DV101: Modulation Transfer Function: Sizing Up Lens Resolution
Okay, bear with me this month. This topic is actually much deeper technically than I normally venture here in DV101, but it’s an important topic to understand—especially since I’ll be writing more columns on lenses in the near future.
This month, I’m going to talk about modulation transfer function (MTF), which is a methodology of measuring the resolving power of a lens or an imaging system.
The Basics of MTF
Let’s break it down:
What are we modulating?
What are we transferring?
No lens is perfect. In fact, it’s physically impossible to have a lens that doesn’t alter the light passing through it to some degree. A phenomenal lens might transfer 95 percent of the light cleanly through its glass; a not-so-good lens might transfer only 70 percent. There will always be some loss of light and, more importantly, loss of contrast.
I already snuck in one of the aspects of MTF: transference. We’re transferring light through a substance—in this case a lens—and that transference will not be complete; there will be a variation of the light that comes out the other side. This variation, generally noted in terms of spatial frequency, is called a modulation. The light transferring through the lens has been modulated, or changed, from its initial state as it passes through the lens.
That’s the basis of modulation transfer function: how is light passing through a lens or lens system (or later, any component of an imaging system) and how is it changed by the lens itself?
An illustration that shows how closely related contrast is to resolution; without contrast, there is no resolution.
Contrast, Contrast, Contrast
Mostly what we’re looking at when we’re measuring MTF is resolution via contrast. MTF test charts are primarily alternating areas of high-contrast black and white lines or wedges at increasing frequency within a given space. It’s important to understand that contrast and resolution are closely related, even dependent on one another. Without contrast, you cannot have resolution. Imagine a white line on a white piece of paper—if there is no contrast between the white line and the paper, there can be no resolution of the image to see the line.
The original resolution chart made for the United States Air Force in 1951.
The design for most MTF or resolution charts is based on a test chart created in 1951 for the U.S. Air Force to measure the resolution of optics for reconnaissance photography. The chart features groupings of black and white lines that get smaller and smaller. In the most simple terms, the lens with the highest resolving power will show the smallest lines clearly as black and white stripes.
Modern versions of the USAF 1951 chart feature the same concept of alternating lines of increasing spatial frequency—as the lines become narrower, we are increasing the spatial frequency of the lines in a given space. Spatial frequency is generally measured in line pairs per millimeter (LP/mm) or sometimes lines per millimeter (L/mm). Those two can be a bit confusing if you’re not paying attention, but they’re two different ways to measure the same thing. If you have 20 lines per millimeter, you have 10 line pairs (a pair being one black and one white line).
One line pair per millimeter is the same as two lines per millimeter. 10 LP/mm = 20 L/mm.
The first thing we note when looking at blocks of lines that increase in spatial frequency is that it gets harder to see individual lines as they get narrower and closer together. You might see those lines close up, but if you step away from the test chart, the higher frequency lines quickly become gray blocks in which you can no longer tell black from white; they blend into one. When you lose contrast, you lose resolution, and vice versa.
The same thing happens when the image of that chart is projected through a lens. There will always be a loss of contrast, meaning that the blacks won’t be quite as black and the whites won’t be quite as white. There is often also a loss of sharpness, where the lines may appear to be out of focus.
No lens is perfect. All lenses will lose some contrast and some sharpness of the image.
Imagine a test chart of high-contrast, low-spatial-frequency black and white lines, maybe 10 LP/mm. Let’s say that this series of lines has a contrast ratio of 600:1 (the white is 600 times brighter than the black). Our lens can only resolve this spatial frequency with a contrast ratio of 500:1, however, so we lose some of the richness of the blacks, and in most cases we lose a bit of the sharpness of the transition between black and white. This is our modulation of transfer. The image being projected by the lens is not exactly what is visible.
It’s important to note that as contrast increases in spatial frequency—as the lines on the chart get closer together—our contrast reproduction drops. This property is common to every lens ever made. Lenses will render lower spatial frequency with better contrast and better resolution than they will images with higher spatial frequency. The lens’ contrast reproduction is defined by how well it transfers low-frequency contrast, and its resolution is defined by how well it transfers high-frequency contrast.
With all lenses, as spatial frequency increases, MTF decreases; the lens will project less contrast and less resolution onto the imaging plane.
Forming the Curve
There are two ways the MTF of a lens can be represented on a chart: by measuring the lens’ performance over increasing spatial frequency, or by measuring a single spatial frequency across the width of the lens. Both measurements are important and reveal a lot about the performance capabilities of that lens.
As we already know, the contrast reproduction of any lens will drop off as spatial frequency increases. If you were to graph that function, you’d start with a percentage scale along the y (vertical) axis to represent transference, the percentage at which the frequency is passed through the lens. The scale would be expressed in decimals from 0 to 100 percent or 0 to 1 (i.e., .1 = 10 percent, .5 = 50 percent and 1 = 100 percent). Ninety-five percent would be extremely efficient and accurate transference with little modulation; 30 percent would be poor transference with a lot of modulation.
Along the x (horizontal) axis we’ll show spatial frequency, increasing as we go off to the right. We’ll start at 10 LP/mm and increase to 50 LP/mm on the x axis. We already know that the modulation will increase as spatial frequency increases, so when we plot the transfer performance on this chart, it starts high and gently slopes off as the spatial frequency increases. The slope on the chart is called the MTF curve; it tells you at a glance how a lens will perform at a given resolution (frequency).
