The True Inverted Digital Negative

John Riches at TruNeg has a method using mathematics to fix the failure of the inverted image to make a workable negative. If nothing else, a great lesson in how digital negative works.

Writer and photography / John Riches at TruNeg

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Many books, articles and videos have been published explaining and illustrating a multitude of approaches to solve the problem of the failure of the inverted image to make an acceptable “photographic” negative.

“However, to the best of my knowledge, there are none that explore the mathematics of the digital image, define a true reversal or digital negative and then find a mathematical solution to the problem.”

This article is based on the TruNeg digital negative program and follows the process through from the original photographic image, what happens when it is digitized and inverted, and goes on to show how it can be corrected to make a genuine “photographic” negative that can be sent straight to the printer.

The Photograph

Photography is an exponential phenomenon. That is, it works in multiples of multiples. The aperture f4 lets in twice the brightness of light of f5.6 which lets in twice the brightness of f8 which is twice f11 etc. This means that f4 is eight times brighter than f11. In a typical photographic subject, the subject brightness that becomes white in the print is 8 stops brighter than the subject brightness that becomes black which means that the subject white is 256 times brighter than the black.

Note how the stop changes at the beginning of the exponential scale are numerically small when compared to the stop changes at the end of the scale.

Digitizing

When the image is digitized the brightness is recorded on a scale between 0 and 255. However, the photographic image being exponential cannot start at zero as zero times any number is still zero. So the eight-stop scale starting at 1 RGB should go 1, 2, 4, 8, 16, 32, 64, 128, 256 ¹.

This is not practical in reality. When viewed on a monitor the stop changes between 1 and 2, 2 and 4, 4 and 8 are so small and so dark they can be hardly seen even though they each represent a change in brightness of one stop.

If an exponential constant of 1.4, or half a stop, is used instead of 2, sixteen steps are generated between 1 and 256. 1, 1.4, 2, 2.8, 4.0, 5.7, 8.0, 11.3, 16, 23, 32, 45, 64, 90, 128, 180, 256.

If RGB 16 is considered to be “just black”, the eight steps below 16 can be considered black and the eight steps from RGB 16 to 255 used to record the eight-stop subject range.

When the step wedge is displayed truly on a monitor each step from 16 to 255 will be the same amount brighter when measured with a light meter in f stops or Exposure Values. The lower the gamma the more likely this is to be true. On my Asus ProArt monitor the difference between each step is very close to one stop at a gamma of 1.8.

Inverting the image

The graph below shows the inputs and outputs of a positive image in stop intervals.

When the file is inverted in RGB Color Mode, Photoshop subtracts the positive value from 255 and the exponential structure of the positive is lost and the inputs flip, or reflect, horizontally.

The above graph shows how the single highlight stop T10 to T9 is now output to the monitor and printer with a difference of four and a half stops plus the eight stops below RGB 16. The four shadow stops T2 to T6 are now output within the one stop between RGB 180 to 255.

To restore the exponential structure of the positive image the inputs of the inverted step wedge need to be output as an exponential function.

Joining the dots creates the correction curve.

However there are still some problems.

Inputs from T10 to T9 are only one stop but the outputs from zero to RGB 16 are eight stops (see explanation above). To correct this T10 to T9 must be output as a stop, that is, RGB 11 to 16. A similar situation occurs between RGB 239 and 255 where the inverted eight stops from RGB 0 to 16 are output as one stop. To correct this the next inverted stop down from 239, RGB 244, has to be output as the stop from 180 to 255.

The image below traces the changes from the positive step wedge to the corrected inverted step wedge. The correction is proved if each step in the corrected image is the same multiple of the step before when measured with the Eyedropper tool.

The negative will appear very dark and clogged up on the monitor as the highlight of the negative is 11 RGB. This points to an important difference between printing a positive and a negative image. In the positive, all the inputs less than 16 are considered black and the printer does not have to accurately resolve these as long as they appear black. In the negative, all the inputs from RGB O have to be resolved in order to print the highlights properly.

The Curve in Practice

While the above curve defines the perfect inverted negative in practice there are three problems, the curve does not account for the effect of the “toe” in the photographic material, the printer and print chemistry have to be adjusted to fit the curve and the printer has to print the step wedge correctly.

On the monitor, each output step between RGB 16 and 255 is the same multiple as the step before. However, photographic materials need a larger step between the first two tones to achieve the equivalent tone displayed on the monitor. This effect is described as the “toe” of the photographic material and varies from process to process and if not accounted for will completely wash out the highlights.

The negative also has to have the exact contrast to make the print “just white” and “just black” at rgb 11 and 180 on the analogue material. This could mean extensive testing and may not be achievable in practice, particularly when the “toe” is added into the mix.

To overcome these problems the TruNeg digital negative program uses a Control Negative. The TruNeg Control Negative is used to find the printer and analogue combination that makes a print with a clean white and satisfactory maximum density. When this is done the “just white” and “just black” values can be determined along with the negative density required to correct for the “toe”. Once these three parameters are known a correction curve is calculated for that particular printer and process combination.

The other consideration in making the “perfect’ negative is whether the printer prints a true exponential tonal scale. Photographers working with alternative processes may not see the need for the absolutely perfect negative while those working with traditional processes are likely to want to know the accuracy of the negative and be able to correct any errors. By analyzing the density or scanning the print of the step wedge TruNeg finds any error between the actual and target value for each step and amends the curve to compensate for any printer error.

Conclusion

The principle demonstrated above of outputting an inverted exponential step wedge as an exponential function can be applied to any exponential step wedge with any number of steps between any two RGB values that make “just white” and “just black” to make a true negative image of any positive.

Notes

1. This is one of several anomalies I came across while doing this exercise.

John Riches is the creator of TruNeg and worked in commercial photography in the 1970s and 80s while pursuing an interest in photographing the Australian temperate rainforest. John taught photography from 1976-94, he designed and built his own home from 1982 to 88 (part-time) which led to working as a building designer in recent years. John Riches started carbon printing in 2018.