Typically the world of real-time rendering uses relatively esoteric metrics for the illumination characteristics of a scene. This is largely intentional as the most commonly used lighting models have very little to do with the actual physics involved. Terms like energy, radiance, reflectance, albedo and so on are used fast and loose without much thought given to the units of measure or the phenomena being modeled. Fortunately (or unfortunately) such days are number.

The ground work for the next generation of rendering engines has started and nearly all of them involve a physically derived reflection model and some type of dynamic, global-illumination scheme that accounts for 2nd and even 3rd order reflection events in combination with this generation’s direct (1st order shadows) and indirect (nth order shadows or ambient) occlusion computations. Understanding, using and augmenting this level of sophistication requires real-time graphics programmers to start talking and coding like their offline counterparts in the ray-tracing world. The prerequisites of this necessary adaptation are a basic grasp on vector calculus, multi-dimensional integrals and the physics of light; specifically, radiometric quantities.

The most important radiometric quantities are a set of four metrics that can be used to model the propagation of light throughout a scene. They are: radiant flux, irradiance, intensity, and radiance and are written as Φ, E I, and L respectively. Radiant flux is often times referred to as flux, a useful abbreviation that shall be used here as it lessens the likelihood of confusing it with radiance.

Flux: Φ

Flux is the simplest of all and is used to quantify the total amount of energy passing through a surface per unit of time and is measured in joules per second or watts. For the purposes of rendering it is useful to restrict the energy being measured to photons of light that fall in the visible spectrum while ignoring wavelengths beyond ultraviolet and below infrared as they (usually) have no effect on the visual appearance of the scene. If you were to place an imaginary sphere around a light source then the flux would be the amount of energy passing through the surface of the sphere; it is therefore a natural parameterization of light sources and should be used as such. Note that the radius of the sphere can take on any arbitrary value and as long as the light source is inside of the sphere the measured flux is constant.

Irradiance: E = dΦ / d**A**^{┴}

The next quantity, irradiance, measures the area density of flux where the area is perpendicular to the direction of flow. Once again we start by placing an imaginary sphere around a light source. We know that the flux through the surface of the sphere is equal to the amount of energy passing through it per unit time; the irradiance can be calculated by dividing the total flux by the area of the sphere. Notice that the radius of the sphere changes the measurement; as the radius increases so too does the area, spreading out the flux and therefore decreasing the irradiance. This is where the familiar inverse square law comes from by which the intensity of a light source is scaled by the inverse of the radius squared. It’s worth noting that there is no such thing as a radial emitting light source with an intensity that does not falloff as a linear function of the inverse of the distance squared; such a phenomenon is physically impossible. While building an engine it may be tempting to include more exotic falloff functions; I highly discourage this. It may be easy to achieve the desired amount of illumination in a scene by faking the physics but doing so will lace the rendered image with a subtle strangeness; indeed, the image may look spectacular overall but it will also have a hard to quantify artificialness to it.

Intensity: I = dΦ / dω

Intensity is very similar to irradiance. Instead of measuring the flux per unit area intensity measures the flux per solid angle. A solid angle is the 2-dimensional counterpart to an angle. Just as an angle can be thought of as the length of an arc traced out on the unit circle a solid angle can be thought of as the area of a circular dome traced out on the unit sphere.

Radiance: L = dΦ / ( dω d**A**^{┴})

While the hardest to conceptualize radiance can be thought of as the most fundamental off all radiometric quantities. It is constant along rays through empty space and if its value is known all the other radiometric quantities can be derived from it by integrating against the appropriate dimensions. These two characteristics make radiance the ideal value for managing the propagation of light throughout a scene. Mathematically speaking radiance is the flux density per solid angle per unit area. Irradiance can be calculated from radiance by integrating across a solid angle; likewise, intensity can be calculated from radiance by integrating across a perpendicular area.

Thank you for the explanation! Great! What’s missing for me is what this means from a visualization standpoint. How do these parameters affect what I might see – e.g. impact on how “real” the image becomes. Thank you!