We can plot MTF a second way to see edge performance of a lens. Again use the y axis as percentage of transference. This time the spatial frequency will remain consistent; instead, the x axis is scaled in millimeters, with the far left (0) representing the middle of the lens and each number representing a distance (in millimeters) away from the center toward the edge of the lens. Performance at the sides of the lens will always be less than at the center of the lens. Generally when we’re looking at performance across the lens, we are looking at multiple lines on the MTF chart, each line representing a different spatial resolution measured across the lens and/or a different aperture at which the test was conducted. Older lenses will nearly always perform better at deeper stops. (For many years the rule of thumb was that the “sweet spot” of a lens was about two stops closed from the lens’ maximum aperture.) Modern cine lenses are generally optimized to perform best closer to their maximum aperture.
When you understand the basics of MTF, you can look at any lens manufacturer’s published specs and get an idea of how that lens will perform in terms of contrast and resolution. There are, of course, many other attributes of a lens that cannot be expressed by MTF charts (color, sharpness, bokeh, etc.), but at least you’ll have an idea of that lens’ contrast and resolution performance.
A very important note: MTF doesn’t stop at the lens. Camera filters have MTF. Sensors have MTF. Filter packs on the sensors have MTF. The recording format often has an MTF. The exhibition format has an MTF. The exhibition medium/display has an MTF. Finally, our eyes have an MTF. All of these elements combine to create the MTF for any given camera and display system. This idea is key to understanding the nuance behind what we loosely refer to as resolution. “Resolution” is not merely a count of the pixels in the final display; it’s a combination of the MTF curves of all the elements within that capture and viewing system.
A look at the results of testing my Canon 50mm EF II f/1.8 and its performance at several different f-stops, spatial frequencies and areas of the lens.
So let’s take a look at one lens. I chose a Canon EF II 50mm 1.8, nicknamed the “nifty 50.” It’s an inexpensive (about $90) and fast EF prime that I picked up a few years back for my Canon EOS cameras and tested with my EOS 7D.
I shot a focus chart from Century Precision Optics (a division of Schneider Optics) at f/1.8, f/2.8, f/5.6 and f/11. Focus was set manually (using focus zoom assist on the Canon) and not changed during the exposures. The chart was set at the proper distance of 50 focal lengths away—in this case, 2,500mm or 8.2’. Let’s look at a couple areas of the chart: the center “L” section (2.5 LP/mm), the center “E” section (28 LP/mm) and the center A/B section (80 and 112 LP/mm, respectively). Then we’ll look at the far upper right-hand corner of the lens at the E and A/B sections.
We can see at a glance that this lens performs best at f/5.6. The contrast representation, while not 100 percent, is good, and we can discern contrast between the lines at A/B, even at the far edge of the lens. At f/1.8, this lens performs pretty poorly—even at the 2.5 LP/mm area, contrast at f/1.8 is pretty low, as is sharpness. That single bar shown here at “L” is not sharp, nor is the black very rich.
Canon’s published MTF chart for the 50mm EF II f/1.8. This chart, and MTF results for any of Canon’s lenses, is available from usa.canon.com.
Canon’s published MTF chart for this lens is pretty accurate. You can ignore the dotted lines in this chart; they represent meridional and sagittal measurements (which are not featured on my resolution test chart). On this chart, black lines describe measurements made with the lens at its maximum aperture and blue lines represent measurements taken at f/8. The thick black/blue lines show measurements at 10 LP/mm; the thin blue/black lines show measurements taken at 30 LP/mm. On my chart, the “E” section represents 28 LP/mm, very close to Canon’s 30. If we look at the thin black line, measuring 30 LP/mm at f/1.8, we’ll see that my findings are pretty close to theirs: the MTF is about 58 percent at the center of the lens in the E area and way down to less than 10 percent at the far edge of the lens.
A Few Parting Blows
At the risk of further complicating a complicated discussion, I should mention a few more things about MTF. First, not everyone measures MTF the same. A popular web site, dpreview.com, which features extraordinary reviews of digital cameras and lenses (primarily the still variety), measures their MTF in MTF50, which means that they give you the spatial frequency number at which the MTF has dropped to 50 percent. Instead of reporting the MTF for a given spatial frequency, they report the spatial frequency for a given MTF (50 percent).
As I stated earlier, MTF doesn’t stop with the lens—it goes through the entire imaging system. When we’re talking about camera MTFs, we’re generally talking about OTF (optical transfer function), which is more descriptive of the kinds of loss to transfer function that can happen through the low-pass optical filter in the camera, through the compression codec, etc.
Many times you’ll see OTF (or camera MTF) measured in line widths per picture height (LW/PH) or line pairs per picture width (LP/PW)—this form of measurement is only slightly different. If we have an image that is 1920 x 1080 lines, then our cycles (one cycle being one pair of lines), or line widths (measuring in pairs), would be 960 wide and 540 tall. Theoretically, a 1920 x 1080 system should be able to resolve 960 line pairs across the image and 540 line pairs along the height of the image.
That, my dear readers, is the basics of MTF. There will be a test later.