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 lamda carries energy

upper Q equals StartFraction h c Over lamda EndFraction comma

where c is the speed of light, 299,472,458 normal m slash normal s , and h is Planck’s constant, h almost-equals 6.626 times 10 Superscript negative 34 Baseline normal m squared normal k normal g slash normal s .

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:

normal upper Phi equals limit Underscript normal upper Delta t right-arrow 0 Endscripts StartFraction normal upper Delta upper Q Over normal upper Delta t EndFraction equals StartFraction normal d upper Q Over normal d t EndFraction period

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

For example, given a light that emitted upper Q equals 200,000 normal upper J 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

normal upper Phi equals 200,000 normal upper J slash 3600 normal s almost-equals 55.6 normal upper W period

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

upper Q equals integral Subscript t 0 Superscript t 1 Baseline normal upper Phi left-parenthesis t right-parenthesis normal d t period

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.

Figure 5.6: Radiant flux, normal upper Phi , measures energy passing through a surface or region of space. Here, flux from a point light source is measured at spheres that surround the light.

Irradiance and Radiant Exitance

Any measurement of flux requires an area over which photons per time is being measured. Given a finite area upper A , we can define the average density of power over the area by upper E equals normal upper Phi slash upper A . 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 squared . (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  r ,

upper E Subscript Superscript Baseline equals StartFraction normal upper Phi Over 4 pi r squared EndFraction period

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 normal p Subscript :

upper E left-parenthesis normal p Subscript Baseline right-parenthesis equals limit Underscript normal upper Delta upper A right-arrow 0 Endscripts StartFraction normal upper Delta normal upper Phi left-parenthesis normal p Subscript Baseline right-parenthesis Over normal upper Delta upper A EndFraction equals StartFraction normal d normal upper Phi left-parenthesis normal p Subscript Baseline right-parenthesis Over normal d upper A EndFraction period

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

normal upper Phi equals integral Underscript upper A Endscripts upper E left-parenthesis normal p Subscript Baseline right-parenthesis normal d upper A period

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 upper A and flux normal upper Phi 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 upper A 1 is equal to upper A . Irradiance at any point inside upper A 1 is then

upper E 1 equals StartFraction normal upper Phi Over upper A EndFraction period

However, if the light is at an angle to the surface, the area on the surface receiving light is larger. If upper A is small, then the area receiving flux, upper A 2 , is roughly upper A slash cosine theta . For points inside upper A 2 , the irradiance is therefore

upper E 2 equals StartFraction normal upper Phi cosine theta Over upper A EndFraction period

Figure 5.7: Lambert’s Law. Irradiance arriving at a surface varies according to the cosine of the angle of incidence of illumination, since illumination is over a larger area at larger incident angles.

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 normal p Subscript ; if we project the shaded object onto that circle, some length of the circle s will be covered by its projection. The arc length of s (which is the same as the angle theta ) is the angle subtended by the object. Planar angles are measured in radians.

Figure 5.8: Planar Angle. The planar angle of an object as seen from a point normal p Subscript is equal to the angle it subtends as seen from normal p Subscript or, equivalently, as the length of the arc s on the unit sphere.

The solid angle extends the 2D unit circle to a 3D unit sphere (Figure 5.9). The total area s is the solid angle subtended by the object. Solid angles are measured in steradians (sr). The entire sphere subtends a solid angle of 4 pi normal s normal r , and a hemisphere subtends  2 pi normal s normal r .

Figure 5.9: Solid Angle. The solid angle s subtended by an object c in three dimensions is computed by projecting c onto the unit sphere and measuring the area of its projection.

The set of points on the unit sphere centered at a point normal p Subscript can be used to describe the vectors anchored at normal p Subscript . We will usually use the symbol omega Subscript to indicate these directions, and we will use the convention that omega Subscript 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 normal upper W slash normal s normal r . Over the entire sphere of directions, we have

upper I equals StartFraction normal upper Phi Over 4 pi EndFraction comma

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

upper I equals limit Underscript normal upper Delta omega Subscript Baseline right-arrow 0 Endscripts StartFraction normal upper Delta normal upper Phi Over normal upper Delta omega Subscript Baseline EndFraction equals StartFraction normal d normal upper Phi Over normal d omega Subscript Baseline EndFraction period

As usual, we can go back to power by integrating intensity: given intensity as a function of direction upper I left-parenthesis omega Subscript Baseline right-parenthesis , we can integrate over a finite set of directions normal upper Omega to recover the intensity:

normal upper Phi equals integral Underscript normal upper Omega Endscripts upper I left-parenthesis omega Subscript Baseline right-parenthesis normal d omega Subscript Baseline period

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, upper L Subscript Superscript . Irradiance and radiant exitance give us differential power per differential area at a point normal p Subscript , 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

upper L left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis equals limit Underscript normal upper Delta omega Subscript Baseline right-arrow 0 Endscripts StartFraction normal upper Delta upper E Subscript omega Sub Subscript Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline right-parenthesis Over normal upper Delta omega Subscript Baseline EndFraction equals StartFraction normal d upper E Subscript omega Sub Subscript Subscript Superscript Baseline left-parenthesis normal p Subscript Baseline right-parenthesis Over normal d omega Subscript Baseline EndFraction comma

where we have used upper E Subscript omega Sub Subscript Superscript to denote irradiance at the surface that is perpendicular to the direction omega Subscript . In other words, radiance is not measured with respect to the irradiance incident at the surface normal p Subscript lies on. In effect, this change of measurement area serves to eliminate the cosine theta 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

upper L Subscript Superscript Baseline equals StartFraction normal d normal upper Phi Subscript Baseline Over normal d omega Subscript Baseline normal d upper A Subscript Superscript up-tack Baseline EndFraction comma
(5.2)

where normal d upper A Subscript Superscript up-tack is the projected area of normal d upper A Subscript on a hypothetical surface perpendicular to omega Subscript (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 normal d omega Subscript becomes very small and as the local area of interest on the surface normal d upper A Subscript also becomes very small.

