Exposure

Exposure in the digital age

By Bob Newman, first published December 2012

Definitions of exposure

There is an almost forgotten sourcebook for photography. First published in 1890 as the ‘Ilford Manual of Photography’ and later simply as the ‘Manual of Photography’ this text contains somewhere within it the answer to practically any question about photography. It is my ‘go to’ reference book for the basics of photography, especially when there are disputes about the real meaning of photographic terms. If something photographic is defined in a book first published in 1890, it is likely that is what it has ‘always’ meant. Such a term is ‘exposure’. The Manual of Photography (1971 edition) says:

The exposure received by a film is governed by the strength of the light falling on it and by the time for which the light is allowed to fall… The light falling on a surface is defined as illumination. The relationship between exposure E, illumination I and exposure time t is expressed by the equation:

E = I.t

So, exposure simply refers to the amount of light incident on the film, or these days the sensor. This fact often comes as a shock to those who have gathered the impression that the term ‘exposure’ refers to the lightness or darkness the final image, but that is not the case.

Illumination is measured by the Systeme Internationale (SI) unit lux and the illumination at the sensor resulting from an object of luminance L (measured in candelas per square metre) is (πL/4).(d/v)2 where d is the diameter of the aperture and v is the distance of the image from the lens. For an object at infinity v is equal to the focal length of the lens, and so d/v is equal to the reciprocal of the f-number of the lens, usually written as N. So, putting the two equations together we get

E = (π/4).(L.t/N2). The only variables here are L, N and t, so exposure is determined entirely by the scene luminance, the f-number and the exposure time (at least for distant subjects). At this stage, some people will be wondering what became of ISO, but we will leave that topic for a later article. Here we want to concentrate on the photographic importance of exposure.

Photons

The above is the classical view of exposure, which has held true since the days of film. However, with digital technology exposure takes on a different significance. That significance is due to the quantum nature of light. Mostly, optics deals with light as a type of wave, but light also behaves as it is made of particles called photons. This behavior is critical for the workings of digital photography (as it was also for film photography) because light is detected by a sensor when a photon strikes a silicon atom in the sensor and frees one of its electrons, generating an electronic charge, which then is detected by the camera electronics. An image can be built up if it is known where each photon strikes. In practice that is not possible – so the compromise is to localize groups of photon strikes by dividing the sensor into a square grid of pixels – the smaller the pixel the more accurately it is possible to know the location of the arriving pixels.

The trajectory of pixels is by nature random. They are generated by random events, they typically undergo reflection and refraction passing through different transmission media. The consequence of this is that when photons build up an image, they do so as a pattern of spots, each one signifying where a photon hit the sensor, and the pattern in these spots is intrinsically random.

Photons and noise

This random pattern of arrival of quantum particles is a well-known phenomenon – it is known as ‘shot noise’. It is apparent in any signal carried by quantum particles, such as radio, electrical signal along wires or, in this context, light. Random independent events, such as the arrival of photons, are described by Poissonian statistics, which allows the characteristic patterns to be analysed statistically. Since this is a statistical analysis, it only applies to collections of multiple events, and describes the differences between similar observations. The interval or size of each individual observation might be anything, so long as it is consistent. In the context of photography, we might choose the smallest visible part of an image, or a pixel, either an output (printer or screen) pixel or an input (camera) pixel. However, if we are to make a valid comparison, we must compare like intervals. If we do that, and count the number of photons counted in a number of intervals, we will find, for an evenly illuminated area, that the level of illumination is given by the mean of those counts. However, due to the randomness of photon arrivals, the counts will not all be the same, there will be a variation, the size of which is given statistically by the standard deviation.

Figure 1: A Poisson distribution of photon counts. Each column shows the proportion of samples containing the number of photons on the x-axis. In this case the mean is 10, and the standard deviation is 3.2.

Visually, this variation looks like an unevenness of tone, or grain, and is usually called ‘noise’ (from its effect aurally when heard in radio communication). The effect of the noise in relation to the mean level, or signal is predicted by the ‘signal to noise’ ratio, and is given by the mean count divided by the standard deviation. The statistics predicts that the SNR is the square root of the mean number of photons counted in each observation interval. So, if we count 4 photons in each, the SNT is the square root of 4, or 2. If we count 100, then it is the square root of 100, or 10. Obviously, to ensure as high a signal to ratio as possible, we need to ensure that we have available for counting as many photons as possible. It should be noted at this point that this shot noise has nothing to do with the electronics of the camera, it is intrinsic to the structure of the light itself, and controlled entirely by how many photons are counted.

Photons and exposure

The next step is to understand how the number of photons counted relates to the exposure. Above it was established that the exposure is derived from illumination, which is measured in the SI unit, lux – so the question is, how does lux relate to number of photons. The lux is what is called a photometric unit, that is it relates to visible brightness. The other way of measuring light would be with radiometric units, which measure power. The radiometric equivalent of the lux is watts per square metre. The two are related by a standardized function called the luminosity function, which weights visible light by the apparent brightness of that wavelength light to the human eye. Thus it is apparent that the translation of lux to watts per square metre depends on the colour of the light. Calculations could be made on the assumption of white light, equal strength at every wavelength, or else the wavelength of the peak of the luminosity function, 555 nanometres (mid green in colour). For light of that wavelength, one lux is equal to 1/683 watts per square metre. In turn, the unit of exposure, the lux second is equivalent to 1/683 Joules of energy per square metre. Since the energy of a photon at 555nm is 3.58 times 10 to the power of -19 Joules, the number of photons per square metre can be calculated, at 4000 million million photons per square metre. To finish off this calculation, the standard white level exposure at 100 ISO is 0.78 lux seconds, and the area of a full 24 by 36 millimetre frame is 0.00086 square metres, so the number of photons we would expect to make up a full frame white image at 100 ISO is 2.7 million million. By the same token, at 6400 ISO the white level exposure will be reduced by a actor of 64, and thus the photon count will be reduced in proportion, resulting in a signal to noise ratio 8 times lower.

Figure 2: The luminosity curve. There are two curves here, the scotopic (green) night vision curve and the photopic day vision curve (black). It is the latter which provides the luminosity function that defines photometric units.

Concluding, the noise in images with a very small exposure, typically low light or ‘high ISO’ images, is produced not by electronic amplification, but by the structure of the light itself. The way to decrease noise is to ensure as large an exposure as is possible – hence the popularity of techniques such as ‘expose to the right’, which seek to maximize exposure.

Figure 3: This is a crop from a much larger image. In low light the image is noticeably noisy at these magnifications. The noise is caused by the low exposure, resulting in very low photon counts.