Modulation Transfer Function or "MTF" is the most widely used scientific method of describing lens performance. What is it exactly, how is it defined, and how is it measured?

The modulation transfer function is, as the name suggests, a measure of the transfer of modulation (or contrast) from the subject to the image. In other words, it measures how faithfully the lens reproduces (or transfers) detail from the object to the image produced by the lens.

First, look at the black and white bars in row A of the test pattern below. This pattern consists of totally black bars on a totally white background. If we assign the number 255 to the totally white areas, and 0 to the totally black areas, and we plot a line profile of the test pattern, we get the graph shown in C. The regions at 0 correspond to the black lines, the regions at 255 correspond to the white lines.

 

[A] is the original test pattern
[B] is the image of the test pattern
[C] is the line profile of the original test pattern where 255=white and 0=black
[D] is the line profile of the image of the test pattern where 255=white and 0= black

If we image Test Pattern A with a lens, we will get something that looks like the pattern shown on Line B — note that these both represent crops of a larger test pattern with longer vertical bars, so blurring in the vertical direction isn't shown. The blurring of the lines isn't because we used a bad lens. It could well be a "perfect" lens. The blurring is due to a phenomenon called "diffraction". Diffraction is the fundamental optical limit on image quality and resolution that results from the wave nature of light and the finite diameter of lenses. A perfect lens is sometimes called a diffraction-limited lens, because the the only thing that is limiting its performance is diffraction.

If we take a line profile of Image Pattern B, we get Graph D above. For the widest spaced set of black and white bars, the plot still goes between 0 and 255. This corresponds to the performance of the lens when it recording low frequency detail. For the next set of patterns we can see that the plot no longer reaches either 255 or 0. The modulation in Target A is no longer faithfully reproduced in Image B.

The formal definition of MTF:

MTF = (maximum intensity - minimum intensity)/(maximum intensity + minimum intensity)

For the first pattern group, the MTF is 1. For the second pattern set, the MTF can be calculated to be 0.8. For the third set, the MTF is 0.5, and for the fourth set, the MTF is 0.1. If there were any finer patterns, with narrower black and white bars, the MTF would be 0 and the image of the pattern would be a uniform gray patch, represented on the plot by a straight line at a value of 127. The point at which you can no longer see any variation in the image is the point at which the MTF is zero, and that's the definition of the "resolution" of the lens. In this case, the final pattern set with an MTF of 0.1 would be classified as "just resolved" by this lens.

Technical Nits

When we speak of a lens's MTF, we have to define the type of test pattern being used. The best pattern is not a step pattern as shown above, where the elements are black and white bars. The preferred pattern is called a "sine wave pattern" where the optical density varies between black and white smoothly, and the line profile looks like a sine wave. However, such patterns are difficult to make and the image of such patterns is hard to measure by eye, so most often a bar-type pattern is used. Bar patterns yield a slightly higher MTF measurement than sine wave patterns, but the difference is small.

When plotting MTF as a function of the pattern spacing, for a bar pattern the horizontal axis is in units of line pairs per mm. For a sine wave pattern, the horizontal axis is in units of cycles per mm. Often this distinction is overlooked.

In addition to the type of test pattern, the light used for illumination and the type of detector used for recording also influence MTF. This is because MTF depends on the wavelength of the light used. Using blue light gives a higher MTF than using red light, for example. Normally white light is used, but even then the results will depend on color temperature. Tungsten light has a stronger long wavelength component (red light component) than daylight, so MTF measured under tungsten light will usually be lower than MTF measured in daylight. The detector can influence MTF. If the sensor is more sensitive to blue light than red light, the result will be a higher MTF than with a detector more sensitive to red light.

The effects of these differences are small enough that they are often not specified in tests of photographic lenses.

Back to the practical stuff...

Here's an example of an actual test pattern (USAF 1951 on the right) recorded on film (on the left). As you can see the modulation in the finer spaced patterns gets gradually more and more difficult to see.

The higher the MTF of the lens, the finer spaced pattern that will be visible. In this example the last pattern fully resolved is group 1, pattern set 2

When describing the performance of a lens using MTF, we typically use a plot of MTF against "spatial frequency". "Spatial frequency" is a fancy way of saying how many line pairs per mm (lp/mm) there are in the image. The more line pairs, the higher the spatial frequency. Note that a line pair is one black line and one white line (space), so the number of line pairs per mm (counting black lines and white lines) is the same as the number of lines per mm (counting only black lines). Generally lp/mm and l/mm are used interchangably, but beware of some authors who make a distinction and count 100 lp/mm as 200 l/mm. Below is a typical plot.

