Figure 1: Sample output of chart orientation estimation |

There is an orange plus-shaped coordinate origin marker (in the centre of the chart), as well as a reticle (the red circle with the four triangles) to indicate where the camera is aimed at. Lastly, we have the three orientation indicators in red, green and blue, showing us the three Tait-Bryan angles: Roll, Pitch and Yaw.

But how do I know that the angles reported by MTF Mapper are accurate?

### The set-up

I do not have access to any actual optics lab hardware, but I do have some machinist tools. Fortunately, being able to ensure that things are flat, parallel or perpendicular is a fairly important part of machining, so this might just work. First I have to ensure that I have a sturdy device for mounting my camera; in Figure 2 you can see the hefty steel block that serves as the base of my camera mount.

Figure 2: Overview of my set-up |

I machined the steel block on a lathe to produce a "true" block, meaning that the two large faces of the large shiny steel block are parallel, and that those two large faces are also perpendicular to the rear face on which the steel block is standing in the photo. The large black block in Figure 2 is a granite surface plate; this one is flat to something ridiculous like 3.5 micron maximum deviation over its entire surface. The instrument with the clock face is a dial test indicator; this one has a resolution of 2 micron per division. It is used to accurately measure small relative displacements through the pivoting action of the lever you can see in contact with the lens mount flange of the camera body.

Using this dial test indicator, surface plate and surface gauge, I first checked that the two large faces of the steel block were parallel: they were parallel to within about 4 micron. Next, I stood up the block on its rear face (bottom face in Figure 2), and measured the perpendicularity. The description of that method is a bit outside of the the scope of this post, but the answer is what matters: near the top of the steel block the deviation from perpendicularity was also about 4 micron. The result of all this fussing with parallelism and perpendicularity is that I know (because I measured it) that my camera mounting block can be flipped through 90 degrees by either placing it on the large face with the camera pointing horizontally, or stood up with the camera pointing to the ceiling.

That was the easiest part of the job. Now I had to align my camera mount so that the actual mounting flange was parallel to the granite surface plate.

Figure 3: Still busy tweaking the mounting flange parallel to the surface plate |

The idea is that you keep on adjusting the camera (bumping it with the tripod screw partially tightened, or adding shims) until the dial test indicator reads almost zero at four points, as illustrated between Figures 2 and 3. Eventually I got it parallel to the surface plate to within 10 micron, and called it good.

This means that when I flip the steel block into its horizontal position (see Figure 4) the lens mount flange is perpendicular to the surface plate with a reasonably high degree of accuracy. Eventually, I will arrange my test chart in a similar fashion, but bear with me while I go through the process.

Figure 4: Using a precision level to ensure my two reference surfaces are parallel |

The logic is as follows: I cannot directly measure if the rotary table's surface is parallel with the granite surface plate, but I can ensure that both of them are level, which is going to ensure that their surfaces are parallel to within the tolerances that I am working to here. This means that I know that my camera lens mount is perpendicular to the rotary table's surface. All I now have to do is place my test chart so that it is perpendicular to the rotary table's surface, and I can be certain that my test chart is parallel to my camera's mounting flange. I aligned and shimmed my test chart until it was perpendicular to the rotary table top, using a precision square, resulting in the set-up shown in Figure 5.

Figure 5: overview of the final set-up. Note the obvious change in colour temperature relative to Figure 4. Yes, it took that long to get two surfaces shimmed level. |

### One tiny little detail (or make that two)

Astute readers may have picked up on two important details:

- I am assuming that my camera's lens mounting flange is parallel to the sensor. In theory, I could stick the dial test indicator into the camera and drag the stylus over the sensor itself to check, but I do actually use my camera to take photographs occasionally, so no sense in ruining it just yet. Not even in the name of science.
- The entire process above only ensures that I have two planes (the test chart, and the camera's sensor) standing perpendicularly on a common plane. From the camera's point of view, this means there is no up/down tilt, but there may be any amount of left/right tilt between the sensor and the chart. This is not the end of the world, since my initial test will only involve the measurement of pitch (as illustrated in Figure 1).

### The first measurements

Note: Results updated on 13/02/2017 to reflect improvements in MTF Mapper code. New results are a bit more robust, i.e., lower standard deviations.

From the set-up above, I know that my expected pitch angle should be zero. Or at least small. MTF Mapper appears to agree: the first measurement yielded a pitch angle of -0.163148 degrees, which is promising. Of course, if your software gives you the expected answer on the first try, you may not be quite done yet. More testing!

From the set-up above, I know that my expected pitch angle should be zero. Or at least small. MTF Mapper appears to agree: the first measurement yielded a pitch angle of -0.163148 degrees, which is promising. Of course, if your software gives you the expected answer on the first try, you may not be quite done yet. More testing!

