8 Radiometry and Photometry
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To rigorously model surface and volume scattering, we must first define a set of physical quantities that characterize light in space, time, angle, and spectrum. The study of these quantities is called radiometry, which operates entirely within the domain of geometric optics, treating light as a collection of photons, each traveling in a particular direction and carrying a specific amount of energy (Section 8.1). Radiometry allows us to model light-matter interactions using the framework of radiative transfer or light transport, which describes how energy is transferred between points in space and is the key tool we will use in rendering both surface and subsurface/volume scattering.
The counterpart of radiometry is photometry (Section 8.2), which deals with the measurement of radiometric quantities and thus is affected by the sensitivity of the measurement device itself (e.g., retina, a photodetector). Reinhard et al. (2008, chap. 6) and Bohren and Clothiaux (2006, chap. 4) are good references for radiometry and photometry.
8.1 Key Radiometric Concepts
We will start by defining a few key quantities in radiometry.
8.1.1 Energy and Power
Each photon carries a certain amount of energy that is determined by its wavelength governed by:
\[ Q = \frac{hc}{\lambda}, \]
where \(c\) is the speed of light, \(\lambda\) is the photon wavelength, and \(h\) is the Planck’s constant.
Power, or more formally in radiometry, radiant flux (or simply flux) is the total amount of energy passing through some surface in space per unit time. Or, taking a calculus perspective, power \(\Phi\) is defined as:
\[ \Phi = \lim_{\Delta t \rightarrow 0}\frac{\Delta Q}{\Delta t} = \frac{\text{d}Q}{\text{d}t}. \]
The way to think about this is that each photon carries a certain amount of energy so if you monitor photons passing across a surface over a period of time \(\Delta t\), you can calculate the average power of that period by dividing the total energy passed by by \(\Delta t\). As \(\Delta t\) approaches 0, we get the instantaneous power.
Of course, energy/power is a function of wavelength, so more rigorously we should be talking about spectral power \(\Phi(\lambda)\), which has a unit of \(\text{W}/\text{nm}\):
\[ \Phi(\lambda) = \lim_{\Delta \lambda \rightarrow 0}\frac{\Delta \Phi}{\Delta \lambda} = \frac{\d \Phi}{\d \lambda}, \]
where \(\Delta \Phi\) is the total power within a wavelength interval \(\Delta \lambda\).
8.1.2 Irradiance
Our power calculation is done with respect to a surface area, but how about the power at each point on the surface area? You can imagine that some points get more photons and others get fewer, so it is useful to characterize the power at any given point. Technically, the answer to the question “how many photons hit a particular point” is zero, since the area of a single point is 01. The meaningful question is: what is the power density of a particular point \(p\)? Irradiance is such a quantity.
Imagine again that you are monitoring photons crossing a surface for a period \(\Delta t\); you can calculate the average power received per unit area by dividing the average power by the surface area, and when you shrink the surface area to an infinitesimal point \(p\), we can calculate the power density, i.e., the irradiance, of \(p\) by:
\[ E(p) = \lim_{\Delta A \rightarrow 0}\frac{\Delta \Phi(p)}{\Delta A} = \frac{\text{d}\Phi(p)}{\text{d}A}. \tag{8.1}\]
Irradiance is a more primitive measure than power: in calculus terms, irradiance is a power density function, which means we can derive the power of a surface by integrating the irradiance over the surface area:
\[ \Phi = \int^{A}E(p)dA. \]
Irradiance has a unit of \(\text{W}/\text{m}^\text{2}\), and spectral irradiance has a unit of \(\text{W}/(\text{m}^\text{2} \cdot \text{nm})\).
8.1.3 Solid Angle
Irradiance is concerned with the power of all the photons incident on a point, but photons hit a point from all directions, so how do we quantify the amount of light coming from a direction?
A direction is a vector, which is invariant to translational transformations, so the two parallel “arrows” \(r_1\) and \(r_2\) in Figure 8.1 (left) represent the same vector/direction. Therefore, conceptually it is easier if we translate all the arrows so that they start from the same origin when we want to reason about a collection of directions.
