Word of Warning: Introduction to Radiometry.

Radiometry is really the science that is concerned with measuring the distribution of electromagnetic radiation being transferred from a source to a receiver (such as for instance the human eye).

You might find this first part abstract. In the following chapter we will show why these units are important, how we can use them and also their relationships with each other.

Radiant Energy $$\scriptsize Q$$: amount of energy produced by a given surface of a given area in a given period of time. Measured in Joules.

Radiant energy is of course a fundamental unit however we will not really use it directly in computer graphics. In our context a joule can be defined as the work required to produce one watt of power per second, or one "watt second". Just look up (as an exercise) for the definition of watt and you will find the relationship between watt and joule. Energy is not an imagination of the mind, you can measure it: imagine for instance that you relate energy to the number of particles hitting the surface of a detector. Remember too that one photon has an energy of $$\scriptsize E = h \nu$$, where $$\scriptsize h$$ is the Plank constant and $$\scriptsize \nu$$ is the photon's frequency. Generally, in the case of light, photons are responsible for energy and radiant energy really expresses the rate at which this energy is transferred from a point (or more exactly a surface) to another point.

Radiant Power or Radiant Flux (or power) $$\scriptsize \Phi$$: time rate of flow of radiant energy. Measured in Watts ($$\scriptsize J.s^{-1}$$).

Figure 1: photons passing through a region of space. Radiant flux measures the amount of energy transferring through a region of space or a surface per unit time.

In some other references this unit is actually defined as the overall amount of radiant energy transferred through a surface or a region of space per unit time. However we prefer to use the term flow rate in the definition because it emphasizes the fact that we are dealing here with a flow of energy over a period of time. Lets explain. As often mentioned in the lessons, light can be seen as a collection of photons traveling through space. Imagine that you define in space a small region as showed in figure 1. What you are really interested in, is to count the number of photons passing through this small region from time $$\scriptsize t$$ to time $$\scriptsize t + \Delta t$$. Generally a term $$\scriptsize \Delta t$$ denotes a small value of t but in this case, the time during which you measure the number of photons passing through the region you defined can be anything you like. In essence what you are trying to do, is get a sense of how much light the region you define is being exposed to (which is a transfer of energy from the region where the photons come from to the region of interest) by counting the number of photons passing through it over a given period of time. Of course, the more photons the greater the energy (by having more photons passing through your region, you would actually increase the radiant energy flow rate). And if you divide that number by $$\scriptsize \Delta t$$, then the number you get expresses a measure of the radiant flux for that region. Radiant flux can be written as:

$$\scriptsize \Phi = {dQ \over dt}$$

The symbol for radiant flux is the greek capital letter $$\scriptsize \Phi$$ (phi). if you are not familiar with this notation, the letter d in the denominator and numerator stands for difference but the way you want to see it is really the amount of radiant energy over time. What is important here is the concept of flow (or rate) which you can see as water moving through a pipe. The concept is not limited to three-dimensional regions but also applies to surfaces of course: generally when we measure radiant flux, we are interested in the total quantity of flux over the entire surface rather than the flux per unit area. Thus, when speaking of radiant flux, it is a good idea to specify the extent (how big is the surface or the three-dimensional region) of the surface or region that is being measured.

Irradiance $$\scriptsize E$$: area density of radiant flux or the radiant flux per unit area in a specified area that is passing incident on, passing through, or emerging from a point in the specified surface. Measured in $$\scriptsize W.m^{-2}$$.

Figure 2: irradiance consists of measuring light coming from all directions in the hemisphere above a small area centred at P. In this figures we have only represented a few light rays, but the transparent hemisphere indicates that light comes from all directions (there should be an infinity of these rays arriving on the small surface element which in computer graphics we often call dA).

