Pseudocolor in Pure and Applied Mathematics, a Free on-Line e-Book with Source Code

From the Youvan Foundation, a nonprofit 501(c)3 educational charity in formation

Pseudocolor is very useful in the representation of computational output that involves complex data and phenomena - including numerical tensors and digital images.  Many of the examples in this e-book are pedagogical in nature and useful for introducing students to higher levels of Mathematics. Our full source code is published in Mathematica ".nb" and Adobe ".PDF" formats; the latter has fully executed graphics embedded.  We have used a very limited vocabulary of Mathematica functions so this code can be read easily and ported to other languages.  As such, the simple syntax of Mathematica can serve as a flow chart for other implementation.  Our 'pure' examples include work in matrix algebra,  fractals, tuples, permutations, transcendental numbers, statistics, computation speed, and P=NP.  Our applied examples include work in biology, molecular biology, biotechnology, chemical kinetics, energy conversion, thermodynamics, spectroscopy, aerodynamics, quantum physics, electromagnetism, artificial intelligence, signal processing, image processing, digital imaging spectroscopy, graphics, meteorology, crystallography, gemology, language, medicine, and law.

As a way of introducing the idea of using pseudocolor to represent mathematical concepts and applications, three exemplary graphics are shown below.  These graphics come from the fields of aerodynamics (Savonius rotor), biology (genetic code), and pure math (Tuples).  Following these examples, you will find a Forward and Table of Contents with hyperlinks to explanations and source code.  Currently, 16 examples are completed, 4 more are in progress, and as many as 60 more examples are planned.  Please contact the author if you have any comments - as this is a work in progress.  Blue text indicates active hyperlinks.

 

       Savonius Waves.

 

 

      Pseudoinverse of the Genetic Code.

 

    

    Tuple Imagery.

Foreword

For JTY

With the advent of inexpensive computers running with gigahertz speed and  gigabyte RAM, it appears that issues of compiled code and concise memory management might become concepts of the past for most applications.  Such statements always look funny in hindsight, after computers become still faster and cheaper.  That makes a high-level language such as Mathematica even more attractive as a future platform.  A developer using a platform such as Mathematica can take advantage of the work of a larger group, such as Wolfram Research, to free themselves from monolithic operating systems.  The developer is then free to code and spend most of their time in logic rather than in ever-changing fundamentals underlying developer studios.  We also anticipate that Mathematica code is easy to read - even without flowcharts - and that it can be easily ported to another language.

It is important to visualize mathematics in order to learn.  For example, calculus is more easily learned when it is combined with analytical geometry.  One sees derivatives as the slope of functions, and one can picture the concept of an integral as simply the area under a functional curve.  Our sense of vision can not be neglected in the path to learning higher math.  That is why physicists are often the people that advance math - they actually see what they are doing.  A century ago, quantum mechanics began to show us the picture of atoms in the form of wave functions that graph as easily understood electron clouds.  Without this visualization, it would be very difficult to appreciate the new, underlying mathematics.  In electromagnetism and optics, theorems are best visualized as vectors pointing away from surfaces.  Many of us have learned math through physics.

Biotechnology is also increasing important in applied and pure mathematics.  One seeks to understand our 3 billion nucleotide genome and the interaction of drugs in terms of combinatorial possibilities.  This opens up the entire field of discrete mathematics and combinations to the applied mathematician for use in biotech. Rapid prototyping of new algorithms in these fields is very important.  That is aided by high level languages that can be used by an individual researcher or a sole practitioner.

Images and source code from this e-book are also made available to the public for any lawful use (worldwide) via our PD-Self deposition at Wikimedia.  As of 2008, the public domain SAGE platform will run Mathematica computer code and display, for example, using the Microsoft Internet Explorer.

Completed Examples:

  1. Temporal Color Changes in a Stack of Images Containing Randomly Generated Colors
  2. Pseudocoloring Ordinary Photographs to Highlight Grayscale Changes
  3. 300 Different Pseudocolor Schemes via One RGB Orthonormalization
  4. Counterfeit Detection Based on Ink Color
  5. Random Planes Passing Through a Schroedinger Cube
  6. Anti-Color, Real Color, and Super-Color in Tuple-Images
  7. Morphological Stability in Anti-Color and Super-Color Diffusion
  8. Synergy and Antagonism for Colors in Diffusion
  9. Chambered Detonation of Super-Colors
  10. Pixel Accretion Yielding a Dissipative Structure
  11. Pseudocolored Stereo-Pair of a 3-D Lissajous Function
  12. Grayscale Rendering of Perpendicular Half-Planes
  13. Colorizing a Fractal with a One Dimensional Fractal
  14. Rendering 3-D Depth in an RGB Image Using Nested Subsets and Permutations 
  15. Similarity (SSD) Sorting of Signals Displayed in Contour Plots
  16. Selecting Pixels from an Image Stack that Match Temporal RGB Functions

