Random Processes and Visual Perception: Stochastic Art

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Random Processes and Visual Perception:

Stochastic Art Jean Constant Hermay.org ABSTRACT The object of this chapter is to explore in visual terms a model of recursive thinking applied to a stochastic problem. 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 Art too depends on elements of uncertainty and chance. To explore the commonality of concern between Science and Art and better understand stochastic processes, I used a graph theory reference model called the ‘shortest route problem’ and added additional elements specific to the art-making process to highlight a specific occurrence of randomness in visual perception. Keywords: recursive thinking, stochastic problem, graph theory, art, visual communication, perception.

INTRODUCTION Randomness by nature is challenging to define and is often associated with unpredictability. The word stochastic is synonymous with “random.” It is of Greek origin and means "pertaining to chance.” The term stochastic art was used to differentiate arts practices such as medicine or rhetoric, in which the knowledge and skill of the practitioner could not be measured simply by the results of their work. The relationship between Mathematics and randomness has always been complex because of the nature of the concept of randomness. Every non-mathematical probabilistic assertion suggests a mathematical counterpart that sharpens the formulation of the non-mathematical assertion and may also have independent mathematical interest (Snell, 1997). Greek axiomatic geometry explores the logic of shape, quantity and arrangement. Mathematicians Richard Courant and Herbert Robbins stated that Mathematics offers Science both a foundation of truth and a standard of certainty based on precision and rigorous proof (Courant & Robbins, 1996). However, more recently, the theory of probability, to which the concept of random processes is attached, opened mathematical research to broader and more complex investigation in the area of applied mathematics, mathematical physics, mathematical biology, control theory, and engineering. There have been some attempts at developing a ‘stochastic movement’ in the 50s and 60s. It was still pre-technology time. Indeed, it would have indeed required tremendous amount of work and dedication to follow on the track of Xenakis or Stockhausen; few were willing to take that route. That’s what motivated me to take a second look into the problem and approach it from a different angle this time. The artists control the process as much as they can and integrate elements of randomness based on scientific data relating to physiology and perception.

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In the visual arts, the perception and appreciation of an artwork depends also on random elements pertaining to light, optical alertness, and various other physical and cultural parameters. To illustrate the mathematical concept of randomness in visual terms, I selected a model used by professor of Management Science Evan D. Porteus for a demonstration of stochastic random processes calculation (Porteus, 2002). I broke down each element of the model (or numbers) into separate objects and recombined them according to the scientific narrative while adding distinct components pertaining to visual communication methodology. Finally, to insure the validity of the process, I informally tested the results with colleagues from the scientific and artistic communities to underscore the common interest that joins scientific and artistic research in this field and gather possible direction for future interdisciplinary collaboration that will benefit from the study of probability.

BACKGROUND A random process, also called a stochastic process is a collection of random variables defined on an underlying probability space. The study of stochastic processes was first attributed to botanist Robert Brown who described the physical trajectories of pollen grains suspended in water. Much of the mathematical models on stochastic processes were developed in the context of studying Brownian motion (Scott, 2013). Mathematics offers Science both a foundation of truth and a standard of certainty. It is a science of pattern and order that uses observation, simulation as means of discovering truth, and relies on logic to demonstrate it (National Research Council, 2000). Mathematics is based on precision and rigorous proof. Its focus throughout history has been exploring the logic of shape, quantity and arrangement. Edoxus, Archimedes, Euclid, and Greek axiomatic geometry were the foundations on which classic mathematical theories were developed. The definition of randomness does not analyze or investigate numbers but focuses on the characteristics of the sequence of digits. One definition is that sequence of numbers is random if it has no shorter description than itself (Chaitin, 1975). The ensemble theory of probability, to which the concept of random process is attached, opened mathematical researches to broader and more complex investigation. In 1905 Albert Einstein, using a probabilistic model, provided a satisfactory explanation of the Brownian motion. From 1930 to 1960 J. L. Doob and A. N. 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 stated in the preface of his 1953 book “Stochastic processes” (Doob, 1953). Art is an intuitive expression of the perceptual environment. It is also significant medium through which the understanding and dynamic of probability can be further developed. The turn of the XX century saw many artists reevaluate the framework and channels of communication in which they operate and integrated probability in new artistic forms, both as producers of artistic material and to open new dimension of esthetic appreciation for the audience their work was intended to be created for. The musical field in particular, maybe because it relates more closely to Mathematics through its system of notation and its exploration of acoustic, vibration, sound waves and other physical elements, opened the door to original and innovative exploration of randomness in art. German composer Karl-Heinz Stockhausen’s series of Klavierstück (compositions for piano) (Henck, 1976), Iannis Xenakis (1971), and John Cage’s indeterminacy (Kohler, 2014) investigated the effect of probability theory and stochastic phenomena in their compositions, with statistical theory and Markov chains computation. In the graphic environment printers were among the first to devise technical means by which they could increase the sharpness of any given material (DiNicolantonio, 2014). Traditionally they used halftone screens that produce a dot pattern. Large dots cover the paper with ink to create dark areas, small dots represent the light areas. If the dots are small enough, they seems to disappear and producing a smooth