To answer the question "how much light is reflected back from P in a given direction (generally the viewing direction), we first need to know how much light arrives at P from all directions in the hemisphere above the point. This is in essence, what irradiance is about. It answers the questions "how much light arrives at P" where rather than considering P as a point we do look at how much light falls in a generally small area around P (a point in geometry has no dimension, no area, no volume or no length. Thus measuring anything at something that has "no part" as Euclid put it, is not possible which is the reason we consider an infinitesimal area around P instead. Why we consider a very small area is explained further down). Light needs to be "collected" from all directions in the hemisphere above the point (or small area of interest). Note though that in some case we will also collect this information for the hemisphere below P (it might be needed when dealing with transparent surfaces for instance). Not matter how small the area around P is, dividing the radiant flux by this area will give a measure of the total amount of radiant energy per unit area (generally meter square). Irradiance can be written as:

$$\scriptsize E = {{d\Phi} \over {ds_0}}$$

where $$\scriptsize d\Theta$$ is radiant flux and $$\scriptsize ds_0$$ is an area in the surface. The unit for irradiance is $$\scriptsize watt.m^{-2}$$ (figure 2). The reason why generally $$\scriptsize ds_0$$ needs to be small is because the radiant flux distribution over a given surface is generally not constant. Imagine for example what happens at the boundary of a shadow cast on a surface (figure 3: a cylinder cast a shadow on the surface around P). Irradiance outside the shadowed area is not the same than irradiance inside the shadowed area. Thus measuring irradiance over a surface that would include both regions would actually give an incorrect result. For this reason we need to consider the smallest region as possible (we used the term infinitesimal before), at least small enough that over this area, we can assume the incident radiant energy to be constant. In computer graphics, a small "differential" area is often denoted dA. Remember that we can't use the point itself, because in geometry a point has no dimension. The same problem applies to "collecting" light falling on the surface. Integrating the total amount of light coming from all directions in the hemisphere is not a simple problem: surfaces visible from P and emitting light in P's direction are continuous however the distribution of emitted light over these surfaces may vary. Similarly to the problem of cutting the surface into an area small enough around P, we need to consider the radiant flux for directions in the hemisphere above P within a collection of the smallest possible solid angles; at least small enough that we can consider the incident radiant flux constant over the set of directions contained within this solid angle (figure 3).

In computer graphics paper the point P is generally denoted with the letter x. We will be using P here for simplicity (L for the light direction and V for the viewing direction). The hemisphere of direction is generally denoted with the greek capital letter $$\scriptsize \Omega$$ (omega).

Figure 3: elements of solid angle and surface elements have to be small enough. In this figure, the cylinder casts as shadow on the plane. If the surface element was too large, the quantity we would try to measure wouldn't be "constant" across the area of the surface element. The same problem applies to element of solid angle: it has to be small enough that light comes from a singular surface.

Like point, lines (which we use to represent directions) have "no part" (Euclid describes them this time as "breadthless lengths"). Thus, we can't use them to measure how much light arrives at P since light is emitted by surfaces which have themselves an area; similarly to the idea of measuring light arriving at P using a small area around P, we can only measure how much light is emitted by a surface for a given direction by considering a small area around the point of intersection between P and the surface emitting light (figure 4).

Figure 4: the concept of surface element can be extended to solid angle. We can define a small cone oriented along the direction of light propagation (in this case imagine light traveling from the green patch for instance to P).

If you draw a line representing the direction of interest L (a line which origin is P and which direction is just one of the possible directions in the hemisphere above P), rather than using this line to find out which object might emit light in the direction of P along this line, we draw a small cone centred around L. This cone is generally called an element of solid angle. As you can see in figure 4, the intersection of the cone with an object S emitting light in the direction of P (light traveling along the line) defines a small surface (lets call it $$\scriptsize dA_s$$). As as we just mentioned, a small surface is what we need to find out how much light is emitted by this object at the intersection point between L and the surface S.

This concept of small differential area around P ($$\scriptsize dA$$) and around the point of intersection between an element of solid angle and a surface emitting light in the direction of P ($$\scriptsize dA_s$$) is very important in shading (it will be used to define radiance which is probably the most important radiometry unit of all). In the next chapter, we will explain why irradiance (the amount of energy received by a point light source) falls off with the square distance to the light.

