Generation of Fractals using Polar Coordinates

The article describes an algorithm for generating fractals using polar coordinates. The classic Julia and Mandelbrot polynomial iteration applied to a complex number is replaced by an iteration with separate functions for distance and angle. A polynomial function is used for an angle and a power function for a distance. Varying the functions parameters allows to create a wide range of attractive pictures. Distance values ​​are used for coloring fractal images.

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

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.

FAMILY OF SHAPE PRESERVING FRACTAL-LIKE BÉZIER CURVES

Subdivision schemes generate self-similar curves and surfaces for which it has a familiar connection between fractal curves and surfaces generated by iterated function systems (IFS). Overveld [Comput.-Aided Des. 22(9) (1990) 591–597] proved that the subdivision matrices can be perturbated in such a way that it is possible to get fractal-like curves that are perturbated Bézier cubic curves. In this work, we extend the Overveld scheme to [Formula: see text]th degree curves, and deduce the condition for curvature continuity and convex hull property. We find the conditions for positive preserving fractal-like Bézier curves in the proposed subdivision matrices. The resulting 2D/3D curves from these binary subdivision matrices resemble with fractal images. Finally, the dependence of the shape of these fractal-like curves on the elements of subdivision matrices is demonstrated with suitably chosen examples.

Fixed Points, Symmetries, and Bounds for Basins of Attraction of Complex Trigonometric Functions

The dynamical systems of trigonometric functions are explored, with a focus on tz=tanz and the fractal image created by iterating the Newton map, Ftz, of tz. The basins of attraction created from iterating Ftz are analyzed, and some bounds are determined for the primary basins of attraction. We further prove x- and y-axis symmetry of the Newton map and explore the nature of the fractal images.

Visual Exploration of Omni-Directional Panoramic Scenes

How do we explore the visual environment around us, and how are head and eye movements coordinated during our exploration? To investigate this question, we had observers look at omni-directional panoramic scenes, composed of both landscape and fractal images, using a virtual-reality (VR) viewer while their eye and head movements were tracked. We analyzed the spatial distribution of eye fixations and the distribution of saccade directions; the spatial distribution of head positions and the distribution of head shifts; as well as the relation between eye and head movements. The results show that, for landscape scenes, eye and head behaviour best fit the allocentric frame defined by the scene horizon, especially when head tilt (i.e., head rotation around the view axis) is considered. For fractal scenes, which have an isotropic texture, eye and head movements were executed primarily along the cardinal directions in world coordinates. The results also show that eye and head movements are closely linked in space and time in a complementary way, with stimulus-driven eye movements predominantly leading the head movements. Our study is the first to systematically examine eye and head movements in a panoramic VRenvironment, and the results demonstrate that a VR environment constitutes a powerful and informative research alternative to traditional methods for investigating looking behaviour.

Turning the (virtual) world around: patterns in saccade direction vary with picture orientation and shape in virtual reality

Research investigating gaze in natural scenes has identified a number of spatial biases in where people look, but it is unclear whether these are partly due to constrained testing environments (e.g., a participant with their head restrained and looking at a landscape image framed within a computer monitor). We examined the extent to which image shape (square vs. circle), image rotation, and image content (landscapes vs. fractal images) influenced eye and head movements in virtual reality (VR). Both the eyes and head were tracked while observers looked at natural scenes in a virtual environment. In line with previous work, we found a bias for saccade directions parallel to the image horizon, regardless of image shape or content. We found that, when allowed to do so, observers move both their eyes and head to explore images. Head rotation, however, was idiosyncratic; some observers rotated a lot, while others did not. Interestingly, the head rotated in line with the rotation of landscape, but not fractal images. That head rotation and gaze direction respond differently to image content suggests that they may be under different control systems. We discuss our findings in relation to current theories on head and eye movement control, and how insights from VR might inform more traditional eye-tracking studies.

An Edge Detection Approach for Fractal Image Processing

Research in the field of fractal image processing (FIP) has increased in the current era. Edge detection of fractal images can be considered as an important domain of research in FIP. Detecting edges in different fractal images accurate manner is a challenging problem in FIP. Several methods have introduced by different researchers to detect the edges of images. However, no method works suitably under all conditions. In this chapter, an edge detection method is proposed to detect the edges of gray scale and color fractal images. This method focuses on the quantitative combination of Canny, LoG, and Sobel (CLS) edge detection operators. The output of the proposed method is produced using matrix laboratory (MATLAB) R2015b and compared with the edge detection operators such as Sobel, Prewitt, Roberts, LoG, Canny, and mathematical morphological operator. The experimental outputs show that the proposed method performs better as compared to other traditional methods.