 Work in Progress:

  1. Singular Value Decomposition (SVD) of Amino Acid Hydropathy in the Genetic Code (PDF; nb)
  2. 106 - Fold Increase in Computation Speed of a Scroll Matrix Pseudoinverse (PDF; nb)
  3. Spatial Traces of the Savonius Rotor (in preparation)
  4. Short Time-Course Kinetics of Doppler Radar Images

Future Work:

  1. Constructing a Stained-Glass Window from Colored Points
  2. Modeling a Cyclone
  3. Graphics and Complexity if N=N! or P=NP in a Designer's Universe
  4. Automated Sorting of Pixels with Noisy Colors
  5. Intelligent Ants and the Traveling Salesman Problem (TSP)
  6. Refining a Color Scheme with a Genetic Algorithm
  7. Pseudocoloring Pascal's Triangle for Increased Depth
  8. Melding Images of Two Faces
  9. Chain Code for Color Dependent Feature Detection
  10. A Convolution Kernel with R, G, B Layers
  11. Simulating Distance (D) Dependencies for D1 to D6 in Energy Transfer
  12. Image Processing at Sub-Pixel Resolution
  13. Simulating a Winogradsky Ecosystem
  14. Graphics of Third-Power Dependence of Wind Velocity for Energy Conversion
  15. 'Images' from Transcendental Numbers
  16. A Temperature Jump Between Two States of Equilibrium
  17. Enzyme Kinetics Limited Only by Diffusion
  18. Simulating a Fluorescence Activated Cell Sorter
  19. Automated Detection of a Crystallographic Unit Lattice
  20. Stress Analysis on an 1850's Stagecoach Wheel
  21. Oscillations of a Conical Parachute
  22. Imaging, Digitization, and Analysis of a Wind Sock in Gusty Wind
  23. An Artist's Charcoal Algorithm for Digital Photographs
  24. Using N! as a Pseudocoloring Scheme
  25. Music in Pseudocolor Representation
  26. Language in Pseudocolor Representation
  27. Pseudocolor Representation of  N0.5 / N 'Shot-Noise'
  28. A 360 Degree Panoramic Image on a Rotating Cylinder
  29. Feature Extraction of a Fingerprint
  30. An Image Between Parallel Mirrors
  31. Simulating Specular and Diffuse Reflectance from a Wavy Pond
  32. Solar Sail Thrust
  33. Extraction of Features in the Complex Plane
  34. Pseudocoloring Feynman's QED Mirror / Grating
  35. Pseudocoloring Gödel Numbers
  36. Medical EKG in Pseudocolor
  37. Chemical EPR and NMR in Pseudocolor
  38. Dispersive Chemical Kinetics
  39. Encoding Physicochemical Properties of Protein into Combinatorial DNA
  40. 2-D Depth Pseudocoloring of the Faces of a Brilliant Cut Diamond
  41. Simulation of Recursive Ensemble Mutagenesis in Directed Evolution
  42. Simulating the View from a Compound Eye
  43. Graphic Art Using Only Reversible Tools
  44. Pandemic Modeling and the Traveling Salesman Problem (TSP)
  45. 2-D Valence Shell Electron Pair Repulsion (VSEPR) Images
  46. Global Warming Predictor
  47. Marcus Theory, Cold Fusion, and Coulombic Explosions
  48. Deutch's Burning Droplets
  49. Maximum Entropy, Boltzmann and Shannon
  50. Constraints on Markov Transitions in DNA Encoding Protein
  51. Electromagnetism: Fermi-Golden, Green, and Stoke
  52. Energy Transduction by Chemiosmotic Membranes
  53. Zeiss Microscopy, Foerster Energy Transfer, and Youvan's Method
  54. Genetic Algorithms and Gene Shuffling
  55. Charge Separation in Photosynthetic Reaction Centers
  56. Excitons and Light Harvesting Antennae
  57. Single and n-Point Failures in Complex Docketing Systems
  58. Quantifying 'Paranoia Factors' in Transactional Negotiations
  59. Heisenberg Limitations to Computer and Enzyme Size

Appendix:

  1. Data Files Required for Import
  2. Selected Export Data
  3. Tuple Images From 8-tuples of Random RGB Numbers Between -1 and +1
  4. Tuple Images From 8-tuples of Experimental RGB Numbers

Epilogue:

Epilogue Image: "The Particle Out of the Box", contributed by Harvey Bialy

Interview with a mentor, Dr. Robert Reed: Part1 ; Part2

About the author ... 

(c) Youvan Foundation, 2006.  Last updated on May 11,  2008.