Another type of irradiance in a volume is scalar irradiance, also known as fluence. This is the flux per unit area that would land on a small spherical detector placed at a given point. The fluence $$\scriptsize \phi(x)$$ (phi) counts all the particles that pass through a neighbourhood of x, without regard to the direction they are traveling.

Radiant Intensity or Intensity $$\scriptsize I$$: angular density of radiant flux or the radiant flux per unit solid angle that is passing incident on, passing through, or emerging from a point and propagating in specified direction. Measured in $$\scriptsize W.sr^{-1}$$.

Figure 5: radiant intensity is a measure of radiant flux coming from a point light source for a given direction or arriving at P from a point light source for a given direction.

$$\scriptsize I = {{d\Phi} \over {d\omega}}$$

Radiance $$\scriptsize L$$: area and angular density of radiant flux or the power flux per unit projected area and per unit solid angle incident on, passing through, or emerging from a point in a specified surface in a specified direction. Measured in $$\scriptsize W.m^{-2}.sr^{-1}$$.

Radiance is probably the most important unit of all because it directly relates to how our eyes perceive the brightness of objects: sensors, whether the eye or a camera are sensitive to radiance. In computer graphics radiance is the quantity that needs to be measured for each pixel of a rendered image (or to say it differently the quantity we compute and assign to pixels). However, it is often the one that is the least properly explained. Radiance combines both the concept of irradiance and radiant intensity; it measures the amount of radiant flux incident on, passing through, or emerging from an arbitrary point in a given direction and for a surface element around P perpendicular to the chosen light direction. We will explain what this means, but for now lets write the equation for radiance which is denoted with the letter $$\scriptsize L$$ (equation 1):

$$\scriptsize L = {{d\Phi d\Phi} \over {dA^{\perp} d\omega}} \rightarrow {{d^2\Phi} \over {dA^{\perp}d\omega}}$$

Figure 6: measure of radiance at point P in direction $$\scriptsize \omega$$. We consider the radiant flux coming from this direction within a small element of solid angle and through a surface element perpendicular to $$\scriptsize \omega$$.

Do not mix up the notation $$\scriptsize L$$ used for radiance with the notation L used to define light direction. To avoid a possible confusion, we suggest to use $$\scriptsize \omega_i$$ for light direction and keep $$\scriptsize L$$ for radiance (we will keep using L for light direction though in this chapter). Lets now explain. Radiance is just a measure of the amount of light passing through (incident on or emerging from) a small surface element which is perpendicular to direction of propagation. To measure a quantity of light we need to consider the direction in which this light is coming from (or the direction it is traveling to); in the case of irradiance, we have been considering all the directions in the hemisphere. In the case of radiance though, we will only be considering an element of solid angle oriented along a given light direction at the point where radiance is measured. This idea is illustrated in figure 6. Note that because the differential area is perpendicular to the light direction we use the up tack () symbol next to dA. Every object that we see is somehow a continuous surface; a surface can be "seen" as being made up of many very small surfaces. The reason why we break down surfaces in small differential areas is because, as we have already mentioned, the properties of the surface and the amount of light incident (and thus the amount of light reflected which is directly proportional to the amount of incident light) on a surface are likely to change from one point on the surface to another. Thus the smaller these surface elements, the more precise is our measure of the amount of light incident on a given point. When it comes to measuring how much light is incident on a point from a given direction, it actually makes sense to apply the concept of small surface element to directions and use a very small element of solid angle (a very small cone if you prefer). We have already explained this concept further up. What we have at the end is a way of measuring how much light is incident on (passing through or emerging from) a point on a surface (for which we will be using a small surface element) from a direction parallel to the normal at that point (for which we will be using an element of solid angle).

Figure 7: to be more useful radiance needs to be measured for surface elements which are not necessarily perpendicular to the direction of propagation (denoted L in this figure).

