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The photographic exposure equation

In photography, this equation governs the fundamental relationship between the scene, the camera, and the captured image:

Image brightnessScene illumination × Subject reflectivity × Lens aperture area × Shutter open time × ISO sensitivity.

A simplified three-factor form is typically what photographers use in practice:

Image brightnessLens aperture area × Shutter open time × ISO sensitivity.

So in other words, the image brightness is proportional to the product of all of these factors. That was quite a mouthful, wasn’t it? Let’s dissect the first equation piece by piece and look at it from different viewpoints to understand how it works.

The factors

Scene illumination

The amount of light that a source casts onto a scene is a factor in the image brightness. For example, an outdoor scene under direct sunlight receives about 10× the light compared to the same scene under a fully overcast sky. In the studio, illumination can be controlled by adding lights; e.g. doubling the number of 60 W light bulbs used will double the illumination of the scene.

This factor is not easily controllable in most photography situations – nearly impossible for outdoor and candid shots, and inconvenient for indoor shots (switching lights on/off, bringing in additional lighting equipment).

Subject reflectivity

Light from a source bounces off a subject and into the camera. The subject reflects anywhere from 0% of the light (pitch black) to 100% (assuming diffuse rather than specular reflection). If a subject being photographed happens to be a light source (such as an LED), then the reflectivity of it is not applicable.

This factor is generally not controllable at all because the subject reflectivity is directly tied to its color and appearance. For example, you wouldn’t ask the subject to change his shirt from black to white to reduce the scene lighting requirements. However, it is useful to pay attention to it for analysis.

Lens aperture area

The rate of light entering the camera is proportional to the area of the lens aperture. If we interpret standard aperture notations like f/5.6 or F5.6 as the number 1/5.6, then the lens aperture area is proportional to the square of this number (e.g. 1/31.36). For example, this means when comparing aperture f/2.8 to f/5.6, the former has 4 times the area of the latter.

This factor is easily controlled by the photographer – during shooting (aperture settings) and when choosing lenses (e.g. max f/4.0 lens vs. max f/1.4 lens, the latter is 8× brighter). Note that there are generally no lenses having relative apertures larger than f/0.9.

Shutter open time

We live in a shower of photon rain, and the camera is a bucket to catch these photons. Clearly, the number of photons gathered is proportional to the time the shutter is open to let the light into the camera.

This factor is easily controlled by the photographer, with typical cameras allowing an exposure time of 1/5000 s to 30 s. Moreover, the exposure time has an effectively unlimited upper bound by shooting in bulb mode or by digitally summing multiple exposures.

ISO sensitivity

Different physical materials have different sensitivities to light. In film photography, large film grains are reactive to smaller amounts of light, while fine grains need more light to react. In digital photography, the image sensor has a configurable analog amplifier that can increase its sensitivity. Doubling the sensitivity means that only half the amount of light is required to produce the same image brightness. However, increasing the sensitivity increases the noise in the image.

This factor is easily controlled by the photographer on a digital camera, and is moderately difficult on film cameras (need to carry and change film rolls for different ISO).

Omitted factors

These factors are less relevant for various reasons:

  • Lens transmittance: Camera lenses strive to transmit 100% of the incoming light, but they are not perfect. Some manufacturers quantify the actual transmittance using T-stops, which are compared to idealized f-stops. For the most part, lens transmittance doesn’t drop below 90%.

  • Lens filters: A popular filter for modifying light is the neutral density (ND) filter – it attenuates the light by a certain factor (such as 1/8×) without any changes in color, depth of field, focus, etc. An example use of this filter is described below in this article. The vast majority of photographs are taken with no filters, so we rarely need to consider this.

  • Image editing: It is of course possible to brighten an image in a digital editing program (“photoshopping”) or in a film darkroom using push processing. However, this factor is not particularly interesting because the degree of control is effectively unlimited – arbitrary changes and redraws can be made to the image, and there need not be any meaningful relationship between the input and output images.

All of these omitted factors can be folded into the constant of proportionality that is implicit in the equation.

Consequences and scenarios

How to use the equation

It might not be immediately clear how the exposure equation can be applied in practice. It’s difficult to quantify every one of the 6 variables in absolute numerical terms; it’s not even clear how to measure image brightness in a standard way.

The approach I recommend is to start with numbers for a known balanced exposure, then analyze or modify based on it. For example, consider a hypothetical scene where an aperture area of 0.7, exposure time of 1/200, and ISO 1600 (we’ll take the illumination and reflectivity for granted) results in an image of brightness 1. If we change the aperture area to 2.1 (triple), then the image brightness will be 3. Next change the exposure time to 1/1200 (one sixth), and the brightness will be 0.5. Finally change the ISO to 3200, and the brightness of the image will be 1 again. (Note that the aperture area is not the same as the f-number, but can be calculated from it.)

