Density curve and film speed
Characteristic density curve
On black-and-white negative film, increasing exposure results in increasing density of the film. The relationship between exposure and densit is not linear. On the one hand, a certain minimum exposure is required to achieve a significant density, while on the other hand, the density cannot exceed a maximum value.
DSince the film is able to process a very wide range of brightness, a logarithmic scale is used to show the relationship between the density and the exposure. The optical density D of the film, defined as the logarithm of the light attenuation when passing through the film is: \begin{equation}D=\log{\left(\frac{I_{in front}}{I_{after}}\right)},\label{dd}\end{equation} if $I$ denotes the light intensity before and after the film. An optical density of D=1 corresponds to a 10-fold decrease in brightness.
If you plot the density $D$ against the logarithm of the exposure $H=I*B$, the product of light intensity $I$ and exposure time $B$, $E=Log(H)$, you get the characteristic blackening curve of the film, shown in Figure 1 in an idealised way..
In the range of no or insufficient exposure, the film has a certain base density $D_0$, usually referred to as fog (or grey level). For black and white films used for pictorial photography, $D_0$ is typically in the range of 0.3, the film attenuates the light by a factor of $10^{0.3} = 2$. In the range of saturation, maximum densities in the range of $D=3$ are reached, the attenuation is $10^3=1000$.
In the middle area of the density curve, it is approximately linear, i.e. The slope of this linear area, i.e. the steepness of the density curve, determines the contrast of the film (or more precisely, the film-developer combination), defined by the coeffient \begin{equation}\gamma=\left(\frac{\Delta D}{\Delta E}\right),\label{gamma}\end{equation}.
For pictorial photography and ‘normal’ subjects, film-developer combinations with a γ between 0.5 and 0.7 are typically used; figure 1 shows a γ of 0.7.
With a typical subject in bright daylight, the difference in brightness between deep shadows and highlights is about a factor of 30 or ΔE=1.5. Usually, the film is minimally exposed so that the darkest parts of the subject just show visible darkening. So you use only a relatively small area of the density curve for the important parts of the image, shown in the image with a grey background. Film densities above D=2 are reserved for highlights.
Figure 2 shows a real example of a densitometrically measured blackening curve and an image taken with the same film-developer combination.
The deepest black, i.e. the lowest film density (point 1), is located approx. ΔD=0.2 above the fog, skin tones (points 2 and 4) in the range D = 1, the brightest points (points 7 and 8) at a film density of D=1.5.
Definition of film speed
Historically, different methods were used to define film speed. Only the two that are important for pictorial photography are presented here. One method, described as early as 1934, is defined today (since 1996) in ISO 6, and replaced the DIN 4512-1 standard. It uses the necessary exposure $H_m=10^{E_m}$ at which the density $ΔD = 0.1$ above the fog is reached to determine the film speed, see Figure 3 left. If the film is developed correctly, the density at point $E_n=E_m+1.3$ is then 0.8 higher than at point $E_n$ . The mean slope, often referred to as the $β$ value, defined by the points $E_m$ and $E_n$ is then 0.8/1.3=0.61. This is the case for the mean density curve shown in the left-hand figure (according to DIN). For the absolute definition of the sensitivity scale, the required exposure at $E_m$ for a film of 1 DIN speed is set to $H_m=10^{E_m}=0.8\;lux\,sec$. This gives a value for the arithmetic sensitivity $S=0.8/H_m$. The ISO sensitivity used in practice is determined by rounding to 1/3 f-stops.
Example: a standard exposure of $H_m=0.007\;lux\,sec$ is measured for a film. Its arithmetic sensitivity would then be $S=0.8/0.007=114.3$. The 1/3-stop ISO scale in this range is 80-100-125, so the closest value is 100, and consequently the film would have a sensitivity of ISO 100.
The standard exposure is defined as an exposure in which the geometric mean of the brightness of the object $H_g=11.38*H_m$. This is equivalent to the requirement that the arithmetic mean of the logarithmic brightnesses (f-stops) is $E_g=E_m+1.06$. As Figure 3 shows, this leads to an average film density of D=1 above the fog.