This is all great but the problem with this definition of radiance is that it only applies for light coming from a direction parallel to the normal at a point on the surface which is quite limiting. What we would like is a more generic equation that would be usable to measure radiance for any given light direction (in the hemisphere above P oriented along the surface normal). In figure 7 we have showed the relationship between the first definition we have been given of radiance and the more general form we would like to come up with; in the former definition, the surface element is perpendicular to the light direction (note that in mathematics a perpendicular surface or direction is denoted with the up tack symbol (⊥) which explains why the symbol appears next to dA in most books). However in the latter, the same surface element is contained within the plane of a given surface and the normal at that point is not perpendicular to the light direction anymore. Lets denote $$\scriptsize \theta$$ the tilt angle between the two surface elements.

At this point we need to explain one of the most important relationships between light and surfaces. When a surface element (which we call $$\scriptsize dA^{\perp}$$) emitting light is parallel and facing another surface (denoted $$\scriptsize dA$$), assuming the two surface elements have the same area, all light emitted by $$\scriptsize dA^{\perp}$$ falls on $$\scriptsize dA$$. However as the angle $$\scriptsize \theta$$ of inclination of $$\scriptsize dA^{\perp}$$ in regard to $$\scriptsize dA$$ increases, some photons miss dA or to say it differently, fewer photons arrive at $$\scriptsize dA$$ compared to the initial case in which the two differential areas were facing each other (figure 8). The greater the angle, the less photons strike $$\scriptsize dA$$'s surface. Eventually, when $$\scriptsize dA^{\perp}$$ is perpendicular to $$\scriptsize dA$$, all photons miss the target area and no light falls on $$\scriptsize dA$$ at all. You can look at this relationship in a different way: as the angle increases, the projected area of $$\scriptsize dA^{\perp}$$ on $$\scriptsize dA$$'s plane becomes larger. However the area of the target element $$\scriptsize dA$$ doesn't change thus the ratio between $$\scriptsize dA$$ and the area over which the photons emitted are spread becomes smaller as this projected area increases.

Figure 8: the projected area of $$\scriptsize dA^\perp$$ increases with the angle $$\scriptsize \theta$$.

This is an extremely important relationship and the equation that links the two surface elements is surprisingly simple: the amount of light emitted by $$\scriptsize dA^{\perp}$$ and arriving at $$\scriptsize dA$$ is proportional to the cosine of the angle $$\scriptsize \theta$$ which is the angle between the light direction (which in that case is the normal of surface element $$\scriptsize dA^{\perp}$$) and the normal of the surface element $$\scriptsize dA$$. When $$\scriptsize \theta$$ = 0, cos($$\scriptsize \theta$$) = 1 thus in this case all light emitted by $$\scriptsize dA^{\perp}$$ is incident on $$\scriptsize dA$$. When $$\scriptsize \theta$$ = 90 degrees, cos($$\scriptsize \theta$$) = 0 thus, in this case, light emitted from $$\scriptsize dA^{\perp}$$ never illuminates $$\scriptsize dA$$.

Lets mark a stop here, and highlight one more time what is probably one of the most important laws in optics. If you need to remember something from this chapter, it is that, assuming the normal at $$\scriptsize dA^\perp$$ is actually pointing towards the center of the surface element $$\scriptsize dA$$ (lets call this point $$\scriptsize x$$), the amount of light exchanged between these two surface elements (actually in this case from $$\scriptsize dA^\perp$$ to $$\scriptsize dA$$) is proportional to the cosine of the angle subtended by the normal at $$\scriptsize dA$$ and the direction between $$\scriptsize x$$ and $$\scriptsize x'$$, the centroid of $$\scriptsize dA^\perp$$. The Lambert's cosine law which you may have heard of (we will introduce it in the next chapter) is somehow connected to this law. In the next chapter, we will also show the relation between this law and irradiance.

Figure 9: the area seen by the observer depends on the angle $$\scriptsize \theta$$. The greater the angle, the smaller the area.

