Cinema Shutter, Blur and the Wagon Wheel

What is modelled

A film or digital cinema camera does not watch the world continuously. It takes a fixed number of pictures a second, and for each picture the shutter stays open for only a fraction of the frame time, described by the shutter angle: a 180 degree shutter is open for half of the frame time. Things that move during that time are smeared into streaks. A wheel with identical spokes looks the same after it has turned by one spoke pitch, so when the spokes pass faster than half the frame rate the camera cannot tell how far the wheel has turned, and it shows the wheel turning slowly, stopped, or even backwards. This is aliasing, the same effect that makes a sampled sound wave change its pitch. A longer exposure averages the flicker of the spokes and lowers their contrast, with a factor that is the sinc function of the exposure time and the passing rate.

Equations

• Exposure time t = angle / (360 · fps); motion blur L = v t.
• Spoke passing rate f = n fw; apparent frequency f − k · fps folded into the range ± fps / 2.
• Contrast through the shutter |sin(π f t) / (π f t)|.
• The wheel seems to stand still when n fw = k · fps.

Checks run in the page

• The blur length from a numerical exposure against speed times exposure time.
• The apparent frequency from a numerical Fourier scan of the sampled frames against the folded frequency.
• The shutter average of a flickering signal against the sinc factor.
• The standing wheel and the signs of the apparent motion just below and above it.

Honest simplifications

An ideal rotating shutter that opens and closes instantly, a perfectly uniform rotation, identical spokes and no rolling shutter. Real digital sensors may read out line by line, and the light of the scene may itself flicker, so a real film shows further effects that are not modelled here. The effect on a screen also depends on the refresh rate of the display on which you watch this page.

• Press Wheel appears to stop, then Wheel creeps forwards, and compare with the first preset where the wheel turns backwards.
• Press 360 degree shutter, spokes fade and watch the spokes lose contrast.
• Raise the object speed and compare the blur for 90 and 360 degrees.