5.4 Radiometry

Radiometry provides a set of ideas and mathematical tools to describe light propagation and reflection. It forms the basis of the derivation of the rendering algorithms that will be used throughout the rest of this book. Interestingly enough, radiometry wasn’t originally derived from first principles using the physics of light but was built on an abstraction of light based on particles flowing through space. As such, effects like polarization of light do not naturally fit into this framework, although connections have since been made between radiometry and Maxwell’s equations, giving radiometry a solid basis in physics.

Radiative transfer is the phenomenological study of the transfer of radiant energy. It is based on radiometric principles and operates at the geometric optics level, where macroscopic properties of light suffice to describe how light interacts with objects much larger than the light’s wavelength. It is not uncommon to incorporate phenomena from wave optics models of light, but these results need to be expressed in the language of radiative transfer’s basic abstractions. (Preisendorfer (1965) has connected radiative transfer theory to Maxwell’s classical equations describing electromagnetic fields. His framework both demonstrates their equivalence and makes it easier to apply results from one worldview to the other. More recent work was done in this area by Fante (1981).)

In this manner, it is possible to describe interactions of light with objects of approximately the same size as the wavelength of the light, and thereby model effects like dispersion and interference. At an even finer level of detail, quantum mechanics is needed to describe light’s interaction with atoms. Fortunately, direct simulation of quantum mechanical principles is unnecessary for solving rendering problems in computer graphics, so the intractability of such an approach is avoided.

In pbrt, we will assume that geometric optics is an adequate model for the description of light and light scattering. This leads to a few basic assumptions about the behavior of light that will be used implicitly throughout the system:

• Linearity: The combined effect of two inputs to an optical system is always equal to the sum of the effects of each of the inputs individually.
• Energy conservation: When light scatters from a surface or from participating media, the scattering events can never produce more energy than they started with.
• No polarization: We will ignore polarization of the electromagnetic field; therefore, the only relevant property of light is its distribution by wavelength (or, equivalently, frequency).
• No fluorescence or phosphorescence: The behavior of light at one wavelength is completely independent of light’s behavior at other wavelengths or times. As with polarization, it is not too difficult to include these effects, but they would add relatively little practical value to the system.
• Steady state: Light in the environment is assumed to have reached equilibrium, so its radiance distribution isn’t changing over time. This happens nearly instantaneously with light in realistic scenes, so it is not a limitation in practice. Note that phosphorescence also violates the steady-state assumption.

The most significant loss from adopting a geometric optics model is that diffraction and interference effects cannot easily be accounted for. As noted by Preisendorfer (1965), this is a hard problem to fix because, for example, the total flux over two areas isn’t necessarily equal to the sum of the power received over each individual area in the presence of those effects (p. 24).

5.4.1 Basic Quantities

There are four radiometric quantities that are central to rendering: flux, irradiance / radiant exitance, intensity, and radiance. They can each be derived from energy (measured in joules) by successively taking limits over time, area, and directions. All of these radiometric quantities are in general wavelength dependent. For the remainder of this chapter, we will not make this dependence explicit, but this property is important to keep in mind.

Energy

Our starting point is energy, which is measured in joules (J). Sources of illumination emit photons, each of which is at a particular wavelength and carries a particular amount of energy. All of the basic radiometric quantities are effectively different ways of measuring photons. A photon at wavelength carries energy

where is the speed of light, , and is Planck’s constant, .

Flux

Energy measures work over some period of time, though under the steady-state assumption generally used in rendering, we’re mostly interested in measuring light at an instant. Radiant flux, also known as power, is the total amount of energy passing through a surface or region of space per unit time. Radiant flux can be found by taking the limit of differential energy per differential time:

Its units are joules/second (J/s), or more commonly, watts (W).

For example, given a light that emitted over the course of an hour, if the same amount of energy was emitted at all times over the hour, we can find that the light source’s flux was

Conversely, given flux as a function of time, we can integrate over a range of times to compute the total energy:

Note that our notation here is slightly informal: among other issues, because photons are actually discrete quanta, it’s not really meaningful to take limits that go to zero for differential time. For the purposes of rendering, where the number of photons is enormous with respect to the measurements we’re interested in, this detail isn’t problematic in practice.

Total emission from light sources is generally described in terms of flux. Figure 5.6 shows flux from a point light source measured by the total amount of energy passing through imaginary spheres around the light. Note that the total amount of flux measured on either of the two spheres in Figure 5.6 is the same—although less energy is passing through any local part of the large sphere than the small sphere, the greater area of the large sphere means that the total flux is the same.

Irradiance and Radiant Exitance

Any measurement of flux requires an area over which photons per time is being measured. Given a finite area , we can define the average density of power over the area by . This quantity is either irradiance (E), the area density of flux arriving at a surface, or radiant exitance (M), the area density of flux leaving a surface. These measurements have units of W/m. (The term irradiance is sometimes also used to refer to flux leaving a surface, but for clarity we’ll use different terms for the two cases.)

For the point light source example in Figure 5.6, irradiance at a point on the outer sphere is less than the irradiance at a point on the inner sphere, since the surface area of the outer sphere is larger. In particular, if the point source is illuminating the same amount of illumination in all directions, then for a sphere in this configuration that has radius ,

This fact explains why the amount of energy received from a light at a point falls off with the squared distance from the light.

