Abstract
The objective of this chapter is to help solve a classic stochastic problem using tools of the graphic environment. Stochastic processes are associated with the concepts of uncertainty or chance. They are a major focus of studies in various scientific disciplines such as mathematics, statistics, finance, artificial intelligence/machine learning, and philosophy. Visual Arts also depend on elements of uncertainly and chance. To explore the commonality of concern between Science and Art and better understand stochastic processes, the authors use a graph theory reference model called the “shortest route problem” and add additional elements specific to the art-making process to highlight the relevance of interdisciplinary studies in the field of randomness and visual perception.
TopBackground
A random process, also called a stochastic process, is a collection of random variables defined on an underlying probability space. The first comprehensive study of a stochastic process according to M. Scott (2013) is attributed to botanist Robert Brown who describes the physical trajectories of pollen grains suspended in water. Many later mathematical stochastic processes models have been developed in the context of studying Brownian motion. Mathematics is based on precision and rigorous proof. It offers Science both a foundation of truth and a standard of certainty. It is a science of pattern and order that uses observation and simulation as means of discovering truth. According to a National Research Council report (2000) it relies on logic to demonstrate it. The focus of mathematical sciences throughout history has been to explore the logic of shape, quantity and arrangement. Edoxus, Archimedes Euclid and Greek axiomatic geometry are the foundation on which classic mathematical theories have been developed. However, randomness does not analyze or investigate numbers but instead focuses on the characteristics of a sequence of digits. G. Chaitin (1975) posits that a sequence of numbers is random if it has no shorter description than itself. The ensemble theory of probability, to which the concept of random processes is attached, opens mathematical researches to broader and more complex investigations. As early as 1905, Albert Einstein, using a probabilistic model, provided a satisfactory explanation of the Brownian motion. From 1930 to 1960 J. L. Doob and Kolmogorov, transformed the study of probability to a mathematical discipline and set the stage for major developments in the theory of continuous parameter stochastic processes. “Probability is mathematics”, Doob clearly states in the preface of his 1953 book ‘Stochastic processes’.
Key Terms in this Chapter
Stochastic Problem: Stochastic problems are mathematical problems where some of the data incorporated into the objective is uncertain. Uncertainty is usually characterized by a probability distribution on the parameters.
Hue: Hue refers to the pure spectrum of colors, red, orange, yellow, blue, green violet. In visual art, all hues can be mixed from three basic hues: red, blue, yellow, known as primaries. When pigment primaries are all mixed together, the result is black.
Graph Theory: Graph theory is about the relationship between lines and points. A graph consists of some points and some lines between them. No attention is paid to the position of points and the length of the lines.
Random Processes: A random process models the progression of a system over time, where the evolution is random rather than deterministic. Random processes are used in a variety of fields including economics, finance, engineering, physics, and biology.
Volume: In the context of this paper, volume references scientific visualization, computer graphics rendering and various set of techniques used to display a 2D projection of a 3D object.
CIELAB: CIELAB is the second of two systems adopted by the CIE. CIE 1931 RGB and CIE 1931 XYZ color spaces are the first mathematically defined color spaces. The International Commission on Illumination (CIE) created them in 1931. CIELAB is an opponent color system based on the earlier system of Richard Hunter. Like all CIE models, it is device independent and is used for color management as the device independent model of the ICC (International Color Consortium) device profiles.
Chroma: Defines the strength or dominance of a hue and its saturation. Variations in chroma can be achieved by adding different amounts of a neutral gray of the same value to alter a color.
Neuroscience: Scientific study of the nervous system. In the context of this paper, Neuroscience finding in the study of brain mechanisms and neural representations in the human visual cortex helped define the parameter by which this work was completed.
Perception: Perception is the ability to understand external stimuli. Visual perception is the ability to detect light and interpret it as the perception known as sight or vision. Vision has a specific sensory system, the visual system. Because what people see is not simply a translation of retinal stimuli, it is the object of constantly evolving studies in the field of neuroscience.
Visual Communication: Visual Communication is a multi-disciplinary field encompassing graphic design, illustration, fine arts (like drawing and painting), multi-media, and photography. Visual Communication, applies the fundamentals of major art forms and art techniques to solve communication problems.
Value: Value defines the intensity of a color in term of lightness or darkness. It helps artists and designers to define form and creates spatial illusions on a two-dimensional surface. Contrast of value separates objects in space; gradation of value suggests mass and contour of a contiguous surface.
Recursive Thinking: The process of solving large problems by breaking them down into smaller, simpler problems that have identical forms.