Equivalent Settings for APS-C and MFT - Explained

Makro on FF

Currently, cameras in various sensor sizes are in use, ranging from the tiny sensors of smartphones to the huge "medium format" sensors. In the old times, film sizes were even bigger than that. 

The author of this blog once used a 6x6 centimeter camera and film rolls with 12 shots. But the practical and smaller 36x24 millimeter format for film rolls with 36 shots was the gold standard for a long time. Everybody used it because the rolls were easy to get and to handle, and photographers could easily switch cameras and find the same handling. The larger formats were for studio work or photographers who needed even more image quality.

Makro on MFT

In this blog posting, I am going to explain the effects of the sensor size on the image, and how to handle it using a simple formula for equivalent settings. The main goal is to show that smaller sensors are capable of producing excellent results, and to point out the limits of those sensors.

You can achieve equivalent results on all reasonable sensors sizes throughout a wide range of scenes.
But there are limits.

Note, that we need to compare equal pixel counts. Different pixel counts can only be compared if the larger one is reduced to the lower one. Otherwise, you zoom in a lot more when viewing the image at 100%. The quality of the algorithm for the reduction matters, of course, in this case. We have now 50MP sensors on FF, while the limit on MFT is currently 24MP. Thus, everything said here applies mainly for comparable megapixel counts, and only with care to vastly different sensors.

The Form Factor ff

The 36mm times 24mm format was called "Kleinbild" in Germany, translating to "small format". Now, we call it "full frame", although "medium format" still exists, and that is larger. Smaller sensors are called "crop sensors" by most photographers with a negative touch. 

The purpose of this posting is to show you that you can get equivalent results on both by learning how to apply the "crop factor".

We call the 36x24mm "full format" FF (better: "film format").
Smaller sensors are characterized by a "crop factor" ff (better: "form factor").

• Currently, the most widespread smaller sensor in cameras is the APS-C format with ff=1.5 or ff=1.6 (Canon). Nikon calls it DX format in contrast to the FX format.
• An even smaller sensor is the MFT ("micro four thirds") with ff=2. It uses a 4:3 aspect ratio instead of the 3:2 ratio of the FF cameras, but the diagonal is roughly half the size of the FF sensor.
• Then there are many compact cameras around ff=2.7 (1'' sensor) and higher (2/3'' etc.). 
• Finally, the smartphone sensors range from ff=5 to ff=7.

You can find a list of form factors here.

The Equivalence Formula

To compare the results of these sensors in terms of 

• field of view (FOV), 
• depth of field (DOF), 
• background blurriness, 
• noise handling,
• diffraction, 

we can use a very simple formula.

To get the equivalent of a crop sensor on full frame,
multiply the focal length of the lens and the F-stop with the form factor ff.

The ISO number has also an equivalent setting, but it is a bit more complicated, and the effects are not as clearcut as with the focal length and the F-stop, because the pixel count is usually different. But let me state it nevertheless.

Multiply the ISO by the square of ff to get the same noise handling,
provided the pixel count is similar.

Before we explain the reasoning behind this formula and show images, let me show some examples.

• The angle of view (AOV) of a 35mm lens on an APS-C camera (ff=1.5) like the Fuji XT-4 is approximately equal to the AOV of a 50mm lens on an FF camera like the Nikon Z5. 50mm is a standard lens for film cameras with a natural AOV. The images with the equivalent settings will show the same view when taken from the same spot. 
• Likewise, the AOV of a 25mm MFT is equivalent to a 50mm FF. If the aperture of the MFT lens is set to f/2, the FF can be at f/4, and will yield almost the same areas of sharpness (DOF) and the same blurriness at infinity. In other words, the images will look the same.
• For another example, we use an MFT (ff=2) for bird photography with a 200mm lens at f/4, ISO400, 1/250. On the FF, we get an equivalent result using a 400mm lens at f/8, ISO1600, 1/250. Not only do we get the same FOV and DOF, we also get a similar noise handling, provided that the pixel count or the print size are similar. Note, that we could increase the ISO by two stops. This is the noise advantage of FF.

It is interesting to find setups for one camera which have no easy equivalent setup on the other camera.

• Very fast lenses, like the 50mm, f/1.2 on FF, are not easy to match on APS-C or MFT, especially not in the same quality. Isolation of subjects is easier on FF.
• Telephoto lenses are much more comfortable and cheaper on MFT. Any 200mm MFT looks tiny in comparison to a 400mm FF lens.

Of course, FF lenses are heavier, larger, and more expensive than APS-C lenses. This is the most important reason for the revival of APS-C or MFT. Moreover, the cameras can be made more compact.

A Practical Test

You don't have to believe in the equivalence formula blindly. You can test it yourself, even if you have only one camera.

On a Nikon Z camera, e.g., you can use a zoom lens and shoot a scene at 50mm focal length, f/6. Then, set the camera to DX mode, zoom out to 35mm to get the same image in DX mode, and use f/4. You will have to increase the ISO by one stop from ISO100 to ISO200 to get the same exposure time. Unfortunately, you will lose quite a lot of megapixels. You will therefore notice the degradation in details, but the noise should be the same on the same output size.

