Angles and the Camera Equation

Imagine a bucolic scene on a clear sunny day at the equator, sand warmed by the tropical sun with a typical irradiance ( $E$ ) of about 1000 watts per square meter. As discussed earlier we could express this quantity as illuminance in lumens per square meter ( $lx$ ) – or as a certain number of photons per second ( $\Phi$ ) over an area of interest ( $\mathcal{A}$ ).

(1) $\begin{equation*} E = \frac{\Phi}{\mathcal{A}} \; (W, lm, photons/s) / m^2 \end{equation*}$

How many $photons/s$ per unit area can we expect on the camera’s image plane (irradiance $E_i$ )?

Figure 1. Irradiation transfer from scene to sensor.

In answering this question we will discover the Camera Equation as a function of opening angles – and set the stage for the next article on lens pupils. By the way, all quantities in this article depend on wavelength, which will be assumed in the formulas to make them more readable.

Spherical Cones and Lambertian Reflectance

Irradiance is not directional, meaning that the photon flux can arrive from any direction or even from multiple/extended sources, for example an overcast sky. What matters in this article is the number of photons incident on the relevant area of the object of interest, so that’s what we will refer to as $E_o$ , the subscript $o$ indicating a quantity referred to the object. Think of it as the output of a photodiode flush with the surface, counting every photon impinging on it, regardless of where it came from.

The object absorbs some photons and reflects some depending on its morphology and physical characteristics. Those reflected propagate in straight lines in air in all directions. Neighboring photons reflected from small uniform portions on the surface of the object ( $\mathcal{A}$ ) form conical volumes. The direction and relative number of photons within each cone depends on the object’s physical characteristics, their distribution the result of so-called specular, diffused or mixed reflection.

Figure 2 shows photons in air reflected from such a small sample area $\mathcal{A}$ , the specific spherical cone chosen because it is the one that points directly towards the lens of the camera, which is what this article is concerned about.

Figure 2. When photons arriving from an arbitrary direction strike an incremental surface area A on an object they are partly reflected in a general proportion and direction based on the properties of the surface. From there they spread out in the shape of conical volumes made up of a right circular cone with a spherical cap (not drawn to scale). The angle between the cone’s axis and the surface normal n is the same beta as in Figure 1.

We will assume that the object presents an ideal Lambertian surface with reflectance $R$ – so a total number of photons per unit area equivalent to $RE_o$ will be reflected diffusely in a predictable fashion, in the set of all directions contained within the hemisphere above the small uniform reference area $\mathcal{A}$ .

For a given conical half opening angle ( $\theta$ ) the number of photons reflected into the relative cone will decrease the more the observed cone axis is tilted away from the direction perpendicular to the surface (its normal $\vec{n}$ ). In fact with Lambertian reflectors the number of encompassed photons decreases with the cosine of the angle ( $\beta$ ) between the surface normal and the axis of the cone – because less and less projected area is visible from the direction of interest as tilt increases. This is known as Lambert’s Cosine Law.

Figure 3. Proportion of photons reflected by a Lambertian surface in each direction according to Lambert’s Cosine Law (central slice through the reflected hemisphere shown, perpendicular to the surface). The longer the red arrow, the more the reflected photons in that direction. Image under license, courtesy of GianniG46, modified by me for this article.

Figure 3 shows the relative number of photons that can be expected in the shown directions within the central slice of the set of all hemispherical directions: the longer the red line, the more the photons reflected in that direction. It turns out that many natural surfaces tend to behave this way when incident light is within about 40 degrees of the normal (angle $\alpha$ in Figure 1). Beyond that reflections become more and more directional, hence mixed.

Photons Reflect into Spherical Caps

Since the lengths of the red arrows in Figure 3 represent a relative number of photons reflected in the given direction from the same small area on the object, clearly the larger the opening angle of the cone of interest (also known as its angular aperture), the larger the proportion of total reflected photons that will be part of it.

