The chapter describes the problem that the thesis aims to solve and describe more in detail the different characteristics of Participating Media.

Radiative Transport Problem

The problem that the thesis aims to solve is a radiative transport problem and the radiometric quantity, that we are searching for, is the radiance $L$. This quantity describes how much light arrives from a direction $\omega$ on an hypothetical differential area perpendicular to that direction $dA^\bot$ .

\(\begin{aligned} L=\frac{d^2\Phi}{d\omega\, \,dA^\bot} =\frac{d^2\Phi}{d\omega\, \,dA\,\cos\theta} \, \label{eq:Ldef} \end{aligned}\)

Where ${\Phi}$, expressed in Watt (Joule/sec), is the radiant flux, i.e time rate of flow of radiant energy. When this light energy interact with a participating medium, than it can scatter or be absorbed. In both cases the light energy is attenuated. This attenuation is usually called extinction (see ).

\[\text{Extintion = scattering + absorption}\]

In this type of problems a distinction is done between solid objects, which have a well defined boundary, and other media, like gases or liquids. In this work, we are concentrating on the first type of media, considering also cases where the object boundary defines a change on the index of refraction of the medium.

Participating Media

We can think of a participating media as an agglomerates of small particles. Each of those particles can be described by the following characteristics identified by :

  • $C_{sca}$ (scattering cross section): is the area of the particle which is scattering light;

  • $C_{abs}$ (absorption cross section): is the area of the particle that is absorbing light;

  • $F$ (scattering diagram of the particle): describes the scattering behavior and it can be used to obtain the more easily manipulable phase function $f_p$.

Given those characteristics, the final medium is described by the density function $\rho(x)$ of these particles for any point $x$ inside the medium. When a photon arrives to a point $x$, inside a medium which contains some particles, four types of event can happen:

Absorption: light is absorbed by the particles in that point $x$, proportionally to their $C_{abs}$; the amount of light absorbed is defined as the absorption coefficient $\sigma_a(x) [m^{-1}]$.
$$ \begin{aligned} \label{eq:absorption} \sigma_a(x) = C_{abs} \cdot \rho(x) \end{aligned} $$ Out-scattering and In-scattering: light can be out-scattered or in-scattered in the point $x$ , based on the $C_{scat}$ of the particles in that point; the amount of light scattered is defined as the scattering coefficient $\sigma_s(x) [m^{-1}]$.
$$ \begin{aligned} \label{eq:scattering} \sigma_s(x) = C_{scat} \cdot \rho(x) \end{aligned} $$ Emission: finally the medium can emit more energy in the point $x$; we will call the radiance emitted by the medium in a point $x$ and direction $\omega$, $L_e(x, \omega)$.

There are also some derived quantities important to describe interactions with a participating media. Those are the extinction coefficient $\sigma_t$, defined as

\(\begin{aligned} \label{eq:extinction} \sigma_t(x) = \sigma_a(x) + \sigma_s(x) \end{aligned}\)

and the albedo of the medium $\alpha$, defined as

\(\begin{aligned} \label{eq:albedo} \alpha = \frac{\sigma_s}{\sigma_t}. \end{aligned}\)

It is important to notice that it is always possible to derive the first two quantities in equations absorption and scattering from the last two equations extinction and albedo, and vice versa. By integrating the extinction coefficient along a line segment $l$, we get the optical thickness of our material $\tau$:

\[\begin{aligned} \tau(l) = \int_l\sigma_t(x)dx. \end{aligned}\]

Now, we have all the elements to completely describe the change of radiance inside a medium using the radiative transport equation (RTE), which can be defined with the following equation:

\(\begin{aligned} \label{eq:RTE} (\omega \cdot \nabla) L(x,\omega) = L_e(x,\omega) + \sigma_s(x)L_i(x,\omega) -\sigma_a(x)L(x,\omega) -\sigma_s(x)L(x,\omega). \end{aligned}\)

where $L_i(x,\omega)$ is the in-scattered radiance at $x$, through the direction $\omega$, coming by other scattered radiance. The equation RTE includes all the light interactions that we have previously described, which are in this order: emission, in-scattering, absorption, out-scattering.

