PART 5 OF A 5-PART SERIES ON UNDERSTANDING HOW MUCH RESOLUTION A HOLOGRAM NEEDS TO APPEASE THE EYE
INTRO
In this final chapter of our 5-part series, we will accommodate the full complexity of the human visual system to finally determine what is required of a holographic display to satisfy the eye’s need for resolution in order to provide an immersive and realistic visual experience. We will see how the old DMV Snellen 20/20 eye test (circa 1862) fits into our modern visual acuity model, and even offer a conversion of Snellen eye standards to measurable visual acuity numbers. But the big payoff is understanding perceived resolution and how it dictates the specifications for an immersive and realistic holographic experience.
CSF + DMV = That pesky vision test (at the DMV)
The DMV vision test, a cornerstone of scientific visual acuity measurement, utilizes the Snellen acuity values where 20/20 (imperial) or 6/6 (metric) vision represents the collective norm. The first number (20) represents the distance in feet from the eye chart that the test is conducted (usually 20 feet). The second number (say, 100) is the distance that a normally sighted person (20/20) could read the chart. So, if you had 20/100 vision you would have to move to within 20 feet of a test chart that normal people could see from 100 feet. The trouble is, the Snellen human-based measurements are not quantified in a mathematically useful form.
The graph in Figure 1 affixes Snellen visual acuity tests to established values measuring visual acuity, neatly “calibrating” Snellen. It converts 20/20 into our now-familiar terms of visual angle in arc minutes vs. spatial frequencies in cycles/degree (c/d). It shows on the Y axis, for example, that normal 20/20 vision can resolve a feature as small as 1 arc minute, intersecting spatial frequencies on the X axis at 30 cycles/degree. Since each cycle consists of two features (a black and white line pair) the 30 cycles represent 60 features per degree, or one feature per arc minute.1
Similarly, 20/100 vision maps to 5 arc minutes which results in about 8-10 cycles per degree (c/d). The interesting thing about this graph is how it hits key reference points of visual acuity that we have been exploring throughout this 5-part series. Case in point — 20/10 vision is very near the Dawes diffraction frequency limit (Part 1), so it is about as sharp as vision can possibly be for the human eye. The “normal” 20/20 vision is actually near the Rayleigh diffraction limit (Part 1), and 20/100 vision occurs near the peak of the Contrast Sensitivity Function (Part 4).
FUN FACT
Hawks’ visual acuity is estimated to be about 20/5, meaning a person with normal vision (20/20) would have to move to within 5 feet of an object to see it as well as a hawk sees it at 20 feet.
EFFECTIVE MTF% OF THE EYE
In Part 4 we met the Contrast Sensitivity Function (CSF) which graphed how the MTF% of human vision falls off at high frequencies as well as low frequencies. Here, we offer a simplified equation to represent the CSF as a percentage,3
CSF% = 2.6 (0.0192 + 0.114 f ) exp(-0.114 f )1.1
where f is frequency in cycles per degree (c/d). It describes an MTF% curve that incorporates the CSF to illustrate the falloff of MTF% contrast sensitivity up and down the frequency range as the human visual acuity falls off in the higher and lower frequencies that we saw in Figure 5 of Part 4 revisited below.
This, in turn, can be plotted against a diffraction-only limited optical system’s MTF% to better understand the human visual system’s actual performance characteristics to aid in the design of holographic display systems. Before you run off to plot a graph of the CSF% equation above we thought we would save time and graph it for you. Figure 2 graphs the CSF normalized MTF% in cycles/degree (red line) against an equivalent optical system’s diffraction-only limits (black line). It reveals that the optimum human visual acuity is around 1-8 cycles/degree, falling off on either side of that. It should be noted that the CSF calculations were based on a best-case scenario of well-illuminated visual information which results in a ~2-3 mm pupil diameter where the eye is the sharpest. Lower illumination, different chrominance and increased age are among the many factors that decrease the eye’s sensitivity.
VIEWING ANGLE AND PERCEIVED RESOLUTION
The Viewing Angle is based on the Display Diameter. While a camera may output a rectangular image with several possible aspect ratios, the lens of the camera is circular and the output image is cropped out of that circle. For purposes of analysis, we can think of displays in the same way. Figure 3 illustrates several common display aspect ratios from the same screen diameter by thinking of the screen diagonal (red lines) as another word for the screen diameter. So, a 60 inch diagonal FHD display can be thought of as a 60 inch diameter spherical display with a 1.78 aspect ratio image fit within it and any calculations using the diameter would be the same as using the diagonal.
The virtue of this spherical model is that resolution calculations in terms of the diameter become independent of the aspect ratio of the displayed image. As Figure 3 makes clear, all three circles have the same diameter but the different aspect ratios produce different horizontal and vertical resolutions. The instinct to base resolution calculations on width and height would require three different calculations, but basing it on diameter requires only one. From the resolution of the diameter, the width and height resolutions are trivial to calculate from the aspect ratio.
What every designer of displays wants to know is how much resolution is required for the eye to see the picture instead of the pixels. Armed with just Snellen’s criteria for 20/20 vision and a bit of trig, we can calculate the Viewer Acuity Distance, the minimum viewing distance required to avoid seeing the pixels for a given display size and resolution. We will, in fact, be exhibiting the math used to calculate the entries in Figure 4. We’ll use the FHD (Full HD) table entry and stipulate a 60 inch HD display as our test case to determine the minimum viewing distance, and from that the Viewing Angle.
