4.1.1.1. Luminous flux

The luminous flux represented by Φ, expressed in watts, is defined as the power emitted by this source: Φ = E/δt, and the spectral luminous flux (expressed in watts per meter (SI unit), or more commonly in watt by micrometer²) is denoted by Φ(λ).

4.1.1.2. Radiant intensity

The radiant intensity of the source is by definition the flux per solid angle surrounding an observation direction θ. It is expressed in watt/steradian and is, therefore:

[4.1]

The spectral intensity is defined similarly:

[4.2]

In the calculation of intensity, the source is considered as a whole. The distribution of the emission on its surface is not distinguished.

4.1.1.3. Irradiance

The irradiance³ of an object is defined as the intensity per area unit in the direction of observation. It is, therefore, the luminous flux emitted by the elementary surface ∂s of the source in a small solid angle ∂ω around the direction θ (Figure 4.1):

[4.3]

The dependence of irradiance with respect to the direction of observation θ is a property of the material. If (θ) is not depending on θ, it is said that the source is Lambertian; Lambertian emitters are good approximations for a large number of matte or rough surfaces in nature photography. Conversely to the Lambertian surfaces, there are monodirectional sources such as lasers and specular surfaces that only reflect in a single direction.

Similarly, the spectral irradiance can be defined from the spectral intensity :

[4.4]

Irradiance is expressed in watts per square meter, and steradian (W.m⁻².st⁻¹) and the spectral irradiance in W.m⁻³.st⁻¹.

When M is a primary source, (θ) is known by its radiation pattern (also called far-field pattern). When M is a secondary source, it is defined using its reflectance (see section 4.1.4).

4.1.1.4. Radiance or exitance

Exitance⁴ (in watts/m²) characterizes the source and expresses the luminous flux emitted per unit area:

[4.5]

and the spectral radiance:

[4.6]

For a Lambertian source ( independent of θ), as dω = 2π sin θdθ:

[4.7]

4.1.1.5. Irradiance

Irradiance is the incident flux per unit area received by the sensor originating from the source. It depends both on the luminous intensity emitted by the source in the direction of the camera and on the orientation of the sensor, given ∂s′ the surface on the sensor around the point M′, described by the beam traveling from M and with a solid angle ∂ω. The surface ∂s′, including the normal, makes an angle θ′ with MM′ and receives the flux ∂²Φ emitted by the source. We have:

[4.8]

if r is the source sensor distance r = d(MM′).

By definition, the irradiance fraction of ∂s′ provided by the surface ∂s of the source on the sensor is given by:

[4.9]

4.1.1.6. Luminous exposure

It is the integral of the flux during the exposure time per area unit of the receiver. In practice, it is expressed by:

[4.10]

and is measured in joule/m².

4.1.1.7. Total irradiance

M′, expressed in watt/m², is given by the integral of all points of the object contributing to the pixel, that is to say the inverse image of the pixel:

[4.11]

Similarly, the spectral irradiance is equal to:

[4.12]

In photography, the observation distance of the various points of the object and the angle of the sensor are almost constant and thus, for small objects, we have:

[4.13]

4.1.2.1. Color temperature

For a given temperature T, Planck’s law (equation [4.14]) gives the maximal emittance that a body emitting light can achieve. In nature, where the emittance of a material is always less than that of the black body, it can be expressed by:

[4.19]

where ε(λ) is the spectral emissivity of the material, between 0 and 1.

If ε(λ) is not dependent of λ, it is said that the body is gray. It uniformly absorbs for all wavelengths a part of the radiation that the black body would reflect. A gray body is fully described by the temperature T^o of the corresponding black body and the value of ε: T = T^o.

If ε(λ) is dependent on λ, a search is launched among the black body curves for the curve that has the same wavelength behavior of the material being studied. If this behavior is reasonably similar, it is then said that the object behaves as the black body at temperature T^o. It yields that T ≠ T^o.T^o is called the color temperature of the object. How is the value of To found? Starting from equation [4.19], ε(λ) is expressed on the basis of 1/λ for all temperatures T using equations [4.17] and [4.19] and allowing the tracing of: log(ε(λ)) = f(1/λ). The most linear curve designates the most suitable temperature T^o because it has a slope proportional to 1/T – 1/T^o.

We will return to the transition from a color temperature to another using Wien’s formula in Chapter 9 where we describe the filters that can equip the objectives. This transition is based on filters which optical density varies inversely to the wavelength and proportionally to the difference between the inverses of the source temperatures (see section 9.7.5).