How do we count the number of directions? In 2D, we use a planar angle to measure the amount of directions. Given an origin \(O\) and a vector, we rotate it to generate an arc. The angle subtended by the arc and \(O\) is a measure of the amount of directions we have just covered. The angle can also be mathematically given by the ratio \(s/r\), where \(s\) is the arc length and \(r\) is the radius of the circle. This matches our intuition that if we increase the radius of the circle, we would get a longer arc but the same angle. A full circle has a planar angle of \(2\pi\).
We can similarly define the size of a set of directions in 3D. We draw a sphere around \(O\), and imagine that we have some area on the spherical surface. Connecting \(O\) to every point on that area represents a direction in 3D. So the spherical surface area is a measure of the amount of 3D directions. Like in the 2D case, we want the measure to be invariant to the spherical radius, so we define solid angle, a measure of the size of a set of 3D directions, as:
\[ \Omega = \frac{A}{r^2}, \tag{8.2}\]
where \(A\) is an area on a spherical surface and \(r\) is the radius. The unit of a solid angle is the steradian (\(sr\)), and the entire sphere subtends a solid angle of \(4\pi\).
Sometimes we want to know the size of the set of directions from a point \(O\) to an arbitrary surface. We would project that surface to a sphere and get a projected spherical area \(A\), using which we can invoke Equation 8.2 to estimate the solid angle subtended by the surface. One useful trick that might help sometimes is to project the surface to the unit sphere (i.e., \(r=1\)), and the solid angle is mathematically equivalent to the projected area on the unit sphere. But the most useful intuition I use whenever I am confused about what a particular solid angle means is to always think of the set of directions/vectors that are represented by that solid angle.
8.1.4 Radiance
We can now ask, what is the amount of flux received by a point from a particular direction? Photons travel in all sorts of directions. Let’s assume that we place an imaginary flux detector with an area \(A\) in the field. The detector is able to receive light from only one direction \(\omega\), as illustrated in Figure 8.1 (c). We can then read out the total flux \(\Phi\) received by the detector, from which we know that the power per unit area along the direction \(\omega\) is simply \(\frac{\Phi}{A}\).
Now imagine that we orient the detector so that its normal subtends an angle \(\theta\) with respect to the light direction \(\omega\). Figure 8.1 (b) explicitly illustrates this angle, where the tilted detector lies in the \(xy\)-plane, and the \(z\) direction is the normal \(n\). In a spherical coordinate system, a direction \(\omega\) can be parameterized by two angles: a polar angle \(\theta\) and an azimuthal angle \(\phi\).
The total flux received by the detector has changed to \(\Phi\cos\theta\), because the area that is available to receive photons is now \(A\cos\theta\). We call this the “effective area”. As a result, the power per area at the direction \(\omega\) remains the same, i.e., \(\frac{\Phi}{A}\). This is not surprising, because we are not changing the radiation field, only how we measure it. When the effective area reaches 0 (i.e., the detector is completely parallel to the light direction), the detector collects no photons, but it certainly does not mean that there is no light in the field.
If we now want to measure light power coming from another direction, we would change the detector so that it receives light from only that direction. In reality, this is, of course, not possible. No detector can screen lights only from one direction. If we place a detector in a radiation field, it is going to receive photons from all sorts of directions. We can limit the directions of photons that the detector collects by placing a baffle that allows only certain directions to hit the detector.
This setup is illustrated in Figure 8.2 (left). The total flux collected by the detector is \(\Delta \Phi\), the detector size is \(\D A\), and the solid angle subtended by the baffle is \(\D \omega\). The average power collected per unit “effective area” per unit direction by the detector is then:
\[ \frac{\D \Phi}{\D A \cos\theta \D \omega}. \]
The baffle does a good job of rejecting many directions that are outside \(\D \omega\), but unless it is infinitely long, the detector will still collect some photons traveling through directions outside \(\D \omega\). But as we reduce the detector size and the baffle size, the baffle becomes a very thin cylinder over a very small detector, which collects light from a very small area along a very small solid angle, visualized in Figure 8.2 (right)2. In calculus terms, when we let the detector size and baffle’s solid angle approach 0, we obtain the quantity called radiance:
\[ L(p, \omega) = \lim_{\D \omega \rightarrow 0} \lim_{\D A \rightarrow 0} \frac{\D \Phi}{\D A \cos\theta \D \omega} = \frac{\text{d}}{\text{d}\omega}\frac{\text{d}\Phi(p)}{\text{d}A \cos\theta} = \frac{\text{d}^2\Phi(p)}{\text{d}\omega\text{d}A \cos\theta}. \tag{8.3}\]
Equation 8.3 is the definition of radiance, and it can be rewritten to Equation 8.4 given the definition of irradiance (see Equation 8.1).