While an HD display has a resolution of 1,920 x 1,080 pixels, we need the 60 inch diameter (diagonal) pixel resolution which is easily calculated with a little Pythagorean theorem. Calculating the number of pixels (p) in the diameter:
Next, we need to convert Snellen’s 20/20 criteria to visual acuity to do the math. With the help of Figure 1 we find 20/20 vision equates to a visual acuity of 30 cycles per degree (c/d). Recalling that a cycle consists of a pair of black and white pixels, so each pixel represents 1/60 of a degree, or one arc minute. Trig goes better with radians, so for convenience, one arc minute is converted to 0.0002908 radians.
Earlier, we established that the 60 inch display was 2,203 pixels in diameter, so each pixel must be 0.0272 inches in size (60 inches / 2,203 pixels). Now the question becomes what viewing distance (D) is required such that a 0.0272 inch pixel (d) subtends an angle of ~0.00029 radians (ø). For the answer, we just have to solve:
If we were standing 7.8 feet away from a 60 inch diagonal display, what would be the viewing angle, the angle that the entire display subtends to the eye? We can use the familiar trig function for calculating a viewing angle θ for a height (h) at a distance (d):
where h = 5 feet (the 60 inch display) and d = 7.8 feet, the viewing distance. Solving for the viewing angle θ we get:
Rounding the answer up to 36 degrees for an FHD display, we find it comports exactly with the FHD entry of Figure 4 including the 30 c/d and 20/20 visual acuity (from above). However, you may have noticed that Figure 4 does not have a viewing distance of 7.8 feet like our example, but rather has “Relative Viewing Distance." That’s because the viewing distance is relative to the size of the display diameter and/or pixel size sufficient to meet the visual acuity requirements. If the display gets smaller, the viewing distance gets shorter in a neat one-to-one relationship, but the Viewing Angle stays constant as it is dependent only on the size and number of the pixels in the display. Make the pixels smaller and you can move closer, increasing the Viewing Angle. Reduce the number of pixels and the display get smaller, decreasing the Viewing Angle.
Figure 4 is a grand encapsulation of all that we have covered in the five parts of this blog which graphs the Viewing Angle to the Relative Viewing Distance for a wide range of flat 2D display formats from a high of 8K to a low of SD (Standard Definition), all calculated from the standard human visual acuity and a fixed screen diameter — meaning as a viewer gets closer to the display surface, the noted resolution in the Format row would be required in order to meet the visual acuity requirements at a constant ~30 c/d. However, as the above illustration highlights, there is an inflection point where the frequency of the resolution format Full HD (FHD in blue) and the visual acuity requirement both converge at 30 c/d. By no accident, this approximate viewing angle is often the relative distance where people naturally view their displays and devices comfortably.
By comparison to the format resolution line, the Frequency row shows the resultant cycles per degree that a particular resolution format relative to this inflection point actually represents — more simply put, for a viewer standing at a comfortable viewing distance (the FHD inflection point in blue), the Frequency row illustrates the effective cycles per degree (c/d) that is then created for other display resolution formats with 8K forming 120 c/d and SD 10 c/d, respectively.
Recalling the Snellen visual acuity test for 20/20 vision at your friendly neighborhood DMV, the Snellen Acuity row in Figure 4 has several acuities posted for different display resolutions relative to the comfortable viewing distance (the FHD inflection distance) telling us the level of visual acuity required to fully resolve the details of the display. Taking UHD as an example, the 20/10 Snellen acuity value means that you would need super sharp 20/10 vision to resolve all those pixels (60 c/d) if you were standing at the inflection point. Recalling the Fun Fact from above that the hawk had 20/5 vision, this is half as sharp, but still twice as sharp as the average 20/20 viewer.
Recalling the MTF% (Modulation Transfer Function) represents the percentage of contrast that a display system can reproduce relative to the inflection distance, if a super high resolution display had microscopic adjacent black and white pixels the basic human eye would simply blend those all together approaching 50% gray resulting in an MTF of less than 9% (the Rayleigh Criterion), meaning that the MTF contrast below this threshold would not be perceivable given 20/20 vision, which we see exceeded starting at QHD resolution (40 c/d) and higher. By comparison, we have to look all the way out to the lower resolution of SD to get an MTF of 75% (for those of you old enough to remember when DVDs all looked fantastic…).
In summary, starting on the left of Figure 4 with very high resolution displays viewed at the visual acuity inflection distance, they have high cycle per degree (c/d) values meaning very small pixels which require very sharp Snellen visual acuity to resolve, but when observed by the normal human eye these fine details blur together into a very low MTF%. Looking at the other end of the graph with a low resolution (SD) display, it exhibits low c/d which can be viewed by even poor 20/70 vision and for normal vision conveys most of the contrast with an MTF of 75%.
CONCLUSION
I’m sure you have enjoyed this trip down Mensa Lane where we have methodically calculated the performance of display systems based on the measured acuity of the average human eye culminating in the grand summary of Figure 4 for a fixed flat 2D screen diameter. These visual acuity calculations taught through these five chapters for a single plane in space are foundational for the ultimate goal — the calculation of the resolution requirements for a true 3D holographic display that meets visual acuity requirements for all distances simultaneously within a target volume.
Reflecting on the arduous path we have taken, so far, to calculate the requirements for just a single plane of resolution, we can only breathe a collective sigh of relief that this series mercifully ended here and did not fall into the abyss of the full 3D resolution requirements — which are orders of a magnitude greater. Such daunting computations are best left to the holographic mathematicians of Light Field Lab, leaving the rest of us to simply marvel at the reality of being able to actually view full motion color holograms projected into a 3D viewing space without any headgear as the 9th wonder of the world.
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1. http://oisc.net/Vision%20Optics.pdf
2. https://www.cg.tuwien.ac.at/research/theses/matkovic/node20.html
3. https://www.audioholics.com/hdtv-formats/how-to-find-the-right-size-tv