Nevertheless, this thermodynamic definition of color temperature is inconvenient because it requires a good knowledge of the sources, a condition hardly satisfied in practice. Therefore, in reality, empirical approaches are frequently used strongly based on spectral distribution hypotheses in the scene. We will encounter these approaches when we discuss white balance (section 5.3).

4.1.2.2. The tungsten case

Tungsten is used a lot for filaments of light bulbs because it has the highest melting point of all the metals (3,422 K). This allows us to create high brightness sources and a relatively high color temperature. The color temperature of tungsten is approximately 50 K higher than its real temperature in its operating range in the vicinity of 3,000 K. It should, however, be noted that the color temperature of tungsten remains low compared to that of daylight. To simulate daylight, this color temperature is altered by inserting a colored filter in front of the source (or the sensor), as we will see in section 9.7.5.

The International Commission of Illumination (CIE) has standardized source A as the prototype of the incandescent sources with tungsten filaments LEG 68. It corresponds to a filament temperature of 2,855.6 K. Standard illuminants denoted by B and C are derived from this source in order to, on the one hand, express the daylight at midday, on the other hand, the average daylight. These standard illuminants are obtained by filtering the illuminant A by filters which are themselves standard. The standard illuminants A, B and C are still found but nowadays are much less used. The family of standard illuminants D, which will be discussed later, is preferred over these.

4.1.2.3. Sunlight

The Sun’s emission is reasonably similar to that of a black body at a temperature of 5,780 K. Its maximum⁵ is roughly the wavelength of 500 nm. Naturally, the atmosphere considerably filters this flux, but in the visible spectrum, the curves remain almost identical (although attenuated) between the top and bottom of the atmosphere; these curves are, however, very different in the infrared, where the absorption bands of water vapor stop the spectrum received at the level of the sea.

Systematic studies on the spectral density of daytime radiation at various moments of the day (related in JUD 64) have been conducted by Judd et al., who have proposed an empirical law of this radiation, derived from a polynomial approximation of the curves thus obtained. Even if they are implicitly dependent of unknown variables, they are often used to determine the whole of the spectrum from a very small number of measurements (therefore, ignoring the hidden variables):

[4.20]

Furthermore, daytime radiation is represented with a very good accuracy as the combination of three terms:

• 1) a term (denoted by ₀) obtained as average, at a fixed wavelength, of all measurements;
• 2) a term (denoted by ₁) which is dependent of the presence of clouds and introduces a yellow-blue antagonism term;
• 3) a term (denoted by ₂) which is dependent on the amount of water vapor in the atmosphere and translates into a red-green antagonism term.

The functions ₀, ₁ and ₂ have been tabulated for specific temperatures: 5,500 K, 6,500 K, etc.

These curves reflect the spectral evolution. How does the solar radiation evolve, in energy, during the year? The flux received by a point on the Earth also depends on the Earth–Sun distance and, therefore, regardless of the climatic conditions, solar irradiance follows an annual periodic law. If the Sun irradiance is referred by ε_solar(j) on day j, it can be derived from the irradiance ε_solar (j = 0) of the 1^st January 1950, taken as a reference, by the formula HAG 14:

[4.21]

where j is the date in days, j₀ is the date on 1/01/1950 and ε_sun(0) = 1,367 W/m².

4.1.2.4. The standardized sources

The International Commission on Illumination has standardized on several occasions light sources according to technological needs. The ISO makes use of several of them as references for photography ISO 02. Their curves are normalized in order to have a value of 100 at the wavelength of 560 nm. These are mainly the ones derived from the standard illuminant A:

• – the standard illuminant A, itself derived from the radiation of the tungsten filament at a temperature of 2,855.6 K, which is used as reference for many studio lightings;
• – the illuminant B, corresponding to direct daylight with the sun at its zenith. Obtained from source A, filtered by a standardized filter B, it has been assigned a color temperature of 4,874 K;
• – the illuminant C corresponds to an average day. In a similar manner as B, it also derives from a filtering from source A. It is assigned a color temperature of 6,774 K.

These are illuminants mathematically defined to match the outdoor and natural lighting. They have been obtained from Judd’s studies that we have cited above. From these remarks, the family of illuminants D has been inferred, which is declined using two numbers which designate the particular temperature associated with a given moment: D50 at 5,000 K, D55 at 5,500 K which is the ISO reference of bright sun, D65 at 6,500 K, D75 at 7,500 K. The illuminants D are not on the Planckian locus, but they are close to it (Figure 4.5).

Furthermore, illuminant E is used as reference (also referred to as W₀) which corresponds to a constant wavelength spectrum. It is also not a black body, but its color temperature is very close to the black body at 5,455 K.