The main point of this equation is to predict the resulting consequence of changing any of these 6 variables. It should be clear that any 5 of the variables determine the value of the remaining one. Changing the value of one variable will always cause at least one other variable to change. For example, doubling the scene illumination will mean the image brightness will double, or one or more other factors will change to keep the brightness constant.

Exposure triangle

All photography tutorials worth their salt will cover how photographic exposure works. They mention the 3 primary factors – aperture, shutter, and ISO – and how to set and balance them to produce the desired image. However, they usually don’t explain the underlying equation that makes it all work. This leaves the reader with a specific skill of balancing exposure without understanding why it works, and is hard to transfer to other systems that have similar behaviors.

There is one notable exception to the exposure triangle, which is reciprocity failure in dim light reaching the film. This is more of a concern for film than digital, though at longer exposure times digital sensors will have problems with fixed pattern noise and thermal noise.

Camera modes

In light of the exposure equation, different camera modes can be seen as choosing which variables the user (photographer) controls and which variables the camera (algorithm) controls:

Mode Exposure/brightness Aperture size Shutter time ISO sensitivity
AutoNeutralCameraCameraCamera
PUserCameraCameraUser/Camera
P*UserLinkedLinkedUser/Camera
A/AvUserUserCameraUser/Camera
S/TvUserCameraUserUser/Camera
MSceneUserUserUser
M*NeutralUserUserCamera

Notes:

  • Full auto mode is also known as “green” mode because it’s usually marked with a green icon/text on every camera. Usually no other mode uses a green icon. A neutral exposure corresponds to an exposure compensation of ±0 EV.

  • In all the semi-manual exposure modes, exposure compensation (image brightness adjustment) is available to the user, and the ISO can be set by the user or can be automatically determined by the camera. Note that in these modes, there is always at least one free variable for the camera to determine.

  • In P-shift mode (P*), the exposure settings start from numbers computed in P mode, and then user can directly trade off aperture and shutter while maintaining exposure and ISO. Both variables always change simultaneously. This mode gives some of the benefits of A/Av and S/Tv mode without having to switch into those modes.

  • In full manual mode (M), all 3 exposure variables are set by the user, so the image brightness is determined by the scene illumination and the subject. The image can be arbitrarily underexposed or overexposed.

  • In manual-ish mode (M*), the aperture and shutter are set by the user, the exposure compensation is locked at ±0 EV (at least in cameras I’ve seen), and the camera determines the ISO value to achieve the proper exposure.

  • This discussion doesn’t even begin to explore the relationship between flash lighting and exposure – which has topics such as auto vs. manual flash exposure, flash exposure compensation, balancing flash light and ambient light, balancing flash output with aperture and ISO, high-speed sync, etc.

Flash lighting

We often don’t have control over the scene illumination, but there is one situation where we have full control over it: Flash-only photography. In this case, all the light falling on a scene comes from flash units that we control, and none from ambient sources.

For example, when shooting macro subjects we might require a lot of depth of field, so a small aperture is requested. If there isn’t enough light to make a proper exposure, we can add more flash units until the scene is sufficiently lit. In this situation, we are increasing the scene illumination to counteract the decreasing aperture area in order to maintain the image brightness.

Taking logarithms

Some of the factors, such as scene illumination, are conventionally measured in logarithmic units like EV (exposure values). As well, the final image brightness is measured in EV in relative terms, with 0 EV meaning “properly exposed” (according to a particular metric), −1 EV meaning underexposed at half the desired brightness, +1 EV meaning overexposed at double the brightness, etc.

Based on the mathematical fact that log(a × b) = log(a) + log(b), we can take the logarithm of the entire equation to get:

log(Image brightness) = log(Scene illumination) + log(Subject reflectivity) + log(Lens aperture area) + log(Shutter open time) + log(ISO sensitivity) + Constant of proportionality.

Another name for EV is “stop”. For example, if we make the aperture larger by 1 stop (doubling) and shorten the exposure time by 1 stop (halving), then the image brightness stays the same. Clearly we see that (+1) + (−1) = 0, which is why the equation predicts the brightness to be unchanged.

A further reason for using logarithmic units is that there is a large range of possible values involved. For example, a camera operating at (f/1.0, 60 s, ISO 102400) produces an image that is 1.1×1012 (1.1 trillion; 40 EV) times brighter than a camera operating at (f/32, 1/8000 s, ISO 50). Using a logarithmic scale allows us to add and subtract numbers in the range 0 to 40, instead of multiplying/dividing numbers in the range 1/10000 to 100000.

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