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gray tone because the human eye is not sensitive enough to see them. In stochastic printing, images are printed by dots spread randomly throughout the image area. The dots are not equally spaced and aligned in a row or grid and they vary according to the tonal value to be reproduced. The lighter areas have few dots, the darker areas have more dots (Fig.1).

Figure 1. Stochastic (left) and conventional (right) printing. (© 2014, J. Constant. Used with permission). [Figure 1 about here]

Randomness is also part of the artistic landscape. The Drip Paintings of Jackson Pollock can be seen as the precursors of artworks where randomness is an inherent component of the creative process. From still image to moving images, this concept was further exploited in film animation combining stochastic paintbrush transformation and motion detection by the like of Levente Kovács and Tamás Szirányi for the Department of Image Processing and Neurocomputing of the University of Veszprém, and the Hungarian Academy of Science (Kovács & Szirányi, 2002). Other qualifying attempt at studying randomness in visual art have tapped neuroscience advance in the studies of subconscious perception, optics, after effect images to develop deeper and more precise

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understanding of the image recognition process. Computer technology has added to the artist’s palette a sophisticated tool to express better-defined visual statements and integrate in the creative process objective elements defined by the physiology of perception and how the interpretation of complex messages shapes and light components directly affect the interpretation of visual messages. Neuroscientist Rob Levy’s series of plates on double stochastic process is a good example of the combination of Science, computerization and artistic rendering on the theme of randomness and visual perception (Levy, 2013).

RANDOMNESS AND VISUAL COMMUNICATION The Shortest route problem To illustrate the relationship between mathematics and the visualization of a stochastic process, I selected a series of models used for a demonstration on the subject by Dr E. Porteus from Stanford University called “The shortest route problem.” The shortest route problem finds its origin in graph theory. More than 200 recent publications on the subjects in mathematics, statistic and physics demonstrate the interest of all major fields of science in exploring random processes and use the methodology to model a broad range of problems going from robot navigation and control of non-deterministic systems to stochastic game-playing and planning under uncertainty and partial information (Bonet & Geffner, 2000). In this example, Dr Porteus approaches the random process from the recursive perspective and decomposes a complex problem into a series of smaller problems, after the Bellman principle called dynamic programming. Porteus’ demonstration breaks the problem into two parts: first, a study of a recursive model, based on a plan of 4 horizontal and 4 vertical squares (Fig. 2A), and second, a study of a similar surface incorporating elements of uncertainty, represented by circles. Square nodes are called decision nodes; circle nodes are named chance nodes (Fig. 2B). Figure 2. The shortest route problem. A) Problem solving by simple recursive calculation. B) Introduction of element of uncertainty defined as circles on the template. (© 2014, J. Constant. Used with permission). [Figure 2 about here]

The shortest route problem visualization Mapping the visualization

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I recreated Dr Porteus’ “Shortest route” model in a graphic editor software program and traced the outline of the 4 succeeding templates illustrating the mathematical reasoning process in a vector-based application to insure line sharpness and clear definition of each individual object. I drafted the initial layout in black and white design to focus on the object placement and shape dynamic. I created 3 different copies of the same pattern in 3 different sizes to draw attention to the recursive aspect of the demonstration (Fig. 3A). The resulting surfaces were allocated a distinct identity and set apart from each other by filling them with multiple variation of gray. Each set of square/circles objects was given specific parameters of shades and texture to differentiate one from another (Fig. 3B). Finally, I slightly turned the surface of the main board at a 30° angle on the ‘problem solved’ template to emphasize Dr Porteus’ recommendation to approach the problem in terms of arc rather than straight line to calculate the shortest route (Fig. 3C). Figure 3. Three different copies of the same pattern. A) B&W outline. B) Separating the surfaces. C) Tilting the central board (© 2014, J. Constant. Used with permission). [Figure 3 about here]

