Given a 2d picture of a rectangle distorted by perspective:

![enter image description here][1]

I know that the shape was originally a rectangle, but I do not know its original size.

If I know the pixel coordinates of the corners in this picture, how can I calculate the original proportions, i.e. the quotient ( width / height ) of the rectangle?

(background: the goal is to automatically undistort photos of rectangular documents, edge detection will probably be done with hough transform)


UPDATE:
=======

There has been some discussion on whether it is possible at all to determine the width:height ratio with the information given. My naive thought was that it must be possible, since I can think of no way to project for example a 1:4 rectangle onto the quadrangle depicted above. The ratio appears clearly close to 1:1, so there should be a way to determine it mathematically. I have however no proof for this beyond my intuitive guess.

I have not yet fully understood the arguments presented below, but I think there must be some implicit assumption that we are missing here and that is interpreted differently.

However, after hours of searching, I have finally found some papers relevant to the problem.
I am struggling to understand the math used in there, so far without success. Particularly the first paper seems to discuss exactly what I wanted to do, unfortunately without code examples and very dense math.

* Zhengyou Zhang , Li-Wei He, &quot;Whiteboard scanning and image enhancement&quot;
&lt;http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf&gt; p.11
&gt; &quot;Because of the perspective distortion, the image of a rectangle appears to be a quadrangle. However, since we know that it is a rectangle in space, we are able to estimate both the camera’s focal length and the rectangle’s aspect ratio.&quot;


* ROBERT M. HARALICK &quot;Determining camera parameters from the perspective projection of a rectangle&quot;
&lt;http://portal.acm.org/citation.cfm?id=87146&gt;
&gt; &quot;we show how to use the 2D perspective projection of a rectangle of unknown size and position in 3D space to determine the camera look angle parameters relative to the plans of the rectangle.&quot;


  [1]: http://i.stack.imgur.com/rI36D.png

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proportions of a perspective-deformed rectangle