We received a request from an optics pro to add MTF and OTF to our online layman's glossary. These are notoriously difficult concepts to simply explain! So, we added a small entry to the glossary and a longer beginner's definition right here. Please leave your comments if this works for you or if something is still unclear. The glossary and these articles are living documents and we're always open to improving them!
Aka “Modulation Transfer Function”
TLDR: MTF is a term describing how well an imaging system can reproduce contrast as details in the image get smaller and smaller (as far as you, the layman, is concerned). It’s part of a more comprehensive description of imaging quality called “OTF” (optical transfer function) -- see below. In a chart of MTF, typically, the higher the line rides on the Y-axis, the better the MTF.
The longer story:
When an imaging system, like a camera, creates an image of, say black and white lines, as the lines get closer together, the image gets lousier. It’s that way with all imaging systems, even your eyeballs. When you have big lines of black, the black is very black. When you have a fat band of white next to that, the white is very white.
But when those lines get very thin and crammed together, the whites in an image become more gray and the blacks become more gray, too. And different shades of gray are much harder to distinguish than black vs. white. Also, the edges of where white ends and black begins become harder to pick out, too.
So if this is the test target we’re trying to make an image of (aka the object):
The image we create of it might look like some kind of hot mess like this:
How fast an image goes to pot as you get thinner and thinner lines varies between imaging systems. So it’s useful to have a measure of how this differs between different lenses or cameras or telescopes or microscopes. That’s where measuring an imaging system’s MTF comes in (or that of a component of the system).
Let’s break down the words in “MTF” - Modulation Transfer Function.
MODULATION: for the sake of these example tests, this is the change as we go from black to white across the test pattern.
TRANSFER: refers to going from an object in the real world to the image your camera (or lens or whatever) creates.
FUNCTION: meaning you can bet your butt there’s some sort of graph involved. And graphs come into play whenever one result changes in a predictable way as other factors change.
How is MTF Measured?
There are different ways to measure MTF along with lots of experts arguing about the best methods. For this explanation, we’re using those simple blocks of black and white. It’s good enough for most rough applications. (If you need something better, you probably already know what MTF is and don’t need to be reading this in the first place.)
So, to come up with a number to represent the contrast we’re seeing in an image, we’ll assign number values to blackest black possible and whitest white possible. Let’s say black = 0 and white = 1. Then, all the shades of gray we’d see in an image will fall between 0 and 1.
Let’s go back to that first example of the fat black and white test pattern. And let’s say the image we got of this reproduced contrast perfectly -- black is all the way black and white is all the way white. So for the black we see here, it will have a value of 0 and white will have a value of 1.
And let’s also say that in this example pattern below, the test target is 1 millimeter wide. Since there are 2 pairs of black-and-white blocks in this 1 mm-wide pattern, we’ll say this target is 2 line pairs per millimeter or “2 LP/mm”.
Now to get contrast, which is really just figuring out how far apart our different our black and white (or gray) values are, we use this equation:
(Highest Value - Lowest Value) / (Highest Value + Lowest Value)
So it’s the difference between our black and white boxes divided by their total. The highest value is our white which equals 1. The lowest value is our black which equals 0. Filling out the equation, we get:
(1-0) / (1+0)
This reduces down to 1/1, which equals 1. So here, we measured the MTF value at 2 LP/mm to be a perfect 1. (Sometimes this would be expressed as a percentage instead, and in that case, we would get 100%.) And that would be just 1 data point for our MTF curve.
Now let’s calculate the contrast we’re seeing in the second example of black bars that had more line pairs per millimeter.
This was the original test pattern we created an image of:
Let’s say that test pattern was also 1 mm long. This time, there are 5 pairs of black and white we can count, so this contrast measurement will be for the resolution of 5 LP/mm. Now let’s look at that image again.
The black here is not a solid black. It’s gray. On a scale of 0-1, let’s say we measure the blackest part to be 0.2.
The white is also gray, but not as gray as the black appears. On a scale of 0-1, let’s say we found the white to be 0.7.
So now let’s crunch those numbers again in the same formula: (max - min) / (max + min)
(0.7-0.2) / (0.7 + 0.2)
This reduces to: 0.5 / 0.9, which equals: 0.55, rounding to 0.6 (with sig figs). So, our second data point for this MTF curve would be 0.6 on the Y-axis for an X-axis value of 5 LP/mm.
Now let’s pretend that we repeated this for a bunch of other test patterns going up to 10 LP/mm in resolution. The resulting example MTF curve of all those data points might look like this:
As we go to the right on the X-axis, we squeeze more and more line pairs into that millimeter of target area. We can see as we do this, the contrast shown on the Y-axis falls. The black and white bars get harder to distinguish the smaller they get.
If we measured these same targets with another lens and got a line that hovered above the green line we created in the graph, that would be a better MTF curve. The closer the curve gets to that top value of 1 (or 100% depending on the graph), the more contrast we can see in the image.
Where Things Might Get Weird