Get 3D coordinates from 2D image pixel if extrinsic and intrinsic parameters are known

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If you got extrinsic parameters then you got everything. That means that you can have Homography from the extrinsics (also called CameraPose). Pose is a 3×4 matrix, homography is a 3×3 matrix, H defined as 

                   H = K*[r1, r2, t],       //eqn 8.1, Hartley and Zisserman

with K being the camera intrinsic matrix, r1 and r2 being the first two columns of the rotation matrix, R; t is the translation vector.

Then normalize dividing everything by t3.

What happens to column r3, don’t we use it? No, because it is redundant as it is the cross-product of the 2 first columns of pose.

Now that you have homography, project the points. Your 2d points are x,y. Add them a z=1, so they are now 3d. Project them as follows:

        p          = [x y 1];
        projection = H * p;                   //project
        projnorm   = projection / p(z);      //normalize

Hope this helps.

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