Fractal Dimension of Color Fractal Images With Correlated Color Components

The nature of the stable points of the general two-dimensional quadratic map is considered analytically, and the boundary equation of the first bifurcation of the map in the parameter space is given out. The general feature of the nonlinear dynamic activities of the map is analyzed by the method of numerical computation. By utilizing the Lyapunov exponent as a criterion, this paper constructs the strange attractors of the general two-dimensional quadratic map, and calculates the fractal dimension of the strange attractors according to the Lyapunov exponents. At the same time, the researches on the fractal images of the general two-dimensional quadratic map make it clear that when the control parameters are different, the fractal images are different from each other, and these fractal images exhibit the fractal property of self-similarity.

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Parametric cortical representations of complexity and preference for artistic and computer-generated fractal patterns revealed by single-trial EEG power spectral analysis

10.31234/osf.io/g3s7r ◽

2021 ◽

Author(s):

Eric Rawls ◽

Rebecca A. White ◽

Stephanie Kane ◽

Carl E. Stevens ◽

Darya Zabelina

Keyword(s):

Spectral Analysis ◽

Fractal Dimension ◽

Power Spectral Analysis ◽

Alpha Band ◽

Single Trial ◽

Power Spectral ◽

Subjective Preference ◽

Fractal Patterns ◽

Fractal Images ◽

Eeg Power

Fractals are self-similar patterns that repeat at different scales, the complexity of which are expressed as a fractional Euclidean dimension D between 0 (a point) and 2 (a filled plane). The drip paintings of American painter Jackson Pollock (JP) are fractal in nature, and Pollock’s most illustrious works are of the high-D (~1.7) category. This would imply that people prefer more complex fractal patterns, but some research has instead suggested people prefer lower-D fractals. Furthermore, research has suggested that parietal and frontal brain activity tracks the complexity of fractal patterns, but previous research has artificially binned fractals depending on fractal dimension, rather than treating fractal dimension as a parametrically varying value. We used white layers extracted from JP artwork as stimuli, and constructed statistically matched 2-dimensional random Cantor sets as control stimuli. We recorded the electroencephalogram (EEG) while participants viewed the JP and matched random Cantor fractal patterns. Participants then rated their subjective preference for each pattern. We used a single-trial analysis to construct within-subject models relating subjective preference to fractal dimension D, as well as relating D and subjective preference to single-trial EEG power spectra. Results indicated that participants preferred higher-D images for both JP and Cantor stimuli. Power spectral analysis showed that, for artistic fractal images, parietal alpha and beta power parametrically tracked complexity of fractal patterns, while for matched mathematical fractals, parietal power tracked complexity of patterns over a range of frequencies, but most prominently in alpha band. Furthermore, parietal alpha power parametrically tracked aesthetic preference for both artistic and matched Cantor patterns. Overall, our results suggest that perception of complexity for artistic and computer-generated fractal images is reflected in parietal-occipital alpha and beta activity, and neural substrates of preference for complex stimuli are reflected in parietal alpha band activity.

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Particle-Like Fractal Images for Testing Algorithms that Measure Boundary Fractal Dimension

Microscopy and Microanalysis ◽

10.1017/s1431927602104454 ◽

2002 ◽

Vol 8 (S02) ◽

pp. 1574-1575

Author(s):

D. S. Bright

Keyword(s):

Fractal Dimension ◽

Fractal Images ◽

Testing Algorithms

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FRACTAL DIMENSION ESTIMATION OF RGB COLOR IMAGES USING MAXIMUM COLOR DISTANCE

Fractals ◽

10.1142/s0218348x16500407 ◽

2016 ◽

Vol 24 (04) ◽

pp. 1650040 ◽

Cited By ~ 8

Author(s):

XIN ZHAO ◽

XINGYUAN WANG

Keyword(s):

Fractal Dimension ◽

Estimation Method ◽

Fractal Dimensions ◽

Color Images ◽

Natural Images ◽

Real World Data ◽

Dimension Estimation ◽

Fractal Images ◽

Partition Method ◽

High Degree

Natural images exhibit a high degree of complexity, randomness and irregularity in color and texture, however fractal can be an effective tool to describe various irregular phenomena in nature. Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately by means of experiments. In this paper, we proposed a fractal dimension estimation method for RGB color images. In the proposed method, we present a hyper-surface partition method which considers the hyper-surface as continuous and divide the image into nonoverlapped blocks. We also defined a counting method in color domain. To validate the proposed method, experiments were carried on two types of color images: synthesized fractal images and natural RGB color images. The experimental results demonstrate that the proposed method is effective and efficient. The behaviors of the proposed method on the rescale images are also shown in the paper. And it can be performed as a reliable FD estimation approach for the RGB color images.