If we assume that all photons spreading out inside the cone towards the lens all depart from the same relatively small area $\mathcal{A}$ on the object at the same time – like pellets spreading out from a shotgun, all traveling at equal velocity in a straight line – by the time they reach the lens they will have taken the shape of a spherical cap.

Figure 3 below shows an (almost) two-dimensional view of a spherical cap, looking like the arc of a circle with radius $r$ subtended by angle 2 $\theta$ . With $\theta$ in radians the length of such an arc is 2 $r\theta$ by definition.

Figure 4. The solid angle associated with spherical cones is the area of the cap after normalizing r to 1. It is computed by rotating the semi-arc r.theta through 360 degrees around the axis.

In three dimensions the area of the spherical cap on the cone would simply be the semi-arc $r\theta$ rotated through 360 degrees – the arriving photons will be spread out evenly over it. Its area is

(2) $\begin{align*} \mathcal{A}_{cap} &= 2 \pi r^2 (1-cos\theta) = 4 \pi r^2 sin^2 (\frac{\theta}{2}) \\ &\approx \pi r^2 \theta^2 \approx \pi r^2 sin^2\theta \end{align*}$

The bottom two approximations are valid for the small angles normally involved in photography (see Appendix II for exactly how valid). The last one is more relevant to well corrected lenses that need to meet the Abbe Sine Condition, which forces working f-number to $N_w=\frac{1}{2sin\theta}$ .

Solid Angles are Normalized Spherical Caps

Just like radians are normalized for $r$ in one dimension it is useful to normalize the area of the cap for $r^2$ , a quantity that is then referred to as a solid angle $\Omega$ :

(3) $\begin{equation*} \Omega = \frac{A_{cap}}{r^2}, \; sr \end{equation*}$

Its units are radians rotated through 360°, formally steradians ( $sr$ ). Per Equation (2)

(4) $\begin{equation*} \Omega \approx \pi \theta^2 \approx \pi sin^2\theta \approx \frac{\pi}{4N_w^2}, \; sr. \end{equation*}$

We can see intuitively why this would be useful in our case, because for a given interacting irradiance $E_o$ the number of photons $\Phi_o$ instantaneously reflected out through the cone from a small area on the object $\mathcal{A}$ towards the lens depends only on the opening angle $\theta$ – hence solid angle $\Omega$ – but stays otherwise the same from when it departs the surface as it spreads out into an enlarging spherical cap on its way to the lens.

Radiance and Luminance

The number of photons reflected per small area from the object per normalized spherical cap (the solid angle) in the direction of interest is denoted radiance ( $L$ , in photometry Luminance). We can write this definition generically as follows

(5) $\begin{equation*} L = \frac{\Phi}{\mathcal{A}\Omega}, \; photons/s/sr/m^2 \; \text{or} \; cd/m^2 \end{equation*}$

Since a Lambertian surface reflects all incident photons per unit area $E_o$ into the hemisphere above itself subject to reflectance $R$ , we can calculate constant $L_o$ by integrating over all possible solid angles (I’ll spare you the gory details^[1]) obtaining

(6) $\begin{align*} L_o = \frac{RE_o}{\pi}, \; photons/s/sr/m^2 \; \text{or} \; cd/m^2 \end{align*}$

the subscript $_o$ indicating quantities at the object as seen from the lens. Radiance/Luminance $L$ from a Lambertian surface is indeed only a function of reflected incident total power density $E_o$ and is therefore the same in all directions and at all distances.

It’s easy to prove to oneself that radiance/Luminance is conserved in radiation transfer, just check whether an image on a monitor appears equally bright from 300mm or from an order of magnitude farther away. Or whether a quasi-Lambertian gray card looks equally bright when tilted at different angles, within its limits of Lambertian diffuse reflection. This property of Luminance comes in handy in the discussion that follows.

