In Part 1, after denying that the eye was a camera, we proceeded to treat it like one anyway to establish some semblance of an optical model that could be measured, manipulated, and characterized. We were then able to introduce the Airy disk and quantify its role in degrading our vision to establish a diffraction-limited model. In Part 2 we focus (no pun intended) on the tools and techniques for measuring and quantifying the resolving power of imaging systems using the Modulation Transfer Function (MTF) and Cycles Per Degree (CPD). Not to worry, the gnarly-sounding Modulation Transfer Function will be gently introduced and carefully explained.
THE MIGHTY MTF (Modulation Transfer Function)
We begin our dive into the mighty MTF by defining spatial frequencies, the “yardstick” used to measure the Modulation Transfer Function. Whenever an imaging system (e.g., camera or human eye) captures a scene, there are inevitable losses in the captured signal because physical imaging systems are less than perfect. One type of loss is the loss of detail as the original scene contains information that is smaller than the imaging system can resolve. We quantify the scene information detail size in terms of “spatial frequency,” a signal processing term used to characterize the size of features such as the black and white line pairs shown in Figure 1. If the features are large, they generate long cycles and are referred to as low frequency, and small features generate short cycles so they are referred to as high frequency. If the frequencies are high enough the details become so small that the imaging system can no longer resolve them due to the diffraction limit (the subject of Part 1 in this series) and will blur them together into a featureless gray field.
The Modulation Transfer Function (MTF) is a very important method of measuring the resolving power of an imaging system. An imaging system is presented with a modulation pattern like the one at the bottom of Figure 2, and then the contrast of the captured image is measured. As the frequency goes up the features get smaller and the captured image loses contrast. How much contrast is retained is the MTF percent. Figure 2 shows a typical response curve of an imaging system. The captured image is altered or “modulated” when it is transferred into the optical system which is why this function is officiously named the Modulation Transfer Function.
The modulation pattern in Figure 2 is a row of sinusoidal line pairs that get progressively closer together towards the right with increasing spatial frequency. The captured image above it has transferred through an optical system and suffered some loss of contrast due to diffraction and other optical defects that we want to measure. Following the red graph line, the captured image has low-frequency detail on the left end so the contrast between the black and white line pairs near 100% in that area. Moving towards the right, as the frequency goes up the line pairs start to blend together lowering the contrast until at the far right the contrast is 0% as it ends in a uniform 50% gray. To summarize, as the spatial frequency increases the contrast decreases.
The Modulation Transfer Function is not just for optics. It has a wide range of signal-processing applications such as electronics, medical MRI imaging, and satellite imaging.
Bottom line, the Modulation Transfer Function is actually a measure of contrast for a given frequency. So how is this contrast calculated? For a given line pair (which is one full cycle), the maximum intensity (brightness) is measured as well as the minimum intensity (Imin) then this equation is used to calculate the MTF contrast percent:
How to calculate the MTF contrast %:
To take a concrete example, using normalized intensity readings where lmax = 0.6 and Imin = 0.4 the equation evaluates as:
The percent contrast becomes MTF 20%. What does this look like when the higher frequencies lose contrast?
Figure 3 illustrates two potential MTF reference points at 90% and 20%. The top image A1 illustrates a low-frequency image resulting in a 90% MTF which indicates the system is retaining an extremely high level of contrast. Image A2 is a much higher frequency resulting in a mere 20% MTF which is often considered the cutoff point for many systems below which it ceases to produce useful results. The graph below them labeled B shows where A1 and A2 fall on the overall MTF plot for the frequencies considered between MTF 0% to 100%.
So how does the system MTF affect the appearance of various displays? This question is illustrated for three generalized display viewing distance examples in Figure 4 where the visual appearance of successively degraded MTFs is shown. Going from left to right, the spatial frequencies may be equated to far viewing (e.g., movie screen), mid-range viewing (e.g., TV or laptop), and close viewing (e.g., cell phone) display environments. The top row (MTF 100%) shows the original theoretical signal. As you look down the list of ever-worsening MTF % values, it demonstrates how the perceived modulation pattern would conceptually appear with decreased contrast until finally the barely noticeable modulation differences of the Rayleigh Criteria (MTF 9%) used for the human eye and the ISO standard 12233 (5%) used for cameras.
