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Airplane propellers imaged with a rolling shutter

Posted 2 years ago

enter image description here

Most cell phone cameras capture an image using a rolling shutter. This means an image is not captured all at once but rather is captured with a scrolling line, either horizontally or vertically.

This scrolling is fast and in most situations is not noticeable. But for a scene with motion rivaling the camera's scroll speed, things will look distorted. A fast moving car might look sheared, squished, or stretched, and a spinning propeller looks quite strange.

To see how the strange pattern in the above image appears, let's start off by idealizing a propeller as a trifolium in polar coordinates:

propeller[θ_] := Sin[3θ + π/2];

PolarPlot[propeller[θ], {θ, 0, 2π}]

Now as the scrolling happens, each scroll line will be looking at a different rotated version of the propeller. In polar coordinates that's just a shift in angle, e.g. propellar[θ + shift]. If we assume the camera scrolls in the x (horizontal) direction at a constant speed, then the shift value will be a constant times x.

For this example we'll assume the propeller makes 2.25 revolutions in the time the camera completes the scrolling. Now that we've mixed polar and Cartesian, we'll switch to RegionPlot:

revs = 2.25;

plot = With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]},
    r <= propeller[θ - π*revs*x],
    {x, -1, 1}, {y, -1, 1},
    Frame -> None,
    PlotPoints -> 100

We can get an intuitive feel for why this happens by simulating the rolling shutter ourselves:

    Graphics[{{EdgeForm[], White, HalfSpace[{-1, 0}, {t, 0}]}, 
      {Thick, Black, InfiniteLine[{{t, 0}, {t, 1}}]}}],
      propeller[θ - π*revs*t], 
      {θ, 0, π}, 
      RegionFunction -> Function[{x, y, r, θ}, x > t], 
      PlotStyle -> Red
  {t, -1, 1}

enter image description here

Lastly, we can simulate what a video of a spinning propeller would look like. Essentially each frame of the video would contribute a constant shift to θ (unlike the how the scrolling's shift depends on x).

  plots = With[{r = Sqrt[x^2 + y^2], θ = ArcTan[x, y]},
        r <= propeller[θ - π*revs* x + α],
        {x, -1, 1}, {y, -1, 1},
        Frame -> None
        PlotPoints -> 100
      {α, 0, 2π, π/20}

Omitting the code for styling and coloring each propeller differently, here's the result: enter image description here

POSTED BY: Greg Hurst
5 Replies

We now have a collection of creative post about “Rolling Shutter”. Here is the full current list for them:

Rolling Shutter III

Imaging a rotating disk with a rolling shutter

Modeling the Rolling Shutter Effect

POSTED BY: Moderation Team

This been quite an excitement! I see we have a following :-) Two new followups inspired by this post:

enter image description here

POSTED BY: Vitaliy Kaurov

This is awesome! Thanks a lot for posting.

For consideration: ToPolarCoordinates.

Dude - this is beautiful. Going to use it to demo rolling shutter artifacts in class.

POSTED BY: Flip Phillips

enter image description here -- you have earned Featured Contributor Badge enter image description here Your exceptional post has been selected for our editorial column Staff Picks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!

POSTED BY: Moderation Team
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