Finally, illuminants F (from F1 to F12) have been defined corresponding to various fluorescent sources (without reference to a temperature). They cover various types of emission with more or less wide lines, more or less numerous, superimposed to a more or less significant continuous spectrum.

4.1.3.1. The case of light-emitting diodes

Light-emitting diodes (LEDs) have become quite quickly accepted as white light sources. Nevertheless, because of these commercial developments, they are still neither stabilized in their technology nor a fortiori standardized.

White LEDs can be designed according to various principles which confer them significantly different spectral properties:

• – diodes formed from three monochromatic red, green and blue diodes exist but are seldom used. It is the proportion of the sources and the position of the lines that secures the quality of the white color obtained. Green LEDs are unfortunately less effective than red and blue;
• – the sources consisting of a single blue diode (of yttrium aluminum garnet (YAG) or gallium nitride (GaN) surrounded by a phosphorus-based material whose luminescence is excited by the LED, provide a fine band (blue) accompanied by a broad band (the luminescence emission), placed around the yellow (complimentary color of blue);
• – sources constituting of a diode in the ultraviolet, exciting a fluorescence covering the entire visible spectrum, therefore white.

LEDs are today characterized by a CRI greater than 80, some exceeding 90.

4.1.4.1. Reflectance

In photography, most of the objects are secondary sources and the flux they emit toward the sensor originates from a primary source (the Sun and a lamp). How is the radiometry of these objects related to the sources? A new quantity is introduced: the reflectance of the object (also referred to as the reflection coefficient).

The reflectance of an object is defined as the ratio of the reflected flux Φ to the incident flux Φ_S (Figure 4.6):

[4.22]

The reflectance is always less than 1. It is an intrinsic property of the material depending, therefore, neither on the source nor on the sensor. It depends on the wavelength λ, on the direction of the incident light ξ and on the direction of observation θ.

Reflectance can also be written depending on the incident illumination ε_S and on the luminance of the object by the equation:

[4.23]

When the dependencies of the reflectance are expressed with respect to all its variables, a function of five variables is built (the wavelength, two angles to identify the source and two angles to identify the sensor), which is then referred to as bidirectional reflectance distribution function (BRDF), which is a tool widely used in image synthesis.

The albedo is defined as the ratio of the spectral irradiance attenuation in the direction θ due to the diffusion in other directions, and the attenuation due to the absorption in the direction θ and to the diffusion in all the other directions. It is especially used in remote sensing and planetology. It is a dimensionless quantity.

4.1.4.2. The case of distributed sources

The case of strongly dominant point sources allows the luminance of the objects in the scene to be determined, as we have just seen. A more complex case, but very common, is met when there is no dominant source in the scene, either because the sources are very numerous, or because they are diffuse. The first case often occurs in night photos, for example in urban areas. The second case is, for example, illustrated by outdoors photographs under overcast skies.

We are then faced with a complex situation which is dealt with using the formalism of radiosity.

In this approach, each point in the scene is considered as receiving a luminous flux that it partly re-emits according to the rules of its reflectance in the whole of the scene. The thermodynamic balance of the scene is then sought, exclusively limited to the wavelengths of the visible spectrum. Such a complex integral problem (where each source point interacts with its neighbors) can only be solved from a very large number of approximations, for example:

• – by considering materials as Lambertians;
• – by subsampling the diffuse sources;
• – by limiting the number of reflections of the beams;
• – by ignoring fine interactions: diffraction, dispersion, etc.

Radiosity is primarily a tool for image synthesis [COH 93, PHA 10, SIL 94] and is only rarely employed to interpret real scenes. It is yet capable of explaining the presence of a signal in the areas of the image which do not see the primary sources. With the developments of cameras that give good-quality signals even with a very small number of photons, radiosity will assume an increasingly important role in the interpretation of images.

4.3.2.1. In the general case

Equation [4.28], taking into account an object which is not at infinity and therefore would not form its image at distance p′ = f but at distance p′ = f + s′ = f − fG (G the transversal magnification being negative in photography where the image is inverted on the sensor) and taking into account the vignetting effect at the edge of the field (angle θ′), a slightly more complex but more general formula is obtained that can be found in numerous formulations [TIS 08] (especially in the standard that establishes the sensitivity of sensors [ISO 06a]):

[4.30]

Assuming that all the photons have the same wavelength λ, starting again from the simplified equation [4.29], the number of photons incident on the photodetector can be calculated (see section 4.1.2):

[4.31]

where h is Planck’s constant, c the speed of light and η the external quantum efficiency of the sensor (see section 7.1.3).

