The subject of image equivalence is often mis-represented as overly complex or (worse yet) a matter of opinion. This article will counter incorrect statements you might read elsewhere, while simplifying the subject for concision. References are provided for those who wish to delve further.
1. When is equivalence useful?
Equivalence is important when comparing different sensors / film sizes / back planes / cartridges. There are three common cases:
- When comparing systems prior to purchase, in order to determine which best suits your needs.
- When you own more than one camera system and wish to make a choice for the task at hand.
- If you are cropping in to use a smaller portion of the available sensor, for example to get a higher frame rate when shooting video.
2. Defining equivalent images
We define equivalent images as those that look identical. The key photographic/cinematographic parameters to we must equate are motion blur, perspective, angle of view, depth of field, and exposure.
Differences of sensor technology and other factors will introduce further distinctions between systems. Equivalence does not claim to equalise all such factors; that is not the aim.
3. Other optical factors
Additional optical factors are critical to how an image looks but aren't included in the definition of equivalence, since they are not (significantly) impacted by the sensor size. These include the amount of detail, sharpness, contrast, vignetting, colour, bokeh, field distortion, image noise, coma, astigmatism, and other aberrations.
Matching lenses between different systems is a difficult, if not impossible, task. Again, equivalence does not claim to accomplish this.
4. Crop factor
The ratio of the diagonal length of two sensor sizes is commonly called the "crop factor" (abbreviated CROP below). For some time 135 film (so-called "full frame") has been held as the standard, defined as the ratio of 1. Some people get quite annoyed by this arbitrary decision, but in fact we are free to compare any two systems and calculate the appropriate ratio.
Way back in 2011 I argued that we should instead use the square root of the sensor area, since this takes into account different aspect ratios. Though I rather doubt that the industry is going to change its mind!
FORMAT SENSOR SIZE CROP 645 56.00 x 41.50 0.62 H3D 48.00 x 36.00 0.72 645D 44.00 x 33.00 0.79 35mm 36.00 x 24.00 1.00 APS-C 23.60 x 15.70 1.53 APS-CC 22.20 x 14.80 1.62 MFT 17.30 x 13.00 2.00 2/3" 8.80 x 6.60 3.93 1/1.63" 8.00 x 6.00 4.33 1/1.7" 7.60 x 5.70 4.55 1/2.3" 6.17 x 4.55 5.64 1/2.5" 5.76 x 4.29 6.02
APS-CC is my abbreviation (not a standard) for Canon's implementation of APS-C, which differs from all other manufacturers.
Now we can consider the key photography parameters in turn.
5. Motion blur
For still photography, motion blur is controlled by our shutter speed. Since shutter speed is not affected by sensor size, we hold this factor invariant to get equivalent images.
(For video, motion blur is a combination of our frame rate and shutter speed/angle, so we must also hold frame rate invariant for equivalence.)
6. Perspective
Perspective is based on the position of the camera relative to subject and nothing else. Perspective is held equivalent by not changing our camera position.
The term compression is sometimes used when describing perspective affects. Since compression is a combination of perspective and angle of view (see next section), it need not be considered independently of those two factors. The term is redundant.
7. Angle of view
The angle of view (AoV) is the angular extent of the imaged scene. The term "field of view" is sometimes used as a synonym, but at other times as a distance measure, so I'll stick with the more correct term. AoV is based on the ratio of focal length (f) to diagonal sensor size (s).
The actual equation is AoV = 2 * arctan(s / 2f). We can match AoV for different systems (indicated using subscripts) by equating the ratios within the function. Hence, s1 / s2 = f1 / f2.
From section 4 the ratio of sensor sizes is the crop factor, so s1 / s2 = CROP. Substituting, we get f1 = CROP * f2.
Conclusion: To find the focal length that results in an equivalent angle of view, multiply the existing focal length by the crop factor.
8. Depth of field
DoF is a subjective description of how much of a subject is in focus. In truth, a given lens can only perfectly focus the subject at one precise distance. At every other distance, objects are out of focus. What matters is how much objects are out of focus. This is measured using the blur circle, specified by the ratio of aperture diameter (the entrance pupil) to subject distance. But since we are maintaining the same distance to subject (for equivalent perspective) the blur circle is proportional only to the aperture diameter.
Photographers rarely know the aperture diameter (a) of their lens. Instead, this quantity is specified by the f-stop (p), calculated by dividing focal length by the aperture (p = f / a).
Rewriting this as a = f / p we can ensure the same blur circle by equating apertures on the two systems. Hence f1 / p1 = f2 / p2. Substituting in the equation from the previous section gives us p1 = CROP * p2.
Conclusion: To find the f-stop that results in the equivalent depth of field, multiply the existing f-stop by the crop factor.
9. Exposure
At this point our two images are shot using the same shutter speed (section 5) but different apertures (section 8). To ensure identical exposure, we must increase the ISO of the larger sensor camera by that same crop factor.
While this might add more noise to our image, it is generally true that larger sensors are better at handling noise. So in practice this difference is not significant. In any case, noise is an image quality factor, not a matter of image equivalence as such.
10. In summary: our recipe
To maintain image equivalence between two sensors:
- Multiply focal length by the crop factor.
- Multiply f-stop by the crop factor. (Or add the crop factor in stops to the f-stop value, since this is the same operation).
- Add the crop factor in stops to ISO.
- Maintain the same perspective (distance to subject).
- Maintain the same shutter speed/angle.
11. Examples
Two examples will make these relationships plain. The following will produce equivalent images, all else being invariant. One can readily find lenses that will allow this equivalence.
MFT: 25mm f/1.4 ISO 200 APS-C: 33mm f/1.8 ISO 320 FF: 50mm f/2.8 ISO 800
What about a standard portrait lens? Fulfilling this criteria with a smaller sensor is quite difficult.
MFT: 42mm f/0.9 ISO 400 APS-C: 56mm f/1.2 ISO 700 FF: 85mm f/1.8 ISO 1600
The following table presents equivalent focal lengths for three systems, including these common focal lengths for each: 18, 24, 28, 35, 50, 85. This can be handy for quick look-ups.
MFT APS-C FF 9 12 18 12 16 24 14 18 28 17 23 35 18 24 36 21 28 42 24 32 48 25 33 50 26 35 53 28 38 56 35 46 70 37 50 75 42 56 85 45 60 90 50 66 100 64 85 128 85 113 170
12. Finally
Equivalence is an intriguing concept that can shed light on how different photographic properties interact. It's essential knowledge in several recurring situations where you need to match stills or video footage shot on different systems. I trust that this article has provided a clear and concise overview of a topic that is often over-complicated.
13. References
Wikipedia articles are not necessarily written for clarity but do contain equations for crop factor, circle of confusion, and image sensor format.
Steve Yedlin, ASC has some clearly articulated reasoning similar to my own.
Joseph James has rather over-long and detailed coverage.
Tony & Chelsea Northrup provide a simple video demonstration, though with some imprecise use of terminology.
Updates
6 July 2024: Significant edits for clarity
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