【計算機圖形學】離線渲染專題 (三)

【Computer Graphics】Offline Rendering

Heskey0 (Bilibili)

December 2021

Based On Mark Pauly's Thesis[1999] and 《PBRT》

Chapter 2 . Light Transport for Global Illumination

2.1. Light Transport in Participating Media

2.1.1. Light Interaction Events

When a photon travels through a collection of microscopic particles, it may either miss all the particles and continue unaffected, or it may interact with some of the particles. The probability that an interaction does occur is related to the extinction coefficient, \(\sigma_t\) (units [l/m]), of the medium. This quantity depends on the density and size of the particles within the medium.

When an interaction occurs, two things may happen: the photon may be absorbed by the particle (by being converted to another form of energy, such as heat), or the photon may be scattered in another direction. The relative probabilities of these two events is given by the absorption coefficient \(\sigma_a\) and the scattering coefficient \(\sigma_s\), and \(\sigma_t=\sigma_a+\sigma_s\) is the extinction coefficient. Either of these two events lead to a change of radiance along the ray.

2.1.2. Extinct

The number of photons entering this beam is proportional to the incident radiance \(L(x→\omega)\) at the start of the beam \(x\). At each small step \(\Delta t\) along the beam, some fraction of the photons will interact with the medium and become absorbed. If the absorption coefficient within the segment is \(\sigma_a(x+ t\omega)\), then a fraction \(\sigma_a(x+t\omega)\Delta t\) of the photons will be absorbed. Hence, the number of photons exiting this segment can be expressed as:

\[L((x+t\omega)→\omega)=L(x→\omega)(1-\sigma_a(x+t\omega)\Delta t) \]

Taking the limit as \(\Delta t→0\) :

\[\lim_{\Delta t→0}(\frac{L((x+t\omega)→\omega)-L(x→\omega)}{\Delta t})=(\omega *\nabla_a)L(x→\omega)=-\sigma_a(x)L(x→\omega) \]

2.1.3. The Radiative Transfer Equation

Given the four scattering events described in the previous sections we can begin to form a complete model of how light behaves in a participating medium.

This equation is known as the integro-differential form of the radiative transfer, or radiative transport equation, or simply the RTE [Chandrasekhar, 1960], and it incorporates the four possible types of interaction events that can occur within the medium.

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2.2. Path Integral Formulation of the Light Transport Equation

2.2.1 The three-point form of the transport equations

we define:

\[L(x→x^\prime)=L(x,\omega) \]

where \(\omega\) is the unit-length vector pointing from \(x\) to \(x^\prime\). The arrow notation \(x→x^\prime\) symbolizes the direction of the light flow.

Similarly, we write the BSDF as a function of the form:

\[f_s(x→x^\prime→x^{\prime\prime})=f_s(x^\prime,\omega_i→\omega_o) \]

The three-point form of the light transport equation can now be written as:

\[L(x^\prime→x^{\prime\prime})=L_e(x^\prime→x^{\prime\prime})+\int_M L(x→x^\prime)f_s(x→x^\prime→x^{\prime\prime})G(x\leftrightarrow x^\prime)dA(x) \]

where

\[G(x\leftrightarrow x^\prime)=V(x\leftrightarrow x^\prime)\frac{|cos\theta_o cos\theta_i^\prime|}{||x-x^\prime||^2} \]

Here \(\theta_o\) and \(\theta_i^\prime\) are the angles between the segment \(x\leftrightarrow x^\prime\) and the surface normals at \(x\) and \(x^\prime\) respectively, while \(V(x\leftrightarrow x^\prime)=1\) if \(x\) and \(x^\prime\) are mutually visible and is zero otherwise.

2.2.2 Path Integral Formulation

We recursively expand the three-point form of the light transport equation to obtain:

\[I_j=\sum_{k=1}^{\infty}\int_{M^{k+1}}L_e(x_0\rightarrow x_1)G(x_0\leftrightarrow x_1)\prod_{i=1}^{k-1}f_s(x_{i-1}\rightarrow x_i\rightarrow x_{i+1})G(x_i\leftrightarrow x_{i+1}) \\*W_e^{(j)}(x_{k-1}\rightarrow x_k)dA(x_0)...dA(x_k) \]

We define:

\[\bar{x}=x_0 x_1...x_k, \]

\[\mu(\bar{x})=\int_DdA(x_o)dA(x_1)...dA(x_k), \]

\[f_j=L_e(x_0\rightarrow x_1)G(x_0\leftrightarrow x_1 )\tau(x_0\leftrightarrow x_1) \\*\prod_{i=1}^{k-1}f_s(x_{i-1}\rightarrow x_i\rightarrow x_{i+1})G(x_i\leftrightarrow x_{i+1})\tau(x_i\leftrightarrow x_{i+1}) \\*W_e^{(j)}(x_{k-1}\rightarrow x_k) \]

\(I_j\) can be expressed as pure integral of the form (path integral formulation): \(I_j\) can be expressed as pure integral of the form (path integral formulation):

\[I_j=\int_\Omega f_j(\bar{x})d\mu(\bar{x}) \]

where \(\bar{x}\) is a light transport path, \(\Omega\) the set of all finite-length transport paths, \(\mu\) the area product measure (a Lebesgue measure on \(\Omega\) ) and \(f_j\) is called the measurement contribution function.

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