Is MTF50 a good proxy for perceived sharpness?   In this article and those that follow MTF50 indicates the spatial frequency at which the Modulation Transfer Function of an imaging system is half (50%) of what it would be if the system did not degrade detail in the image painted by incoming light.

It makes intuitive sense that the spatial frequencies that are most closely related to our perception of sharpness vary with the size and viewing distance of the displayed image.

For instance if an image captured by a Full Frame camera is viewed at ‘standard’ distance (that is a distance equal to its diagonal), it turns out that the portion of the MTF curve most representative of perceived sharpness appears to be around MTF90.  On the other hand, when pixel peeping the spatial frequencies around MTF50 look to be a decent, simple to calculate indicator of it, assuming a well set up imaging system in good working conditions.

Psychophysical Subjective Tests

The explanation starts with the human Contrast Sensitivity Function (aka the visual system’s ‘MTF’).  It looks as follows when expressed in cycles per degree on the retina, log scale on the left (also often seen in log-log as the outline of a Campbell-Robson chart) and the less familiar linear scale on the right:

The two graphs show average Human Visual System Contrast Sensitivity as a function of cycles per degree on the retina for persons with normal vision in normal conditions. As a result of measurements such as these, 20/20 human vision has been chosen to correspond to 30 cycles per degree and maximum acuity is often assumed to be around 50 cycles/degree.

Granger and Cupery (1972)* performed a number of controlled psychovisual ranking experiments with people of normal vision and concluded that the spatial frequencies on the retina that are most representative of the subjective perception of sharpness are those between 3 and 12 cycles/degree (they worked off data in log scale, as Bob Atkins explains).

From Retina to Photograph to Frequency on Sensor

The distance covered by one degree on the retina can be converted to the equivalent distance on a displayed photograph as follows:

(1)  

with

• the period of detail on the photograph, typically in mm
• viewing distance in the same units as above
•  the angle subtending one period on the retina

In fact for the tiny angles involved in the representative range we can simplify things by approximating the tangent by the angle itself so , with in radians.

In order to obtain the corresponding distance on the sensor, the distance covered by period on the photograph needs to be multiplied by the inverse of the output enlargement (aka magnification): the ratio of the lengths of the diagonals of the camera’s sensor and the displayed photo

(2)  

with the period in the same units as viewing distance .   Using the simplified expression for and since frequency is one over the period, the corresponding spatial frequency on the sensor can then be expressed as follows**:

(3)  

with the spatial frequency on the retina in cycles/ degree and , , in mm.  Cycles and line pairs (lp) are interchangeable in this context.

Critical Frequency Range for Perceived Sharpness

So back to Granger and Cupery and their results suggesting that the frequencies on the retina most representative of the perception of sharpness are those between 3 and 12 cycles per degree.  Applying Equation (3) to a Full Frame sensor with a diagonal of 43.3mm and = 3 cycles per degree on the retina we get about 4 lp/mm times the ratio of the dispayed photograph diagonal and viewing distance; = 12 cycles per degree correspond to about 16 lp/mm on the sensor times the same factor.

Generalizing to other formats, the range of frequencies on the sensor representative of subjective perceived sharpness suggested by Granger and Cupery is then about

(4)  

with CF sensor crop factor  relative to full frame cameras (1 for FF, 1.5 for APS-C, 2 for mFT etc.) and the length of the displayed photo diagonal in the same units as viewing distance .

MTF50 not Relevant at Standard Distance

Based on (4), if we assume standard viewing distance to be equal to image  diagonal , the spatial frequency range on the sensor reduces to about

(5)  

For instance the range of interest would be about 8->32 lp/mm for a Micro Four Thirds image viewed at standard distance, because its crop factor is 2.

On the other hand you can see where the 4->16 lp/mm range falls on the MTF curve of a full frame Nikon D610 with crop factor 1 below

Granger’s most sensitive frequencies are highlighted below on the measured MTF of a Nikon D4 – but the same region of interest applies to the graph above and to that of any camera with a sensor of about 24mm height – when the photograph is viewed at standard distance:

Since MTF curves in those low spatial frequencies tend to be around 90% in current Full Frame imaging systems, if we mostly view our images at standard distance we should really focus on their combined MTF performance at around MTF90 +/-10 as far as expected perceived sharpness is concerned, according to Granger.   That’s the portion of the MTF curve that photographers normally relate to the property they typically call contrast in a final photograph.  It may come in handy to know when evaluating lenses for purchase.

