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Let's assume we have a linear coordinate system on a plane. If we make a 2d perspective projection of that plane from 3d, we get a "skewed" coordinate system.

For example, one axis would look like this in the original plane:

|.........|.........|.........|.........|.........|

but will look something like this in the projection:

|.|...|.....|..........|..........................|

I don't have any information at all about the focal length, the viewing angle, the distances involved, anything. The only information I have are the distances between the "markings" of the new coordinate system, and the knowledge that it was linear before the projection. this is why I can't find a way to use the well-known formulas, matrix notations, etc. of the perspective projection.

If I knew what form this function can take, I would be able to calculate its parameters from the points I have. For example, if it were quadratic, in the form of $ax^2 + bx + c$ I could calculate $a$, $b$ and $c$ by using the data points I have. I know it's not quadratic, so what can it be?

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Where does this problem come from? According to your description, is it from 3D vision field? If yes, there are a mature mathematical system on 3D vision and of course perspective projection. –  Shiyu Dec 1 '12 at 16:07

1 Answer 1

up vote 1 down vote accepted

A perspective projection of a plane is a special case of a projective transformation. So you can model it as a $3\times 3$ matrix operating on homogenous coordinates. Such a transformation is fully defined if you know the preimage and image coordinates for four points.

For points on a single line, like the ones in your illustration, three points are sufficient. The cross ratio is a particularly useful tool to reconstruct a scale on a projectively transformed line.

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