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 upper L Subscript normal s from Section 11.1.4: it gives the change in radiance at a point normal p Subscript in a particular direction omega Subscript due to emission and in-scattered light from other points in the medium:

upper L Subscript normal s Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals upper L Subscript normal e Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis plus sigma Subscript normal s Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis integral Underscript script upper S squared Endscripts p left-parenthesis normal p Subscript Baseline comma omega prime Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega prime Subscript Baseline right-parenthesis normal d omega Subscript Baseline Superscript prime Baseline period

The source term accounts for all of the processes that add radiance to a ray.

The attenuation coefficient, sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis , accounts for all processes that reduce radiance at a point: absorption and out-scattering. The differential equation that describes its effect is

normal d upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline plus t omega Subscript Baseline comma omega Subscript Baseline right-parenthesis equals minus sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis normal d t period

The overall differential change in radiance at a point normal p prime along a ray is found by adding these two effects together to get the integro-differential form of the equation of transfer:

StartFraction partial-differential Over partial-differential t EndFraction upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline plus t omega Subscript Baseline comma omega Subscript Baseline right-parenthesis equals minus sigma Subscript normal t Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis plus upper L Subscript normal s Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis period

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

upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Subscript Baseline right-parenthesis upper L Subscript normal s Superscript Baseline left-parenthesis normal p prime comma minus omega Subscript Baseline right-parenthesis normal d t comma

where normal p prime equals normal p Subscript Baseline plus t omega Subscript (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.

Figure 15.1: The equation of transfer gives the incident radiance at point upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis accounting for the effect of participating media. At each point along the ray, the source term upper L Subscript normal s Superscript Baseline left-parenthesis normal p prime comma minus omega Subscript Baseline right-parenthesis gives the differential radiance added at the point due to scattering and emission. This radiance is then attenuated by the beam transmittance upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Subscript Baseline right-parenthesis from the point normal p prime to the ray’s origin.

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 ( normal p Subscript , omega Subscript ) intersects a surface at some point normal p 0 at a parametric distance t along the ray, then the integral equation of transfer is

upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals upper T Subscript r Baseline left-parenthesis normal p 0 right-arrow normal p Subscript Baseline right-parenthesis upper L Subscript normal o Superscript Baseline left-parenthesis normal p 0 comma minus omega Subscript Baseline right-parenthesis plus integral Subscript 0 Superscript t Baseline upper T Subscript r Baseline left-parenthesis normal p prime right-arrow normal p Subscript Baseline right-parenthesis upper L Subscript normal s Superscript Baseline left-parenthesis normal p prime comma minus omega Subscript Baseline right-parenthesis normal d t Superscript prime Baseline comma

where normal p 0 equals normal p Subscript Baseline plus t omega Subscript is the point on the surface and normal p prime equals normal p Subscript Baseline plus t prime omega Subscript are points along the ray (Figure 15.2).

Figure 15.2: For a finite ray that intersects a surface, the incident radiance, upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis , is equal to the outgoing radiance from the surface, upper L Subscript normal o Superscript Baseline left-parenthesis normal p 0 comma minus omega Subscript Baseline right-parenthesis , times the beam transmittance to the surface plus the added radiance from all points along the ray from normal p Subscript to  normal p 0 .

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 upper L Subscript normal o Superscript 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 normal p 0 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 n

upper P left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis equals ModifyingBelow integral Underscript upper A Endscripts integral Underscript upper A Endscripts midline-horizontal-ellipsis integral Underscript upper A Endscripts With bottom-brace Underscript n minus 1 Endscripts upper L Subscript normal e Baseline left-parenthesis normal p Subscript Baseline Subscript n Baseline right-arrow normal p Subscript Baseline Subscript n minus 1 Baseline right-parenthesis upper T left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis normal d upper A Subscript Baseline left-parenthesis normal p 2 right-parenthesis midline-horizontal-ellipsis normal d upper A Subscript Baseline left-parenthesis normal p Subscript Baseline Subscript n Baseline right-parenthesis comma

where the throughput upper T left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis was defined as

upper T left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis equals product Underscript i equals 1 Overscript n minus 1 Endscripts f Subscript Baseline left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline right-arrow normal p Subscript Baseline Subscript i Baseline right-arrow normal p Subscript Baseline Subscript i minus 1 Baseline right-parenthesis upper G left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline left-right-arrow normal p Subscript Baseline Subscript i Baseline right-parenthesis period

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 upper A Superscript n . Now, we’ll need a formal way of writing down an integral that can consider an arbitrary sequence of both 2D surface locations upper A and 3D positions in a participating medium upper V . First, we’ll focus only on a specific arrangement of n surface and medium vertices encoded in a binary configuration vector bold c . The associated set of paths is given by a Cartesian product of surface locations and medium locations,

script upper P Subscript n Superscript bold c Baseline equals times Underscript i equals 1 Overscript n Endscripts StartLayout Enlarged left-brace 1st Row 1st Column upper A comma 2nd Column if bold c Subscript i Baseline equals 0 2nd Row 1st Column upper V comma 2nd Column if bold c Subscript i Baseline equals 1 period EndLayout

The set of all paths of length n is the union of the above sets over all possible configuration vectors:

script upper P Subscript n Baseline equals union Underscript bold c element-of StartSet 0 comma 1 EndSet Superscript n Baseline Endscripts script upper P Subscript n Superscript bold c Baseline period

Next, we define a measure, which provides an abstract notion of the volume of a subset upper D subset-of-or-equal-to script upper P Subscript n 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.