I decided to shim the base of the plywood board that the test chart was mounted on. The board is 20 mm thick, so the 180 micron shim (0.18 mm) that I happened to have handy should give me a tilt of about 0.52 degrees. I also had a 350 micron (0.35 mm) shim nearby, which yields a 1 degree tilt. That gives me three test cases (~zero degrees, ~zero degrees plus 0.52 degree relative tilt, and ~zero degrees plus 1 degree relative tilt). I captured 10 shots at each setting, which produced the following results:

- Expected = 0 degrees. Measurements ranged from -0.163 degrees to -0.153 degrees, for a mean measurement of -0.1597 degrees and a standard deviation of 0.00286 degrees.
- Expected = 0.52 degrees. Measurements ranged from 0.377 to 0.394 degrees, for a mean measurement of 0.3910 degrees with a standard deviation of 0.00509 degrees. Given that our zero measurement started at -0.16 degrees, relative angle between the two test cases comes down to 0.5507 degrees (compared to the expected 0.52 degrees).
- Expected = 1.00 degrees. Measurements ranged from 0.814 to 0.828, for a mean measurement of 0.8210 degrees with a standard deviation of 0.00423 degrees. The tilt relative to the starting point is 0.9806 degrees (compared to the expected 1.00 degrees).

I am calling that good enough for government work. It seems that there may have been a small residual error in my set-up, leading to the initial "zero" measurement coming in at -0.16 degrees instead, or perhaps there is another source of bias that I have not considered.

### Compound angles

Having established that the pitch angle measurement appears to be fairly close to the expected absolute angle, I set out to test the relative accuracy of yaw angle measurements. Since my set-up above does not establish an absolute zero for the yaw angle, I cheated a bit: I used MTF Mapper to bring the yaw angle close to zero by nudging the chart a bit, so I started from an estimated yaw angle of 0.67 degrees. At this setting, I zeroed my rotary table, which as you can see from Figure 5 above, will rotate the test chart approximately around the vertical (y) axis to produce a desired (relative) yaw angle. At this point I got a bit lazy, and only captured 5 shots per setting, but I did rotate the chart to produce the sequence of relative yaw rotations in 0.5 degree increments. The mean values measured over each set of 5 shots were 0.673, 1.189, 1.685, 2.211, 2.717, and 3.157. If we subtract the initial 0.67 degrees (which represents our zero for relative measurements), the we get 0.000, 0.5165, 1.012, 1.538, 2.044, and 2.484, which seems pretty close to the expected multiples of 0.5.

In the final position, I introduced the 0.18 mm shim to produce a pitch angle of 0.5 degrees. Over 5 shots a mean yaw angle of 3.132 degrees was measured (or 2.459 if we subtract out zero-angle of 0.67). I should have captured a few more shots, since at such small sample sizes it is hard to tell if the added yaw angle has changed the pitch angle, or not. It is entirely possible that I moved the chart while inserting the shim. That is what you get with a shoddy experimental procedure, I guess. Next time I will have to machine a more positive mechanism for adjusting the chart position.

### Discussion

Note that MTF Mapper could only extract the chart orientation correctly if I provided the focal length of the lens explicitly. My previous post demonstrated why it appears to be impossible to estimate the focal length automatically when the test chart is so close to being parallel with the sensor. This is unfortunate, because it means that there is no way that MTF Mapper can estimate the chart orientation completely automatically --- some user-provided input is required.

The good news is that it seems that MTF Mapper can actually estimate the chart orientation with sufficient accuracy to aid the alignment of the test chart. Both repeatability (worst-case spread) and relative error appears to be better than 0.05 degrees, or about three minutes of arc, which compares favourably with the claimed accuracy of Hasselblad's linear mirror unit. Keep in mind that I tested under reasonably good conditions (ISO 100, 1/200 s shutter speed, f/2.8), so my accuracy figures do not represent the worst-case scenario. Lastly, because of the limitations of my set-up, my absolute error was around 0.16 degrees, or 10 minutes of arc; it is possible that actual accuracy was better than this.

How does this angular accuracy relate to the DOF of the set-up? To put some numbers up: I used a 50 mm lens on an APS-C size sensor at a focus distance of about 1 metre. If we take the above results, and simplify it to say that MTF Mapper can probably get us to within 0.1 degrees under these conditions, then we can calculate the depth error at the extreme edges of the test chart. I used an A3 chart, so our chart width is 420 mm. If the chart has a yaw angle of 0.1 degrees (and we are shooting for 0 degrees), then the right edge of our chart will be 0.37 mm further away than expected, or our total depth error from the left edge of the chart to the right edge will be twice that, about 0.73 mm. If I run the numbers through vwdof.exe, the "critical" DOF criterion (CoC of 0.01 mm) yields a DOF of 8.95 mm. So our total depth error will be around 8% of our DOF. Will that be enough to cause us to think our lens is tilted when we look at a full-field MTF map?

How does this angular accuracy relate to the DOF of the set-up? To put some numbers up: I used a 50 mm lens on an APS-C size sensor at a focus distance of about 1 metre. If we take the above results, and simplify it to say that MTF Mapper can probably get us to within 0.1 degrees under these conditions, then we can calculate the depth error at the extreme edges of the test chart. I used an A3 chart, so our chart width is 420 mm. If the chart has a yaw angle of 0.1 degrees (and we are shooting for 0 degrees), then the right edge of our chart will be 0.37 mm further away than expected, or our total depth error from the left edge of the chart to the right edge will be twice that, about 0.73 mm. If I run the numbers through vwdof.exe, the "critical" DOF criterion (CoC of 0.01 mm) yields a DOF of 8.95 mm. So our total depth error will be around 8% of our DOF. Will that be enough to cause us to think our lens is tilted when we look at a full-field MTF map?

Only one way to find out. More testing!

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