How Many Pixels Do We Need


The rise of the megapixel count in cameras is undoubtedly the result of their technological development. This furthermore leads to the inevitable discussions about the necessity or sufficiency of individual figures. The arguments about the sufficiency or even overabudance of megapixels range from the capabilities of the human eye to the fidelity of modern optics to the difficulty of storing large files, especially when dealing with RAWs. Additionally, when dealing with digital technologies, increasing the sampling frequency appears to be an effective solution to a number of technical issues and leads to the improvement in quality of the received signal. This applies to both one-dimensional signals, such as sound, and two-dimensional signals, such as images. The aim of this article is to explore the real limitations that modern optics impose on the sampling frequency of images—that is the megapixel count. We will do so by examining the behaviour of various lenses when combined with two cameras: the Sony A7r, as the 36 megapixel full frame camera with one of the highest megapixel counts as of the time of this writing (January 2015), and a “440 megapixel full frame camera” described below. This research thus differentiates itself from a large number of other articles that evaluate the quality of lenses by comparing them among each other using cameras with “standard” resolution.

Method

The measurement method is described in [1]. The images of the target were taken from the distance allowing 150x size reduction. For examine, this is 7.5 metres when using a 50mm lens. This results in the target’s scale corresponding to a 3-300 line pair range per millimetre (lp/mm). The target was lit by a 400 watt lamp aimed at the ceiling, thereby providing dispersed illumination of the target. The cameras were fixed to a tripod and set off using a two second timer. The focusing was performed by hand with bracketing. The target’s profile was measured using a program called ImageJ. In order to calculate the MTF, a program comparable to the script from [1] was written.

36 megapixel Sony A7r

CZ Planar 50/1.4 ZF was used as a reference lens, as it offers, according to the manufacturer, a maximum resolution of 300 lp/mm. Below are the MTF curves at optimum apertures of F4 and F5.6 (ISO 100).
Sony A7r, CZ Planar 50/1.4 ZF
The shots were taken as RAWs and converted with C1Pro v.8 without sharpness amplification. The camera is capable of registering up to 100 lp/mm, which corresponds to a 34.5 megapixel resolution of the final image. Aliasing was observed at high lp/mm values. The MTF curve for a 50mm lens found in [2] (pp. 8) is likely to be that of CZ Planar 50/1.4 ZF at a F5.6 aperture. Furthermore, as demonstrated in [3], the sensor’s MTF can be described as the MTF of a low frequency filter,
MTF(f) = |SINC(fd)|
where f is the spatial frequency and d is the effective pixel size. The pixel size of the Sony A7r is 4.9 μm. However, as demonstrated in [4], [5] and [6] the effective size is affected by the presence of the Bayer interpolation as well as the pixel’s shape and its fill factor. Let’s us calculate the MTF of the sensor and the combined MTF of the lens–sensor system, by assuming the effective pixel size being equal to the physical pixel size multiplied by the correction coefficient k.
Sony A7r, CZ Planar 50/1.4 ZF
The closest fit to the conducted test appears at k=1.8. Let us consider the effect the increase of sharpness via the unsharp mask (USM) has on the MTF.
Sony A7r, CZ Planar 50/1.4 ZF, USM
The USM thus results in the middle of the curve going up and has very little effect at higher frequencies. Needless to say, USM is not capable if create details, where none are present.

“440 megapixel full frame camera" BenQ_GH20X

As a high resolution camera, a compact 14 megapixel camera BenQ_GH20X equipped with a CMOS was used. Its pixel size is 1.4 μm.
BenQ_GH20X
If we were to fill the 24x36mm sensor with pixels of the aforementioned size, we would have a 440 megapixel full frame sensor. The sensor was removed from the camera’s case and put on a metal mount. The lens’ cap was attached to the other side of the mount. This functioned as a bayonet E-mount and allowed to attach lenses of various systems using appropriate adapters. The antialiasing filter was removed. The apparatus is displayed below.
BenQ_GH20X
Unfortunately, the camera neither supports RAWs nor USM control.
A shots were taken at ISO 64.
Below are the results of measurements using the CZ Planar 50/1.4 ZF lens. The resolution inside of the circles is scaled to 200%
CZ Planar 50/1.4 ZF
As seen in F4, the lens is capable of resolving 225 lp/mm. This corresponds to a final resolution of 175 megapixels. Also note that the curves are distorted in a typically USM manner. Below you can see two diagrams: the full target pixel profile with MTF and a section thereof limited to the boundary resolution area.
The CZ Distagon 25/2.8 ZF lens reached a slightly lower resolution of 200 lp/mm, which corresponds to 138 megapixels.
CZ Distagon 25/2.8 ZF
The Leica Summicron-M 35/2 lens also reached only 165 lp/mm or 94 megapixels. The measurements performed with Sony A7r furthermore produce a lower curve with the aforementioned lens when compared to CZ Planar 50/1.4 ZF.
Leica Summicron-M 35/2
Up until this point, only prime lenses were considered. However, a large number of photographers prefer zoom lenses. What would be the situation in this field? The Nikkor 28-80/3.5-5.6 lens was tested. This lens with a plastic bayonet and the price tag of around $100 is included with the film camera Nikon F80.
Nikkor 28-80/3.5-5.6 with zoom positioned at 50mm
The lens is capable of resolving up to 150 lp/mm, which corresponds to final image of 78 megapixels.

