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

Abstract

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.

Viewing Art in Different Contexts

“…Images containing coarse spatial structures with lack of fine spatial detail are associated with low FD values, whereas images with high levels of intricate and fine spatial detail would have high FD values. FD is inversely related to the slope a of the Fourier amplitude spectrum (higher a values are equivalent to low FD values and vice versa) and the relationship between them has been both established mathematically and validated empirically (Graham and Field, 2008;Forsythe et al, 2011;Spehar and Taylor, 2013;Bies et al, 2016).…”

Computational and Experimental Approaches to Visual Aesthetics

“…Interestingly, the amplitude spectrum of many uncomfortable visual stimuli contains an excessive energy at medium spatial frequencies and thereby deviates from the linear spectral properties of natural scenes and images of artworks that are perceived as pleasant (Fernandez and Wilkins, 2008 ; O'Hare and Hibbard, 2011 ). The Fourier spectral slope of images correlates with measures of image complexity (Table S1 in Redies et al, 2017 ), in particular with the fractal dimension (Bies et al, 2016b ). A shallower slope indicates more power in the high-frequency part of the spectrum; consequently, the images show more fine detail and thus higher complexity.…”

Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique

“…Bies et al [31] proposed an indirect method based on spectral decay rate for the FD calculation of n-dimensional data. They presented a generalized relationship among FDs and power spectrum decay-rate parameters (β) using empirical validation.…”

“…The calculated FD of nine images using different methods and relative errors. Manual method, M1: Simple Canny edge detection, M2: This study, M3: Bies et al[31].…”