The Camera’s Perspective

We now understand that the number of photons reflected towards the lens is by the definition in Equation (5) above

(7) $\begin{equation*} \Phi_o = L_o \mathcal{A}_o \Omega}_o, \; photons/s \end{equation*}$

with subscript $_o$ indicating quantities on the object side and

$L_o$ Radiance/Luminance from the object ( $= \frac{RE_o}{\pi}$ for a Lambertian reflector);
$\mathcal{A}_o = \mathcal{A} cos\beta$ because the small reflecting area $\mathcal{A}$ is ‘foreshortened’ by the cosine of $\beta$ in the direction of propagation since the recipient of those photons (say the camera) sees less of it the more it is tilted with respect to the surface normal;
$\Omega_o \approx \pi \theta^2 \approx \pi sin^2\theta \approx \frac{\pi}{4N_w^2}$ is a function of opening angle $\theta_o$ from the object plane to the edge of the pupil of the lens.

At typical photographic distances, all photons from uniform small reflecting area of interest $\mathcal{A}$ in the neighborhood of the vertex of the lens-subtending cone will end up in the same spherical cap, in the limit this assumption is exact.

Figure 5. Directionless irradiating energy E is funneled by a Lambertian reflector to the sensing plane through the lens via the two shown spherical cones and relative solid angles.

Assuming no losses through the optics, the same number of photons that arrive at the lens ( $\Phi_o$ ) will leave it on the way to the image plane ( $\Phi_i$ ), with subscript $_i$ indicating quantities on the image side.

In air, the setup there is the reciprocal (the conjugate) of that on the object side, with $L_i=L_o$ being constant and the solid angle $\Omega_i$ a function of opening angle $\theta_i$ . Per geometrical optics, area $\mathcal{A}_i$ on the image plane near the vertex of the cone of arriving photons will be a (de)magnified version of area $\mathcal{A}_o$ on the object from which they left, with lens magnification $|m|$ typically less than one. Appendix I has some further details on magnification.

Bringing it All Together: the Camera Equation

So with an ideal lens with no losses, the total number of photons per second arriving on the sensing plane in air $\Phi_i$ is

(8) $\begin{equation*} \Phi_o = \Phi_i = L_i \mathcal{A}_i \Omega_i, \; photons/s \end{equation*}$

because of conservation of energy and reciprocity. Since radiance/Luminance is conserved, $L_i = L_o = L$ and the number of photons per unit area near the vertex of the cone on the image plane (sensor irradiance $E_i$ ) is from the definitions in Equations 1 and 8

(9) $\begin{equation*} E_i = \frac{\Phi_i}{\mathcal{A}_i } = L \Omega_i, \; photons/s/m^2 \end{equation*}$

Equation (4) tells us that $\Omega_i$ is to a good approximation equal to $\frac{\pi}{4N_w^2}$ , with $N_w$ the lens working f-number, so irradiance on the image plane is

(10) $\begin{equation*} E_i \approx L \frac{\pi}{4N_w^2} \; photons/s/m^2. \end{equation*}$

This is known as the Camera Equation with a perfect lens. Since Exposure $H_i = E_it$ , with $t$ exposure time in seconds

(11) $\begin{equation*} H_i \approx \frac{\pi}{4}L\cdot\frac{t}{N_w^2} \; photons/m^2 \end{equation*}$

a formula photographers know well and apply daily since it is the basis of the EV system. The approximation is pretty good at typical working f-numbers as shown in Appendix II.

With Lambertian reflectors the last two equations can be expressed as a function of incoming interacting irradiance $E_o$ instead of $L$ thanks to Equation (6), answering the question posed at the beginning of this article. Therefore, ignoring transmission losses, the photon flux per unit area at a particular spot on the sensor is

(12) $\begin{equation*} E_i \approx \frac{RE_o}{4N_w^2} \; photons/s/m^2 \end{equation*}$

with $R$ surface reflectance, $E_o$ photon flux per unit area interacting with the object (irradiance/illuminance) and $N_w$ the lens working f-number. As mentioned up top these are typically spectral quantities valid both in radiometric and photometric contexts. A discussion of pupils is next.

