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Computer Vision Geometry - Collinear Points In A Pinhole Camera Model

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that a line in 3D space is imaged to a line on the image plane in a pinhole camera model.

    2. Relevant equations
    A 3D point give by (X,Y,Z) will be imaged on the image plane at
    x = f([itex]\frac{X}{Z}[/itex])
    y = f([itex]\frac{Y}{Z}[/itex])
    where f is the focal point.

    3. The attempt at a solution
    My first thought was a more intuitive one. If you have a line in 3D space and the point which is the pinhole, you have a plane on which both the pinhole and line lie. Where this plane intersects the image plane it forms a line and this is where the 3D line is mapped to on the image plane. However, this seems like too much hand waving.

    Instead I decided to try saying that for the line in 3D there must be a parametric equation given by
    X = at + d
    Y = bt + e
    Z = ct + g
    x = f[itex]\frac{at+i}{ct+k}[/itex]
    y = f[itex]\frac{bt+j}{ct+k}[/itex]
    From here I know that for a line to exist on the image plane there must be a q and m such that
    y = mx + q
    Yet, this doesn't seem to lead me in the right direction.

    Any suggestions? Thank you for your time!
  2. jcsd
  3. Feb 20, 2012 #2
    imaging a plane containing the line and the pinhole; this plane intersects with the image plane, the intersection
    is obviously a line, isn't it just the image of the original line? I apologize for giving the answer directly, I simply can't think of any more disguised form ...
  4. Feb 20, 2012 #3
    Like I said, that was my first thought, but it seemed to hand wavy. Maybe I'm just worrying to much and that's a fine answer.

    Thank you.
  5. Feb 20, 2012 #4
    You're welcome, it's good to worry a bit more than others, as long as you're not obsessive :)
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