35mm, F5, DX-Mode

Above, you see the DX mode and below the FX mode, with equivalent settings. Everything looks almost identical. The exposure was slightly different, but I corrected that in Lightroom. Some details of the texture of the bowl are lost.

53mm, F8, FX-Mode

We are now going to explain the reasoning behind the equivalence formula.

Equivalence of Focal Lengths

This is the easiest equivalence to explain. The angle of view is determined by the ratio of the sensor diagonal and the focal length. That should be simple geometry, because the focal length is the distance of the sensor when the camera focusses to infinity. In fact, in modern lenses the focal length is determined by the angle of view more or less. Now, the factor ff applies to both, the focal length and the sensor diagonal, if we use the equivalent focal length. Thus, it cancels out.

By the way, to get the AOV in degrees, you need to apply the arcus tangent function. But here is a list of AOV in degrees.

You might have noticed that the ratio mentioned above only applies when the camera is focused to infinity. Older lenses did indeed produce different AOV for nearer objects. This is called "focus breathing". Modern lens designs avoid this effect.

Equivalence of F-Stops

This is harder to explain. I don't want to derive the math for an ideal lens here, as I did in this EMT notebook. It is just so, that ff cancels out correctly when going from a smaller sensor to FF. 

We explain this for an MFT camera to simplify the thinking, and compare a 25mm MFT at f/4 with a 50mm FF at f/8.

• The focal length L affects the circle of diffusion at infinity in squared form. Thus, we gain four times more blurriness at infinity on the FF.
• The aperture goes linear into the circle of diffusion. Thus, we lose half blurriness by taking F8.
• The MFT image is magnified twice for the same output size. Thus, the FF loses another half of the blurriness.

The end result is that the images are equivalent.

For the depth of field DOF, the equivalence is not 100% exact, but very close on similar images. In the macro range, it is a bit off. But DOF is very small in that case anyway.

Equivalence of Diffraction

Diffraction occurs, because the light rays are bent around the edges of the aperture. It needs to be viewed in proportion to the light going through the opening of the lens. This is solely determined by the diameter of the aperture. 

The F-stop A is the quotient of the focal length L and the diameter of the aperture D, A=L/D. We get for the diameter of the aperture D=L/A. If we use the equivalent focal length and the equivalent F-stop the form factor ff cancels out. Thus, diffraction is also taken care of by the equivalence formula.

Equivalence of Noise Handling

Noise is generated by electronic circuits and by the nature of light measurement itself. It cannot be avoided completely due to laws of physics. Modern cameras are very close to the physical limits.

For a signal (like the texture of a photographed object) to be useful, it must exceed noise by a certain amount. This is called the "signal to noise ratio". It can be stretched only by AI methods which reconstruct the signal using experience. But that has nothing to do with good old photography.

Now, our sensor consists of sensor pixels, each one producing noise. For a start, we assume that pixel count on our smaller sensor is the same. So, we see the same noise on our image as on the FF camera.

On the other hand, our signal is the light on the sensor, measured in light per area. This is solely determined by the F-stop on each and every camera. F-stops are defined as they are because of this independence. But the area of the smaller sensor is smaller by a factor of ff^2, and so is the signal falling on each sensor element.

Consequently, the signal to noise ratio is ff^2 times better on FF. This allows one stop higher ISO compared to APS-C, and two stops compared to MFT. I described the effect of ISO here.

But the better ISO handling does not help if we need to increase the F-stop to get the same DOF.

The increase in ISO just balances out the higher equivalent F-stop on the FF camera.

We have assumed that the pixel count is the same. If it is not, things are not that easy to grasp. Today, we see 24MP APS-C cameras and 48MP FF cameras. These cameras have approximately the same area per pixel, gathering half the light per pixel for equivalent F-stops on the FF camera. However, we use a software to average out pixels since we never look at 100% magnification on the output, but always compare the same output size. How this averaging is done determines the result.

Resolution

Here is one point where smaller sensors are in a disadvantage. 

To get the same resolution on an MFT systems as on a FF lens,
you need a lens that resolves twice as good. 

After all, the image needs to be magnified twice as much. This is very unlikely to happen on real lenses. Consequently, larger sensors have an advantage in terms of image clarity and resolution. In the end, this part of the equation depends on the lens, however. It may be cheaper to produce good small lenses than good large glass.

Summary

Smaller sensors can be practical. They allow smaller cameras and especially smaller lenses, resulting in cheaper, lighter and more compact equipment. For hikers and backpacks, this is a huge factor. In the telephoto range, it is a clear advantage. We also learned, that they can produce equivalent images in many everyday scenes.

Larger sensors are superior in low light and in wide lenses, because they allow better object isolation in that range with the available lenses. It should also be mentioned that larger lenses can be made to be optically superior. The Nikon Z mount proves that impressively. There are physical limits for making tools smaller.

Smartphone cameras are at the extreme end. Their thin housings do not allow for telephoto lenses on the main sensor. So, the main advantage of small sensors is gone. Object isolation is consequently generated by software with all shortcomings. Noise handling is done by AI methods, moving away from photography to AI generated images. It is a different world. If everything in the image is to be sharp, the difference may be marginal, however.

Low Light Advantage of FF