A good illustration of the utility of Bart's target. A nice feature of the method is that the object distance is not important.

Yes, this is deliberate. One of the easiest mistakes to make is to shoot targets that are sensitive to shooting distance (=magnification) at the wrong distance. Besides, it is not always clear how to measure that distance when setting up for a test. Do we need to measure from the film/sensor plane, or from the front of the lens (and where, entrance pupil, nodal point, filter threads, etc.). We can only figure out the actual resulting magnification after shooting, by measuring the size of the target on the sensor (microscope on film, or pixels times sensel pitch for digicams or scanners) divided by the size of the original. When there are fixed markings on the target, then we need to re-adjust the focusing distance iteratively to arrive at the intended magnification. This is such boring work that most either skip this calibration step of don't do the test in the first place.

For reasons I don't understand, it seems as if the Nyquist limit is always at 92 pixels, regardless of the pixel pitch of the sensor. Perhaps Bart can explain.

I'll give it a try. At any diameter, the target always has 144 full cycles per circumference. The only thing that changes with distance is the magnification factor. Well, some lenses perform a bit better at some distance than at others, but in the suggested range of 25x to 50x focal length the differences are not likely to be significant.

We can know how high the spatial frequency is along the circumference at any given diameter, the circumference of a circle is 2 x Pi x radius (or Pi x diameter), and there are always 144 cycles at that circumference. So by dividing 144 cycles by the circumference we know the number of cycles per pixel. Since Nyquist is at 0.5 cycles per pixel (or 1 cycle per 2 pixels), the equation becomes 144 / (Pi x Diameter), and when the diameter is expressed in pixels we multiply by 2 to find the Nyquist frequency.

BTW, I prefer to write the formula as Cy/px = (144 / Pi) / Diameter, which is the same, but it allows to pre-calculate 144 / Pi once and change the diameter as we take different measurements. A very basic calculator suffices.

Now, as to why we also find similar values regardless of shooting distance. As shown, we can calculate the Cy/px for any diameter (and thus circumference) on the target, or a projection of it. The diameter is expressed in pixels (1 sensel or sample per output pixel), and the sampling frequency in pixels is constant for any sensor. The only thing we change with shooting distance is magnification, but the sampling density remains constant. Therefore we shoot a different radius on the target itself (e.g. larger radius at longer distance, but also with equally smaller magnification, thus with the same resulting diameter or circumference), and the frequency we can resolve per pixel remains the same. So the distance and magnification factor result in a constant projected blur diameter, although the origin is sampled at different diameters on the target. The Nyquist frequency of a sensor is always at 2 sensels per cycle, and thus remains constant between comparisons of different sensors when expressed in pixels.

The importance of this fact is that one does not have to go through the equations Bart posted if one does not need the actual resolution in cycles/mm. One can simply measure the blur diameter.

That's right, expressed in cycles/pixel, the diameter is all that's needed. Only when one wants to calculate things like magnification potential does it make sense to add the physical sensel pitch into the equation, by changing the diameter to pixels x sensel pitch in mm. When we e.g. know that our output medium can resolve 5 cycles/mm, and our optical system resolves 78 cycles/mm, then we know we can magnify our sensor size by 78/5= 15.6x to find the uncompromised maximum output dimensions, fit for reading distance inspection (5-8 cycles/mm at a normal reading distance is at the verge of human visual acuity). Larger output should be viewed at a proportionally larger distance for the same quality impression.

Cheers,

Bart