Would a 50mm lens on a Canon APS-C crop produce the exact same image as an 80mm lens on a full frame camera?

Question

Scenario:

• I put a 80mm lens on a FF camera and take a picture
• I put a 50mm lens on a Canon APS-C crop camera (1.6x crop) and take a picture

Going by the 1.6x crop, I understand the fields of view of these two photos would be the same, due to the effective focal length of the 50mm lens on a crop body being 80mm.

Would these two photos be identical (i.e. depth of field, perspective etc.)?

Answer 1

Would these two photos be identical (i.e. depth of field, perspective etc.)?

• Depth of field: not exactly, but very similar if you also multiply the f-number used by the same crop factor. If you use f/2.8 with your 50mm lens on an APS-C camera, you would use f/4.48 on the 80mm lens with a FF camera to get the same overall DoF. You might have the same amount of DoF, assuming both images will be viewed at the same display size and from the same distance, but the distribution in front of and behind the actual focus distance would be slightly different. On the other hand, if you use f/1.4 with the 80mm lens on a FF camera, you would need a 50mm lens at f/0.875 on the 1.6X crop body to get the same DoF.
• Perspective: Yes, as long as the camera position is identical, the perspective is the same regardless of focal length, sensor size, or the resulting field of view. Perspective is determined by one thing only: The position of the optical center of the camera relative to the various objects in the scene.
• Geometric distortion: No. Some people confuse perspective and geometric distortion, but they are two entirely different things. The first is determined by the position of the camera relative to the scene. The latter is determined by the lens' projection of a three-dimensional world onto a two-dimensional recording medium. Even two different designs of 50mm lenses for the same sensor size can use different projection models and have different distortion characteristics.
• Background blur: The number and shape of aperture blades have a profound influence on the appearance of background (and foreground) blur. We call the aesthetic qualities of the background blur bokeh. It is influenced by the lens' optical design, including how much correction is done for several of the classic aberrations presented any time a lens with real thickness is used to refract light of more than one wavelength. It's also influenced by the shape and number of aperture blades that may be in the optical path. Two lenses used on the same camera with the same focal length and very similar optical formulae can render out of focus areas differently and their bokeh will look very different if their aperture irises are made differently.
• ISO, shutter time, etc.: No. If you change the aperture to get the same DoF, you need to change one of the other exposure parameters to maintain the same exposure. Either shutter time or ISO or some of both must be changed to compensate.

In the end, there's no such thing as exact equivalence when using different sized sensors. In addition to the points above, DoF equivalence breaks down at macro distances¹ and when computing hyperfocal distances (which are based on DoF and how much of the DoF is in front of and how much is behind the actual point of focus).

_{¹ As lenses are focused to very close "macro" distances in order to give magnification approaching 1:1, their effective focal lengths change enough that using the "stated" focal length of a lens non longer accurately computes the depth of field one can expect.}

Answer 2

Images from different lenses will look different because the lenses are different. However, depending on your tolerance for difference, you may be able to achieve results that are close enough.

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For the purpose of discussion, assume that the same lens can produce the same image, even though it may be physically impossible. For those who are concerned that images won't perfectly match, "identical images" will be defined as "images whose difference approaches zero as the difference in the conditions in which they were taken also approach zero."

This answer will also focus on recreating images on larger formats that were taken on smaller format to avoid issues with lenses that don't exist. However, going from larger to smaller wouldn't be an issue if someone makes the required lenses.

• Field of View – Same after multiplying focal length by crop factor.

• Perspective – Same when lens (entrance pupil) is in the same position.

• Depth of Field – Same as long as aperture also multiplied by the crop factor. Here's the formula for depth of field:

  DOF = 2 u² N C / f²

  N = aperture F-number
  C = circle of confusion
  u = distance to subject
  f = focal length

  Note that when focal length is multiplied by the crop factor, it is squared in the denominator. To keep DOF constant, the numerator needs to be multiplied by the square of the crop factor as well. The places to do that are in the aperture F-number and the size of the circle of confusion. Since we're going from crop sensor to full frame, we can multiply each by the crop factor to get the same DOF.

• Background Blur – Can also be kept the same when both focal length and aperture are multiplied by crop factor. Here's a formula for background blur:

  b = f m_s x_d / (N (s ± x_d))

  b = blur
  f = focal length
  N = aperture F-number
  m_s = subject magnification (what's this?)
  x_d = distance between subject and background
  s = subject distance 

However, the formulas ignore lens distortion, aberration, and other factors related to lens design. So while it's "possible" for different lenses to produce images that share the same parameters and appearance when switching between crop sensor and full frame, they usually won't look the same. Even lenses with the "same" design have enough small differences that they will look different.

Answer 3

Assuming the same number of pixels, they would be identical in shape and depth of field giving the same aperture (not aperture number but the full f:x.y ratio) and using an ISO value larger by a factor of 2.56 on the crop camera. When you use the same ISO and the same aperture number, the depth of field grows by the crop factor.

There is a reason people use larger sensors for more light and/or less depth of field.

