Sensor Size and Depth of Field

You see a lot of hand waving explanations of the effect of sensor size on depth of field.  I thought I'd do a little math on the subject.  Let's make three hypothetical pictures of a subject with three hypothetical cameras and keep the camera parameters as similar as we can.

The perspective of our picture depends on the distance from the camera to the subject.  The field of view of our picture depends on the focal length of the lens and the size of the image sensor.  A full frame camera (36x24mm sensor), an APS-C camera (24x16mm or 22.5x15mm sensor) and a Micro 4/3rds camera (17.3x13mm sensor) will have the same perspective of a subject 10' (3m) away.  A full frame camera with a 50mm lens, an APS-C camera with a 35mm lens, and a Micro 4/3rds camera with a 25mm lens have about the same field of view.

The maximum tolerable amount of blur on our final print usually is given as a circle about 0.25mm in diameter.  To make an 8x10 (20cm x 25cm) print the images produced by each camera have to be enlarged by different amounts, and the tolerable blur circle on the sensor depends on the amount of enlargement.  The image on a full frame digital camera will need to be enlarged 8 times, making the tolerable blur circle on the sensor 0.25 / 8 or about 0.031mm.  An image from a DX digital camera needs to be enlarged 12 times, making the tolerable blur circle 0.25 / 12 or about 0.021mm.  The Micro 4/3rds camera needs to be enlarged 16 times, making its tolerable circle 0.25 / 16 or about 0.016mm.

The hyperfocal distance is the distance a lens can be focused and have objects in sharp enough focus from infinity to 1/2 of the hyperfocal distance.  The hyperfocal distance depends on the focal length of the lens, the focal ration of the lens, and the tolerable blur circle (also called the circle of confusion).  A formula for calculating the hyperfocal distance of a lens is H = f^2/(N * c).  H is the hyperfocal distance, f is the focal length of the lens, N is the focal ratio of the lens, and c is the diameter of the circle of confusion.

The near and far limits of acceptable focus depend on the hyperfocal distance (H) and the distance the lens is focused (u).  The distance to the nearest point in focus is given by R = H * u / (H + u).  The distance to the farthest point in focus is given by S = H * u / (H - u).  The total depth of field is T = S - R.

Let's take pictures of a subject 3m (10 ft) away and calculate the depth of field for each camera.  We'll set the lenses at f/2.8.

Full frame digital:

H = 50^2 / (2.8 * 0.031) = 28,802mm or 28.8m

R = 28.8 * 3 /(28.8+3) = 2.7m

S = 28.8 * 3 /(28.8-3) = 3.3m

T = 3.3 - 2.7 = 0.6m

APS-C digital:

H = 35^2 / (2.8 * 0.021) = 20,833mm or 20.8m

R = 20.8 * 3 / (20.8 + 3) = 2.6m

S = 20.8 * 3 / (20.8 - 3) = 3.5m

T = 3.5 - 2.6 = 0.9m

Micro 4/3rds (17.3 x 13 mm) digital:

H = 25^2 / (2.8 * 0.016) = 13,951mm or 14.0m

R = 14.0 * 3 / (14.0 + 3) = 2.5m

S = 14.0 * 3 / (14.0 - 3) = 3.8m

T = 3.8 - 2.5 = 1.3m

The Micro 4/3rds camera has about twice the total depth of field as the full frame camera.  The APS-C camera has about 1-1/2 times the total depth of field as the full frame camera.  In general, to get the same depth of field as a full frame camera the aperture of the Micro 4/3rd camera needs to be twice that of the full frame camera (for example f/1.4 instead of f/2.8).