Old lenses - “Made in USSR”

Perhaps the owners of old lenses do not need cameras with high pixel counts? Let us examine Industar 50/3.5—a very simple and inexpensive lens, going for 7 Rubles in 1980s, which corresponded roughly to 2 kilograms of butter. At its optimal aperture of F5.6, the lens is capable of resolving 195 lp/mm. This corresponds to 131 megapixels.
Industar 50/3.5
A different lens, dating to roughly the same time, is the ultra wide angle lens Mir-20M 20/3.5. It is capable of 135 lp/mm or 62 megapixels.
Mir-20M 20/3.5

Comparison with a pre-calculated curve

As with Sony A7r, let us now compare the acquired MTF with the pre-calculated curve. Unfortunately, USM severely skews the results, making any comparison very problematic. This applies in particular at lower frequencies. However, at higher frequencies we have a qualitative agreement between the experimental and the pre-calculated curves for pixels with the effective size of 1.8 * 1.4 μm = 2.5 μm.

Diffraction

Could diffraction then be considered the limiting factor of this camera? It could be and it is.
CZ Planar 50/1.4 ZF, the dotted lines represent diffraction limits for various apertures for the 500 nm wavelength
Regardless, only the F16 aperture diminishes the quality of the 440 megapixel camera to the level beneath that of the 36 megapixel camera Sony A7r.

Open apertures

As is to be expected, open apertures display a degradation of image quality due to aberration.
CZ Planar 50/1.4 ZF
However, the change is different than the one previously observed in the case of diffraction. While diffraction causes a reduction in resolution and maintains a high contrast ratio, aberration reduces the contrast ratio and maintains a reasonably high resolution. Examples of similar MTF behaviour can be found in [7] pp. 29, fig. 2.6

Moiré pattern

Would other lenses when combined with the Sony A7r produce moiré pattern? Considering the aforementioned boundary resolutions, we can safely conclude that it would.
Sony A7r, 200% magnification
CZ Planar 50/1.4 ZF, F5.6 CZ Distagon 25/2.8 ZF, F5.6 Leica Summicron-M 35/2, F5.6 Nikkor 28-80/3.5-5.6, ~50mm, F5.6 Nikkor 18-70/3.5-4.5 AF-S DX, ~50mm, F5.6 Industar-50 50/3.5, F5.6 Mir-20M 20/3.5, F5.6 Uniter-9 85/2, F5.6
Sony A7r with open apertures, 200% magnification
CZ Planar 50/1.4 ZF, F2.8 CZ Planar 50/1.4 ZF, F2 CZ Planar 50/1.4 ZF, F1.4 Leica Summicron-M 35/2, F2

Reality

Finally, can any of this be seen on a real photograph or is this something only visible on specially-made targets? Yes, of course, you can. Photographs made with the CZ Planar 50/1.4 ZF lens taken from the same location at a distance of roughly 2 metres.
Sony A7r, 340%. Hover your cursor over the image to see BenQ_GH20X at 100%
It isn’t difficult to spot that the image taken with the 400 megapixel camera is not only more detailed, but it also doesn’t have any digital artefacts, such as the moiré pattern on the reflective surface of the watch.

Conclusion

The increase in the megapixel count is a completely natural process in the development of digital technologies. It allows for the removal of the artificially-included “analogue” elements, e.g. AA filter, which are always a bit of a compromise. This increase is beneficial to the sensors both with an array of colour filters and those of the Foveon type, with their inherent aliasing [8]. The state-of-the-art megapixel technology today still cannot bottleneck the now 30-year-old, entry-level optics. Unsurprisingly, all of the lenses described in this article produce a colourful moiré pattern when combined with the Sony A7r camera at the frequencies of around 100 lp/mm. At this point, it is important to note that the final image, resulting from the image of a higher resolution, will look better than an image, whose original resolution corresponds to that of the final product. Thank you for reading. If you liked this article and would like to support this project, consider donating. :-)

References:
  1. Lens testing by Norman Koren
  2. How to Read MTF Curves by H. H. Nasse
  3. Detector Footprint Modulation Transfer Function
  4. "Modeling and Measurement of Image Sensor Characteristics" Karel FLIEGEL
  5. Section 4 Digital Imaging Systems Sampled Imaging Systems Pixelated Imaging Systems
  6. Fast MTF measurement of CMOS imagers using ISO 12233 slantededge methodology
  7. IMAGE SENSORS and SIGNAL PROCESSING for DIGITAL STILL CAMERAS
  8. Foveon inside

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