Answer 4

Simple answer No. It could create a similar field of view for framing purposes given the same distance to subject. Also, not all glass is created equal, so one might have an edge over the other in clarity and sharpness. One advantage of the crop sensor is that it captures just the center of the lens if your using a full frame lens on a crop sensor body. In this case, it is possible that corner sharpness on the 50mm would be better than the 80mm on the full frame.

If everything was completely equal, then the crop sensor would be gathering less light in the exact same conditions so would have to push the ISO up or slowing the shutter (considering the lens was already fully opened).

Answer 5

The near-equivalence relations between FF and cropped camera are the following:

FocalLength_FF = FocalLength_crop * CropFactor

Fstop_FF = Fstop_crop * CropFactor

ISO_FF = ISO_crop * CropFactor²

where most people know the first equation, but many forget the second and last equations.

The first equation explains how focal length needs to be modified to take into account the crop factor to maintain the same field of view. This is what most people know.

Now, if you make the focal length smaller but maintain the same F-stop, it means the lens will collect less light, because the diameter of the aperture opening is:

ApertureOpening = FocalLength / Fstop

...and to keep the fraction the same, both the numerator and denominator need to be changed by multiplying by the crop factor. This ensures the aperture opening and thus the ability to collect light stays the same.

Now, exposure and ISO are defined in such a manner that exposure is:

Exposure = ISO * ExposureTime / Fstop²

Exposure time is obviously the same on FF and crop cameras if you want to take the equivalent picture. Now, as I explained that to maintain the light-collecting ability, you must multiply Fstop by CropFactor. To maintain the exposure, you must therefore multiple ISO by CropFactor². Is this a problem? No, because FF sensors are physically larger in terms of area by a factor of, you guessed it, CropFactor², so you can multiply ISO by CropFactor² without having any adverse noise effects, assuming the pixel size becomes larger, i.e. megapixel count is the same.

So, let's check:

1. Field of view: maintained
2. Light-collecting ability: maintained
3. Noise level: maintained
4. Exposure: maintained

Now, there are two other factors which might affect your choice of equipment. They are the depth of field (DOF) and background blur.

As @xiota explained, DOF formula is:

DOF = 2 SubjDistance² Fstop CoC / FocalLength²

Subject distance stays the same, Fstop is multiplied by CropFactor, CoC (circle of confusion) is multiplied also by crop factor because sensor dimensions are larger by a factor of CropFactor. The denominator is multiplied also by CropFactor², so depth of field (DOF) stays the same.

However, there's also another aspect, the background blur. My understanding is that background blur is:

Blur = FocalLength SubjMagnification BgDistance / (Fstop (SubjDistance ± BgDistance))

If SubjMagnification is unitless, the numerator has units of length squared. The denominator has units of length. So, blur has units of length.

Let's check what happens for an FF camera. FocalLength is multiplied by crop factor, but also Fstop is multiplied by crop factor too. Subject magnification is apparently size of sensor divided by size of subject. Size of subject stays the same, but size of sensor is smaller or larger by a factor of CropFactor. So, on FF, SubjMagnification is multiplied by CropFactor. So, Blur is multiplied by CropFactor. Thus, the blur disc size becomes larger, but so does the sensor size, so the blur disc occupies the same percentage of the sensor!

So, let's check the background characteristics:

5. Depth of field: maintained
6. Background blur: maintained

So, yes, the photos would be identical if you use an equivalent lens. However, do note that you can probably find a 80mm f/1.2 lens very easily (well, ok, it might be 85mm but close enough), but finding a 50mm f/0.75 lens might be a bit challenging. So, if you want to have lots of background blur, shallow depth of field and low noise, there's some benefit in using full frame: you probably can't find the lens you want for a crop camera!

If we go smaller still, and consider mobile phone sensors (crop factor of 7-8), you would need 10-11mm lens with F-stop of approximately f/0.15 - f/0.17. I'm sure you won't find such a lens!

Let's do a quick check on the validity of the near-equivalence relations. The Canon 17-55mm f/2.8 IS USM zoom weighs 645 grams. On full frame, it would be 27-88mm f/4.5. You can find 24-70mm f/4 IS USM lens that weighs 600 grams, and 24-105mm f/4 IS USM lens that weighs 669 grams. Filter thread size is 77mm on all lenses. So I guess they must be nearly equivalent, having approximately the same amount of glass.

However, the 24-70mm f/2.8 non-IS USM weight is 953 grams, so it clearly has more glass in it.

Also, consider for example Coolpix P1000. It is advertised as having 125x zoom, 4.3 - 539 mm, equivalent to 24-3000 mm. The F-stop is f/2.8 - f/8, but there is no "equivalent to" spec for F-stop, which the manufacturer conveniently forgot. Have you seen a 3000mm f/8 lens? I haven't, but it would be honking huge, being at least 3000mm/8 = 375mm in diameter. The manufacturer should have remembered to say that f/2.8 - f/8 is f/15.6 - f/44.5 equivalent. This demonstrates that people typically forget the equivalence relation for the F-stop, remembering just the relation for the focal length.