PART 4 OF A 5-PART SERIES ON UNDERSTANDING HOW MUCH RESOLUTION A HOLOGRAM NEEDS TO APPEASE THE EYE
INTRO
In Part 3 of this series, we introduced the Aberration Transfer Function as a method of characterizing the degrading effects of wavefront deformations that naturally occur with all physical lenses. An ATF equation was proffered that concatenated the long list of wavefront error types (astigmatism, coma, chromatic aberration, etc.) into a single variable ω (omega) representing the RMS wavefront error, which in turn could be used to calculate the approximate MTF% degradation due to wavefront errors. In this chapter, our resolution model will be refined to include the Contrast Sensitivity Function, a human perception issue.
ATF vs. f/n
In Part 3, Figure 1 is a table that lists the diffraction-limited lp/mm performance for a range of f-stops at selected MTF% values. We offer an updated version of that table Figure 1 that factors in the wavefront deformations covered in Part 3. They are inherent in physical optics and apply to both refractive and diffractive1 technologies, bringing us one step closer to the real world. These calculations limited the Dawes frequency (MTF 0%) to positive values starting at 0 Cycles per Degree, where applicable.
The table in Figure 1 summarizes the tradeoff between aperture size, light-gathering ability, and resolution limits for optical systems. It shows that for small f-stops starting at f/1.0, their large apertures cause wavefront errors to be the limiting factor for resolution. As the f-stop increases in value and the aperture gets smaller, at around f/2.0 (highlighted in green) an inflection point is reached where the limiting factor switches from wavefront errors to diffraction. This switch is due to the progressively smaller apertures as the f-stop increases above f/2.0, making diffraction the dominant factor in limiting resolution.
Figure 2 is a graph of the table data in Figure 1 for MTF 50% using a 50mm lens to illustrate the inflection point outlined in green for both. The bottom of the graph is marked out in f-stops with their associated aperture sizes in mm across the top. The Y axis is the maximum resolution in line pair per mm (lp/mm) for the MTF 50% data column from Figure 1 which has been plotted in the graph. The shape of the curve clearly shows the optimal resolution to be around f/2.0. We can see this born out with test data from real lenses.
Figure 3 is a sample of two of the highest resolution 35 mm format lenses ever measured, which produced results closely aligned to the calculated MTF% limits presented in Figure 1, and not similar to a diffraction-limited model only. In both of these examples, f/1.4 – f/2.0 optical systems are found to have the highest resolving ability when stopped down to f/4 to balance between resolution and wavefront errors — meaning these systems have more wavefront errors and an inflection point (f/4) above the more idealized f/2 inflection point in figure 1. These high-performance lenses also closely match the Figure 1 limits at MTF 50% and MTF 80%.
MTF vs. DYNAMIC RANGE
The optics of an imaging system, including the human visual system, place an upper limit on the dynamic range resolvable by that imaging system. Those upper limits may be estimated using known diffraction limits, wavefront errors, and physical optics. Recalling that the MTF% is a normalized function, meaning it is in terms of the percentage of the maximum possible preserved contrast for each spatial frequency, we can plot MTF% against the number of stops (log2) of dynamic range that an optical system can preserve. This is analogous to characterizing how well a speaker can play at different volumes, with the volume levels representing the stops, and the MTF% representing the sound quality.
Returning to our optical case, recalling from Part 2 that MTF% is simply a normalized measurement of potential contrast, a simple mathematical conversion can be performed to translate between MTF% and dynamic range, such that:
Using the same concrete example of MTF 20% from Part 2:
That was disappointing. An MTF of 20% yields less than one stop of dynamic range, and as we shall see in Figure 4, higher MTF values yield equally disappointing results.
Figure 4 is a graph of MTF% vs stops of dynamic range for an optical system with an aperture diameter of 3 mm and f/5.6, which is typical of a human eye accommodated to normal daylight lighting. The takeaway here is how little dynamic range is actually available with each MTF%. As calculated above, an MTF of 20% did not even get one full stop of dynamic range, which is confirmed in this graph. Even the extremely high MTF of 90% only gets us about 4 stops, and to get 8 stops we would need a whopping MTF 99%! The graph emphasizes how important it is to maintain high MTF values in lens design in order to preserve dynamic range, not just visual acuity.
Circling back to the design of a holographic display system, if the accommodated eye can only see a contrast ratio of about 100:1 (~6.6 stops) then why would you waste time and money building a display that exceeds 7 or 8 stops of dynamic range?