Randomness and visual perception To insert an additional element of randomness into the visual statement, I used a variation of an afterimage effect first discovered by German scientist L. Hermann in 1870, which highlights the inhibition that neighboring neurons in the brain pathways have on each other (Fig. 4A). I increased the size of square #1 (left-center line) of the final model “shortest route problem solved” by 5 points and slightly changed the opacity of color to prompt the eye to subjectively assert what is the minimum distance solution of the stochastic problem. The ensuing design induces the eye to fill black space with white dots and cover white dots black. This effect varies depending on various parameters having to do with visual perception, alertness of the nervous system as well as in this specific case, inference of color size and shape elements (Fig. 4B). Figure 4. A) The Hermann grid. Dark patches appear in the section crossings, except the ones that the viewer is directly looking at. B) The shortest route problem – volumes & visual cues. (© 2014, J. Constant. Used with permission). [Figure 4 about here]

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Color scheme I transferred the design into a pixel-based environment to add color to the composition. For that purpose, I selected a specific palette based on Korean traditional color and used the finding of a survey conducted for the School of Design of the University of Leeds by the Changwon National University (Shin, Westland, Moore, & Cheung, 2012) to determine the spatial positioning and dynamic interaction of each color in the overall composition (Fig. 5). Figure 5. Shin-Westland spatial color perception survey: preferred (circles), traditional (squares) and trend (diamond) colors in CIELAB space. (© 2014, J. Constant. Used with permission). [Figure 5 about here]

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I did several tests to adjust the Shin Westland color scheme to the existing black and white set up. One issue commonly known to designers using color has to do with the unintentional effects caused by the interaction of contrasting colors. It can be visually distracting and diminish the readability of the composition. Our perception of hues, values, and chroma depends upon their interaction with adjacent hues, values and chromas. Also the value of a background color affects the perception of all other colors values: a color looks lighter on a dark background, darker on a light background. The size of an area of color appears to change, depending on its value contrast with its background color. I adjusted the levels of contrast between the hues, values, chromas of its various areas of color size and placement of areas of color to achieve balance among them and be consistent with the main purpose of the work which had to do with readability and balance to encourage both an esthetic and educational experience. Figure 6. The ‘Shortest route problem solved’. Final composition, color scheme. (© 2014, J. Constant. Used with permission). [Figure 6 about here]

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EVALUATION AND RECOMMENDATION I built a multimedia short film recreating the build-up steps it took to complete the artwork and posted it online (Constant, 2014). I also conducted an informal test with several colleagues, mathematicians, architects and art historians. The consensus on both venues was that once the purpose was explained, it made the problem solving challenging, intriguing, but easier and fun to conduct. As a visual statement, several mentioned a similarity to the kinetic and op art, which also deal with issues of light, color, and random after-images effect. It is worth mentioning I encountered significant problems in the course of this project. Introducing what was for me a new and foreign color thematic palette was extremely challenging at first. It made me realize how deeply art creators are conditioned by their cultural, professional background and physical location when it comes to solve new challenges relating to visual communication issues relating volumes, color, location, and space dynamic. Also in regard of the object of this exercise, it was extremely difficult to be observant and patient and develop a consistent and meaningful picture. Fortunately I was able to rely on mechanical data and computer aided technology to confirm what my eyes could not comprehend. Only by testing the results with an outside help was I able to bridge perception, intuition, and correctness to complete my goal, which was an objective statement that we all can comprehend and react to in a similar way regardless of background or physical location

FUTURE RESEARCH DIRECTIONS The objective of this project was to construct a model that meet classic visual communication requirements, be educational in terms of presenting complex mathematical theories in a different context – and create a work of art that if anything, would intrigue, entertain and somehow open the dialog between the viewer and the creator based on an experience all can appreciate.