The statement above is theoretically valid whether you look at your FF captures as 5x7s from 218mm away, 8x12s from 366mm away, fit to your 24″ monitor from 611mm away or 10x15m billboards from 18m away: they are all standard distances equal to the diagonal of the displayed image, hence project the same cycles per degree on the retina.

MTF50 Relevant when Pixel Peeping

Of course the pixel peeper in me would want to choose a lens that will also provide good perceived sharpness when watching a final photograph from closer than standard distance, say when pixel peeping on a monitor.

When pixel peeping at 100% zoom, current consumer monitors are typically only able to show a portion of the full captured image – and that’s ok because when we pixel peep we concentrate on detail and sit so close to the monitor that we are typically not able to take it all in at once anyways.  Therefore to calculate the  diagonal of the full-sized photograph as displayed we pretend that the monitor is of the size required to display every captured pixel: simply take the ratio of the number of captured to monitor pixels on the diagonal times the physical size of the monitor’s diagonal.

For example my Nikon D610 4000×6000 pixel FF camera has 7211 pixels on the diagonal and my Dell U2410 1200×1920 24″ monitor has 2264 pixels on the diagonal so is equal to 7211/2264*24″ = 76.4″ or almost two meters.  I take pixel peeping distance to be that viewing distance at which I just cannot see individual pixels on the screen, which for me is about 400mm (16″) so works out to about 76.4/16 = 4.8.

Plugging those values into Equation (4) above the frequencies of interest for perceived sharpness when pixel peeping are then in the range of about 19 to 76 lp/mm on the full frame Nikon D610 MTF chart shown below:

For a  4000×6000 pixel APS-C camera like the Sony a77 mark II the range in the same conditions becomes about 29 to 115 lp/mm, and this is how the area of interest would look like for it when pixel peeping:

On the other hand the 3280×4928 pixel Full Frame Nikon D4 produces a smaller overall image when pixel peeping at 100% on a monitor because it has fewer pixels.  The ‘pixel peeping’ photo diagonal to viewing distance ratio would therefore be about 4 for it on my monitor at my pixel peeping distance.  Below in blue are what Granger would call critical frequencies for perceived sharpness then (D4s MTF shown):

And here is a 4.85um pitch, 4912×7360, Full Frame, D810 mounting an AF-S 85mm f/1.4G in the same conditions, this time with spatial resolution expressed in cycles per pixel (divide by pitch and multiply by 1000 to get lp/mm, see here for how to convert between spatial resolution units)

We can see that when evaluating images up close MTF50 looks like it is right in the thick of things with current kit. Keeping in mind that these curves are derived from unsharpened raw data linear in intensity, perhaps when pixel peeping at 100% zoom on a 24″ monitor from 400mm (16″) away MTF50 is not such a bad proxy for perceived sharpness after all.  The underlying assumption here is that the equipment is being set up to obtain a sharp working image.

MTF50 a Good Proxy for SQF too

In fact as you can see above MTF curves of decent setups captured with good technique in typical photographic working ranges tend to be approximately linear around MTF50, the value of which can then often be considered to be proportional to the area under the curve weighted by the HVS Contrast Sensitivity Function.  Such a weighted area in various forms  is the the basis for other, better accepted sharpness metrics with names like Subjective Quality Factor (SQF), SquareRoot Integral (SQRI), CMT Acutance***.

The following graph shows how well MTF50 correlates to SQF (not normalized to the same peak values):

These D610+AFS50mm/1.8D captures of a backlit razor edge were shot 1 stop apart by varying f-number, with file 1 at f/2.8 and file 7 at f/22.  Note how both MTF50 and SQF obtained off the well corrected green and blue raw channels effectively provide similar relative information.

Resizing an image also results in changed magnification and similar effects as discussed above.  This article deals with it from a downsizing perspective.

 

* “An optical merit function (SQF), which correlates with subjective image judgments”, Granger and Cupery (1972).

** The precise version of Equation (3) is

(6)  

but the difference is immaterial in the range of interest here.

***  The I3A Camera Phone Image Quality initiative adopted a middle of the road Contrast Sensitivity Function and uses weighted data to Nyquist only, as described in this paper on the development of CPIQ Acutance metric.