To determine the relative sensitivity, based on a ‘standard film’, it is sufficient to know that for a halving (doubling) of the sensitivity, $E_m$ changes by $± 0.3$. The sensitivities I determined all refer to Tmax 400 as standard film developed in Kodak Tmax developer, which results in a practical sensitivity of ISO 400 at a constant slope of 0.6.
In addition to the DIN definition, the American Standards Association (ASA) has proposed a definition based on requiring a minimum contrast for shadow details. This is done as follows: the normal object range is assumed to be ΔE=1.5 (= 5 f-stops). Within a range $[E_1,E_1+1.5]$ the mean slope $β$ is defined and then $E_m$ is determined in such a way that at point $E_1$ a slope of $γ(E_m)=0.3 β$ is achieved. The procedure is shown in Figure 3 on the right (according to ASA). The mean $β$ determined in this way is referred to below as the contrast index ‘CI’.
Alongside these academic definitions, there is Ansel Adams' zone system, which is based on practical experience. Here, too, Zone I is defined such that it produces a grey value of $ΔD = 0.1$ above the fog. For normal development (N±0), the sequence of zones then follows an increase in density with a steepness of $γ$ = 0.6, and thus corresponds to the above-mentioned definition of speed.
All these definitions are based on a point of minimal density and do not take into account the individual steepness of the density curve. They are well suited if the films are developed ‘according to standard’. For reasons of image composition, it may be desirable to develop the film to a greater or lesser steepness. In this case, it is useful to adjust the film speed so that an average density of ‘standard exposure’ of D=1 is achieved. Figure 4 illustrates this. In the left-hand image, the film is developed at a higher contrast; nominally, the ISO speed does not change. If you expose according to this, the entire density range of a normally developed film centred around D=1 is at high to very high film density (red symbols). By reducing the exposure time, i.e. effectively increasing the film speed (in the example from ISO 100 to ISO 320), the entire density range is centred around D=1 again (green symbols).
The image on the right shows the opposite case, where the film is developed softer than nominal. In this case, negatives with low density throughout are obtained without adjusting the film speed. Increasing the exposure time, i.e. effectively lowering the film speed (in the example from ISO 100 to ISO 40), brings the density back into the central area.
Fig.4: Illustration of how the film speed is adjusted according to the steepness of the density curve and definition of D1-ISO sensitivity.
This is also the basis of the so-called ‘push’ developments to increase film speed: by extending the development time or increasing the activity of the developer (e.g. less dilution). With a higher steepness of the density curve, the mid-tones and highlights can be placed in the area of medium film densities with less exposure. However, this inevitably results in the loss of shadow detail, as the dark areas of the image slip into the haze. This can be illustrated very well using the greyscale values of the zone system:
Fig.5 : Steepness and sensitivity in push (N+) development in the example of the zone system on the negative and the resulting positive.
BIn a normal development (N±0), the zones 0 - X on the negative are reproduced with a density of 0.1 over fog to a maximum density. If the film is exposed 2 f-stops below at the same development parameters, the drawing in the first two zones (shadows) is lost; at the same time, the maximum density is not reached in the negative, and in the positive, the highlights remain grey. This can be compensated for by developing at N+2. Now midtones and highlights are rendered correctly again, but shadow detail cannot be restored in this way. The sequence of greyscale levels is compressed during push development, i.e. the steepness of the density curve increases. In the example of an N+2 development by a factor of about 1.3, i.e. to $γ$ = 0.8 if we start from $γ$ = 0.6 for standard development. However, even with this push development, more than the range (zones) necessary for a ‘normal’ contrast range of 5 apertures is available. The greater negative hardness can be compensated for in the positive process.
Fig.3: Illustration of the definition of film speed according to DIN (ISO 6 standard) on the left and according to the ASA method on the right.
Fig.2: Densitometrically measured density curve on the left and image example with plotted density points on the right.
Fig. 1: Idealised density curve