\[ L(p, \omega) = \frac{\text{d}E(p)}{\text{d}\omega\cos\theta}. \tag{8.4}\]
Radiance is an intrinsic property of the radiation field, and the reason we have the \(\cos\theta\) term in the definition is merely due to the way we have chosen to measure the property (using a detector that is \(\theta\)-oriented). Radiance has a unit of \(\text{W}/(\text{m}^\text{2}\cdot \text{sr})\), and spectral radiance has a unit of \(\text{W}/(\text{m}^\text{2}\cdot \text{sr} \cdot \text{nm})\).
Looking at the effective area in Figure 8.2, if the irradiance at the infinitesimal area \(p\) is \(\d E(p)\), the irradiance at the (infinitesimal) effective area (projected from \(\d A\) along \(\omega\)) is \(\frac{\d E(p)}{\cos\theta}\), which we denote \(\d E_\bot(p)\). Combining this with Equation 8.4, radiance \(L(p, \omega)\) can also be defined as:
\[ L(p, \omega) = \frac{\text{d}E_\bot(p)}{\text{d}\omega}. \tag{8.5}\]
Equation 8.5 and Equation 8.4 each corresponds to a concrete way of measuring the radiance. Equation 8.5 places the detector perpendicular to the direction of light that we care to measure, and the detector used by Equation 8.4 is \(\theta\)-oriented with respect to the direction of interest. They give us an identical radiance result because radiance, again, is an inherent property of the radiation field invariant to how we measure it.
Radiance is a density function: the density of power at a point along a direction. As with any density function, it is useful when it gets integrated to compute some other quantities. For instance, given the radiance \(L(p, \omega)\), the irradiance at \(p\) is given by:
\[ E(p, \Omega) = \int^{\Omega}L(p, \omega)\cos\theta\text{d}\omega. \tag{8.6}\]
Here we write the irradiance as \(E(p, \Omega)\) to explicitly signify that the irradiance depends not only on the specific position \(p\) but also the solid angle \(\Omega\) over which the lights are coming.
Using the interpretation of radiance in Equation 8.5, we can also give a more operational interpretation of Equation 8.6: we first calculate the infinitesimal irradiance \(\d E_\bot(p) = L(p, \omega)\do\) made by lights at the direction \(\omega\), then “transfer” that to the infinitesimal irradiance at the detector surface through the \(\cos\theta\) factor, and then repeat this for all the directions to accumulate the contributions from all directions.
8.1.5 Radiant Intensity and Lambert’s Cosine Law
A Lambertian emitter or an ideal diffuse emitter is a flux-emitting point whose emitted radiance is constant regardless of the outgoing direction. A related concept is a Lambertian scatterer or an ideal diffuse surface, which is a surface point where the scattered radiance is independent of the scattering direction.
It might come as a surprise that the flux emitted by a Lambertian emitter through a fixed solid angle is different for different emission directions. Consider a setup where a Lambertian emitter has an infinitesimal area \(\d A\). The power emitted by \(\d A\) toward its normal direction in an infinitesimal solid angle of \(\do\) is \(\d\Phi_0 = L\do\d A\), where \(L\) is the radiance. The power emitted toward an oblique direction \(\omega\) through the same solid angle is \(\d\Phi_\theta = L \do \cos\theta \d A\).