This shows three plots. The blue trace is that expected from a perfect (diffraction limited) lens operating at f4, though few, if any, lenses would be diffraction limited at f4. The black trace is that expected from the same lens operating at f8. At f8 many lenses are close to diffraction limited in the center of the optical field, so you could obtain a trace like this in practice. Note that the "resolution" at f4, defined as the spatial frequency at which the MTF drops so low that you can't see any modulation in the image, is around 450 lp/mm, while that of the perfect f8 lens is exactly 1/2 that, 225 lp/mm. For a perfect lens the resolution as defined in these terms (MTF -> 0) is linearly related to the aperture and is given by:

Resolution (spatial frequency@MTF=0) = 1800/fstop

The red trace shows what you would expect from an f4 lens with 1/2 wavelength of wavefront error (not an unreasonable amount to find in a typical camera lens). As you can see at f8 a lens may actually be better (i.e. have a higher MTF) than the same lens at f4, at least for the region from 0 to about 175lp/mm. Since you rarely get over 100 lp/mm on film the region above 175lp/mm is of no importance and the lens at f8 would give better performance on film. This is the basis of the old rule that lenses are often best when stopped down a couple of stops from maximum aperture. It also shows why the simple measure of maximum resolution when measured opically in the aerial image prduced by the lens (rather than on film) isn't a good predictor of practical lens performance, since as you can see both the "perfect" and aberrated f4 lenses show the same limiting resolution.

What do real lenses look like?

Real lenses rarely, if ever, come close to the theoretical maximum MTF at apertures below f8. A few (expensive) lenses may be an exception to this rule. Performance at the maximum theoretical MTF is called "diffraction limited" performance, since diffraction is the reason why MTF falls with increasing spatial frequency, even for a "perfect" lens.

Below is a plot of the MTF of 6 different 50mm camera lenses from a study published in 1960 (modern 50mm lenses should be similar). The horizontal axis reads 0 to 1; this is the "normalized spatial frequency". What this means is that the horizontal axis is different for each aperture. The scale expressed in lp/mm would be from 0 to approximately 1800/f, where f is the f-stop. So for the f2 trace, the scale runs from 0 to 900 lp/mm, for  f4 it runs from 0 to 450 lp/mm and for f11 it runs from 0 to 164 lp/mm. By using the normalized spatial frequency you can see how the lenses perform relative to their maximum theoretical performance at each aperture. It's pretty obvious that none of these lenses comes anywhere close to diffraction-limited performance wide open, and that most of them will show best results when stopped down to the f5.6 to f8 region.

Data from K. Rosenhauer and K.J. Rosenbruch, "Die optischen Bildfehler und die Ubertragungsfunktion", Optik 17, 249-277 (1960)

Though we normally think of focusing errors as simply "blurring" the image, we can look at the effect of defocus on MTF. Below is a plot showing the MTF of a perfect, diffraction-limited f2.8 lens. Traces are shown for various amounts of defocus (measured in units of wavelengths of wavefront error). Wavefront error is simply a measure of how far the image formation is from perfect. Various aberrations, such as spherical aberration, could also be represented in terms of wavefront error and plots for such aberrations would appear somewhat similar to the traces shown.

The red region represents an MTF less than zero. In reality what this means is that black areas appear as white and white areas appear as black, so although a pattern may appear to be resolved (insofar as you may see black and white areas), in reality it isn't.Resolution above the point at which the MTF first reaches zero is known as spurious resolution

MTF Maps and Single Value "ratings"

We've now defined MTF, but realize that we've only defined MTF at a single point in the image. The MTF curve at the center will be normally higher then the curve at the edge, which will itself normally be higher then the curve at the corner. Some authors give an "MTF" value to a lens which is some sort of weighted average across the whole image. While this may have some value, it's about the same as describing the Mona Lisa by it's average color!

Now it's not quite that bad since the MTF "map" will usually be symmetrical and peak in the center of the frame, but a badly assembled lens may well show asymmetric behavior that a single number misses and even if the lens is symmetric a single number doesn't tell you how MTF varies across the frame. A lens with high center MTF and low edge MTF may have the same "average" MTF as a lens that has a medium MTF value all across the frame.

The complexity of measuring a lens's MTF is well-illustrated by the plot below, of a (now discontinued) Canon EOS 20-35/2.8L lens at 20mm. This graph shows the MTF measured at various distances from the center of the frame at different spatial frequencies, apertures and target orientations. Start by looking at the heavy solid black line, which represents the saggital MTF (lines pointing towards the center of the frame) at f8 and a spatial frequency of 10 lp/mm. You can see the MTF varies between about 0.93 in the center of the frame to about 0.87 at the top and bottom (12 mm from the center) to about 0.4 at the sides of the frame (18mm from the center) to about 0.02 in the corners of the frame (21.5mm from the center). Imagine trying to come up with a single number which describes this plot, or even just one of the lines on this plot! Wide angle lenses such as this typically have the most complex MTF vs. position plots, but all lenses show significant variations across the frame and different plots at different apertures and with different target orientations.

What's SQF?

SQF stands for "Subjective Quality Factor". SQF takes the quality of the lens into account, but also factors in the acuity of human vision (the "MTF" of the human visual system).  An important observation is that the human visual system has an MTF which peaks in the 10-20 cycles/mm (lp/mm) on the retina. SQF factors this into the viewing equation, so that. for example, for a given lens at a given aperture, each print size (viewed from a constant distance) would have its own SQF value. The system was developed at Kodak by E.M.Granger around 1970 and a modified form is used by Popular Photography for their lens tests. SQF defines the subjective quality of an image (print) rather than defining the quality of a lens.

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