The amount of light arriving at $$\scriptsize dA$$ from $$\scriptsize dA^\perp$$ can also be seen as the amount of light emitted by $$\scriptsize dA^\perp$$ scaled down by how "big" $$\scriptsize dA^\perp$$ appears to be from $$\scriptsize dA$$. Imagine that you look at a sheet of paper located some distance away from you (you can see yourself as being $$\scriptsize dA$$): if the sheet (which is playing the role of $$\scriptsize dA^\perp$$) is facing you, the area of the paper you are looking at is simply the width of the sheet times its height. If you now turn the sheet by a certain angle, the "projected" area is reduced by the cosine of the angle subtended by the line of sight and the normal defined by the surface of the paper; the absolute area of the sheet is always the same but how much of this area you see, changes with this angle; it actually decreases as the angle $$\scriptsize \theta$$ increases (figure 9). At 90 degrees, the area is 0. Thus rather than projecting $$\scriptsize dA^\perp$$ onto the plane in which lies $$\scriptsize dA$$ (as we did in figure 8), since $$\scriptsize dA$$ has a fixed area, we can instead reduce the area of $$\scriptsize dA^\perp$$ by the cosine of the angle $$\scriptsize \theta$$. Figure 10 illustrates this idea in two different ways.

Figure 10: $$\scriptsize dA^\perp$$ is the projection of $$\scriptsize dA$$ along the light direction and is called the projected area.

Remember from the lesson 4 (Spherical Coordinates and Trigonometric Functions), that:$$\scriptsize cos(\theta) = {hypothenuse \over adjacent}$$Thus adjacent, which in the context of this lesson is $$\scriptsize dA^\perp$$ can be defined as $$\scriptsize adjacent = cos(\theta) \times hypothenuse$$ or $$\scriptsize dA^\perp = dA cos(\theta)$$. You can also say that we projected the hypothenuse onto the adjacent side; which is the reason we say in the context of this lesson, that $$\scriptsize dA^\perp$$ is the projected area of $$\scriptsize dA$$.

Figure 12: radiance is likely to change drastically for all the direction in the hemisphere. For instance in this example, light coming from the window is much strong from light coming from the plant in the background.

We can now write the following equation (2):

$$\scriptsize dA^{\perp} = dA cos(\theta)$$

The term $$\scriptsize dA^{\perp}$$ is a quantity called the projected (or foreshortening) area; it corresponds to the projection of $$\scriptsize dA$$, the surface element which contains the point where the radiance is measured, projected in the light direction onto a plane perpendicular to this direction. The term $$\scriptsize d\omega$$ is the element of solid angle oriented along the light direction and $$\scriptsize\theta$$ is the angle between the light direction and the surface normal at P (or x depending on which convention you use). All we need to do now to get a more generic definition of radiance, one in which the element of differential area at which the radiance is measured is not necessarily perpendicular to the light direction is to substitute $$\scriptsize dA cos(\theta)$$ to $$\scriptsize dA^{\perp}$$ in equation 1:

$$\scriptsize L(x, \overrightarrow \omega) = {{d^2\Phi(x, \overrightarrow \omega)} \over {cos(\theta)dA(x)d\omega}}$$

Note that $$\scriptsize cos(\theta)$$ can also be noted as the dot product between $$\scriptsize N$$ and $$\scriptsize \overrightarrow \omega$$ ($$\scriptsize N \cdot \overrightarrow \omega$$). And this becomes our final equation for radiance.

Question from a reader: "When the denominator in the definition of radiance approaches 0, radiance goes to infinity! I don't understand why?". This is often a source of confusion indeed and the confusion comes from the fact that in this form, radiance defines the radiant flux per projected area per unit solid angle. We are not looking really at the radiant flux of a small surface element around P or x (depending on which convention you use) but how much flux goes through a projection of that surface in the direction of incidence. If you consider 1 unit of projected area, as the angle $$\scriptsize \theta$$ increases, the area of the surface element corresponding to the projection of $$\scriptsize dA^\perp$$ on the horizontal plane increases (and goes to infinity in the limit of $$\scriptsize \theta$$ approaching 90 degrees). Thus what you measure here in way is that amount of "light" that is incident or emerging from this very large area (increasing in size) back into a very "projected" area of fixed size (1 unit in our example). You can now hopefully better understand why it goes to infinity? You can see it that way: you gather light from a much larger area which you compact in a projected surface of fixed area. The larger the area over which light is gathered, the higher the amount of light packed into the projected area. This number goes to infinity as the "collecting" area goes to infinity as well.

