Monday, December 04, 2023

Photographic Equivalence

The problem of photographic (or cinematic) equivalence is a subject that rears its ugly head every so often. Many people simply misunderstand this concept. For my own benefit I am providing the following summary. Details can be found in the references, but the purpose of this article is to clearly state the objective facts. After this, people can argue as much as they like about personal preferences.

The sections will be numbered for ease of reference.

1. Equivalent images

We define equivalent images as those that look identical on two different camera systems. Hence the concept of equivalence only has meaning when we are comparing cameras with different sensors / film sizes / back planes / cartridges. When shooting on one camera system, without varying how much of the sensor is used, there is no need to consider equivalence. 

But there are cases where equivalence can be useful. First, if you are comparing camera systems prior to purchase, in order to determine which is more suitable for you. Second, if you have more than one system and likewise wish to choose one for the task at hand. Third, if you are required to "crop in" and use only a portion of the entire available sensor, for example to get a higher frame rate when shooting video. 

2. Key parameters

The key photographic/cinematographic parameters that we wish to balance are exposure, perspective, motion blur, angle of view, and depth of field. If we do this, we get images that look exactly the same, everything else being equal.

Everything else is never equal, at least not on different camera systems, but we can get close.

3. Image quality factors

Additional factors are not included in the definition of equivalence, since they are not (significantly) impacted by the sensor size. Detail, sharpness, contrast, vignetting, colour, bokeh, distortion, magnification across the field, coma, astigmatism, and other factors will vary with each lens. 

If we need to match two lenses on different systems, this is indeed a difficult task. But that's a different subject.

4. Crop factor

The ratio of two sensor sizes is commonly called the "crop factor". For some time 135 film has been held as the standard, termed "full frame". Of course there are sensors larger than this, but for convenience, 35mm is defined as 1.0 and other sensors are compared to it. 

Some people get quite annoyed by this, for reasons I fail to understand. It's a ratio. It's arbitrary. You are free to compare between the two systems you most care about. In fact, it's essential to the process.

Typically the value we use in the ratio is the diagonal length of the sensor. Way back in 2011 I argued instead for using the square root of the sensor area, since this takes into account different sensor aspect ratios. I rather doubt that the industry is going to change its mind, but here's that original table for reference. Note that the difference between diagonal and root area is usually quite small, so unless you need to be precise, it's a moot point which you use. 

   645  56.00 x 41.50      0.62   0.37       0.61
   H3D  48.00 x 36.00      0.72   0.50       0.71
  645D  44.00 x 33.00      0.79   0.60       0.77
  35mm  36.00 x 24.00      1.00   1.00       1.00
 APS-C  23.60 x 15.70      1.53   2.33       1.53
APS-CC  22.20 x 14.80      1.62   2.63       1.62
   MFT  17.30 x 13.00      2.00   3.84       1.96
  2/3"   8.80 x  6.60      3.93  14.88       3.86
1/1.63"  8.00 x  6.00      4.33  18.00       4.24
1/1.7"   7.60 x  5.70      4.55  19.94       4.47
1/2.3"   6.17 x  4.55      5.64  30.78       5.55
1/2.5"   5.76 x  4.29      6.02  34.97       5.91

Note that APS-CC is my abbreviation (not a standard) for Canon's implementation of APS-C, which differs from all other manufacturers.

Note also that particular cameras may have sensors that vary from these standards. To maintain precision -- I repeat -- calculate your own crop factor.

Now we can consider the key photography parameters in turn. 

5. Motion blur

For still photography, motion blur is controlled by our shutter speed. For video, motion blur is a combination of our frame rate and shutter speed/angle. Since shutter speed is not affected by sensor size, we hold this factor invariant to get equivalent images. (And naturally for video we also hold frame rate invariant.)

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. But compression is a combination of perspective and angle of view (see next section) and so need not be considered independently of those two factors.

7. Angle of view

AoV is the angular extent of the imaged scene. The term field of view is also used, but this is sometimes confused with 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 (numbered using subscripts 1 and 2) by equating the ratios within the function. Hence, s1 / s2 =  f1 / f2

But the ratio of sensor sizes is simply the crop factor, which I'll abbreviate CROP. If s1 / s2 = CROP then f1 = CROP * f2.

In other words, to find the focal length that results in an equivalent angle of view, multiply the existing focal length by the crop factor. A longer focal length is required on the larger sensor. 

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 they are out of focus.

Focus blur is objectively measured using the blur circle, specified by the ratio of aperture diameter (entrance pupil) to subject distance. Since we are maintaining the same distance to subject for equivalent perspective, the blur circle is proportional only to the aperture diameter.

But photographers rarely know the aperture diameter (a) of their lens. Instead we use the f-stop (p) as a relative quantity. The f-stop is calculated by dividing focal length by the aperture. So, p = f / a.

Rewriting this as a = f / p we can then ensure the same blur circle by equating apertures on the two systems. Hence f1 / p1 = f2 / p2. Or, using the definition of crop factor, p= CROP * p2.

In other words, to find the new f-stop that renders 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 but are two stops apart in aperture. To ensure the same exposure, we must increase the ISO of the larger sensor camera by that same two stops.

While this might add more noise to our image, as a rule larger sensors are better at handling noise. So in practice this is not significant. Besides, noise is an image quality factor, not a matter of image equivalence as such.

10. In summary: our recipe

To maintain image equivalence across two different sensor sizes (or crops):

  • 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, if that's more convenient).
  • 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 quite 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

The following are also equivalent, starting with a standard portrait lens on 35mm. Here we can see the difficulty in fulfilling the criteria on a smaller sensor. 

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.

  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

Image equivalence is not subjective, but a matter of optics. I trust that this article provides a clear and concise overview of a topic that is often over-complicated.

It's disappointing to still read posts (and view videos) where authors without understanding state that the concept is nonsense. These presentations get a lot of support from others who also don't understand. 

Equivalence is actually an intriguing and useful 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. 

13. References

Wikipedia articles are not necessarily written for clarity but do contain equations for crop factorcircle 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.


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