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Computer aided technology helped me get a better, more accurate control of the space and color parameters that constitute a visual statement. Resource and finding from the field of neuroscience, in particular in neural physiology and optic were also a determining factor in helping the decision making process and minimize any subjective interference that could affect the outcome of the design. As a visual artist, the recurring thread binding the stages of this project development emphasizes collaboration and cross pollination as a mean to develop better, more all-encompassing statements that can be interpreted in their own right in many different circumstances. One may raise the concern that computer aided technology or mechanical means may take over any creative impulse – which I doubt after completing this series. As long as the mind direct the project, all tools available come to the service of the goal. Mechanical can’t create by themselves self-standing answer to any given problem. Science gives us a better deeper understanding of processes, and provides an objective acuity that all artists and visual communicators can put to good use to reach a set goal. It also opens the door to new esthetic expression and avenues that were not previously available and grounded on solid, proven foundations. It also put the light on the axiomatic situation of the science researcher or the artist as a lone individual, unconnected and isolated in the object of his research. As access to information becomes more prevalent in today’s environment, it becomes undeniable that whether in Science or in Art, strong collaborative efforts lead to deeper, more meaningful statements.

CONCLUSION Applications of mathematics in the field of random processes have emerged across the landscape of natural, behavioral, and social sciences, from medical technology to economic planning (input/output models of economic behavior), to genetics and geology (locating oil reserves). More academic and scientific studies will ensure a better comprehension of the process. It will also help artists get a better handle on the tools they use, to convey their concepts, and provide the viewer with a uniquely distinct aesthetic experience.

REFERENCES Bonet, B., & Geffner, H. (2000). Planning with incomplete information as heuristic search in belief space. Proceedings of AIPS-2000, Breckenridge, CO, USA. Chaitin, G. J. (1975). Randomness and Mathematical Proof. Scientific American 232(5), 47-52. Constant, J. (2014). Stochastic Art. Retrieved June 12, 2014, from: http://www.hermay.org/jconstant/stochasticart/ Courant R. and Robbins H. (1996). What Is Mathematics? Second Edition Oxford University Press. DiNicolantonio, J. (2014). Stochastic vs. Conventional Printing. Millcraft Paper Company, Retrieved June 10, 2014, from http://millcraft.blogspot.com/ Doob, J. L. (1953). Stochastic Processes. John Wiley & Sons. Henck, H. (1976). Karlheinz Stockhausens Klavierstück IX: Eine analytische Betrachtung. Bonn-Bad Godesberg: Verlag für Systematische Musikwissenschaft.

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Kohler, E. (2014). Indeterminacy, an archive of John Cage's one-minute stories from Indeterminacy. Retrieved June 12, 2014, from http://www.lcdf.org/indeterminacy Kovács, L., & Szirányi, T. (2002). Creating animations combining stochastic paintbrush transformation and motion detection. 16th International Conference on Pattern Recognition Proceedings (Volume 2), IEEE online. Levy, R. (2013). Doubly Stochastic Processes. Retrieved June 12, 2014, from http://scientificabstractart.com/2013/04/30/rob -levy-doubly-stochastic-processes/ National Research Council (2000). Everybody Counts: A Report to the Nation on the Future of Mathematics Education, National Academy Press. Washington DC. Porteus, E. L. (2002). Foundations of Stochastic Inventory Theory. Stanford University Press. Scott, M. (2013). Applied Stochastic Processes in Science and Engineering. University of Waterloo Press. Shin M. J., Westland, S., Moore, E., & Cheung, V. (2012). Colour preferences for traditional Korean colors. Journal of the International Colour Association vol. 9. Snell, J. L. (1997). A conversation with Joe Doob. Statistical Science 12(4). Xenakis, I. (1971). Formalized Music: Thought and Mathematics in Composition. Bloomington and London: Indiana University Press.

ADDITIONAL READING SECTION Mathematics Cover, T. M. and Thomas, J. A. (1991). Kolmogorov Complexity. John Wiley & Sons. Newman, C. M. (2006). The work of Wendelin Werner. Courant Institute of Mathematical Sciences. Ren, Y., & Hu Lanying. (2012). Backward doubly stochastic differential equations driven by Levy Processes. Retrieved from http://www.paper.edu.cn Zhou, X. W., & Li, D. (2000). A Stochastic LQ Framework. Applied Math- Optimization. Springer-Verlag Music Irizarry, R. (2001). Musical and Stochastic Processes. Retrieved September 15, 2014, from http://bsmith.mathstat.dal.ca Maurer, J. A. (1999). A Brief History of Algorithmic Composition. Retrieved September 15, 2014, from https://ccrma.stanford.edu/~blackrse/algorithm.html