Figure 5.10: Radiance upper L Subscript Superscript is defined as flux per unit solid angle normal d omega Subscript per unit projected area normal d upper A Subscript Superscript up-tack .

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 upper L 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

StartLayout 1st Row 1st Column upper L Superscript plus Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis 2nd Column equals limit Underscript t right-arrow 0 Superscript plus Baseline Endscripts upper L left-parenthesis normal p Subscript Baseline plus t bold n Subscript normal p Sub Subscript Subscript Baseline comma omega Subscript Baseline right-parenthesis comma 2nd Row 1st Column upper L Superscript minus Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis 2nd Column equals limit Underscript t right-arrow 0 Superscript minus Baseline Endscripts upper L left-parenthesis normal p Subscript Baseline plus t bold n Subscript normal p Sub Subscript Subscript Baseline comma omega Subscript Baseline right-parenthesis comma EndLayout
(5.3)

where bold n Subscript normal p Sub Subscript is the surface normal at normal p Subscript . 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 normal p Subscript 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 upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis (Figure 5.11). The function that describes the outgoing reflected radiance from the surface at that point is denoted by upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis . Note that in both cases the direction vector omega Subscript is oriented to point away from normal p Subscript , but be aware that some authors use a notation where omega Subscript is reversed for upper L Subscript normal i Superscript terms so that it points toward  normal p Subscript .

Figure 5.11: (a) The incident radiance function upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis describes the distribution of radiance arriving at a point as a function of position and direction. (b) The exitant radiance function upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis gives the distribution of radiance leaving the point. Note that for both functions, omega Subscript is oriented to point away from the surface, and, thus, for example, upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis gives the radiance arriving on the other side of the surface than the one where omega Subscript  lies.

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

StartLayout 1st Row 1st Column upper L Subscript normal i Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis 2nd Column equals StartLayout Enlarged left-brace 1st Row 1st Column upper L Superscript plus Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis comma 2nd Column omega Subscript Baseline dot bold n Subscript normal p Sub Subscript Baseline greater-than 0 2nd Row 1st Column upper L Superscript minus Baseline left-parenthesis normal p Subscript Baseline comma minus omega Subscript Baseline right-parenthesis comma 2nd Column omega Subscript Baseline dot bold n Subscript normal p Sub Subscript Subscript Baseline less-than 0 EndLayout 2nd Row 1st Column upper L Subscript normal o Superscript Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis 2nd Column equals StartLayout Enlarged left-brace 1st Row 1st Column upper L Superscript plus Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis comma 2nd Column omega Subscript Baseline dot bold n Subscript normal p Sub Subscript Baseline greater-than 0 2nd Row 1st Column upper L Superscript minus Baseline left-parenthesis normal p Subscript Baseline comma omega Subscript Baseline right-parenthesis comma 2nd Column omega Subscript Baseline dot bold n Subscript normal p Sub Subscript Subscript Baseline less-than 0 EndLayout EndLayout

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), upper L is continuous, so upper L Superscript plus Baseline equals upper L Superscript minus , which means

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

In other words, upper L Subscript normal i Superscript and upper L Subscript normal o Superscript 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 upper V left-parenthesis lamda right-parenthesis , 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 upper Y Subscript ; it related to spectral radiance upper L left-parenthesis lamda right-parenthesis by

upper Y Subscript Baseline equals integral Underscript lamda Endscripts upper L left-parenthesis lamda right-parenthesis upper V left-parenthesis lamda right-parenthesis normal d lamda Subscript period Baseline

Luminance and the spectral response curve upper V left-parenthesis lamda right-parenthesis are closely related to the XYZ representation of color (Section 5.2.1). The CIE upper Y left-parenthesis lamda right-parenthesis tristimulus curve was chosen to be proportional to upper V left-parenthesis lamda right-parenthesis so that

upper Y Subscript Baseline equals 683 integral Underscript lamda Endscripts upper L left-parenthesis lamda right-parenthesis upper Y left-parenthesis lamda right-parenthesis normal d lamda Subscript period Baseline

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

Table 5.1: Representative Luminance Values for a Number of Lighting Conditions.

ConditionLuminance (cd/m squared , or nits)
Sun at horizon 600,000
60-watt lightbulb 120,000
Clear sky 8,000
Typical office 100–1,000
Typical computer display 1–100
Street lighting 1–10
Cloudy moonlight 0.25

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

Table 5.2: Radiometric Measurements and Their Photometric Analogs.

RadiometricUnit PhotometricUnit
Radiant energy joule ( normal upper Q ) Luminous energy talbot ( normal upper T )
Radiant flux watt ( normal upper W ) Luminous flux lumen ( normal l normal m )
Intensity normal upper W slash normal s normal r Luminous intensity normal l normal m slash normal s normal r = candela ( normal c normal d )
Irradiance normal upper W slash normal m squared Illuminance normal l normal m slash normal m squared = lux ( normal l normal x )
Radiance normal upper W slash left-parenthesis normal m squared normal s normal r right-parenthesis Luminance normal l normal m slash left-parenthesis normal m squared normal s normal r right-parenthesis equals normal c normal d slash normal m squared equals normal n normal i normal t