How does background blur (bokeh) relate to sensor size?

Question

This is a somewhat theoretical question.

Suppose I first take a photo of a subject using a full frame sensor DSLR, with a given lens (say 50mm prime lens at f/3.5).

Now suppose I exchange the camera for an APS-C sensor DSLR (with a 1.6× crop factor). I keep the same lens (same focal length, same aperture) and I step back a few meters in order to maintain the field of view (at least keep the same magnification of the subject). I now take a second photo.

Clearly depth of field will have increased between the two photos. But what about background blur (for example, trees at infinity)? Will I have the same amount of background blur, or will that have changed?

I have read somewhere that background blur depends on the physical aperture size. In this case the physical aperture (physical focal length divided by f/stop) stays the same. But should this number be taken in relation to the sensor size? In which case with the smaller APS-C sensor, the physical aperture will be relatively bigger, which would mean more background blur. This would be rather counter-intuitive as we usually consider it to be more difficult to get background blur on an APS-C camera.

Please give the reasoning behind the answer. I would answer the question myself using this background blur calculator but I can't manage to get it to run on my computer.

Answer 1

A lot has been said in the previous answers, and I just want to add a visual comparison of the specific lens settings you talk about in your question. Like said before, the amount of background blur is also dependent on the subject size. This plot is for a head and shoulders portrait.

As can be seen the FF camera will have more background blur, however their values converge in the end.

Image source: http://howmuchblur.com/#compare-1x-50mm-f3.5-and-1.6x-50mm-f3.5-on-a-0.9m-wide-subject

Here you can also set another subject size if you want.

Answer 2

Yes, bokeh is actually proportional to the physical width of the lens opening.

Say you focus on a near-field object at a finite distance = Z and have a camera/lens combo that gives you a field of view (FOV) with angular half-width = Q degrees. If you define bokeh as the ratio of the diameter of the blur circle B (blurred image of a background point at infinity) to the width of the image frame W, then

                     bokeh   =   B / W    ~    R / ( Z  * tanQ )

where R is the radius of the lens opening - ie half the diameter (Note: In the above equation, Z should technically be Z - F, where F is the lens focal length, but you can usually ignore the F when looking at a far-away object).

So if you have two cameras, a large DSLR and a small point-and-shoot, both with the same angular FOV (ie, lenses are same 35mm-equivalent), then the camera with the larger diameter lens will give you more bokeh. This is independent of the camera sensor size.

Answer 3

Depth of field depends on two factors: distance to subject and physical aperture size (calculated by the focal length divided by the f-number). Depth of field increases as you move farther from your subject and decreases as you increase the physical aperture size. The sensor size does not directly affect bokeh as the image projected by the lens does not change when it is used on a different sensor format; different sensor formats simply use different portions of the image circle. Larger sensors enable shallower depth of field because a longer focal length is required to achieve the same field of view, and a longer focal length results in a larger physical aperture and therefore shallower depth of field.

As such, the same lens at the same f-stop at the same focusing distance on two different sensor formats will not affect the degree of background blur. It is the adjustments necessitated by the different sensor formats (decreased distance to subject or increased focal length on full frame relative to APS-C) that produces the difference in depth of field.

Answer 4

In theory, you will have exactly the same background blur in both cases. In practice, this only works if the background is very far (a lot further than your subject) as pointed out by Jerry Coffin. If this condition is not met, then the APS-C body will give you slightly less background blur.

The reasoning... I'm to lazy to make a drawing now, try to make it yourself... The key idea is to model a background light as a point source at infinity, which will be rendered as a “bokeh disc” on the image. The level of background blur can be measured by the ratio of the diameter of this disc to the total frame size. This ratio happens to be the same as the ratio between the diameter of the entrance pupil and the size of the field of view at the distance the lens is focused at (here a small geometrical drawing would be useful...).

Edit: Here is my crappy schematic. Hope this makes things clearer. Consider that the image you get is just a scaled down version of what you have in the plane of focus. The beam in red is the beam of light coming from the point source and going through the entrance pupil. The stuff I labeled “bokeh disc” is where this beam intersects the plane of focus. It has exactly the same diameter as the entrance pupil, provided the source is far enough, and it is the object-side counterpart of the bokeh disc. The actual bokeh disc lives in image space, and is the image of the disc drawn here.

Answer 5

Background blur depends on your depth of field. Depth of field (DOF) is the distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image (wikipedia). Thin depth of field allows you to isolate your subject: the subject is in focus, and the background is blurred. Depth of field depends on several factors:

1. Lens focal length (35mm, 200mm, 50mm)
2. Lens Aperture (f1.8, f5.6, f8)
3. Sensor size (APS-C, 35mm, medium format, large format)
4. Subject distance

With (1), the longer the focal length, the thinner the DOF. With (2), the larger the aperture (smaller number) the thinner the DOF With (3), the larger the sensor, the thinner the DOF.* With (4), the closer the subject, the thinner the DOF.

Example: If you have a 200mm lens, at say, f2.8, on a 35mm full frame sensor, and the subject is near you (2-3m), you can blur the background quite a lot.
Inversely, if you have a 35mm lens, at f8, on a cropped dslr (APS-C), and the subject is 6m from you, the background won't really be blurred out.

* I am not absolutely sure if this is correct in theory, but in practice, with the same setup on APS-C sensor and Full Frame, the FF picture has shallower DOF.

Read more: http://en.wikipedia.org/wiki/Depth_of_field

Answer 6

It depends on how large your "infinite" distance really is. When you move back from the subject to maintain the same magnification, the relative distance to a background object becomes smaller, so it'll be less blurry.

Just for example, let's assume you start with the subject 10 feet away, and the "infinite" distance background is really 100 feet away. When you switch to the APS-C camera, you back up to 15 or 16 feet (depending on camera brand). In the first case, the trees were 10 times farther away than the subject (and focus point). In the second case, the subject is at 15 feet and the background at 115, so the background is less than 8 times as far away as the focus point.

If your "infinite" distance really is a lot larger, this effect can become too small to care much about though. If you start out with a background 10000 times farther away than the subject, then move so it's only 9999 times farther away, the difference will probably be so small you won't be able to see or even measure it.