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McCune, D. (2013) Introduction to Randomness, Chance, and Stochastic Composition. Georgia Tech Research Corp. Physics Bellman R. (1964). On a Routing problem. The Rand Corporation. Bryant, P. (1989). Methods of stochastic finite element analysis. AFIT Gardiner, C. W. (2004). Handbook of stochastic methods. Springer. Sankar, A., & Lee Chin-Hui (1996). A Maximum-Likelihood Approach to Stochastic Matching for Robust Speech Recognition. AT&T Bell Laboratories. Scott, M. (2013). Applied stochastic processes in Science and Engineering. University of Waterloo, CND. Printing Friedlander, J. (2010/2014). The Book Designer. Retrieved September 15, 2014, from http://www.thebookdesigner.com Miller, R. (2007). The Tone System. Pibloktok Productions Visual Arts AIC (2013). Colour 2013. 12th Congress of the International Colour Association. Brenneman, K. (1994). Chance in Art. Retrieved September 15, 2014, from http://www.dartmouth.edu/~chance/course/student_projects/Kristin/Kristin.html Cotton, S. D’O. (1996). Colour, colour spaces and the human visual system. School of Computer Science, University of Birmingham, England. Eun-Hee, Yang; Hyung-Kun, Yoo; Kyung-Ja, Kim. (2003). A Study on the Colors in Korean Traditional Wedding Dress at the period of Chosun Dynasty. 6th Asian Design Conference. Li, Jia & Wang, J. Z. (2004). Studying Digital Imagery of Ancient Paintings by Mixtures of Stochastic Models. IEEE. Meong Jin Shin, Westland, S., Vien Cheung, & Cassidy, T. (2012). Colour Preferences of the Korean Young Generation. School of Design, University of Leeds. Pabini G.-P. (2006). Color Theory for Digital Displays. Retrieved September 15, 2014, from http://www.uxmatters.com/mt/archives/2006/01/color-theory-for-digital-displays-a-quickreference-part-i.php

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Prum, R. O. (2013). Coevolutionary aesthetics in human and biotic artworlds. Retrieved September 15, 2014, from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3745613/ Physclips – Light (2014). Eye-colour-vision. Retrieved September 15, 2014, from http://www.animations.physics.unsw.edu.au/light/eye-colour-vision/ Rogowitz, B. E. (1996). Why Should Engineers and Scientists Be Worried About Color. IBM Thomas J. Watson Research Center. Sziranyi, T., & Toth, Z. (2000). Random paintbrush transformation. 15th ICPR, Barcelona, Spain. Tkalcic, M., & Jurij F. Tasic J. F. (2003). Colour spaces - perceptual, historical and applicational background. Ljubljana, Slovenia: EUROCON. UNSW School of Computer Science and Engineering. Retrieved September 15, 2014, from http://www.cse.unsw.edu.au/~cs9314/07s1/lectures/Jian_CS9314_References/Color_spa_per _his_and_app_BG.pdf KEY TERMS AND DEFINITIONS 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. 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. They were created by the International Commission on Illumination (CIE) 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. 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. 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. 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.

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Probability. The mathematical theory of probability deals with patterns that occur in random events. 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. Recursive thinking. The process of solving large problems by breaking them down into smaller, simpler problems that have identical forms. 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. Visual communication. Visual communication is a multi-disciplinary field encompassing graphic design, illustration, fine arts (like drawing and painting), multimedia, 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. 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.

ACKNOWLEDGMENTS I would like to thank the following for their advice and support through the development of this project: Leslie Bollig, FreeForm Art director Hubert Heldner, prof. Levante Kovacs, prof. Herve Lehning, Ist Mile Institute director Richard Lowenberg, LTC director Noreen Masaki, Nikos Milonas, Aya Muramatsu, Justin Smallwood, The Musee Guimet Korean section, Paris FR and the Mathematics Institute of Oberwolfach, DE. I also would like to thank the following for making their material available online and helping develop further Science and Art collaboration: Dr Rob Levy, Drs M. Hann, M. J. Shin, T. Cassidy and Moore of the School of Design Theory, University of Leeds, UK; Dr E. Porteus, School of Management, Stanford University.