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Open AccessArticle

Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals

1
Department of Psychology, University of Oregon, Eugene, OR 97405, USA
2
Department of Physics, University of Oregon, Eugene, OR 97405, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Marco Bertamini
Symmetry 2016, 8(7), 66; https://doi.org/10.3390/sym8070066
Received: 11 April 2016 / Revised: 7 June 2016 / Accepted: 12 July 2016 / Published: 19 July 2016
(This article belongs to the Special Issue Symmetry in Vision)
Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D) and power spectrum decay rate (β). Although a relationship between these measures has been derived mathematically, empirical validation across measurements is lacking. Here, we determine the relationship between D and β for 1- and 2-dimensional fractals. We find that for 1-dimensional fractals, measurements of D and β obey the derived relationship. Similarly, in 2-dimensional fractals, measurements along any straight-line path across the fractal’s surface obey the mathematically derived relationship. However, the standard approach of vision researchers is to measure β of the surface after 2-dimensional Fourier decomposition rather than along a straight-line path. This surface technique provides measurements of β that do not obey the mathematically derived relationship with D. Instead, this method produces values of β that imply that the fractal’s surface is much smoother than the measurements along the straight lines indicate. To facilitate communication across disciplines, we provide empirically derived equations for relating each measure of β to D. Finally, we discuss implications for future research on topics including stress reduction and the perception of motion in the context of a generalized equation relating β to D. View Full-Text
Keywords: fractal patterns; scale-invariance; fractal dimension; spectral scaling; midpoint displacement; Fourier noise; Fourier decomposition fractal patterns; scale-invariance; fractal dimension; spectral scaling; midpoint displacement; Fourier noise; Fourier decomposition
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Bies, A.J.; Boydston, C.R.; Taylor, R.P.; Sereno, M.E. Relationship between Fractal Dimension and Spectral Scaling Decay Rate in Computer-Generated Fractals. Symmetry 2016, 8, 66.

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