How to transform a projected 3D rectangle into a 2D axis aligned rectangle

I have an image of a 3D rectangle (which due to the projection distortion is not a rectangle in the image). I know the all world and image coordinates of all corners of this rectangle.

What I need is to determine the world coordinate of a point in the image inside this rectangle. To do that I开发者_开发问答 need to compute a transformation to unproject that rectangle to a 2D rectangle.

How can I compute that transform?

Thanks in advance

This is a special case of finding mappings between quadrilaterals that preserve straight lines. These are generally called homographic transforms. Here, one of the quads is a rectangle, so this is a popular special case. You can google these terms ("quad to quad", etc) to find explanations and code, but here are some sites for you.

Perspective Transform Estimation

a gaming forum discussion

extracting a quadrilateral image to a rectangle

Projective Warping & Mapping

ProjectiveMappings for ImageWarping by Paul Heckbert.

The math isn't particularly pleasant, but it isn't that hard either. You can also find some code from one of the above links.

If I understand you correctly, you have a 2D point in the projection of the rectangle, and you know the 3D (world) and 2D (image) coordinates of all four corners of the rectangle. The goal is to find the 3D coordinates of the unique point on the interior of the (3D, world) rectangle which projects to the given point.

(Do steps 1-3 below for both the 3D (world) coordinates, and the 2D (image) coordinates of the rectangle.)

1. Identify (any) one corner of the rectangle as its "origin", and call it "A", which we will treat as a vector.
2. Label the other vertices B, C, D, in order, so that C is diagonally opposite A.
3. Calculate the vectors v=AB and w=AD. These form nice local coordinates for points in the rectangle. Points in the rectangle will be of the form A+rv+sw, where r, s, are real numbers in the range [0,1]. This fact is true in world coordinates and in image coordinates. In world coordinates, v and w are orthogonal, but in image coordinates, they are not. That's ok.
4. Working in image coordinates, from the point (x,y) in the image of your rectangle, calculate the values of r and s. This can be done by linear algebra on the vector equations (x,y) = A+rv+sw, where only r and s are unknown. It will boil down to a 2x2 matrix equation, which you can solve generally in code using Cramer's rule. (This step will break if the determinant of the required matrix is zero. This corresponds to the case where the rectangle is seen edge-on. The solution isn't unique in that case. If that's possible, make special exception.)
5. Using the values of r and s from 4, compute A+rv+sw using the vectors A, v, w, for world coordinates. That's the world point on the rectangle.