Measuring Lens "Sharpness" across the image field: MTF

[Bernard Delley]

A scientific way to describe imaging system sharpness, or imaging component, like lens, sharpness, is is by the optical transfer function. Usually it is more handy to present only its module, the modulation transfer function (MTF), and this is usually done as a function of the radius off the optical axis in meridional and sagittal direction to describe the spatial frequency response for tangential and radial lines.  Manufacturers provide such plots to set the right expectations for their lenses. Nikon shows, presumably calculated, MTF at 10, 20, and 30 cycles/mm for fully open lens aperture. Zeiss shows MTF, presumably emulated from measurements, at 10, 20 and 40 cycles/mm for open and a stopped down aperture. Other manufacturers often use presentations like Nikon and some more detailed like Zeiss.

I got fascinated by the idea of quantifying sharpness definitively by the contrast loss for fine image detail already as a young student and amateur photographer. Unfortunately, such measurements were way too challenging, and requiring high tech equipment, and thus out of reach.  This changed dramatically with the advent of the open source program mtf_mapper written by Frans van den Bergh starting about a decennia ago for slanted edge spatial frequency response (SFR) analysis of test images.  This started a long research process on my side culminating in a scientific paper together with Frans in 2021 describing all the details, how with this method lens MTF can be extracted with better than about 0.02 % accuracy across the image field.  The content of the publication is freely accessible via this link:
https://www.dora.lib4ri.ch/psi/islandora/object/psi%3A37151
there is also a link to the extensive supplementary material at the original publisher.

Attached below  is an image of my measurement station for normal to ultrawide lenses, with a special "tripod" with camera and a huge chart assembled from 4 DIN B0 size parts.

[Bernard Delley]

For each of the black trapezoids, appearing in the raw image, one can extract a meridional and a sagittal MTF function. This is a mass of information 2 x 864 tabulated functions.
The figure, appearing also as fig 3 in the publication illustrates 3 ways of cutting it.  The points in 3a show all the sagittal MTF at their radial position to the left, while the meridional MTF is drawn to the right. The function values re taken at 10 cycles/mm (red), 30 (green) and 50 (blue).  The points show quite a spread, the lines show the average MTF at these three spatial frequencies as a function of radius from the image center. This averaging abstracts a bit from the individual properties of the lens sample and shows something that is hopefully not too far from the expectations posted by the manufacturer. The cloud of points does not give a good hint to the why of this spread of values. The maps Fig 3b chosen at MTF for 40 cycles/mm show how MTF for this lens sample varies across the image field, this is shown again separatly a map for the sagittal and one for the meridional direction.  This is near optimal focus for my lens sample of the AF-S 300mm f/2.8 lens at f/2.8 measured in the Bayer green channel, using the A0 sized target. This results in an imaging ratio of ~ 1:33  onto the FX sensor.

[Bernard Delley]

My sample of the AF-S 105mm f/1.4  at f/1.4 appears to be close to the expectations. Fig 10 from the supplementary material shows a comparison of my measurement a) at imaging ratio 1:33 in the Bayer green channel at 10 (red) 20 (orange) 30(green) 40 (aqua blue) 50 cycles/mm (deep blue).  Meridional MTF are the full lines, sagittal MTF dashed lines.
b) is drawn from measurements at Lens Rentals. They use an optical bench nicknamed "Olaf" and probably do a single radial sampling at 10 or so points but for 10 samples in their store. The standard collimators in an optical bench are for infinity focus. The spectrum used for the measurement was not mentioned, be it white or weighted white, perhaps like eye sensitivity. I have dropped their 20 and 40 measurement to spare copying work.
c) MTF posted by Nikon, presumably calculated from lens design. It is not specified at what distance this is: portrait ? or just standard infinity. The spectrum is also not mentioned. The computational lens simulations are probably done neglecting diffraction, however this is about negligible for an f/1.4 lens fully open.

[Bernard Delley]

Here is an example where I have assembled the MTF maps through focus and stopped down for my old AIS 20mm f/3.5 lens. The left part is starting at front focus going to back focus to the right. Fully open aperture are the bottom meridional and sagittal maps with maps from the increasingly stopped down lens towards the top. All mapes are at 40 cycle/mm for Bayer green at imaging ratio ~ 1:33. The camera D850 matters only marginally in that its directionally dependent pixel aperture MTF is divided out. Conveniently there is no AA to be corrected out. The directional chart (edge) MTF has also been divided out to obtain the MTF of just the lens sample. This is salvaged from my 2019 thread at dpreview.

[Bernard Delley]

It is clear in the previous graphic that optimal focus for the little AI 20mm f/3.5 lens that wide open optimal focus appears around the bottom center of the figure. Meridional contrast gets low outside a not sol big inner circle. It takes stopping down to f/8 or  f/11 to improve this, and there optimal setting is towards the left of the figure.

[Bernard Delley]

MTF at 40 cycles/mm for the AF-S 105mm f/1.4, Bayer green, front to back focus and stopped down.

[MEPER]

With this you can get an "image" of the focus plane if plane is not flat field?
Many vintage lenses have a bit of field curvature?
As I remember a lens like the Nikkor-H 50/2 if you focus at center then the corners are sharp at a closer distance.

[Bernard Delley]

indeed, the AF-S 105mm f/1.4 is a nice example of filed curvature. The mild decenterings and higher order aberrations mask things somewhat at f/1.4. stopped down it is quite clear, with the most clear perhaps at f/2.8: maximum micro contrast happens at the center at about 32mm front focus. At about 32mm back focus you find maximum micro contrast far out in the image field. The achievement for this lens design is that meridional and sagittal maps track almost perfectly, thus there is negligible astigmatism ! As the field curvature in object space goes towards the back far out in the image field, it is interesting to realized that the curvature of field at the sensor side is opposite, concave towards the lens. Because of the imaging ratio ~ sqrt(1000) it is that the 32mm in the object side translate to 32 micrometers at the sensor side.