Figure 4.9 shows the number of photons reaching the photosite for several luminance values according to the size of the photosite and for defined experimental conditions: a wavelength λ = 0.5μm, an f-number N = f/4, an exposure time dt= 1/100 s, a quantum efficiency η = 0.9 and an optical transmission τ = 0,9, leading to the very simple relation:

[4.32]

with a in micrometers and in Wsr⁻¹m⁻² or _υ in lm × sr⁻¹m⁻² (because V_p(500 nm)= 1,720 lm/W).

4.3.2.2. Discussion

From these equations, we can draw a few elements of photography practice:

• – as expected, for a given scene, the received energy depends on the ratio dt/N². In order to increase the energy, the choice will have to be made between increasing the exposure time or increasing the diameter of the diaphragms;
• – when magnification G is small, that is under ordinary photography conditions, the distance to the object has a very small role in the expression of energy. In effect, the magnification G of equation [4.30] is then significantly smaller than 1 and will only be involved in the energy balance in a very marginal manner which is almost independent of the distance to the object if it is large enough. The energy received from a small surface ds through the lens decreases in inverse ratio of the square of the distance, but on a given photosite the image of a greater portion of the object ∫ ds is formed, collecting more photons and virtually compensating this decrease;
• – this observation does not hold for micro- and macrophotography where G is no longer negligible compared to the unit and must imperatively be taken into account. Recalling that G is negative, the energy balance is always penalized by a stronger magnification;
• – from these same equations we can see that, if we maintain an f-number N constant, then the focal length f has no influence on the exposure time. With N constant, it is therefore possible to zoom without changing the exposure time;
• – it is nevertheless difficult to keep N constant for large focal lengths because N = f/D. If the diaphragm D is limiting the lens aperture, it can be seen that f can only be increased at the expense of a longer exposure time dt. We can then expect to be very quickly limited in the choice of large focal lengths by motion defects due to long exposure times associated with large magnification. This is the dilemma of sports and animal photography.

4.3.2.3. More accurate equations for thick systems

We have conducted our calculations with a thin lens and small angles. The lens objectives utilized are more complex and allow the small angles hypothesis to be abandoned. They lead to more accurate results at the price of greater complexity.

It is then important to take image formation into account in the thick system. The source then sees the diaphragm through the entrance pupil, as an image of the field diaphragm in the object space (see section 2.4.4), while the sensor sees it through the image pupil. These elements then define the etendue [BIG 14].

If the system is aplanatic⁶, then equation [4.28] becomes:

[4.33]

where γ_max is the maximum angle of the rays passing through the exit pupil that arrive on the sensor. For symmetric optical systems (that is to say, having optical components symmetrically identical in their assembly), the exit pupil is confused with the diaphragm, and if the angles are small, it yields sin(γ_max) ~ D/f = 1/N. If this is not the case, formula [4.33] may differ from formula [4.28] in a ratio of 0.5 to 2.

4.3.2.4. Equations for the source points

If the object is very small, its geometric image will not cover the whole of the photosite and the previous equations are no longer valid (that is, for example, the case of pictures of a star). It is then essential to abandon the geometrical optics image formation model to take into account the diaphragm diffraction which defines a finite source size and expresses the manner in which its energy is distributed over the photosites (see section 2.6).

4.5.1.1. In analog photography: film sensitivity curve

The sensitivity of photographic films is governed by the chemistry of their composition and developing processes. History has imposed DIN sensitivities (German standard in logarithmic scale) or American linear standard (ASA) to characterize this sensitivity. It is now standardized by the ISO sensitivity [ISO 03] which is in practice aligned with the ASA standard and unified for films and solid sensors. Generally, the price of high sensitivity comes at the expense of a low film resolution⁷ which presents a coarser “grain” (lower limit of the film resolution). A film has a fine grain for a sensitivity of 100 ISO (or 20 DIN) or below. It is then known as “slow”. It is known as “fast” for a sensitivity of 800 ISO (or 30 DIN) and beyond. The film has a strongly nonlinear behavior which results in particular by the lack of response at low energies and by saturation when it is overexposed. Doubling the ISO sensitivity is equivalent to exposing the film twice, either by opening the diaphragm by a ratio or by exposing with twice the time.

