## Equivalence of sensor formats

Imagine taking a head-and-shoulders portrait in a studio using a full-frame DSLR and a 50mm lens. In the article on depth of field, we saw that the depth of field (DOF) is given by:

where N is f/number, c is acceptable circle of confusion, f is focal length and s is distance to subject.

What happens if we now switch to an APS-C camera and wish to frame the photo in the same way, i.e. using the same field of view and the same distance from the subject? The APS-C camera (Canon) has a smaller sensor - 22.4mm wide vs 36mm, giving a crop-factor of 1.6. So we need to use a 31mm lens.

Refer here for an explanation of magnification with cropped sensors.

At this stage, let’s consider the effect of the diameter of the lens aperture (d). We know that the f/number is the focal length divided by the aperture diameter, i.e. N = f / d. (But see footnote below.) So we can replace N in the formula for DOF as follows:

Which simplifies to:

Now, as we switch from the full-frame to the APS-C camera, the circle of confusion and the required focal length to maintain the same field of view both reduce by the crop factor (1.6). So the term ( c / f ) is unchanged. As we haven’t moved the camera, the distance (s) is unchanged and we see that the depth of field depends only on the diameter of the lens aperture (d). The bigger the diameter of lens, the smaller the depth of field.

So, when switching from one camera to another, if we maintain the same field of view and the same aperture diameter, we will also maintain the same depth of field. The 2 cases are optically “equivalent”.

The relationship between aperture diameter and depth of field is quite interesting. In effect, if we want a shallow depth of field to separate subject from background, we need a physically big lens, regardless of camera format. We need to think in terms of diameter rather than f/number.

As many people switch from full-frame DSLRs to mirrorless APS-C or Micro 4/3rds format, they generally benefit from an increase in depth of field. However, if we want to retain a shallow depth of field for artistic reasons, then we need a physically big lens, which partially defeats the point of using a small format: small size, less weight.

Referring this back to f/number, the following table shows 3 lenses that would have the same diameter and are "equivalent" in angle of view and depth of field characteristics.

Micro 4/3 | APS-C Canon | Full frame |
---|---|---|

25 mm | 31 mm | 50 mm |

f/1.4 | f/1.8 | f/2.8 |

Notice that in this table I have started with a Micro 4/3rds "standard" 25mm f/1.4 lens and shown the APS-C and full-frame equivalents. However, if we started with a "standard" full-frame 50mm f/1.4, we could not find equivalent lenses of the same diameter at the smaller sensor sizes; there aren't any Micro 4/3rds 25mm f/0.7 lenses.

**Noise.** Taking this a step further, let’s look again at the table just above. Let’s say that the Micro 4/3rds is set at f/1.4, ISO 100 and some given shutter speed. To take exactly the same picture on the full-frame camera, we would keep the shutter speed the same but as the f/number would be f/2.8, we will need to use ISO 400. As the diameter of the lens aperture will be the same for both cameras and the shutter speed is the same, the same amount of light (photons) will enter the lens and fall onto the sensor. Referring to the section on sensor noise above, we know that shot noise depends on the number of photons falling on the sensor. So in our example, we can expect the effect of shot noise on the image to be the same for both the Micro 4/3rds and the full-frame sensor. The notion of equivalence can thus be extended to include image noise as well as depth of field.

This is a little surprising at first sight - as surely one of the advantages of a bigger sensor is less noise and better image quality. The point here is that if we now open the full-frame lens to f/1.4 to match the Micro 4/3rds lens, then the full-frame lens will let in much more light and the noise will correspondingly be less. However, when we stop the full-frame lens down to match the diameter and DOF of the Micro 4/3rds lens, we also limit the light entering the lens, increasing both ISO and shot noise.

**Footnote** - definition of f/number. Strictly, the f/number is the focal length divided by the lens entrance pupil diameter, N = f / d, where the entrance pupil is the image of the mechanical aperture as seen through the front of the lens. However, for simplicity we can loosely refer to lens aperture instead of lens entrance pupil diameter.