Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1997 Modelling polarized light for computer graphics Sairam Sankaranarayanan Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Computer Sciences Commons, and the Optics Commons Recommended Citation Sankaranarayanan, Sairam, "Modelling polarized light for computer graphics " (1997). Retrospective Theses and Dissertations. 12029. https://lib.dr.iastate.edu/rtd/12029 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. 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Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell lafonnation Company 300 North Zed) Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 I Modelling polarized light for computer graphics by Sairajn Sankarajiaxayanan A dissertation submitted to the graduate faculty in peixtial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Co-majors: Physics; Computer Engineering Major Professors: John Gustafson, Kai-Ming Ho, Charles Wright Iowa State University Ames, Iowa 1997 UMl Number: 9814690 UMI Microform 9814690 Copyright 1998, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii Graduate College Iowa State University This is to certify that the Doctoral dissertation of Sairam Sankaranaxayanan hcis met the dissertation requirements of Iowa State University Signature was redacted for privacy. Co-major Professor Signature was redacted for privacy. o-major Professor Signature was redacted for privacy. Co-major Professo Signature was redacted for privacy. Signature was redacted for privacy. Signature was redacted for privacy. For the Graduate College Ill DEDICATION For all the illiterates of the world. TABLE OF CONTENTS ACKNOWLEDGEMENTS vii 1 INTRODUCTION I Rendering Techniques 1 Radiometric Definitions 2 The Rendering Equation 4 Raytracing 6 Radiosity 7 Dissertation Structure 11 2 POLARIZATION OF LIGHT 12 Historical Background 12 The Wave Equation 13 The Polarization EUipse 14 Degenerate Forms 14 The Stokes Parameters 15 Properties of the Stokes Paraxneters 17 3 POLARIZING ELEMENTS 23 Mueller Matrices 23 Mueller Matrix of a Polarizer 24 Mueller Matrix of a Phase Shifter 25 Mueller Matrix of a Rotator 27 V Mueller Matrices for Rotated Elements 30 4 REFLECTION OF LIGHT AT AN AIR-DIELECTRIC INTERFACE 32 Fresael's Equations for an Air-dielectric Interface 32 Non-specular Reflection 37 5 PHOTON 38 Monte Carlo Methods 38 Some Definitions from Probability Theory and Statistics 38 Monte Carlo Integration 40 Monte Carlo Simulation 40 Monte Carlo Methods in Computer Graphics 41 6 SURFACE REFLECTION MODEL FOR PHOTON 46 Rough Surface Scattering 47 Statistical Description of a Normally Rough Surface 51 7 MONTE CARLO IMPLEMENTATION OF THE REFLECTION MODEL 57 Random Variate Generation 57 Inverse Transform Method 58 Rejection Method 59 Reflection Model 60 Specular Term 64 Diffuse Term 64 Computational Issues 67 Effective Roughness Factor 67 The Fresnel Term 68 8 CONCLUSIONS AND FUTURE DIRECTIONS 71 BIBLIOGRAPHY 74 VI LIST OF FIGURES Figure 1.1 Definition of radiance 4 Figure 1.2 Rayt racing 7 Figure 1.3 In variance of radiance along line of sight 9 Figure 2.1 Polarization ellipse 1.5 Figure .3.1 Polarizing element 24 Figure 3.2 Rotation of the field components by a rotator 28 Figure .3.3 Rotated polarizing element 31 Figure 4.1 Specular reflection 33 Figure 4.2 Plot of degree of polarization vs. incident angle for reflection of unpolarized light incident on glass of refractive index 1.4 35 Figure 5.1 Photon: Domain definition 43 Figure 5.2 Photon rendering; Harpsichord room 45 Figure 6.1 Geometry for Kirchoff diffraction integral 48 Figure 6.2 Approximate evaluation of solid angle 50 Figure 6.3 Non-specular reflection 52 Figure 7.1 Inverse transform method 58 Figure 7.2 Computation of the Fresnel term 69 Figure 8.1 Renderings for polarization comparison 72 Vll ACKNOWLEDGEMENTS I still remember the misgivings with which I approached Dr. John Gustafson, with no background in graphics whatsoever. I am grateful to him for having given me the opportunity to work with him and I thank Dr. James Oliver for having directed me to him. I am grateful to Dr. Kai-Ming Ho for his support, especially his crucial support during the funding crisis in the final stages of completion of my degree. Nan, Erlene and Lori, I will no longer trouble you but I am afraid there will soon be other students who will perhaps be as troublesome as I was. I thank you for all your help. I would like to thank the Systems personnel and the other staff of the Scalable Computing Laboratory for all their help axid in providing a very comfortable work en­ vironment. There are many other people who have helped not only in the completion of my degree but in learning many valuable lessons in life and I thank all of them. I thank my parents and the Divine for what I am today and the heights I have scaled so far. This work was performed at Ames Laboratory imder Contract No. W-7405-Eng-82 with the U.S. Department of Energy. 1 1 INTRODUCTION The field of Computer Graphics encompasses a myriad of issues both in the hardware and software domain. But the original goal of creating a photo-realistic synthetic image to be displayed on a computer console, still remains one of the central areas of Com­ puter Graphics. Keeping pace with the advance of computer technology, the methods employed for image synthesis have progressively developed from simplistic algorithms to complex ones. Subsequently, the visual quality of the synthesised images is progressively approaching the quality of a photograph. The computation of a synthetic image involves the simulation of light transport, the interaction of light with matter. The visual realism of the computed image is largely determined by the reflection model or shading model, the model used to simulate the interaction of light with matter. The simplistic empirical shading models used in early attempts have been progressively replaced by more complex physically based models, incorporating much of the physics involved in light transport. Within the Computer Graphics community, computation of synthetic images is re­ ferred to as rendering. Various rendering algorithms have been developed which attempt to compute images using various reflection models and approximations. Rendering Techniques Some of the terms used in describing the basis of various rendering techniques de­ rive largely from Radiometry, which is the science of measurement of the production and propagation of electromagnetic radiation. Before proceeding to discuss rendering algorithms, it would be wise to go over the definitions of some of the radiometric terms. Radiometric Definitions The fundamental physical quaxitity which is in some sense the primitive concept in Radiometry, is the radiant flux. Radiant flux is the amount of radiant energy flowing across a given surface in unit time. In other words, it is the power flow across a given surface. Irradiance is the ajea density of the radiant flux and hence has the dimensions of power per area. Radiance is an important concept in Radiometry and is closely related to the sensa­ tion of brightness as perceived by the human eye when visual attention is focussed in a particulgu: direction. It is the power per area per solid angle incident normally on a surface. Since the rendering problem is one of computing the radiance, the reflection model typically is faced with the problem of computing the reflected radiance as a function of the incident radiant flux. The often encountered term BRDF conceptualizes this. BRDF or Bidirectional Reflectance Distribution function is the ratio of the reflected radiance in any given direction to the flux density (irradiance) incident on to the surface at a given direction. It is a function of the incident and reflected directions and symmetric with respect to each, hence the term bidirectional. The distinctions between some radiometric terms are geometrical, distinguished from each other by the particular geometry channelling the flow of radiant energy, that a particular term describes. The term intensity is ubiquitous in the literature and has been cause for much confusion. Authoritative texts [25, 6] use the term in place of the radiance.