15.1 The Equation of Transfer
The equation of transfer is the fundamental equation that governs the behavior of light in a medium that absorbs, emits, and scatters radiation. It accounts for all of the volume scattering processes described in Chapter 11—absorption, emission, and in- and out-scattering—to give an equation that describes the distribution of radiance in an environment. The light transport equation is in fact a special case of the equation of transfer, simplified by the lack of participating media and specialized for scattering from surfaces.
In its most basic form, the equation of transfer is an integro-differential equation that describes how the radiance along a beam changes at a point in space. It can be transformed into a pure integral equation that describes the effect of participating media from the infinite number of points along a ray. It can be derived in a straightforward manner by subtracting the effects of the scattering processes that reduce energy along a beam (absorption and out-scattering) from the processes that increase energy along it (emission and in-scattering).
Recall the source term from Section 11.1.4: it gives the change in radiance at a point in a particular direction due to emission and in-scattered light from other points in the medium:
The source term accounts for all of the processes that add radiance to a ray.
The attenuation coefficient, , accounts for all processes that reduce radiance at a point: absorption and out-scattering. The differential equation that describes its effect is
The overall differential change in radiance at a point along a ray is found by adding these two effects together to get the integro-differential form of the equation of transfer:
With suitable boundary conditions, this equation can be transformed to a pure integral equation. For example, if we assume that there are no surfaces in the scene so that the rays are never blocked and have an infinite length, the integral equation of transfer is
where (Figure 15.1). The meaning of this equation is reasonably intuitive: it just says that the radiance arriving at a point from a given direction is contributed to by the added radiance along all points along the ray from the point. The amount of added radiance at each point along the ray that reaches the ray’s origin is reduced by the total beam transmittance from the ray’s origin to the point.
More generally, if there are reflecting and/or emitting surfaces in the scene, rays don’t necessarily have infinite length and the first surface that a ray hits affects its radiance, adding outgoing radiance from the surface at the point and preventing radiance from points along the ray beyond the intersection point from contributing to radiance at the ray’s origin. If a ray (, ) intersects a surface at some point at a parametric distance along the ray, then the integral equation of transfer is
where is the point on the surface and are points along the ray (Figure 15.2).
This equation describes the two effects that contribute to radiance along the ray. First, reflected radiance back along the ray from the surface is given by the term, which gives the emitted and reflected radiance from the surface. This radiance may be attenuated by the participating media; the beam transmittance from the ray origin to the point accounts for this. The second term accounts for the added radiance along the ray due to volume scattering and emission but only up to the point where the ray intersects the surface; points beyond that one don’t affect the radiance along the ray.
15.1.1 Generalized Path Space
Just as it was helpful to express the LTE as a sum over paths of scattering events, it’s also helpful to express the integral equation of transfer in this form. Doing so is a prerequisite for constructing participating medium-aware bidirectional integrators in Chapter 16.
Recall how in Section 14.4.4, the surface form of the LTE was repeatedly substituted into itself to derive the path space contribution function for a path of length
where the throughput was defined as
This previous definition only works for surfaces, but using a similar approach of substituting the integral equation of transfer, a medium-aware path integral can be derived. The derivation is laborious and we will just present the final result here. Refer to Pauly et al. (2000) and Chapter 3 of Jakob’s Ph.D. thesis (2013) for a full derivation.
Previously, integration occurred over a Cartesian product of surface locations . Now, we’ll need a formal way of writing down an integral that can consider an arbitrary sequence of both 2D surface locations and 3D positions in a participating medium . First, we’ll focus only on a specific arrangement of surface and medium vertices encoded in a binary configuration vector . The associated set of paths is given by a Cartesian product of surface locations and medium locations,
The set of all paths of length is the union of the above sets over all possible configuration vectors:
Next, we define a measure, which provides an abstract notion of the volume of a subset that is essential for integration. The measure we’ll use simply sums up the product of surface area and volume associated with the individual vertices in each of the path spaces of specific configurations.
The generalized path contribution can now be written as
Due to the measure defined earlier, this is really a sum of many integrals considering all possible sequences of surface and volume scattering events.
In this framework, the path throughput function is defined as:
It now refers to a generalized scattering distribution function and geometric term . The former simply falls back to the BSDF or phase function (multiplied by ) depending on the type of the vertex .
A generalized form of this geometric term is given by
where the term now also accounts for transmittance between the two points, and
only incorporates the absolute angle cosine between the connection segment and the normal direction when the underlying vertex is located on a surface.