Radiometry for CG

Radiometry is at the core of practically every rendering algorithm out there. Each pixel in the frame buffer is just a small surface on which light reflected by objects in the scene falls onto. The unique goal of a rendering algorithm is to compute the amount of light passing through every pixel in the frame buffer. For this reason, I think it is important to have, at least, a basic understanding of some of the concepts studied by radiometry. I’ve written this article basically as a reference for myself, but, hopefully, others might find it useful although this stuff is well covered in any good computer graphics text.

1. Radiometric quantities

1.1 Flux (Φ)

The formal definition for flux, also know as radiant flux or power, is the total energy passing through a surface or region of space per unit time. Imagine that you define a region in space, or a region in a surface (which is nothing more than a region in space). To calculate the flux in that region, you have to “count” the number of photons passing through that region from time t to time t + δt.

flux_formula

Obviously, Q must be defined as a function of time in order for the idea of differentiating it to make sense. As Q is energy, it’s defined in Joules, and time is defined in seconds, the unit for flux will be J/s. J/s is often called Watts. Flux can be seen as a function of position and direction. We can measure the flux at point x by measuring the flux at a small differential area around x. We can also measure the flux generated by photons coming from a certain direction by “counting” the number of photons going through the surface from that direction.

1.2 Irradiance (E) and Radiant excitance (M)

Irradiance (E) is a function which describes the area density of flux arriving at any given point on the surface, and randiant excitance (M) describes the area density of flux leaving a surface. Irradiance will allow us to calculate how much energy is passing through any point in the surface per unit area, and Radiant excitance how much energy is leaving from any point in the surface per unit area. Basically, irradiance answers the question “How much light arrives at point p?” and radiant excitance “How much light leaves from point p?” In general, flux will not be constant over the surface, that’s the reason why irradiance is generally defined as: irradiance_formula

Which means that irradiance at a point x equals the flux at point x over the differential area around x. δA needs to be small because the radiant flux distribution over a given surface is generally not constant. We need to consider the smallest region possible, at least, small enough that we can assume the incident flux to be constant over that area ( i.e evaluating Φ(x) for any point x in that area will produce the same result )

1.3 Radiant Intensity

Radiant intensity is flux density per solid angle. It describes the directional distribution of light. Radiant flux will be arriving to a surface from any direction, radiant intensity describes how much flux is arriving per unit solid angle. It answers the question How much light is arriving at the surface from any given direction? If we know how much light passes through the surface per unit solid angle, we can compute how much light is arriving from any cone of directions.

1.4 Radiance (L)

Radiance is the flux density per unit projected area, per unit solid angle. It is a combination of the irradiance and radiant intensity concepts. We can think of radiance as irradiance per unit solid angle ( or as radiant intensity per unit area ). Radiance will answer the question “How much light arrives/leaves at/from point X from/in direction ω?”  It “counts” the number of photons arriving at a unit area of a surface from a unit solid angle. With that measure, we can compute the flux passing through a small patch of a surface coming from a given cone of directions. Radiance is important to us because that is what an optical system will perceive, it is an indicator of how bright a surface will appear. This is the value we have to calculate for every pixel in the screen! It is also important since all the other quantities can be derived from it as we will see later.  Radiance is defined as:

radiance_formula

where δA * cos(ω) is the projected area of δA on a hypothetical surface perpendicular to ω. As you can see,  we are differentiating flux against area and direction. Obviously, flux has to be defined as a function of position and direction in order to be differentiable with respect to area and direction. Another way of writing the previous formula would be this:

radiance2

Radiance is a function of position and direction L(x,ω), making it a 5D function, also know as plenoptic function ( three coordinates for point x and two angles for direction ). The units of radiance are W/(sr*m²).

2. Using radiometric quantities

Say we want to calculate flux passing through a surface, and all we know is how to calculate irradiance for any point in that surface. We know that irradiance ( E ) is defined as:

irradiance_formula

So, if we integrate over area we get

1

Which means that Φ = ∫ E(x) δA. In other words, if we integrate irradiance over the area, we get the flux passing through that area.

As I said, radiance is an important quantity, not only because that’s what our eyes ( and camera sensors ) perceive, but also because it is possible to derive any other radiometric quantity from it. Lets try to derive irradiance from radiance. We know that:

radiance_formula

So,2

If we integrate respect solid angle on both sides:

3

By definition, irradiance is δΦ / δA, so, finally

4

Which means that irradiance is the integral ( infinite sum ) of radiance coming from all the directions over the surface.

Similarly, if we integrate radiance over area, we would get the radiant intensity at the surface, that is, the total flux passing through the surface from any given direction.

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