In radiometry, the ratio of infinitesimal power and infinitesimal solid angle is called the radiant intensity3, denoted \(I\):
\[ I(\omega) = \frac{\d\Phi}{\do}. \tag{8.7}\]
\(I\) is a meaningful measure only for a point source (e.g., our infinitesimal Lambertian emitter here). We can see that for a Lambertian emitter, the radiant intensity decays by a factor of \(\cos\theta\): \(\frac{\d\Phi_\theta}{\d\omega} = \frac{\d\Phi_0}{\d\omega}\cos\theta\). This is usually called the Lambert’s cosine law, named after Johann Heinrich Lambert, from his Photometria (Lambert 1760). Similarly, if we have a Lambertian scatterer, its scattered radiant intensity will also decay by \(\cos\theta\) as the polar angle \(\theta\) of the viewing direction \(\omega\) increases.
Figure 8.3 compares the radiance distribution and radiant intensity distribution of a Lambertian emitter/scatterer. Both distributions are over the entire hemisphere, but we show only a cross section. The distributions are visualized as two lobes, and the distance of a point on the lobe to the origin is proportional to the value at that point. The radiance distribution is constant regardless of \(\omega\) but the radiant intensity is proportional to \(\cos\theta\). This difference stems from the fact that intensity is defined with respect to the power at the detector/emission area (\(\d A\)) while radiance is defined with respect to power at the effective area (\(\d A \cos\theta\)).
8.2 Photometric Quantities
Spectral radiant flux (power), irradiance, radiant intensity, and radiance are all radiometric quantities. They all have a photometric counterpart, which weighs the radiometric quantity by the luminous efficiency function (LEF). The LEF, as we have discussed in Section 4.3.2, at a particular wavelength is inversely proportional to the radiometric quantity at each wavelength needed to produce the same level of perceptual brightness.
For instance, given a spectral radiant flux \(\Phi(\lambda)\), the corresponding photometric counterpart is then:
\[ \Phi_v(\lambda) = K \Phi(\lambda) V(\lambda), \]
where \(\Phi_v(\lambda)\) is the spectral luminous flux, \(V(\lambda)\) is the LEF, and \(K\) is a constant that, for historical reasons, takes the value of 683.002. The total luminous flux is then:
\[ \Phi_v = \int_\lambda K \Phi(\lambda) V(\lambda) \d \lambda. \]
Luminous flux has a unit of lumen (\(\text{lm}\)), so \(K\) has a unit of \(\text{lm}/\text{W}\). We can also weigh the radiant power by the scotopic LEF, in which case \(\text{K} = 1700\) (\(\text{lm}/\text{W}\)).
Other radiometric quantities can be similarly converted to the photometric counterparts. Specifically:
- the photometric counterpart of irradiance is illumination, which has a unit of \(\text{lx} = \text{lm}/\text{m}^2\), which is also called the lux;
- the photometric counterpart of radiance intensity is luminous intensity, which has a unit of \(\text{cd} = \text{lm}/\text{sr}\), which is called the candela;
- the photometric counterpart of radiance is luminance, which has a unit of \(\text{lm}/(\text{m}^2\text{sr}) = \text{cd}/(\text{m}^2)\), which is also called the nit.
Sometimes radiometric vs. photometric quantities are also called the radiant vs. luminous quantities. The way to interpret the photometric quantities is that they take into account the spectral sensitivity of a particular photodetector, which in our discussion above is the photoreceptors on the retina, so the spectral sensitivity function of the detector is the LEF. If we use other detectors, such as an image sensor, we will have a different spectral sensitivity, and the corresponding photometric measurements will be different. We will study the spectral sensitivity of image sensors in Section 16.5.
A radiometer measures the absolute radiometric quantities, whereas a photometer reports photometric quantities. An image sensor and our retina can both be thought of as a photometer but the spectral sensitivities in the two cases are different, so the raw pixel readings and the photoreceptor responses are different even under an identical illumination.
A similar question is: imagine you are throwing a dart at a wall, what is the probability of hitting a particular point \(p\)? The answer is 0. The meaningful question to ask is: what is the probability density of hitting \(p\)?↩︎
It is just a visualization convention, but visualizing \(\do\) as a cylinder rather than a cone makes it easier to imagine what \(\d A \cos\theta\) is like.↩︎
Or simply, the “intensity”, which is an extremely overloaded term, so we will be verbose and use “radiant intensity” when we mean it.↩︎