Occlusions contribute to scaling in natural images

Spatial power spectra from natural images fall approximately as the square of spatial frequency, a property also called scale invariance (scaling). Various theories for visual receptive fields consider scale invariance key. Two hypotheses have been advanced in the literature for why natural images obey scale invariance. The first is that these images have luminance edges, whose spectra fall as frequency squared. The second is that scale invariance follows from natural images being essentially a collage of independent, constant-intensity regions, whose sizes follow a power-law distribution. Recently, an argument by example was made against the first hypothesis. Here we refute that argument and show that the first hypothesis is consistent with the scaling under a wide variety of distributions of sizes. There are two reasons for this: first, for every frequency, the log-log slope of the rotationally averaged power spectrum of an image is the weighted mean of the log-log slopes from the independent regions of the image formed by objects occluding one another. Second, the log-log slopes of the spectrum envelope for a constant-intensity region are 0 and -3 for frequencies corresponding to periods much larger and much smaller than the region's size, respectively. Therefore, it is not surprising that natural images have log-log slopes between -1.5 and -3, with a mean near -2.