proportions of a perspective-deformed rectangle

Question

Given a 2d picture of a rectangle distorted by perspective:

I know tha

Answer 1

Update

After reading your update, and looking at the first reference (Whiteboard scanning and image enhancement), I see where the missing point is.

The input data of the problem is a quadruple (A,B,C,D), AND the center O of the projected image. In the article, it corresponds to the assumption u0=v0=0. Adding this point, the problem becomes constrained enough to get the aspect ratio of the rectangle.

The problem is then restated as follows: Given a quadruple (A,B,C,D) in the Z=0 plane, find the eye position E(0,0,h), h>0 and a 3D plane P such that the projection of (A,B,C,D) on P is a rectangle.

Note that P is determined by E: to get a parallelogram, P must contain parallels to (EU) and (EV), where U=(AB)x(CD) and V=(AD)x(BC).

Experimentally, it seems that this problem has in general one unique solution, corresponding to a unique value of the w/h ratio of the rectangle.

Previous Post

No, you can't determine the rectangle ratio from the projection.

In the general case, a quadruple (A,B,C,D) of four non collinear points of the Z=0 plane is the projection of infinitely many rectangles, with infinitely many width/height ratios.

Consider the two vanishing points U, intersection of (AB) and (CD) and V, intersection of (AD) and (BC), and the point I, intersection of the two diagonals (AC) and (BD). To project as ABCD, a parallelogram of center I must lie on a plane containing the line parallel to (UV) through point I. On one such plane, you can find many rectangles projecting to ABCD, all with a different w/h ratio.

See these two images done with Cabri 3D. In the two cases ABCD is unchanged (on the gray Z=0 plane), and the blue plane containing the rectangle is not changed either. The green line partially hidden is the (UV) line and the visible green line is parallel to it and contains I.