**Introduction**

Miljenko’s chart of his measurements of the Modulation Transfer Function of the Sigma 70mm lens:

http://www.photomacrography.net/forum/v ... hp?t=22379

prompted me to wonder what the measurements mean.

This post is an invitation for anyone to give me a sanity check because I am not confident of my conclusions.

If someone can tell me how to calculate a diffraction limited curve: that would be even better.

**Slanted Edge Software**

The post is about the results of software like Imatest which use the slanted-edge method to calculate the MTF of digital systems [lens + sensor + processing], and particularly about understanding the MTF curve produced by such software for a single point.

This is an example of such an MTF curve - for the Sigma 70mm lens used at 1x magnification at a nominal aperture of f/5.6.

With my copy of the lens this aperture produced the highest MTF at 1:1 on its axis. The lens has a pupil ratio of about 1.13 at 1x [it varies slightly with magnification] so the effective aperture is about 9.2 when adjusted for the shortened focal length of the lens [c. 61mm] at 1:1

Most MTF charts published by online testers use software based on ISO12233 which uses the image of a sharp grey/white edge inclined at about 5 degrees to the sensor’s rows and columns to calculate the MTF. The analysis uses tens of rows of several tens or even hundreds of pixels. Because of the slanted edge the several hundred pixels used are all at slightly different distances from the edge enabling the software to determine the edge profile and MTF value with a much greater spatial resolution than the size of the phototites.

The effective resolution of the slanted edge software is much greater than the resolution of the sensor when the senor is used to take pictures.

Imatest is probably the most well known, but has long evolved away from photographers to industrial users and is now very expensive. There are two free applications which can produce the same charts: Image J’s Slanted Edge MTF Plugin and Frans van den Bergh’s MTF Mapper [though a trick is needed to catch the CSV files MTF Mapper generates before it closes].

**Sensor Resolution**

If you have got this far you probably know that the resolution of a sensor is limited by its Nyquist frequency: the fact that you need a minimum of two rows or columns to records a dark/light cycle. That’s the spatial frequency of 0.5 cycles per pixel marked on the chart.

It practice it’s worse, because the two rows would have to be aligned with the detail and the detail would have to be of high contrast to stand any chance of being detected. There is also the colour filter array which most cameras use, which means that each pixel only samples one colour channel so the missing information has to be interpolated by demosaicing. Tests on video cameras which come in single-senor [with a colour filter array] and three-sensor versions [no colour filter array or demosaicing needed] indicate that the CFA reduces linear resolution by about 30 percent.

To sum up, to detect randomly oriented detail at moderate contrast, in practice you need an absolute minimum of about 3 pixels and probably a bit more.

**Interpreting the MTF Chart**

Returning to the MTF chart, my first thought is that only the data to the left of point A: i.e. spatial frequencies of below 0.33 cycles per pixel or perhaps even less is of any photographic significance. The rest of the curve to the right of point A is real data - but its of no direct photographic significance because these frequencies cannot be detected by the sensor.

By “no direct” significance I am implying that it is important indirectly: Its important for a lens to out-resolve a sensor by a substantial margin [e.g. a factor of two in limiting resolution] if possible in order to make sure that the MTF at point A is high enough to be seen as detail.

Is a lens merely matches the resolution of a sensor [that seems to be the reference point for some cellphone designs] the image won’t look sharp because the MTF at the limiting frequency will be too low. This eventually happens when a lens is opened out [due to aberrations] or closed down [due to diffraction].

My other thought - and it’s only a suggestion, is that to the right of Point A the slanted edge method is recording real data, not just noise or aliasing, at least until somewhere about B when noise does take over. I suggest that the curve is mainly recording lens performance as attenuated by any low pass filter, but that this performance can be recorded beyond the Nyquist limit because of the super sampling of the slanted edge method despite the fact that it cannot be captured by a conventional image.

**Adding a Diffraction Curve ?**

Finally, a question: Does anyone know how to calculate a diffraction-limited MTF curve to add to a chart like this to illustrate how closely the performance of the lens is to its physical limit ?

I have tried using the formulae I have found but the results show this lens to be working so far below its diffraction-limited performance that I can’t believe I have it right.

I tried this formula which is said to give MTF vs spatial frequency in clycles/mm

Using this data:

Pupil Ratio at 1:1 magnification : P = 1.13

Nominal aperture fr = f/5.6

Magnification m = 1.0

Effective Aperture, fe = fr (m/P + 1) = 5.6 (0.885 + 1) = f/10.6

... then adjusting for the fact that at 1:1 the focal length of the lens has reduced from 70mm to c. 61mm, which, for a fixed entrance pupil diameter would change the focal ratio in proportion to the focal length to f/9.2

Cut-off frequency νc =1 / (λ fe)

Assume λ = 0.55 micons (i.e. 550 nm)

νc =1 / (.000550 x 9.2)

νc =1 / (.00506)

νc =197.6 cycles/mm.

Each pixel is 0.00641 mm so the cut-off frequency is 1.267 cycles/pixel

I got this blue calculated curve:

I can’t believe it.

The published results for other lenses suggest that at its optimum aperture for sharpness the Sigma should be very close to the calculated curve.

Where have I gone wrong ?

Henry