A very interesting video of Tony Northrup shows the problematic crop factor considerations when we compare lenses focal lengths, images angle of view, depth of field and noisiness. His main argument is that for an exact comparison of the behavior of a lens in different bodies, you have to multiply the crop factor, not only with the focal length as usual for a similar angle of view, but also you must do the same with the aperture, and the ISO settings (with the squared crop factor). Only with this abstract adjustment you get a more approximate image quality: angle of view, depth of field and noise with different sensors.

An important factor of his arguments is the context of the discussion. He is talking in the context to obtain a comparable full-frame image from a cropped sensor using a particular focal length. It’s trivial that any cropped image might be obtained from a full frame sensor. Then your image should be equally in all the previous measures. The problem is the other way. How you can obtain the same image that a full frame camera with a cropped body? Then, you have necessarily to deal with the focal lengths or distance to the subject to obtain an equal angle of view, bringing as a consequence the changes in the aperture settings and the ISO.

His detractors in comments and other videos are numerous, but I think that the concepts he describes have started to gain roots.

First, you have to start with the following obvious presumptions:

1. Physical distance (focal distance or focal length) from the middle of the lenses to the sensor (focal plane) doesn’t change with different camera sensor sizes. This is obvious, but paradoxically must be stated beforehand to avoid misunderstandings with the application of the crop factors.
2. Aperture settings of the lens doesn’t change either when changing camera sensors sizes. The real aperture of f/2 of a 50 mm lens, 25 mm, will be the same 25 mm in any crop factor application. This is already obvious.
3. Amount of light that enters the lens to the camera doesn’t change for the same aperture settings. That’s different of the sensor whole capability to gather the light.
4. We can obtain any cropped image from a full-frame image at the same subject, since the former is a fragment of the last one. The issue is in the other direction: how to get equivalent images to a full frame version from a cropped sensor body.

That is: the lens constant is present in all the measures when applying the crop factor to get similar or same image. The application of the crop factors is a help tool to compare equivalence at the image level (in all aspects), not the physics of the lens.

But then, what is to get the same image? There are various factors you should consider when pointing to same or equivalent images. In this case, we are concerned only with those that could be affected by the sensor size. Those are the angles (or the field) of view, the depth of field, and noise levels.

Angles of view

Almost since the birth of the digital era, there’s what camera and lens manufacturers call the «crop factor» to distinguish the different angle and field of view you get with the same focal length in a full frame sensor camera (factor of 1) and other sensors sizes. For example, the APS-C sensors have a factor of 1.5 for most of the camera manufacturers (except Canon, which have 1.6). There are hundreds of sources to get the crop factors you might check. The crop factor calculation indicates that to compare—and obtain— the same content, the same angles of view (vertical and horizontal), with the same lens and focal length, but different cameras—a full frame and an APS-C, for example—you should multiply the focal length of the lens for 1.5, or 1.6 for Canon. If your lens is of 50 mm in a full frame, you get the same angles of view of a lens 75 mm in a full frame, when you change the body to an APS-C camera sensor. The little sensor «sees» less content of the image the lens passes to the focal plane, «cropping» the image from the perspective of a full frame sensor, which is the base of all the measurements.

From the camera lens industry, this is the only side they specify when marketing their lenses. But, by this same reason, the implications are more than the angle of view changes. Let’s see all the panorama. For example, to get the same content of an image—the same field of view—from different sensors you have to zoom in or out, with the lens or with your feet. In the example above, you have to use a 75 mm focal length to get the same image from an APS-C camera in a full frame camera. Or vice versa, if you want to get the same content of a 50 mm focal length from a full frame, you need to get 50/1.5=33 mm lens (in the market is more probably to get a 35 mm fixed lens, or use a zoom lens) for an APS-C camera. You have to use different focal lengths, or move close, or far, from the subject to reach the same content (field of view) with the changed angle of view.

Depth of field and Noise

But what the video above shows—igniting the discussion—is that this only consideration is insufficient to get the same image overall. What about the depth of field and the image quality (noisiness)? By changing the focal length or the distance you impact the depth of field of the image. A shot with f/8 aperture (or any other) with a focal length of 75 mm in a full frame and a 50 mm in an APS-C cameras (and equally other exposure settings), wouldn’t achieve the same depth of field. By the effect of increasing focal length or the distance from the subject, the «cropped sensor» truly shows more depth of field than the full frame sensor. As the video clearly shows, you will get very approximate results only if you apply the same crop factor to the aperture setting also.

Those differences increases the value of full frames cameras because, to this day, it is impossible to get a cropped sensor camera that could deliver a nice blurred «bokeh» of a 1.4 or 1.2 lens in a full frame with the same angle of view. To achieve that you should have a lenses with an aperture less than 1. In those extreme situations the full frame has a great advantage.

But not all disfavor the little sensors. From a different point of view, you get a better depth of field when you use the same angle of view and the same aperture (other settings equal) with a cropped sensor. For some type of photography (landscape, street), this capability is of a great advantage.

But, again, the discussion became problematic. How about exposure settings? If I change aperture to get the same DoF, I have to change ISO or the shutter speed. Of course you have. Have you notice that, even when you get the same angle of view (to reach the same field) and the same aperture —apart of the different DoF—the noisiness of the crop factor image is more evident than the image of the full frame camera? ISO setting to 100, from different cameras, seems to deliver different noise levels. Is that true? Of course, that the sensors quality has a strong influence on image quality, if you compare big differences with very old sensors, or pixels quantities (a.k.a. pixel density). But let’s forget that for a minute. Any modern camera have a decent sensor quality within the same price range and total pixels. The importance of all it’s not the sensor manufacturer, but the light information that the sensors could gather to build an image. It is known that sensor size do have an impact in the noise levels of images at the same ISO, because the effective light collection is bigger in larger sensors. This is what Mr. Northrup calls the total collection or amount of light the sensor captures. Big sensors would obtain more light data or signals, than smaller sensors, decreasing the noise. His proposal is to compensate this lesser capability of getting light information from little sensors using the crop factor.

Physic specification or discussion apart, the value of the recommendation for photographers is that if we want to get an identical image, you should adjust the aperture settings AND your ISO level with the crop factor and the squared crop factor. In other videos Tony specifies his crop factor values for the ISO adjustments. For example, APS-C equivalency for ISO 400 should be 400/2.25=178 in a full frame camera.

With all of this, we could conclude the following:

1. Within the same angle of view, the same aperture settings deliver different DoF from different camera sensor sizes. Cameras with bigger sensors will be more shallow because the different focal length or distance you need to set for the same field.
2. Same ISO measure produce different noisy levels from different sensor sizes for the total difference in light gathering. Larger sensors are less noisier than smaller sensors.
3. To achieve same angle of view, DoF, and noise levels in a cropped sensor, you should make adjustment in the focal length, and as a consequence, the aperture, and the ISO, using the crop factors:
   • focal length * the crop factor
   • f-stop * the crop factor
   • ISO number * the crop factor square (1.5², for example)

To the standards that reign the gear specifications there are relevant objections. The more important to me are the value of focal length or distance for lens specifications and the ISO standard for sensitiveness of sensors. For the first specification, we should probably expect another complementary value that indicate the angle-of-view to set differences between focal lengths views and sensor sizes.

Also it seems that there is an opportunity to establish an objective image quality standards or conventions that include—apart from the lens or sensor specifications—the angle of view, the DoF, the noise level, and other features, for better image comparisons.

There are interesting resources below:

Crop Factor: Why you multiply the aperture by the crop factor when comparing lenses

Depth of field and crop factor misconceptions

Photography tips – Everything you need to know about sensor size