factor for each pixel in the high-dynamic range image based on the local adaptation luminance of the pixel and the high-dynamic-range value of the pixel. The resulting value produced by tone-mapping operators is typically an RGB value in the range from 0 to 1 that can be displayed on the output device.
2.3 Light Transport of Subsurface Scattering
With rendering equation we can computed accurate radiance for each pixel and generate color image by applying tone-reproduction operator. Before simulating translucency, we must have accurate BSSRDF model. In this section, we deals with the light transport theory for translucent materials, and the construction of BSSRDF model.
In light transport theory, photons traveling in a medium will collide with the medium, caus-ing them to be absorbed or change directions (scattercaus-ing). Absorption means the energy carried by photons is converted into other types of energy, for instance, the energy could be converted from radiation to kinetic energy of particles in the medium.
The probability that a photon gets absorbed in a medium, per unit of distance along its direc-tion of propagadirec-tion, is called the absorpdirec-tion coefficient σa(x) and the probability that a photon gets scattered in a medium is called the scattering coefficient σs(x). These two coefficients are all measured in the unit of length, e.g., 1/m or 1/mm. This means that a photon traveling a distance ∆x in a medium has a chance σa∆x and σs∆x of being absorbed and scattered, respectively.
The change in radiance L in the direction w due to out-scattering could be modeled as the following equation:
(ω·∇)L(x, ω) = −σs(x)L(x, ω), and the change due to absorption could be modeled as:
(ω·∇)L(x, ω) = −σa(x)L(x, ω).
2.3 Light Transport of Subsurface Scattering 14
We often combine the two coefficients to the extinction coefficient σt(x) as σt(x) = σs(x) + σa(x),
and the combined loss in radiance is given by:
(ω·∇)L(x, ω) = −σt(x)L(x, ω). (2.4)
On the other hand, when photons move through the media, there will also be a gain in radiance due to in-scattering of light from other directions. The change due to in-scattering is modeled by:
(ω·∇)L(x, ω) = σa(x)
Z
Ω
p(x, ωi, ω)Li(x, ωi)dωi, (2.5) where the incident radiance, Li, is integrated over all possible directions. p(x, ω, ωi) is the phase function describing the distribution of the scattered light. We assume that the phase function is normalized, RΩp(x, ωi, ω)dωi = 1, and is a function only of the phase angle, p(x, ωi, ω) = p(x, ωi·ω). The mean cosine g of the scattering angle is defined as g = RΩ(ωi·ω)p(ωi·ω)dωi. It indicates the type of scattering in the medium. If g is positive, the medium is predomi-nantly forward scattering; if g is negative, the medium is predomipredomi-nantly backward scattering;
if g equals zero, the phase function is constant and the medium results in isotropic scattering.
Most translucent materials are strongly forward scattering with g ¿ 0.7. Such strongly peaked phase functions are costly to simulate in media with multiple scattering since the probability of sampling in the direction of the light source will be low in most situations. In this case we can benefit from a powerful technique known as the similarity of moments, which allows us to change the scattering properties of the medium without significantly influencing the actual distribution of light [9, 10]. Specifically, we can modify the medium to have isotropic scattering by changing the scattering coefficient to σ0s = σs(1 − g), where σs0 is the it reduced scattering coefficient. The absorption coefficient remains unchanged, and we get the reduced extinction coefficientσt0 = σs0 + σa.
2.3 Light Transport of Subsurface Scattering 15
In addition to the gain in radiance due to in-scattering, there is also a gain in radiance due to volume emission Le from the medium (i.e., a flame), and it is given by:
(ω·∇)L(x, ω) = σa(x)L(x, ω). (2.6)
By combining Equation 2.4,2.5, and 2.6 we get a linear integro-differential equation, which is the so called light transport equation, or radiative transport equation:
(ω·∇)L(x, ω) = σa(x)L(x, ω) − σt(x)L(x, ω) + σa(x)
Z
Ω
p(x, ωi, ω)Li(x, ωi)dωi. (2.7)
2.3.1 The Volume Rendering Equation
In computer graphics, the light transport equation is often represented in another form, which is derived by integrating Equation 2.7 on both sides for a segment of length s subject to the appropriate boundary conditions [3, 10]. the so called volume rendering equation, which is much more complicated than the rendering equation because the light is influenced by light at every point in space, not just the points on other surface.
2.3.2 The Diffusion Approximation
The light transport equation, either in the form of Equation 2.7 or Equation 2.8, is a five-dimensional equation with integrals, which is very difficult to solve even when the light is scattered isotropically in the medium. It also difficult to construct analytic BSSRDF model from Equation 2.8. Therefore, we need to use the diffusion approximation to make it feasible
2.3 Light Transport of Subsurface Scattering 16
to solve the light transport equation [10, 20]. The diffusion approximation is based on the ob-servation that as the number of scattering events increases, the angular dependence tends to be smoothed out, i.e., the light distribution in highly scattering media tends to become isotropic.
The diffusion approximation begins by dividing the radiance into two components: the un-scattered radiance (or reduced radiance) Lu and the scattered radiance (or diffuse radiance) Ld. The un-scattered radiance is the radiance that reaches point x directly from a light source, or from the boundary of the participating medium. It decreases exponentially with the distance traveled through the medium :
Lu(x + ∆x, ω) = e−σ0t∆xLu(x, ω).
The diffuse radiance is radiance scattered one or more times in the medium. As stated above, after many scattering events, the angular dependence of diffuse radiance tends to be smoothed out, so the diffusion approximation can use the first four terms of the spherical harmonic expan-sion to represent Ld:
where phi(x) = RΩLd(x, ω0)dω0 is the 0th-order spherical harmonic, called the radiant fluenceand−→
E (x) = RΩLd(x, ω0) · ω0dω0 is the 1st-order spherical harmonic, called the vector irradiance.
Substituting the diffusion approximation (Equation 2.9) to the light transport equation (Equa-tion 2.7) yields the classic diffusion equa(Equa-tion:
D∇2φ(x) = σaφ(x) − S0(x) + 3D∇S1(x), (2.10) where D = 1/3σt0; and S0(x) and S1(x) represents the 0th-order and the 1st-order spherical harmonic expansions of the source term, respectively.
The diffusion equation can be solved analytically for special cases [10], or numerically by using a multi-grid method [20]. However, in the case of translucent materials, we are only