We will choose the example of the black and white negative film⁸, but the reasoning would be similar with a positive film, with paper or with a color emulsion: only a few changes of vocabulary about the quantities being manipulated need to be introduced. Film is characterized by its sensitivity curve, H&D curve (Hurter–Driffield) [KOW 72], which expresses its blackening degree according to the amount of light it receives. Such a curve is shown in Figure 4.13: it expresses the logarithm (in base 10) of the transmittance (which is called optical density) with respect to the logarithm of the exposure ξ the film has received.

The blackening curves depend on many factors (in particular on the conditions for developing film: bath temperatures, product concentrations, etc.). The standard ISO 9848 [ISO 03] defines the ISO index which allows the ordering of the film sensibilities.

Starting from the family of the blackening curves of a given film, the minimum transmittance value of the film is first determined (when it has been developed without receiving any light): that is D_min. The curve that passes through the two points M and N, known as standard contrast points, is then selected (it should be noted that there is only one for a given film) (see Figure 4.13), identified by:

• – M has a density D_m = D_min + 0.1 (and an abscissa log ξ_m);
• – N has for coordinates: log ξ_n = log ξ_m + 1.3 and D_n = D_m + 0.8.

The ISO sensitivity is then defined by: S = 0.8/ξ_m (with ξ_m expressed in lux.second), rounded to the nearest integer of the ISO table.

4.5.1.2. In digital photography: sensor sensitivity

The situation is very different in digital photography because, once the architecture of the sensor is defined (site geometry, choice of materials and coatings), the gray level corresponding to a given pixel count can be affected in two ways: by the analog gain of the conversion of the charge into current, and then by the gain of the digital processing chain downstream (see Chapter 3).

Nevertheless, in order to maintain the same quantities as those that were familiar in analog photography, the ISO has tried to transpose (with more or less success) the existing standards. Thus, a digital ISO standard [ISO 06a] has been redefined which allows the use of familiar acronyms in adjustments that can now be performed on an image-basis and on any camera.

This standard considers three slightly different but nevertheless very similar quantities to characterize the camera response to a flux of light. They are needed because the cameras have very different operating: some do not give access to essential for a precise definition settings. Their differences are generally ignored by the users and manufacturers willingly misinterpret them for commercial purposes under the same name of ISO sensitivity in commercial documents intended for the general public:

• – the standard output sensitivity (SOS), which takes into account the entire image manufacturing chain in a monolithic manner;
• – the recommended exposure index (REI), which translates the nominal operating conditions depending on the manufacturer, but do not guarantee image quality;
• – the ISO speed that specifies exactly what is obtained on the sensor output when it is exposed to a given luminous flux. The ISO speed is the closest to what is expected of a standardized sensitivity.

For these quantities, the standard is always concerned with conditions as fixed as possible regarding the temperature, the type of illumination, the camera settings (exposure duration, amplifier gain and white balance), as well as the experimental conditions. It is usually proposed to define default values for a daylight illuminant (D55) or for a tungsten illuminant (in which case, the letter T is added to the given value).

Practical considerations about the precise determination of the sensor sensitivity can be found in [BOU 09] in Chapter 9.

4.5.1.3. ISO speed

The standard provides two different ways to determine the ISO speed S when taking pictures, either by promoting the high values of the image or by considering its low values, and in this second case, it offers two different quality levels. These various definitions will apply to very different cameras because between a professional camera and an affordable cell phone, it is not possible to find common operating ranges that allow the definition of a standard protocol. The ISO speed S is defined as the weighted inverse of an exposure ξ judiciously chosen and expressed in lux.second. The formula type is:

[4.36]

where the value 10 is chosen for compatibility with existing standards.

The two definitions are the result of the following procedures:

• – the saturation method quantifies the sensitivity that characterizes the best image quality possible. It applies equation [4.36] to an irradiance ξ_sat which is just below the saturation. The sensitivity is then defined by: S_sat = 78/ξ_sat. The value 78 is determined such that a picture of an 18% gray⁹ gives a desaturated image of 12.7%, that is to say . The value is introduced to assume a semi-diaphragm below saturation. It is this method which guarantees the best reproduction of very bright areas;
• – the method based on noise concerns on the one hand, very good quality images and in order to achieve this a signal-to-noise ratio of 40 to 1 is required, and on the other hand images of acceptable quality to which it imposes a 10 to 1 ratio. To this end, the curve expressing the signal-to-noise ratio depending on the illumination level is used (Figure 4.14) and the operation points N and M and the corresponding exposure are considered: ξ₄₀ and ξ₁₀. Therefore, it can be derived that:

[4.37]

For a color sensor, ξ is determined by weighting the irradiance noise: Y = 0.21R + 0.72V + 0.07B and in the chrominance channels Y − R and Y − B.