The Volume Rendering Equation (VRE)

Scenes, containing solid participating media, are usually modeled as a volume $\mathcal{V}$ and a boundary $\partial \mathcal{V}$, where $\mathcal{V} \cap \partial \mathcal{V} = \emptyset$. As shown by , the transport can be described differently, based on the position considered on the volume. If the point is on the surface $\delta\mathcal{V}$, we use the classic Rendering Equation

\(\begin{aligned} \label{eq:rendering} L(x,\omega) = L_{e,\delta{\mathcal{V}}}(x,\omega) + \int_{S^2}f_s(\omega,x,\omega')L(x,\omega')|cos\theta_x|d\sigma(\omega'): \end{aligned}\)

where $S^2$ is the set of all directions (in this work we are using only the hemispherical formulation), $f_s$ is the bidirectional scattering distribution function, which describes the scattering behavior at a point $x$ on the surface (in particular is the fraction of incident differential radiation reflected into the direction $\omega’$). Finally, $cos\theta_x$ is the cosine of the angle between direction $\omega’$ and the surface normal at $x$ (the reader interested on more informations about this equation can find its complete definition in the work of ). On the other hand, if the point is inside the participating media we have to consider the Volumetric Rendering Equation (VRE). This equation is obtained integrating the radiative transport equation, that we have described in the previous paragraph, along straight light rays until the next intersection point $x_s$ of the ray with a surface. The equation is:

\(\begin{aligned} \label{eq:volumetric-rendering-equation} L(x,\omega) = \int_0^s T(x,x_t) \cdot L_v(x_t, \omega) dt + T(x,x_s) \cdot L(x_s,\omega) \end{aligned}\)

where

\(\begin{aligned} \label{eq:volumetric-source} L_v(x, \omega) = L_{e,{\mathcal{V}}}(x,\omega) + \sigma_s(x) \int_{S^2} f_p(\omega,x,\omega')L(x,\omega')d\sigma(\omega') \end{aligned}\)

and $T(x,x_t)$ is the transmittance function defined by the Beer-Lambert-Bougeuer law as

\[\begin{aligned} T(x,x_t) = e^{-\tau(||x-x_s||)} \end{aligned}\]

with $\tau$ the previously described optical thickness of the medium.

A more general equation, which combines both the equations rendering and volumetric-rendering-equation, is the path integral formulation. In this formulation the space of integration is the union of spaces containing paths with specific length, i.e $\mathcal{P} := \bigcup_{k\in\mathbb{N}} \mathcal{P}_k$, where $P_k$ is the space of paths ${\overline{x}}$ of length $k$. The reader interested can find its derivation in the work of .

From a mathematical point of view, the volume rendering equation is a Fredholm integral equation of the 2nd kind. Solving this type of equation analytically is very difficult, even applying simplifications on medium and light transport. Many numerical estimations methods have been studied in the literature. Those methods can be divided in two groups: deterministic methods and stochastic methods. The first ones are based on a discretization of the domain of integration and the solution of large linear systems. However, this type of system is usually affected by some limitations and the recent research has focused more on the second types of methods. In the next chapter, we are going to cover exactly this latter type of methods, giving an overview on some of the most important ones.

Porting to GPU

GPU-based programs have a number of limitations on when and how memory can be accessed. Computation speed increases at a much faster rate than memory access speeds, which means that, to improve time performance, special care should be taken to maximize bandwidth usage. Simulating light transport, inside participating media, can also be not obvious. For this scope, one of the most used techniques in the industry of computer graphics is the Monte Carlo path tracing. This is a stochastic method which uses a random process to correctly estimate the light interactions in the medium. The random nature of this method makes the utilization of GPU difficult in this context. GPU is best suited to well predefined tasks with the minimum control decisions made at runtime. Random behavior of different GPU threads can lead to decreased utilization of the device. If a GPU is not utilized at its maximum processing power, the overhead of transferring data and control to it can make the GPU implementation slower than a CPU one.

Limitations

This work is subject to different restrictions which limits the generality of the results obtained:

  • we will not consider possible changes of refraction index inside the medium itself;

  • our tested objects are rendered in the vacuum, not considering participating media outside the volume;

  • inherited from the scattering theory used, we will consider only independent scattering, meaning that we will consider only participating medium which have well-defined separate particles, and single scattering, which means that the concentration of particles of the medium considered is proportional to the total light intensity scattered (further explanation inside the work of  );

  • we will not consider wave effects and only consider ray optics;

  • we have tested our implementation only on one GPU architecture and we address only the usage of CUDA enabled GPU.

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