Note that radiance is a function of both position (the point P or x) and direction (the light direction). We sometimes say that radiance which we can write as $$\scriptsize L(x, \overrightarrow \omega)$$, is a 5D function which is also known as the plenoptic function (three coordinates for point $$\scriptsize x$$ and two for the direction $$\scriptsize \omega$$). Thus when speaking of radiance, the point where radiance is measured, the surface in which lies the point and the direction of propagation need to be specified. If you watch around you, take a point anywhere on any object you can see and trace in your mind a few lines from that point in a few possible directions, you are likely to realise that the amount of light arriving from these directions to the point you observe varies a lot from direction to direction. In most cases, radiance is a strongly varying function of direction

Figure 13: exitant (left) and incident (right) radiance. Note that the direction vectors point away from x (or P).

Figure 14: exitant (left) and incident (right) radiance.

It is important to remember that radiance applies to both light arriving (incident on) at a point on the surface as well as to light leaving (emerging from) a given point on a surface and traveling in any given direction in the hemisphere oriented about the surface normal at that point. What works for incident light also work for light reflected from $$\scriptsize x$$ to any direction in the sphere; all you need to do is just flip the direction. We have illustrated this idea with two different figures (13 and 14). Note that by convention light directions when it comes to radiance always point away from x (or P) regardless of the light flow (arriving $$\scriptsize \omega_i$$ or leaving $$\scriptsize \omega_o$$). Note also that the Bidirectional Reflectance Distribution Function we talked about in the previous chapter, describes the relationship between incident (noted $$\scriptsize L(x \leftarrow \overrightarrow \omega$$)) and exitant (noted $$\scriptsize L(x \rightarrow \overrightarrow \omega$$)) radiance and is a function of both an incoming and outgoing direction vector. Finally we also can look at radiance as irradiance per unit solid angle, or intensity per unit area.

The radiance of the sun at its surface is about $$\scriptsize 2.3 \times 10^7 W \cdot m^{-2} \cdot sr^{-1}$$ and the apparent radiance of the sun from the earth's surface is about $$\scriptsize 1.4 \times 10^7 W \cdot m^{-2} \cdot sr^{-1}$$.

Don't forget that colors and more generally speaking light, are normally defined as a spectrum. All the units we have introduced in this chapter are (technically) wavelength dependent. However, remember that mainly for speed reasons, most 3D applications use a RGB color model. Thus, we generally ignore the wavelength dependency of these units but it is important to keep this in mind.

To help you remembering all the symbols we have been using so far, we have listed them again in the following table:

 Symbol Unit Description P or $$\scriptsize x$$ The point at which the measure is done. N or $$\scriptsize n$$ The normal at P or x. L or $$\scriptsize \omega_i$$ The incident light direction. V or $$\scriptsize \omega_o$$ The outgoing viewing direction. d$$\scriptsize A$$ A surface element surrounding P. $$\scriptsize \Omega$$ Hemisphere of direction (above P and oriented along the surface normal N). $$\scriptsize \omega$$ A direction. d$$\scriptsize \omega$$ A element of solid angle. $$\scriptsize Q$$ $$\scriptsize J$$ Radiant Energy $$\scriptsize \Phi$$ $$\scriptsize J.s^{-1}$$ or $$\scriptsize W$$ Radiant Flux $$\scriptsize E$$ $$\scriptsize W.m^{-2}$$ Irradiance $$\scriptsize I$$ $$\scriptsize W.sr^{-1}$$ Radiant Intensity $$\scriptsize L$$ $$\scriptsize W.m^{-2}.sr^{-1}$$ Radiance

What's Next?

In this paragraph we have introduced the most important units used in radiometry. In the next chapter we will show the relationships between these units. Again do not worry if this page seems too much about theory and if you can't make much sense of what these units are used for and what is the relationship between them. As you keep reading the lesson, hopefully things will start to make more sense to you (we actually suggest you read the next chapter and come back to this chapter again after that).

Chapter 4 of 8