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How do I reverse-project 2D points into 3D?

Asked 2008-09-16T19:42:33.250
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I have 4 2D points in screen-space, and I need to reverse-project them back into 3D space. I know that each of the 4 points is a corner of a 3D-rotated rigid rectangle, and I know the size of the rectangle. How can I get 3D coordinates from this?

I am not using any particular API, and I do not have an existing projection matrix. I'm just looking for basic math to do this. Of course there isn't enough data to convert a single 2D point to 3D with no other reference, but I imagine that if you have 4 points, you know that they're all at right-angles to each other on the same plane, and you know the distance between them, you should be able to figure it out from there. Unfortunately I can't quite work out how though.

This might fall under the umbrella of photogrammetry, but google searches for that haven't led me to any helpful information.

3d geometry 2d photogrammetry reverseprojection
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This is the Classic problem for marker based Augmented Reality.

You have a square marker (2D Barcode), and you want to find its Pose (translation & rotation in relation to the camera), after finding the four edges of the marker. Overview-Picture

I'm not aware of the latest contributions to the field, but at least up to a point (2009) RPP was supposed to outperform POSIT that is mentioned above (and is indeed a classic approach for this) Please see the links, they also provide source.

(PS - I know it's a bit old topic, but anyway, the post might be helpful to somebody)

answered 2012-12-18T16:21:50.147
2

Assuming that the points are indeed part of a rectangle, I'm giving a generic idea :

Find two points with max inter-distance: these most probably define a diagonal (exception: special cases where the rectangle is almost paralell to the YZ plane, left for the student). Call them A, C. Calculate the BAD, BCD angles. These, compared to right angles, give you orientation in 3d space. To find out about z distance, you need to correlate the projected sides to the known sides, and then, based on the 3d projection method (is it 1/z?) you're on the right track to know distances.

answered 2008-09-16T19:58:35.413

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