Speaking of Rayleigh Criteria--- when you take the eye test at the DMV, the Rayleigh Criteria -- is effectively the limit being evaluated. The letter size and distance to the test card are set such that a person with average vision (20/20) will have their vision measured down to an MTF of 9% which is the limit where contrast is just barely detectable by the human visual system. 20/20 vision means that you see at 20 feet what a person with normal vision would see at 20 feet. With 20/50 vision you see at 20 feet what a normal person would see at 50 feet so you would need glasses to pass the test at DMV.
While 20/20 vision is normal, the sharpest human vision measured is 20/10. However, eagles are known to have up to 20/5 vision.
THE ENIGMATIC CPD (Cycles Per Degree)
Note that the graph in Figure 3 is labeled at the bottom with frequencies in line pairs per millimeter (lp/mm). This is a common specification for modulation targets that are used for imaging systems (such as) for film or television. For microscopes, the units will have to be much smaller, such as microns (lp/um) while for interstellar telescopes the units are appropriately vast. If a target with 10 lp/mm fills the frame of an imaging system, the number of “cycles” (black and white line pairs) can be counted. If there were, say, 1000 cycles in the frame and the viewing angle were 35 degrees then we could characterize the resolution of that setup as 1000 cycles / 35 degrees = 28.57 Cycles Per Degree (CPD), a very important metric for characterizing resolution.
The dot pitch of a monitor, scanner, printer, or any pixel-based device refers to the size of its pixels. Most modern monitors have pixels of around 0.30 mm or less.
Figure 5 illustrates the uses of CPD for two types of imaging systems. Case 1 (constant cycles) shows three targets, each with eleven cycles (line pairs) subtending a 35-degree angle. This is the case of projecting the pattern on a screen and then moving the projector further away. Regardless of how far the projector is moved, the cycles get larger so there are still only 11 cycles. Dividing 11 cycles by 35 degrees results in a constant 0.31 CPD for each target regardless of the distance.
Case 2 (constant lp/mm), however, shows a camera setup where the lp/mm is held constant as the camera is moved further away. This case is analogous to printing the pattern on a wall, then as the camera is moved back, progressively more wall is revealed placing progressively more cycles in the frame. With more cycles in the frame, the CPD number increases from 0.31 to 1.08 CPD resulting in decreasing MTF values like in Figure 2, eventually reaching the limits of the system’s resolution where the MTF approaches 0%.
Case 1 illustrating constant cycles explains why the actual resolution of displays large and small can be the same and present the same effective visual quality. In the theatre, the movie screen will be far away and may have pixels that are several millimeters in size, while a much closer TV will have pixels of hundreds of microns, but a hand-held mobile device’s pixels will only be tens of microns. Yet all three extreme examples will have essentially the same CPD. The takeaway from the CPD discussion above is that even though the display may be large and far away or close and small, the actual content resolution will be about the same.
The Modulation Transfer Function is a very powerful tool for quantifying the quality of an imaging system and is a measure of the contrast of the captured image. It characterizes the system based on spatial frequency from large to small features to provide an overall performance metric, may apply reference points to establish the upper and lower limits of quality, and identifies the point where the system has reached its resolvable limit.
Cycles Per Degree (CPD) quantifies how many cycles of black and white line pairs appear within one degree of a field of view and is an important measure of an imaging system’s resolution relative to the viewer. Displays of different sizes - movie theater screens, TVs, laptops, and cell phones - can all have a very similar appearance of resolution even with their very different sizes by maintaining approximately the same CPD.
In Part 3, the Airy disk is back as we expand our view of the limitations of imaging systems from just diffraction limited to include optical aberrations. This will additionally require consideration of the Aberration Transfer Function (ATF). We will then see how the MTF and ATF interact to produce the final resolving power of an imaging system and then relate that to the fabrication tolerances of lenses. These analytical techniques are then turned to the most important topic of all - the limitations of the human visual system which in turn sets the baseline for the required resolution of a holographic display, our ultimate goal.
 For readability of the illustration, the number of cycles was limited to 11.