PS. A similar result was obtained straight from a light source’s Radiant Exitance in the Appendix of this article on Photons Emitted From a Light Source.

Appendix I – Throughput and Magnification

As we are assuming no losses, the number of photons reflected by the object towards the lens is the same that hits the sensing plane, that is $\Phi_i=\Phi_o$ or by definition

(13) $\begin{equation*} L_i \mathcal{A}_i \Omega_i = L_o\mathcal{A}_o \Omega_o , \; photons/s \end{equation*}$

Since radiance/Luminance is conserved, $L_i = L_o$ cancels out and we are left with

(14) $\begin{equation*} \mathcal{A}_i \Omega_i = \mathcal{A}_o \Omega_o \end{equation*}$

The quantities on either side of the Equation above are known as Throughput (or Etendue). Throughput is obviously also conserved in radiation transfer. Rearranging the terms we get

(15) $\begin{equation*} \frac{\mathcal{A}_i}{\mathcal{A}_o} = \frac{\Omega_o}{\Omega_i} \end{equation*}$

and if we assume that the projected area of the object is circular or square with height $h_o$

(16) $\begin{equation*} \frac{\mathcal{A}_i}{\mathcal{A}_o} = \frac{h^2_i}{h^2_o} = m^2 \approx \frac{\pi\theta^2_o}{\pi\theta^2_i} \approx \frac{\pi sin\theta^2_o}{\pi sin\theta^2_i} \end{equation*}$

in other words lens magnification $m$ is equal to

(17) $\begin{equation*} |m| = \frac{h_i}{h_o} \approx \frac{\theta_o}{\theta_i} \approx \frac{sin\theta_o}{sin\theta_i} \end{equation*}$

with $\theta$ in radians. The last fraction to the right takes into consideration the Abbe Sine Condition, which well corrected lenses need to fulfill in order to mitigate aberrations like coma near the optical axis. This is the reason why the preferred expression for working f-number for such lenses in air is

(18) $\begin{equation*} N_w = \frac{1}{2NA_i} \approx \frac{1}{2sin\theta_i} \end{equation*}$

with $NA_i$ the image-referred numerical aperture. Then

(19) $\begin{equation*} |m| = \frac{h_i}{h_o} \approx \frac{NA_o}{NA_i} \approx \frac{N_w}{N_o}. \end{equation*}$

with $N_o$ the object referred working f-number.

Appendix II – Accuracy of Solid Angle Approximations

As we have seen in this article a solid angle is the area of the relative conical spherical cap per Equation (2), divided by the radius squared:

$\begin{align*} \Omega &= 2 \pi (1-cos\theta) = 4 \pi sin^2 (\frac{\theta}{2}) \\ &\approx \pi \theta^2 \approx \pi sin^2\theta \end{align*}$

The last two are approximations valid only for small $\theta$ .

With a circular Exit Pupil of diameter $D$ , the lens focused at infinity and a small opening angle $\theta$ – the spherical cap can be assumed to be almost flat, therefore a circular disk. In that case its area is the area of the aperture $\pi (\frac{D}{2})^2$ and the radius of the cone can be considered equal to published focal length $f$ , so another approximation often used in photography is

$\begin{align*} \Omega \approx \frac{\pi D^2}{4 f^2} = \frac{\pi}{4N^2}. \end{align*}$

with published f-number $N=\frac{f}{D}$ . When can the small angle approximation be used? Figure 6 shows the error in stops for each solid angle formulation as a function of working f-number for a well corrected lens, per Equation (18).

Solid Angle Approximations in Photography — Figure 6. Error in various approximations of the solid angle associated with a spherical dome.

Pretty good in practice over typical working f-numbers, since most photographers are content with 1/3 of a stop Exposure controls in-camera. You can check my math in the Matlab routine that produced the plot.

Notes and References

_{1. For a clear derivation of radiance/Luminance from Lambertian sources see Dr. Robert A. Showengerdt’s class notes at the University of Arizona, 2000.}

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