Radiative transfer equation

In the previous section, we assumed that there was no participating media in the scene. This implies that the radiance leaving a surface remains unchanged

2.4 ● Radiative transfer equation 23

(a) Smallσa(x), zeroσs(x) (b) Smallσa(x), smallσs(x) (c) Smallσa(x), largeσs(x)

(d) Largeσ_a(x), zeroσ_s(x) (e) Largeσ_a(x), smallσ_s(x) (f) Largeσ_a(x), largeσ_s(x) Figure 2.4: Renderings of a glass filled with liquid modeled using homogenous media with varying absorption coefficent,σa(x), and scattering coefficient,σs(x).

Upper row: Renderings using a small absorption coefficient anda) no scattering, b) a small scattering coefficient, andc) a large scattering coefficient.Lower row:

Renderings using a large absorption coefficient andd) no scattering,e) a small scattering coefficient, andf) a large scattering coefficient.

until it hits another surface. In reality, however, surfaces of interest are often located in different forms of participating media, such as air, water or fog.

For optically thin media, such as clean air, the assumption that light travels unobstructed between surfaces serves as a reasonable approximation for short distances. However, over longer distances even clean air scatters light (the sky appears blue due to such scattering), and for photorealistic rendering of scenes with denser media such as water, smoke, fire etc, it is necessary to consider models that take into account the effects of how light interacts with the participating media in the scene.

In computer graphics, and in many other fields of science such as neutron transport [190] and medical physics [10], the media is modeled as a large number of microscopic scattering particles that the light can interact with. As the sheer number of these particles makes deterministic models infeasible, we instead make use of Linear Transport Theory that, similar to other statistical models used in physics [129], considers the aggregated behavior of a large

number of randomly distributed particles. The main insight in these approaches is that we do not need to represent the exact position of each individual particle as long as their average effect on the light propagation through the media can be accounted for. To further simply the models, light-particle interactions in the media is assumed to be independent, that is if the light interacts with a particle in the media, this interaction is statistically independent from the outcome of subsequent interaction events (in other word, a random photon trajectory can be characterized by a Markov process).

In computer graphics we are interested in simulating the interactions between particles in the media and photons with relatively low energy (visible light).

This allows us to model interactions using two type of events, either a photon is absorbed (for example converted to heat) or it collides with a particle in the medium and scatters in another direction. In other fields, considering photons with higher energy, such as radiation dosimetry [10], more complex collision events, such asCompton scatteringandpair production, have to be considered as well [185]. The relative probability of a particle being absorbed or scattered per unit length is described by theabsorption coefficient,σa, and thescattering coefficient,σ_s, respectively. These quantities generally depend on the density of particles in the medium, and are often allowed to vary spatially. Media where σ_a(x)andσ_s(x)are constant for allxis referred to ashomogeneous, otherwise, if the coefficients vary spatially, the media isheterogenous. The absorption and scattering coefficient can have a profound effect on the appearance of the media, an illustration is given in figure2.4. The sum ofσ_a(x)andσ_s(x)constitute the probability that an interaction takes place per unit length, and is described by theextinction coefficientσ_t(x) =σ_a(x) +σ_s(x). Both absorption and scattering can reduce the radiance along a ray in the medium, as photons traveling along the ray can be absorbed or scattered into different directions, referred to as out-scattering. Similarly, the radiance along a ray can also increase due to emission of photons in the media, or fromin-scatteringof photons originating from other directions.

The net effect of the change of radiance along a ray in directionωfrom a point x, is modeled by an integro-differential equation known as theradiative transfer equation(RTE) [36] as:

2.4 ● Radiative transfer equation 25

Figure 2.5: The radiative transport equation describes the radiance reaching a pointxfrom directionω,L(x←ω)as a sum of the attenuated radiance from the nearest surface,L(y→ −ω), and the accumulated (integrated) in-scattering, L_i(x_t → −ω), and emission, L_e(x_t → −ω), for points x_t along the ray in the medium.

describes the in-scattering, given by an integral over the unit sphere,S², defined using aphase function,ρp(x,ω,ωi)that models the angular distribution of light scattering at a point x in the medium. L_e(x → ω) represents the radiance emitted in the direction of the ray from the medium, given in units of radiance per unit length.

Using the rendering equation (2.8) as a boundary condition, the RTE can be formulated in integral form [14,95], describing the radiance reaching a pointx from directionω, illustrated in figure2.5, as:

L(x←ω) = T(x,y)L(y→ −ω)

radiance originating from closest surface +

∫₀^dT(x,x_t) (Le(xt→ −ω) +L_i(xt→ −ω))dt

radiance from accumulated in-scattering and emission in the volume

(2.15)

whereyis the first point on a surface in the directionωfromx,dis the distance fromxtoy,x_t=x+tω t∈ (0,d)are points along the ray, and,T(x,y)is the transmittance between the pointsxandx_tgiven by:

T(x,x_t) =exp( − ∫₀^{∣∣x−xt∣∣}σ_t(x_t^′)dt^′), (2.16) whereσt(x)denotes the extinction coefficient atxdescribing the loss of light due to absorption and out-scattering per unit distance. The integral form of the RTE is also commonly referred to as thevolume rendering equation[62].