Why using 50mm lens on FF gives more DOF than using 50mm lens on APS-C?

Question

Provided both lenses are 50mm focal length never change why using 50mm lens on FF gives more DOF than when using same lens on APS-C.

• 50mm lens on FF f/2.0 focus distance 2m gives DOF = 18cm
• 50mm lens on APS-C f/2.0 focus distance 2m gives DOF = 12cm

If you want to answer like "50mm lens on APS-C is same as using 75mm lens on FF" then can you also explain why

• 75mm lens on FF f/2.0 focus distance 2m gives DOF = 8cm (and not 12cm)

I know about crop factor and that angle of view is different on different format cameras, my question is about DOF.

Answer 1

It is due to magnification/enlargement... a smaller sensor/negative requires more magnification/enlargement in order to create the same sized output image; so a point of blur will be enlarged more and therefore be more apparent (less DoF). I.e. a 36mm wide FF sensor has to enlarged ~7x to make a 10" wide print, whereas a 24mm wide APS sensor has to be enlarged ~ 10.6x; the size of the blur point starts out the same on both, but ends up larger with the greater APS enlargement.

The minimum blur radius is dictated by the lens' aperture diameter/setting (diffraction), but the blur radius may also be made larger by optical aberrations (stopping down for sharpness is a tradeoff between eliminating optical errors and increasing diffraction).

Similarly a longer focal length results in more magnification of the blur radius, thereby making it more apparent by recording it larger on the sensor. This is why when using a longer FL on the FF sensor, in order to record the same composition, the FF ends up with less DoF. And, in fact, viewing an image larger/closer is also more magnification; thereby making any blur more apparent and reducing the DoF (DoF is not a fixed characteristic of an image; it is based upon the viewer's perception of "acceptable sharpness").

Answer 2

It has to do with the circle of confusion, that is, a spot of light that looks acceptably sharp. Smaller sensors have a correspondingly smaller circle of confusion for an acceptably sharp image which is why the DOF is different between full-frame and APS-C sized sensors.

DOF ≈ 2u²Nc/f²

Where u is the distance, N is the f-number, c is the circle of confusion, and f is the focal length of the lens.

The circle of confusion given in the wikipedia article for a full frame 35mm sensor is 0.029mm, 0.018mm for a APS-C sensor.

Answer 3

Introduction

The depth of field on a digital camera is a delicate subject because there are two different approaches.

The first, which has no equivalent in the field of analog photography, corresponds to the best possible image taking into account the characteristics of the sensor, i.e. the optical image on a sensor of a point in the scene must be smaller than the size of a pixel on this sensor.

The second, which is the one that will be discussed next, s related to the perception of a person looking at the image, the observer, looking at the image on a print, or possibly a screen assuming that the definition of this screen is such that the observer does not perceive the pixels of that screen.

Magnification / circle of confusion

On the one hand, an observer has a limited visual acuity, in the sense of his capacity to distinguish two close points. On the other hand, the prints are generally viewed at a distance close to the length of their diagonal. This makes it possible to define a value for the size of the luminous point on the sensitive surface which can be the sensor of a digital camera or a film for an analog camera. This value is the circle of confusion. However, this value is necessarily a function of the size of the sensitive surface, when the size of the sensor decreases, the size of the circle of confusion must decrease because to obtain a similar image, the rate of magnification must be increased.

It is generally accepted that a size of the circle of confusion corresponding to d/1500 (with d being the diagonal of the sensitive surface expressed in millimeters) seems to be admissible, but one finds other rules for example d/1750. Thus it gives for the size of the circleof confusion, approximately, 0.03 mm for a 24 * 36 mm camera and 0.02 mm for an APS-C camera

Depth of field

Now back to the original question.

From the above, we can directly conclude that if for a 50mm lens at f/2.0 and a focus distance of 2m the depth of field (DOF) is equal to 18cm for a 24 x 36mm camera then necessarily it will be 12cm for an APS-C camera given the ratio between the sizes of the circles of confusion.

Answer 4

Why using 50mm lens on FF gives more DOF than using 50mm lens on APS-C?

As others have already answered in greater depth, because the magnification ratio from the two sensors is different if we want to view both images at the same display size. Some blur that is just barely small enough to still look like sharp points to our eyes when magnified less can be seen as blurry when magnified a bit more.

Whether implicitly or explicitly, all depth of field (DoF) calculators assume a specific display size and viewing distance. They even assume a specific acuity of the viewer's vision! If they don't say, most DoF calculators assume a "standard" viewing condition of an 8x10 or 8x12 inch print viewed from 10-12 inches by a person with 20/20 vision.

Some DoF calculators will let you enter a different display size, viewing distance, and assumed viewer vision. This one from Cambridge in Colour offers such a feature if you click the yellow show advanced text in the upper right corner fo the calculator.

If you want to answer like "50mm lens on APS-C is same as using 75mm lens on FF" then can you also explain why 75mm lens on FF f/2.0 focus distance 2m gives DOF = 8cm (and not 12cm)?

The main reason your 75mm lens @ f/2 on a FF camera example breaks down compared to a 50mm lens @ f/2 when both are shot from a distance of one meter is because you need to multiply the f-number by the crop factor as well, since the focal length is longer but the entrance pupil needs to be the same diameter for a meaningful comparison.

That is, if you use a 50mm lens on a 1.5X crop APS-C camera at f/2, the valid comparison is to a 75mm lens on 1.0X FF camera at f/3. Both a 50mm lens at f/2 and a 75mm lens at f/3 have the same entrance pupil diameter: 25mm.

Since most DoF calculators don't have a choice of f/3, we use the nearest f-number available: f/2.8. At f/2.8 on a FF camera a 75mm lens focused at 2m gives a DoF of... 0.12 meters, or 12cm just as a 50mm lens at f/2 yields on an APS-C 1.5X camera.