StackOverflow2013

Note that there are some explanatory texts on larger screens.

plurals

POproportions of a perspective-deformed rectangle
text
Body
copied!<p>Given a 2d picture of a rectangle distorted by perspective:</p> <p><img src="https://i.stack.imgur.com/rI36D.png" alt="enter image description here"></p> <p>I know that the shape was originally a rectangle, but I do not know its original size.</p> <p>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?</p> <p>(background: the goal is to automatically undistort photos of rectangular documents, edge detection will probably be done with hough transform)</p> <h1>UPDATE:</h1> <p>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.</p> <p>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.</p> <p>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.</p> <ul> <li><p>Zhengyou Zhang , Li-Wei He, "Whiteboard scanning and image enhancement" <a href="http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf" rel="noreferrer">http://research.microsoft.com/en-us/um/people/zhang/papers/tr03-39.pdf</a> p.11</p> <blockquote> <p>"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."</p> </blockquote></li> <li><p>ROBERT M. HARALICK "Determining camera parameters from the perspective projection of a rectangle" <a href="http://portal.acm.org/citation.cfm?id=87146" rel="noreferrer">http://portal.acm.org/citation.cfm?id=87146</a></p> <blockquote> <p>"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."</p> </blockquote></li> </ul>

Querying!

Guidance

An individual column

Larger individual text columns get their own page to allow for proper reading.

SQuiL has stopped working due to an internal error.

If you are curious you may find further information in the browser console, which is accessible through the devtools (F12).

Reload