# The middle of a print

Imagine the following experiment. You take a source of white light and put it behind a broad white diffusing screen. Put this above a viewing surface. On the viewing surface you place three pieces of paper. One is a pure white. One is as black as can be. The other is gray. You show these three pieces of paper to a subject. You ask the subject if the gray paper is exactly half-way in tone between the white and the black, or, it is closer to the white or the black. You have a supply of gray pieces of paper of varying tone. Depending on the subject’s answer, you replace the current gray paper with another until you get as close to a middle gray as you can.

Repeat this experiment over with many subjects. What do you think that the result would be? Oddly enough, the answer would be that, on average, people report that a gray surface that reflect 18% of the incident light back is halfway between white and black.

Let’s change up the experiment a little. Instead of having white, gray, and black pieces of paper, let us use pieces of film. Now, we illuminate a matte white surface in front of our subject. We interpose three pieces of film. One is completely transparent, allowing the subject to see the white surface. Another is completely opaque: it appears black. The middle piece of film is gray. Now we ask the subject to identify the gray film that is exactly mid-way between the white and black films. Sure as shooting, the answer turns out to be a piece of film that allows 18% of the incident light to pass through.

Let’s look at this situation a little more closely. Here is a diagram of what is going on.

Light and stuff

The material in the diagram might be either the paper or the film. Some light is incident on the material. This light energy must either be absorbed by the material (and turned into heat energy), transmitted through the material, or reflected back from the material. The total energy in must equal the total energy out.

What we are observing is a rather unexpected result that our human visual perception reports surfaces that reflect 18% back to an observer on the same side as the light source as middle gray, even though the truth is that this is closer to the black surface that reflect 0% than it is to the white surface that reflects 100%. (Of course, real blacks reflect maybe a few percent and real whites reflect somewhere between 92% and 96% or so; but ignore that practical detail for now.) A similar situation arises if the observer is on the far side of the material: then a gray film that transmits only 18% of the light through is reported as a middle gray. Again, the truth is that this material is closer to the black (which lets nothing through) than to the white (which lets everything through).

In the late 1920s, William David Wright and John Guild performed experiments along these lines, but with much more detail. They were actually after the details of human color perception. Their work went, first, into the definition of what is called the CIE RGB color space. In 1931, this lead to what is called the CIE XYZ color space. By 1976, the CIE developed the L*a*b* color space, often called CIELAB. Photoshop allows you to work in the LAB color space; and for B&W work, LAB can be instructive, if not useful in its own right.

If you are unfamiliar with LAB, or L*a*b*, its beauty in the context of B&W is that the L-channel carries all of the luminous intensity information, while the color is in the a-channel and the b-channel. For the record, the a-channel is essentially a green-magenta axis and the b-channel is a blue-yellow axis. The L-channel runs from 0 (for black) to 100 (for white). And interestingly enough, middle gray is right where you might expect it to be, at L=50. Finally, something that makes sense.

Here is an equation for the L value:

$L = 116 f(\frac{Y}{Y_n}) - 16$

I don’t expect this to make sense yet, because we don’t know what those $Y$s are and what the function $f()$ is. First, for the $Y$s. Think of $Y$ as a measure of the amount of green light that illuminates the scene. Our eyes are most sensitive to green light, and assuming that the scene is illuminated with white light, just working with the green component turns out to be OK. [If you want more detail on this, read the Wikipedia article on the CIE XYZ color space. You will find that $Y$ was designed to represent luminance, while X and Z carry color information.] Next, the $Y_n$ is the maximum value of the green light that illuminates the scene. We do not expect to get anything greater than $Y_n$. So, $Y/Y_n$ is just the fraction of green light from some object relative to the total amount we’re hitting it with.

Now what about that function $f()$? Here it is folks. Hang on to your hats.

