Statistical Invariances in Artificial, Natural, and Urban Images

Abstract

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 ensembles of artificial and real-world greyscale images to find different invariance properties: translation invariance by determining an average pair-correlation function, scale invariance by investigating the power spectrum and the coarse graining of the images, and a new hierarchical invariance recently proposed [D. L. Ruderman, Network 5, 517 (1994)]. The results of our work indicated that the assumption of translational invariance can be taken for granted. Our results concerning the scale invariance are qualitatively the same as those found by Ruderman and others. The deviations of the distributions of the logarithmically transformed images from a Gaussian distribution cannot be seen as clearly as stated by Ruderman. This results from the fact that for a correct determination of the deviations the non-linear transformation must be considered. Depending on the preprocessing of the images the results concerning the hierarchical invariance differed widely. It seems that this new invariance can be confirmed only for logarithmically transformed images.

A Survey of Image Statistics Relevant to Computer Graphics

“…The first two images are reasonable examples of natural images, whereas the third image is an example of an image taken in a built‐up area. Built environments tend to have somewhat different natural image statistics compared with natural scenes [ ZL98 ]. Figure taken from [ RKAJ08a ], courtesy AK Peters, Ltd.…”

Scaling of visual acuity with body size in mammals and birds

“…It has been suggested that natural scenes tend to be fractal (‘self‐similar’; Field 1987, 1989, 1993; Ruderman & Bialek 1994; Párraga et al . 1998; Ziegaus & Lang 1998) and that vertebrate visual systems are adapted to deal economically with such images (Swindale 1990; Van Essen, Anderson & Felleman 1992; Virsu & Hari 1996). When images are fractal, visual detail does not change appreciably with viewing distance (Mandelbrot 1983).…”