that the assumption of translational invariance can be taken fo r granted. Our results concerning the scale invariance are qualitatively th e same as those found by Ruderman and others. The deviations of the distrib utions of the logarithmically transformed images from a Gaussian distributi on cannot be seen as clearly as stated by Ruderman. This results from the f act that for a correct determination of the deviations the non-linear trans formation must be considered. Depending on the preprocessing of the images the results concerning the hierarchical invariance differed widely. It seem s that this new invariance can be confirmed only for logarithmically transf ormed images.