Photographic density is the logarithm of the ratio of incident to transmitted light through the processed film: D = log10(I_in / I_out). A density of 2.0 passes one hundredth of the light, 3.0 one thousandth. The relationship between exposure and density is the characteristic curve, and it is emphatically not linear – it is an S shape, flat in the toe, steep through the useful working range, flattening again at the shoulder.

Because the curve is steep in the working range, correcting density is not a matter of scaling exposure by the density ratio. Between density 2.0 and 3.0 the film gradient is around 4 to 5, so a full unit of density costs only about 0.2 in log exposure – a factor of 1.6 to 1.7 in time, not 1.5 times as a naive ratio would suggest. Below density 1.5 the film is in the toe, the gradient collapses, and contrast sensitivity is lost even though the image looks acceptable on a bright viewer.

Codes fix a working range for exactly this reason. ASME BPVC Section V, Article 2, T-282.2 requires a density between 1.8 and 4.0 for single-film X-ray radiographs and between 2.0 and 4.0 for gamma radiographs, measured through the area of interest. ISO 17636-1 requires a minimum of 2.0 for class A and 2.3 for class B. A radiograph outside the range is rejected regardless of how good the IQI reading is, because the contrast the code assumes is not present.

The correction here uses a representative class C4 to C5 characteristic curve. For accurate work on a specific emulsion and processor, expose a step wedge and plot your own curve – film speed drifts with developer age and temperature.

Worked example

Exposure time used	12 min
Density achieved	1.5
Density wanted	2.5
Radiation source	gamma
Relative exposure at density achieved	0.73
Relative exposure at density wanted	1.33
Exposure correction factor	1.822
Corrected exposure time	21.86 min

From the characteristic curve, density 1.5 corresponds to relative exposure 0.73 and density 2.5 to 1.33. The correction factor is 1.33 / 0.73 = 1.822, so t2 = 12 x 1.822 = 21.86 min. Note that the correction is not the naive density ratio of 2.5/1.5 = 1.67: relative exposure is a logarithmic scale, and over this span the curve is climbing out of the toe, so the exposure actually has to rise by more than the density ratio suggests.