Re-project rotated object in an image

Question

I have a rotated (3D) rectangle object in real-world and I'm trying to re-project this object into the image.

Suppose we have the following parameters:

• The four corners of the rectangle in real-world and the image [(xw1, yw1,zw1) , (xw2, yw2,zw2) , (xw3, yw3,zw3) , (xw4, yw4,zw4) ] ; [(x1, y1) , (x2, y2) , (x3, y3) , (x4, y4) ].
• The distance between the camera and the object (e.g. D).
• Intrinsic camera parameters (focal length and principle point).
• Extrinsic camera parameters (the rotation matrix R (3x3) and the translation matrix t (3x1)).
• The 3D surface normal vector (<v1; v2; v3>) of the rectangle object with respect to the camera.
• Check this image: image

How can we re-project the rectangle from the "real world" to the image using the given information above. So, the object in image must look like the real world (oriented).

Answer 1

You can compute a projective transformation from the images of four points. So the first point you mentioned, the coordinates of the rectangle in world and image coordinates, is sufficient.