Mean amplitude spectra

December 06, 2012
Natural scenes exhibit regularities in their spatial statistics. For example, the spatial frequency amplitude spectrum of natural scenes is well characterized by the function 1 / f^k, where f is spatial frequency and k is approximately 1. This characteristic spectrum arises from the scale invariance of spatial structures in natural scenes [1].
 
Graham et al. has shown that a sample of art images, both representational (realistic) and nonrepresentation (abstract) works, show similarities to a sample of natural scenes in terms of their spatial frequency amplitude spectra, but the art images and natural scenes have significantly different mean amplitude spectrum slopes k, as shown below [2]. When plotted on log-log coordinates, the 1 / f^k relationship between amplitude and spatial frequency corresponds to a straight line with slope equal to -k.
 
blog 2012_12_06
 
I measured the spatial frequency amplitude spectra of the photographs in this gallery. The results are shown below. The slope of the best-fit line to the amplitude spectra plotted on log-log coordinates is -1.26 (R^2 = 0.97) where is the correlation coefficient for the fit. The 1D spectra measurements are shown as dots and the mean is shown as a dashed curve.
 
blog 2012_12_06
 
To generate these 1D spectra, I extracted a centered square region of maximal size from each photograph, applied a Welsh apodization function and a 2D Fourier transform and measured mean amplitude within annular bins of width 0.15 log cycles per picture.
 
The amplitude spectrum of these photographs is also well modeled by the function 1 / f^k. With a slope of -1.26, these photographs are more similar to Graham et al.'s sample of art images than to the sample of natural scenes. The reason for this similarity is unclear. Graham et al. suggests that the spatial statistics of art images may be due in part to the nonlinear luminance compression required to utilize the small dynamic range available in art materials. Astronomical photography also requires nonlinear luminance compression.
 
The plot below shows the amplitude spectra of the gallery photographs without nonlinear luminance compression. The slope of the best-fit line is -0.69 (R^2 = 0.76). Nonlinear luminance compression, when applied to these photographs, steepens the amplitude slope (i.e. makes its slope more negative) and decreases its deviation from a 1 / f^relationship.
 
blog 2012_12_06
 
[1] Burton et al., "Color and spatial structure in natural scenes", Applied Optics, 26(1):157-170, 1987 January.
[2] Graham et al., "Statistics regularities of art images and natural scenes: Spectra, spareness and nonlinearities", Spatial Vision, 21(1-2):149-165, 2007.

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