Foundations of realistic rendering : a mathematical approach

Die vorliegende Dissertation ist keine gewöhnliche Abhandlung, sondern sie ist als Lehrbuch zum realistischen Rendering für Studenten im zweiten Studienabschnitt, sowie Forscher und am Thema Interessierte konzipiert. Aus mathematischer Sicht versteht man unter realistischem Rendering das Lösen der stationären Lichttransportgleichung, einer komplizierten Fredholm Integralgleichung der 2tenArt, deren exakte Lösung, wenn überhaupt berechenbar, nur in einem unendlich- dimensionalen Funktionenraum existiert. Während in den existierenden Büchern, die sich mit globaler Beleuchtungstheorie beschäftigen, vorwiegend die praktische Implementierung von Lösungsansätzen im Vordergrund steht, sind wir eher daran interessiert, den Leser mit den mathematischen Hilfsmitteln vertraut zu machen, mit welchen das globale Beleuchtungsproblem streng mathematisch formuliert und letzendlich auch gelöst werden kann. Neue, effzientere und elegantere Algorithmen zur Berechnung zumindest approxima- tiver Lösungen der Lichttransportgleichung und ihrer unterschiedlichen Varianten können nur im Kontext mit einem vertieften Verständnis der Lichttransportgleichung entwickelt werden. Da die Probleme des realistischen Renderings tief in verschiedenen mathematis- chen Disziplinen verwurzelt sind, setzt das vollständige Verständnis des globalen Beleuch- tungsproblems Kenntnisse aus verschiedenen Bereichen der Mathematik voraus. Als zen- trale Konzepte kristallisieren sich dabei Prinzipien der Funktionalanalysis, der Theorie der Integralgleichungen, der Maß- und Integrationstheorie sowie der Wahrscheinlichkeitstheo- rie heraus. Wir haben uns zum Ziel gesetzt, dieses Knäuel an mathematischen Konzepten zu entflechten, sie für Studenten verständlich darzustellen und ihnen bei Bedarf und je nach speziellem Interesse erschöpfend Auskunft zu geben.The available doctoral thesis is not a usual paper but it is conceived as a text book for realistic rendering, made for students in upper courses, as well as for researchers and interested people. From mathematical point of view, realistic rendering means solving the stationary light transport equation, a complicated Fredholm Integral equation of 2nd kind. Its exact solution exists|if possible at all|in an infinite dimensional functional space. Whereas practical implementation of approaches for solving problems are in the center of attentionin the existing textbooks that treat global illumination theory, we are more interested in familiarizing our reader with the mathematical tools which permit them to formulate the global illumination problem in accordance with strong mathematical principles and last but not least to solve it. New, more eficient and more elegant algorithms to calculate approximate solutions for the light transport equation and their different variants must be developed in the context of deep and complete understanding of the light transport equation. As the problems of realistic rendering are deeply rooted in different mathematical disciplines, there must precede the complete comprehension of all those areas. There are evolving principles of functional analysis, theory of integral equations, measure and integration theory as well as probability theory. We have set ourselves the target to remerge this bundle of fluff of mathematical concepts and principles, to represent them to the students in an understandable manner, and to give them, if required, exhaustive information

Efficient Many-Light Rendering of Scenes with Participating Media

We present several approaches based on virtual lights that aim at capturing the light transport without compromising quality, and while preserving the elegance and efficiency of many-light rendering. By reformulating the integration scheme, we obtain two numerically efficient techniques; one tailored specifically for interactive, high-quality lighting on surfaces, and one for handling scenes with participating media

Virtual light fields for global illumination in computer graphics

This thesis presents novel techniques for the generation and real-time rendering of globally illuminated environments with surfaces described by arbitrary materials. Real-time rendering of globally illuminated virtual environments has for a long time been an elusive goal. Many techniques have been developed which can compute still images with full global illumination and this is still an area of active flourishing research. Other techniques have only dealt with certain aspects of global illumination in order to speed up computation and thus rendering. These include radiosity, ray-tracing and hybrid methods. Radiosity due to its view independent nature can easily be rendered in real-time after pre-computing and storing the energy equilibrium. Ray-tracing however is view-dependent and requires substantial computational resources in order to run in real-time. Attempts at providing full global illumination at interactive rates include caching methods, fast rendering from photon maps, light fields, brute force ray-tracing and GPU accelerated methods. Currently, these methods either only apply to special cases, are incomplete exhibiting poor image quality and/or scale badly such that only modest scenes can be rendered in real-time with current hardware. The techniques developed in this thesis extend upon earlier research and provide a novel, comprehensive framework for storing global illumination in a data structure - the Virtual Light Field - that is suitable for real-time rendering. The techniques trade off rapid rendering for memory usage and precompute time. The main weaknesses of the VLF method are targeted in this thesis. It is the expensive pre-compute stage with best-case O(N^2) performance, where N is the number of faces, which make the light propagation unpractical for all but simple scenes. This is analysed and greatly superior alternatives are presented and evaluated in terms of efficiency and error. Several orders of magnitude improvement in computational efficiency is achieved over the original VLF method. A novel propagation algorithm running entirely on the Graphics Processing Unit (GPU) is presented. It is incremental in that it can resolve visibility along a set of parallel rays in O(N) time and can produce a virtual light field for a moderately complex scene (tens of thousands of faces), with complex illumination stored in millions of elements, in minutes and for simple scenes in seconds. It is approximate but gracefully converges to a correct solution; a linear increase in resolution results in a linear increase in computation time. Finally a GPU rendering technique is presented which can render from Virtual Light Fields at real-time frame rates in high resolution VR presentation devices such as the CAVETM

Monte Carlo Methods in Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography (QPAT) is a hybrid biomedical imaging technique that derives its specificity from the wavelength-dependent absorption of near-infrared/visible laser light, and its sensitivity from ultrasonic waves. This promising technique has the potential to reveal more than just structural information, it can also probe tissue function. Specifically, QPAT has the capability to estimate concentrations of endogenous chromophores, such as the concentrations of oxygenated and deoxygenated haemoglobin (from which blood oxygenation can be calculated), as well as the concentrations of exogenous chromophore, e.g. near-infrared dyes or metallic nanoparticles. This process is complicated by the fact that a photoacoustic image is not directly related to the tissue properties via the absorption coefficient, but is proportional to the wavelength-dependent absorption coefficient times the internal light fluence, which is also wavelength-dependent and is in general unknown. This thesis tackles this issue from two angles; firstly, the question of whether certain experimental conditions allow the impact of the fluence to be neglected by assuming it is constant with wavelength, a `linear inversion', is addressed. It is demonstrated that a linear inversion is appropriate only for certain bands of illumination wavelengths and for limited depth. Where this assumption is not accurate, an alternative approach is proposed, whereby the fluence inside the tissue is modelled using a novel Monte Carlo model of light transport. This model calculates the angle-dependent radiance distribution by storing the field in Fourier harmonics, in 2D, or spherical harmonics, in 3D. This thesis demonstrates that a key advantage of computing the radiance in this way is that it simplifies the computation of functional gradients when the estimation of the absorption and scattering coefficients is cast as a nonlinear least-squares problem. Using this approach, it is demonstrated in 2D that the estimation of the absorption coefficient can be performed to a useful level of accuracy, despite the limited accuracy in reconstruction of the scattering coefficient