mu Subscript n Baseline left-parenthesis upper D right-parenthesis equals sigma-summation Underscript bold c element-of StartSet 0 comma 1 EndSet Superscript n Baseline Endscripts mu Subscript n Superscript bold c Baseline left-parenthesis upper D intersection script upper P Subscript n Superscript bold c Baseline right-parenthesis where mu Subscript n Superscript bold c Baseline left-parenthesis upper D right-parenthesis equals integral Underscript upper D Endscripts product Underscript i equals 1 Overscript n Endscripts StartLayout Enlarged left-brace 1st Row 1st Column normal d upper A left-parenthesis normal p Subscript i Baseline right-parenthesis comma 2nd Column if bold c Subscript i Baseline equals 0 2nd Row 1st Column normal d upper V left-parenthesis normal p Subscript i Baseline right-parenthesis comma 2nd Column if bold c Subscript i Baseline equals 1 period EndLayout

The generalized path contribution ModifyingAbove upper P With caret left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis can now be written as

ModifyingAbove upper P With caret left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis equals integral Underscript script upper P Subscript n minus 1 Baseline Endscripts upper L Subscript normal e Baseline left-parenthesis normal p Subscript Baseline Subscript n Baseline right-arrow normal p Subscript Baseline Subscript n minus 1 Baseline right-parenthesis ModifyingAbove upper T With caret left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis normal d mu Subscript n minus 1 Baseline left-parenthesis normal p 2 comma ellipsis comma normal p Subscript normal n Baseline right-parenthesis period

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 ModifyingAbove upper T With caret left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis is defined as:

ModifyingAbove upper T With caret left-parenthesis normal p Subscript Baseline overbar Subscript n Baseline right-parenthesis equals product Underscript i equals 1 Overscript n minus 1 Endscripts ModifyingAbove f With caret left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline right-arrow normal p Subscript Baseline Subscript i Baseline right-arrow normal p Subscript Baseline Subscript i minus 1 Baseline right-parenthesis ModifyingAbove upper G With caret left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline left-right-arrow normal p Subscript Baseline Subscript i Baseline right-parenthesis period

It now refers to a generalized scattering distribution function ModifyingAbove f With caret and geometric term ModifyingAbove upper G With caret . The former simply falls back to the BSDF or phase function (multiplied by sigma Subscript normal s ) depending on the type of the vertex normal p Subscript i .

ModifyingAbove f With caret left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline right-arrow normal p Subscript Baseline Subscript i Baseline right-arrow normal p Subscript Baseline Subscript i minus 1 Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column sigma Subscript normal s Baseline p Subscript Baseline left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline right-arrow normal p Subscript Baseline Subscript i Baseline right-arrow normal p Subscript Baseline Subscript i minus 1 Baseline right-parenthesis comma 2nd Column if normal p Subscript normal i Baseline element-of upper V 2nd Row 1st Column f Subscript Baseline left-parenthesis normal p Subscript Baseline Subscript i plus 1 Baseline right-arrow normal p Subscript Baseline Subscript i Baseline right-arrow normal p Subscript Baseline Subscript i minus 1 Baseline right-parenthesis comma 2nd Column if normal p Subscript normal i Baseline element-of upper A period EndLayout

Equation (14.14) in Section 14.4.3 originally defined the geometric term upper G as

upper G left-parenthesis normal p Subscript Baseline left-right-arrow normal p Superscript prime Baseline right-parenthesis equals upper V left-parenthesis normal p Subscript Baseline left-right-arrow normal p Superscript prime Baseline right-parenthesis StartFraction StartAbsoluteValue cosine theta EndAbsoluteValue StartAbsoluteValue cosine theta Superscript prime Baseline EndAbsoluteValue Over parallel-to normal p Subscript Baseline minus normal p Superscript prime Baseline parallel-to EndFraction period

A generalized form of this geometric term is given by

ModifyingAbove upper G With caret left-parenthesis normal p Subscript Baseline left-right-arrow normal p Superscript prime Baseline right-parenthesis equals upper V left-parenthesis normal p Subscript Baseline left-right-arrow normal p Superscript prime Baseline right-parenthesis upper T Subscript r Baseline left-parenthesis normal p Subscript Baseline right-arrow normal p Superscript prime Baseline right-parenthesis StartFraction upper C Subscript normal p Sub Subscript Subscript Baseline left-parenthesis normal p Subscript Baseline comma normal p Superscript prime Baseline right-parenthesis upper C Subscript normal p Sub Superscript prime Subscript Baseline left-parenthesis normal p prime comma normal p Subscript Baseline right-parenthesis Over parallel-to normal p Subscript Baseline minus normal p Superscript prime Baseline parallel-to EndFraction comma

where the upper T Subscript r term now also accounts for transmittance between the two points, and

upper C Subscript normal p Sub Subscript Baseline left-parenthesis normal p Subscript Baseline comma normal p Superscript prime Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartAbsoluteValue bold n Subscript normal p Sub Subscript Subscript Baseline dot StartFraction normal p Subscript Baseline minus normal p Superscript prime Baseline Over double-vertical-bar normal p Subscript Baseline minus normal p Superscript prime Baseline double-vertical-bar EndFraction EndAbsoluteValue comma 2nd Column if normal p Subscript Baseline is a surface vertex 2nd Row 1st Column 1 comma 2nd Column otherwise EndLayout

only incorporates the absolute angle cosine between the connection segment and the normal direction when the underlying vertex normal p Subscript is located on a surface.