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Fractal Dimension of Color Fractal Images

IEEE Transactions on Image Processing ◽

10.1109/tip.2010.2059032 ◽

2011 ◽

Vol 20 (1) ◽

pp. 227-235 ◽

Cited By ~ 65

Author(s):

Mihai Ivanovici ◽

Noël Richard

Keyword(s):

Fractal Dimension ◽

Fractal Images

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THE AESTHETICS AND FRACTAL DIMENSION OF ELECTRIC SHEEP

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408020975 ◽

2008 ◽

Vol 18 (04) ◽

pp. 1243-1248 ◽

Cited By ~ 8

Author(s):

SCOTT DRAVES ◽

RALPH ABRAHAM ◽

PABLO VIOTTI ◽

FREDERICK DAVID ABRAHAM ◽

JULIAN CLINTON SPROTT

Keyword(s):

Fractal Dimension ◽

Fractal Dimensions ◽

Iterated Function Systems ◽

Two Dimensional ◽

Computer Scientist ◽

Large Community ◽

Iterated Function ◽

Fractal Images ◽

The Relationship ◽

Aesthetic Judgments

Physicist Clint Sprott demonstrated a relationship between aesthetic judgments of fractal images and their fractal dimensions [1993]. Scott Draves, aka Spot, a computer scientist and artist, has created a space of images called fractal flames, based on attractors of two-dimensional iterated function systems. A large community of users run software that automatically downloads animated fractal flames, known as "sheep", and displays them as their screen-saver. The users may vote electronically for the sheep they like while the screen-saver is running. In this report we proceed from Sprott to Spot. The data show an inverted U-shaped curve in the relationship between aesthetic judgments of flames and their fractal dimension, confirming and clarifying earlier reports.

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Use of fractals to describe microstructures of porous thick-film platinum electrodes

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100121971 ◽

1992 ◽

Vol 50 (1) ◽

pp. 314-315

Author(s):

Steven D. Toteda

Keyword(s):

Fractal Dimension ◽

Power Plants ◽

Electrical Response ◽

Cubic Phase ◽

Platinum Electrodes ◽

Fine Grained ◽

Impedance Response ◽

Platinum Particles ◽

Energy Surfaces ◽

Highly Porous

Zirconia oxygen sensors, in such applications as power plants and automobiles, generally utilize platinum electrodes for the catalytic reaction of dissociating O2 at the surface. The microstructure of the platinum electrode defines the resulting electrical response. The electrode must be porous enough to allow the oxygen to reach the zirconia surface while still remaining electrically continuous. At low sintering temperatures, the platinum is highly porous and fine grained. The platinum particles sinter together as the firing temperatures are increased. As the sintering temperatures are raised even further, the surface of the platinum begins to facet with lower energy surfaces. These microstructural changes can be seen in Figures 1 and 2, but the goal of the work is to characterize the microstructure by its fractal dimension and then relate the fractal dimension to the electrical response. The sensors were fabricated from zirconia powder stabilized in the cubic phase with 8 mol% percent yttria. Each substrate was sintered for 14 hours at 1200°C. The resulting zirconia pellets, 13mm in diameter and 2mm in thickness, were roughly 97 to 98 percent of theoretical density. The Engelhard #6082 platinum paste was applied to the zirconia disks after they were mechanically polished ( diamond). The electrodes were then sintered at temperatures ranging from 600°C to 1000°C. Each sensor was tested to determine the impedance response from 1Hz to 5,000Hz. These frequencies correspond to the electrode at the test temperature of 600°C.

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