It is also interesting to look again at the comparison of radial MTF traces, which I attached before.  The MTF from Nikon shows quite a dimple at the center. This means that the focus was chosen a bit to the backfocus side to even out MTF radially. MTF stays quite high nevertheless. Also the separation of meridional and sagittal in the Nikon MTF is symptomatic for minor residual astigmatism.

This and other things, like longitudinal color aberration, defocus control in MTF etc,  are discussed in the supplementary material linked in my opening post.

[Bernard Delley]

One can use such MTF measurements to compare lens choices. Below is an example how a selection of long F lenses compare when all are stopped down to f/5.6 for imaging ratio 1:32 .   The Lens Rentals Olaf style radial plots at 10, 20, 30, 40 and 50 cycles/mm generalise somewhat from the mild decentereing specifics of my lens samples. The plots may help to set the right expectations for the lens types. See how the AF-S 300mm f/2.8 with TC-2.0 at 600mm has slightly less micro-contrast than the zoom lenses at their long end. See also how uniform micro contrast across the image field is provided by the AF 135mm f/2 DC lens at F/5.6 .  Al;so the AF-S 80-400mm f/4.5-5.6 G lens shows impressive micro contrast across the image field at 80 mm.

[Jack Dahlgren]

I’m a bit surprised by the 135DC. I guess I usually use it cranked way out of where it delivers peak sharpness. I’ll have to give it a try with middle settings

[John Geerts]

One can use such MTF measurements to compare lens choices. Below is an example how a selection of long F lenses compare when all are stopped down to f/5.6 for imaging ratio 1:32 .   The Lens Rentals Olaf style radial plots at 10, 20, 30, 40 and 50 cycles/mm generalise somewhat from the mild decentereing specifics of my lens samples. The plots may help to set the right expectations for the lens types. See how the AF-S 300mm f/2.8 with TC-2.0 at 600mm has slightly less micro-contrast than the zoom lenses at their long end. See also how uniform micro contrast across the image field is provided by the AF 135mm f/2 DC lens at F/5.6 .  Al;so the AF-S 80-400mm f/4.5-5.6 G lens shows impressive micro contrast across the image field at 80 mm.

Thanks for the very interesting information with diagrams, Bernard. 

When shooting your lenses only wide open, I presume this is not the best method as comparison only takes place at f/5.6?

[Erik Lund]

I’m a bit surprised by the 135DC. I guess I usually use it cranked way out of where it delivers peak sharpness. I’ll have to give it a try with middle settings

Yes it looks really good, most of the fixed focal length Nikkors are really good/peaking at around f/5.6 so not a huge surprise that this stellar vintage optic is close to the 105mm f/1.4

[Bernard Delley]

When shooting your lenses only wide open, I presume this is not the best method as comparison only takes place at f/5.6?

I chose f/5.6, as this is a good setting for sharp images and all lenses in the comparison can take images at f/5.6. It is also true that reasonably well centered lens samples get also reasonably close to axial symmetry by f/5.6 : so radial plots of MTF represent reality closely, and can show how MTF changes with spatial frequency: 10 20 30 40 and 50 cycles/mm meridional and sagittal can be shown in the same plot frame.  At f/5.6 it is a story mainly of sharpness falloff radially out and of astigmatism, how well meridional and sagittal micro contrast comes to match around optimal focus.

The story is different wide open. More complex, higher order aberrations play a big role.
The maps at 40 c/mm shown above for the 300mm f/2.8 at f/2.8 suggest primarily uneven polishing  with a complicated angular behavior around the image circle. 
The maps for the 105mm f/1.4 show a progression from mildly decentered to well centered stopped down. The achievement for this lens is that astigmatism is so well controlled that meridional micro contrast matches maximum sagittal micro contrast closely.  I call this mildly decentered as optimal focus wide open stays all in the green+blue coloring. I would call frame 0318 optimally focused wide open. It is focused 16mm back of where the Z7II happened to put focus for this series of fixed lens takes.
The wide open maps for the AF 135 f/2 DC sample (not shown here) demonstrate impressive centering with some cameras. The decentering must come from the flange with some others. -- A skewness of 25 micro meters of image sensor to flange shows clearly with such MTF maps ! And after seeing the disassembly/reassembly for the shutter, it seems plausible that this may happen -- Another issue with the 135 DC is flare and longitudinal color aberratio, which causes purple (<==-->green) fringe at high contrast wider open. The longitudinal color aberration can be quantified from the full set MTF measurements. The published supplementary material discussed the Tamron 45mm f/1.8 as example.

[Jack Dahlgren]

Yes, the color aberration on the135dc is strong. Can you post the wide open chart for it?

[MEPER]

There are some talk about decentered lens elements and that it is quite visible shooting wide open in MTF charts.
Are most Nikkors decentered?
Or how can you be sure to get a "perfect" lens?
Probably a dealer will not give you 10 samples to select from?
Maybe some "famous" persons can make special agreements with Nikon to get "hand picked" lenses?

Here I can buy a lens and I can return it if I don't like it within.....I think 14 days if I deliver it back in original box etc.
But I would not be able to detect if a lens is decentered unless it is quite bad wide open. Else I would not know if the performance wide open is "normal" or it cold be better. Then I would need some samples to test and select from.