# Linear or nonlinear?

07/06/19 20:36

This question might surprise you. But there are good arguments to discuss this mathematical concept in the photographic environment. Assume you have a 24 Megapixel sensor, like the one in the current M series. The image quality (a very elusive concept) should be replaced by the information capacity, but for now it are the Image Engineering calculations that dominate the discussion. The Image Engineering analysis of the sensor performance (always including a lens) has a good correlation with perceived image quality. The German magazine Color Foto is a true believer of this software. The recent issue has a report of the Leica M10-D. The results are, for ISO100, 1931 line pairs per image height (LP/IH). One would assume that when doubling the number of pixels on the sensor, the lp/im would also double. This is the linear aproach: when x=y the 2x would be 2y. As it happens, there is also a report of the Nikon Z7 with 45.7 Megapixels. Almost twice the amount of the pixels on the Leica sensor. The result? At ISO 100 it is 2822 lp/ih. The Z6 (with 24.5 Mp) has 1988 lp/ih. The lenses used are different of course. This might have some influence, as is the selection of the JPG format. The results of the Z7 are intriguing. There is a 90% difference between the pixel amount of the Leica M10 compared to the pixel amount of the Z7, but only a 46% increase in resolution. A non-linear result! So a roughly 2 times the amount of pixels results in only 1.5 times resolution.

Another comparison: the Leica has pixel pitch of 6 micron, the Z6 of 5.9 and the Z7 of 4.3 micron. The APS-C sensor of the Ricoh GR III has a pixel pitch of 3.9 micron with an APS-C sensor of 24 Mp and a resolution of 2075 lp/ih. Presumably the pixel size is more important than the sensor size. The Leica M8 is a living proof for this argument!

If and when Leica will decide on an increase of the amount of pixel for the next generation of the M camera it will be somewhere between the 24 Mp of the current model and the ±65 Mp of the Leica S models. Not being in the position to being allowed to compete with the SL in future versions (let us assume 45 Mp) the final amount would be somewhere between 24 and 45: 34.5, which happens to be neatly between both extremes. Then the increase in amount of pixels will be ±40%. The predicted increase in resolution will be around 0.5*40% and 0.75 * 40% = between ± 20 and ±30% or 1900 *1.25 = 2375 lp/ih, not the result to be really happy with. Assuming the usual tolerance of 5% for the bandwidth of the measured results, these figures only give the direction of thinking. The exact values are less important.

The same argument can be found in the discussion about film emulsions that can record 200 lp/mm and film emulsions that can ‘only’ record 80 lp/mm. With 80 lp/mm almost every detail, that is visually relevant in a scene, can be captured. But the price for the higher resolution is slow speed, careful focusing and the use of a tripod. In handheld shooting, the increase in resolution can not be exploited. Again, assuming that the M camera will be the champion of handheld snapshot style of photography, the current level of resolution that is supported by the sensor is more than adequate for the task. Leica could improve the imaging chain and especially the demosaicing section for enhanced clarity and best results.

UPDATE June 10: There is some confusion here:

Let us first get the basic figures about the measurements, based on the IE software, related to the fixed ISO 100

Ricoh GR-III: 24 Mp and 2075 lp/ih (pixel pitch 3.9 micron)

Leica M10-D: 24 MP and 1911 lp/ih (pixel pitch 6 micron)

Nikon Z6: 24 Mp and 1988 lp/ih (pixel pitch 5.9 micron)

Nikon Z7: 46 MP and 2822 lp/ih (pixel pitch 4.3 micron)

It is universally assumed that in order to double the resolution, one needs a four times increase in area: to double the resolution of the Leica sensor (24 Mp) one needs a sensor size of 4 * 24 = 96 Mp. This increase in size would (theoretically!) elevate the resolution from 1900 to 3800 lp/ih.

The Nikon Z7, which has only twice the area of the Leica sensor and therefore its resolution would be less: it is in fact 2800 lp/ih. The sensor of the Ricoh with 24 Mp has 2075 lp/ih with a comparable pixel pitch. This is important to note, because the Nyquist limit is related to the pixel pitch. For a pixel pitch of 4 micron, the Nyquist frequency is 0.008 mm per mm (or cycle) = 125 lp per mm. (application of the Kell factor of 0.7 gives 87.5 lp per mm). 2000 lp/ih is 64 lp/mm for a 15.6 mm image height. So there is some room for improvement, at least theoretically. The pixel size of 6 micron for the Leica would produce 0.012 lp per mm or 83,3 lp per mm. The 1900 lp are for the image height of 24 mm, which is 79 lp/mm. Including the Kell factor which says that you can only reliably resolve 70% of the Nyquist frequency, the practical resolution limit of the Leica sensor would be .7* 83 = 58 lp for every mm. The Leica imaging chain is better than that of the Ricoh! Or one could also claim that the JPG demosaicing of the Leica is more aggressive and that spurious resolution is spoiling the results.

The Nikon Z7 with its 4 micron pixel pitch would be able to resolve 0.7*125 lp/mm = 87.5 lp per mm. The image height is 24 mm and the resolution is 2800 lp/ih. This would result in 117 lp per mm compared to a Nyquist frequency of 125 lp per mm. Compare the measured resolution with the calculated Nyquist limit and the Kell factor:

Leica M10-D: 79 lp/mm; 83.3 lp/mm; 58 lp/mm

Nikon Z7: 117 lp/mm; 125 lp/mm; 87.5 lp/mm

The measured resolution is quite close to the Nyquist number. This is not surprising because the IE software uses the Nyquist calculation as the limiting factor in their calculations. This limiting value would result at the point where the contrast is almost zero. Not very useful! The Kell factor is used because there is a contrast level below which there is no visual difference between two adjacent lines. A contrast difference of 15% is the minimum and the Kell factor is in many cases too conservative.

Now the calculation. Doubling the size (from 24 Mp to 46 Mp) produces an increase in resolution of 1900 lp/ih to 2800 lp/ih. That is an increase of 47% or a factor of about 1.5. This is indeed a one-dimensional relation. It compares only one direction and not the area. But here is the confusion. The resolution is measured one dimensional in line pairs per mm. This resolution is identical in the horizontal and the vertical direction.The pixel pitch is a square measure (the 6 micron length of the Leica are the same for both directions. The pixel has a square area!) Now an example: assume that we would like to have the resolution of the Z7 for a new Leica sensor. Going from 1900 to 2800 lp per mm and increasing the resolution in both directions would require that the amount of pixels for the same size of the sensor grows2800/1900*24 = 35.4 Mp to get the same resolution of 2800 lp/ih. This value is less than expected. But the Leica processing chain might be more effective. The 35 Mp number would require a pixel pitch of 4.1 micron. This would result in a Nyquist value of 122 lp/mm or 2920 lp/ih. If we require to double the resolution of 1900 lp per mm, we would need a decrease in pixel pitch from 6 to 3 micron to get a resolution of 166.7 lp per mm. (Nyquist limit). This would imply an increase in amount of pixels to 96 Mp or 4 times the actual Mp or 24 Mp.

Mixing the concept of the number of pixels in a certain area and the concept of the resolution of the pixel itself (in a one dimensional line) may be the reason for much confusion. The Nyquist frequency is a one dimensional measure, assuming a square sized pixel, and will calculate the resolution of the system. The resulting pixel pitch will define the number of pixels per sensor area.

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