More generally, we can define irradiance and radiant exitance by taking the limit of differential power per differential area at a point :

We can also integrate irradiance over an area to find power:

The irradiance equation can also help us understand the origin of Lambert’s law, which says that the amount of light energy arriving at a surface is proportional to the cosine of the angle between the light direction and the surface normal (Figure 5.7). Consider a light source with area and flux that is illuminating a surface. If the light is shining directly down on the surface (as on the left side of the figure), then the area on the surface receiving light is equal to . Irradiance at any point inside is then

However, if the light is at an angle to the surface, the area on the surface receiving light is larger. If is small, then the area receiving flux, , is roughly . For points inside , the irradiance is therefore

Solid Angle and Intensity

In order to define intensity, we first need to define the notion of a solid angle. Solid angles are just the extension of 2D angles in a plane to an angle on a sphere. The planar angle is the total angle subtended by some object with respect to some position (Figure 5.8). Consider the unit circle around the point ; if we project the shaded object onto that circle, some length of the circle will be covered by its projection. The arc length of (which is the same as the angle ) is the angle subtended by the object. Planar angles are measured in radians.

The solid angle extends the 2D unit circle to a 3D unit sphere (Figure 5.9). The total area is the solid angle subtended by the object. Solid angles are measured in steradians (sr). The entire sphere subtends a solid angle of , and a hemisphere subtends .

The set of points on the unit sphere centered at a point can be used to describe the vectors anchored at . We will usually use the symbol to indicate these directions, and we will use the convention that is a normalized vector.

Consider now an infinitesimal light source emitting photons. If we center this light source within the unit sphere, we can compute the angular density of emitted power. Intensity, denoted by I, is this quantity; it has units . Over the entire sphere of directions, we have

but more generally we’re interested in taking the limit of a differential cone of directions:

As usual, we can go back to power by integrating intensity: given intensity as a function of direction , we can integrate over a finite set of directions to recover the power:

Intensity describes the directional distribution of light, but it is only meaningful for point light sources.

Radiance

The final, and most important, radiometric quantity is radiance, . Irradiance and radiant exitance give us differential power per differential area at a point , but they don’t distinguish the directional distribution of power. Radiance takes this last step and measures irradiance or radiant exitance with respect to solid angles. It is defined by

where we have used to denote irradiance at the surface that is perpendicular to the direction . In other words, radiance is not measured with respect to the irradiance incident at the surface lies on. In effect, this change of measurement area serves to eliminate the term from Lambert’s law in the definition of radiance.

Radiance is the flux density per unit area, per unit solid angle. In terms of flux, it is defined by

where is the projected area of on a hypothetical surface perpendicular to (Figure 5.10). Thus, it is the limit of the measurement of incident light at the surface as a cone of incident directions of interest becomes very small and as the local area of interest on the surface also becomes very small.

Of all of these radiometric quantities, radiance will be the one used most frequently throughout the rest of the book. An intuitive reason for this is that in some sense it’s the most fundamental of all the radiometric quantities; if radiance is given, then all of the other values can be computed in terms of integrals of radiance over areas and directions. Another nice property of radiance is that it remains constant along rays through empty space. It is thus a natural quantity to compute with ray tracing.

5.4.2 Incident and Exitant Radiance Functions

When light interacts with surfaces in the scene, the radiance function is generally not continuous across the surface boundaries. In the most extreme case of a fully opaque surface (e.g., a mirror), the radiance function slightly above and slightly below a surface could be completely unrelated.

It therefore makes sense to take one-sided limits at the discontinuity to distinguish between the radiance function just above and below

where is the surface normal at . However, keeping track of one-sided limits throughout the text is unnecessarily cumbersome.

We prefer to solve this ambiguity by making a distinction between radiance arriving at the point (e.g., due to illumination from a light source) and radiance leaving that point (e.g., due to reflection from a surface).

Consider a point on the surface of an object. There is some distribution of radiance arriving at the point that can be described mathematically by a function of position and direction. This function is denoted by (Figure 5.11). The function that describes the outgoing reflected radiance from the surface at that point is denoted by . Note that in both cases the direction vector is oriented to point away from , but be aware that some authors use a notation where is reversed for terms so that it points toward .

There is a simple relation between these more intuitive incident and exitant radiance functions and the one-sided limits from Equation (5.3):

Throughout the book, we will use the idea of incident and exitant radiance functions to resolve ambiguity in the radiance function at boundaries.

Another property to keep in mind is that at a point in space where there is no surface (i.e. in free space), is continuous, so , which means

In other words, and only differ by a direction reversal.

5.4.3 Luminance and Photometry

All of the radiometric measurements like flux, radiance, and so forth have corresponding photometric measurements. Photometry is the study of visible electromagnetic radiation in terms of its perception by the human visual system. Each spectral radiometric quantity can be converted to its corresponding photometric quantity by integrating against the spectral response curve , which describes the relative sensitivity of the human eye to various wavelengths. ^†

Luminance measures how bright a spectral power distribution appears to a human observer. For example, luminance accounts for the fact that an SPD with a particular amount of energy in the green wavelengths will appear brighter to a human than an SPD with the same amount of energy in blue.

We will denote luminance by ; it related to spectral radiance by

Luminance and the spectral response curve are closely related to the XYZ representation of color (Section 5.2.1). The CIE tristimulus curve was chosen to be proportional to so that

The units of luminance are candelas per meter squared (), where the candela is the photometric equivalent of radiant intensity. Some representative luminance values are given in Table 5.1.

All of the other radiometric quantities that we have introduced in this chapter have photometric equivalents; they are summarized in Table 5.2. ^†