MTF and the CSF: The Contrast Sensitivity Function
We are all familiar with our hearing limitations where we can’t hear audio frequencies below roughly 20Hz or above 20Kz. We have a similar thing here for the visual system, called the Contrast Sensitivity Function (CSF), with upper and lower frequency limits beyond which our perception falls off. Instead of audio frequency limitations for the ear, it is angular frequency limitations for the eye. Angular frequency represents how many vertical black and white line pairs are present per degree of field of view, and is measured in cycles per degree (c/d).
The CSF is graphed in Figure 5 with contrast sensitivity on the Y axis and the angular frequency on the X axis. The X axis is logarithmic, with low frequency cycles per degree on the left (10-1 or 0.1 c/d) and high frequencies on the right (102 or 100 c/d). These represent vertical line pairs that get progressively closer together towards the right. The CSF along the left edge of the graph ranges from 0 to 1, where 1 is the maximum sensitivity to contrast. Striking a line horizontally anywhere on the graph, the contrast remains constant the full width of the graph, even as the angular frequency increases to the right. However, our perception of the contrast falls off outside the curve on the left and the right, producing the upside-down U-shaped graph typical of a band-pass filter.
The greater the c/d, the finer the line pairs are. Being a measure of our visual acuity at various degrees of detail, it is an essential consideration when designing display systems. The graph shows that the visual system’s highest sensitivity to contrast would be in the range of 1 to 8 cycles per degree, depending upon illumination conditions and viewer acuity. The “DC removed” dashed line represents backing out the background scene illumination, so it represents just the changes to contrast sensitivity as the angular frequency increases to finer and finer line pairs.
The falloff of the curve on the right at the high frequency end is due to the limits of our visual acuity, just like any optical system. But the falloff on the left at the low end is a neural processing limitation in that it is caused by a failure of the image processing in the retina and visual cortex. Details outside of this curve become invisible to the eye. With low angular frequency content such as the sky or distant mountains, our visual system cannot detect subtle contrast changes over these broad areas like it can with smaller, high-frequency objects. If the angular frequencies are too high, the MTF crashes to 0% as shown in Figure 4 and we don’t see any detail at all.
We have an interesting visual experiment for you that further demonstrates the band-limited performance of the human visual system:
Figure 6 - Exploiting the band-limited nature of the human visual system with information hiding in plain sight. [Hint: it’s the name of an amazing company!]
There is a hidden message in Figure 6 that will illustrate the band-limited nature of your visual system. Viewed from a normal distance, it appears to be just a bunch of diagonal stripes; however, if you move away from the screen far enough, the hidden message will appear. Move away from your screen until the frequency between line pairs reaches approximately 6-8 cycles per degree. Depending on screen size, this is roughly an angle of view of +/- 6 degrees (typically towards the back of a conference room). As the stripe pattern shifts to the “sweet spot” of angular frequency (6-8 cycles/degree), the increased sensitivity will reveal the hidden message.
If you don’t yet see the message, keep moving away from the screen until you do, being mindful of the safety hazards involved in walking backward while you stare at a computer screen.
CONCLUSION
We are progressively refining our model of the human visual system to eventually achieve absolute confidence in our ability to determine the required resolution of a holographic system that would completely satisfy the perceptual system requirements of the typical human. In the earlier chapters, we started with the simple diffraction-limited model dictated by the aperture sizes and f/n of the eye where we met the mighty MTF (Modulation Transfer Function). We further refined the model by incorporating the wavefront distortions of real lenses with the ATF (Aberration Transfer Function).
In this chapter, we saw how the resolution is wavefront-limited with small f-stops, transitioning to diffraction-limited with larger f-stops with an inflection point at around f/2.0 (Figure 1). Using a little math on the MTF%, we found they returned a disappointingly low dynamic range (Figure 4). We then incorporated the bandwidth limiting role of the CSF (Contrast Sensitivity Function) that puts both an upper and lower limit to our contrast sensitivity, depending on the angular frequency of the image (Figure 5). This rounded out our model, while at the same time satisfied our insatiable appetite for acronyms.
You would think that this would do it, but no. This is why we need Part 5 of this series, to refine our model even further to include the effective MTF% of the eye and to understand why the angle of view changes the perceived resolution of a display. I promise no new acronyms, but we will refine some familiar ones further to hone our human perception model. See you there.
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1. Diffractive optical elements (DOEs) are passive devices that redirect and focus light through the division and mutual interference of a propagating electromagnetic wave. This is in contrast to refractive elements, which redirect and focus light through variations in indices of refraction
2. Modified with updated wavefront error approximations from prior chart by Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision,” University of Houston, TX, USA
3. https://www.lensrentals.com/blog/2017/07/experiments-for-ultra-high-resolution-camera-sensors/
4. https://www.normankoren.com/Tutorials/MTF.html