$f(x)=\begin{cases} \sqrt[3]{x} & \text{if } x>\left(\frac{6}{29}\right)^3 \\ \frac{1}{3}\left(\frac{29}{6}\right)^2x+\frac{4}{29} & \text{otherwise} \end{cases}$

Happy? Well, maybe not. But this mathematical expression is a description of how the human eye responds to light. It also tells us why $Y/Y_n = 0.18$ corresponds to L=50. In other words,

$L = 116 \sqrt[3]{0.18} -16 = 50$

[OK, it’s actually more like 0.1841865…. Let’s call it 18%.] Here is what this relationship between $L$ and $Y/Y_n$ looks like:

L versus luminance

You can look at this graph and see that when $Y/Y_n = 0.18$ then, $L=50$. The mathematics tells us directly that a value of 18% of reflected light relative to the illumination corresponds to our visual system reading 50%. So this L thing is a proxy for how we humans see light. The graph above shows us that we are very sensitive to low values of light: at $Y/Y_n = 0.03$, $L = 20$; but as the intensity of light increases, our perceptual response tapers off quickly. This makes all kinds of sense, really. It has great survival value to see what’s moving in the shadows.

What does this curve suggest about how someone would look at a print? First, it says that they would be quite adept at picking detail out of the shadows. Second, it suggests that the darker parts of a print are extremely important for a balance with the lighter parts. If people are so adept at seeing detail in shadow, then perhaps prints should give them plenty of shadow to look at. Hmmm… lot’s of rich shadows with a few profound bright patches. Who used to make prints like that? Let me think.

If you pause to think about this curve, it is quite amazing. It tells you how you see scenes, how your clients see your prints, how folks on the web see images on screens. What it doesn’t tell you though is how DSLRs work, how printers work, how screens work, how negatives work, or how digital files work. These are all different. We would like, more or less, to get a print or an image on a screen that works this way. Here is an example:

L-channel scale in LAB

I created this L-channel scale in Photoshop using increments of 10 in L-Value from 0 to 100. You might think that this would be a candidate scale for a Digital Zone System, and something more or less identical to this has been proposed before. However, the truth is a bit more complicated than this. My guess is that this scale looks differently for me on my monitor than it does for you on yours. The reason is that I calibrate my monitor to the L* curve, and I believe that this is rather rare.

But my earlier image above that shows reflection and transmission of light is not telling the whole truth. That single ray of light appears to be reflected in a mirror like manner; that is, the incident angle equals the reflected angle. That’s like a spot source of light, say a naked bulb, reflecting off of a mirror surface. We could have a diffuse source of light and we could have a matte surface. In fact, there are four ways that this could go: spot light/reflective surface; diffuse light/reflective surface; spot light/matte surface; and diffuse light/matte surface.

What I’ve been talking about so far, certainly in the context of a print, makes the most sense in the context of a diffuse light source and a matte surface. Let’s consider this situation.

Diffuse light – matte surface

The reason that the light reflects in all sorts of random directions from the matte surface is that it is random itself. This case would be like a print on fine art paper illuminated by a soft box. The same scattering of light in random directions is true for some liquids, like milk, which contain suspensions of small particles that scatter light within the volume of the liquid. More of that a little later.

What about a spot source and a matte surface? That looks more like this:

Spot light – matte surface

This would be like looking at a fine art print with an intense beam of light. Only part of the paper would be illuminated, say. Not a great viewing method.

Finally, how about a diffuse source and a mirrored surface?

Diffuse light – reflective surface

This would be like illuminating a mirror with a soft box. In the real world, you’d just see the image of the soft box in the mirror.

Well, so far so good, but what does it have to do with a print? (See, this is the conversation part…) Well, it should be clear, I hope, that the ideal conditions for viewing a print are diffuse lighting from a matte surface. Now, by matte surface, I am including all kinds of print paper: fine art, luster, glossy, everything. If a print paper has a glossy surface, the last thing you want is some direct light beaming off it into the viewer’s eyes. By mirrored surface, I really do mean (here) a mirror, literally. Any real surface will be some combination of these idealized cases. So sure, a glossy print is somewhere between a matte surface and a mirrored surface; but while you can see some sheen on it, you can’t shave with it either.

My main goal for this page was to explain why an 18% diffuse surface has been chosen as the reference standard for middle gray in prints, screens, negatives, and scenes. The reason has to do with human vision. So, we start from that reference standard and work outwards to black, on the one hand, and white on the other.

For my next topic, I have to explain why most of what photographers are taught about how light falls off with distance is plain wrong.