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I am currently searching for a way to reconstruct a set of 3D lines from a 2D projection.

I have an image of a projection of 3 coordinate system axes, camera parameters etc are known.

The position of the 2D points is given, also it is known that the angles between red/green, green/blue and blue/red are 90DEG (a cartesian coordinate system).

The position of the camera is also known, so it is possible to calculate the line of sight lines depicted in the image.

Is there an easy way to find a set of corresponding 3D points describing the object that lead to the given projection? (given that the solution can have an unknown scale factor?)

projection

Thanks!

taalas
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  • Do you need the 3D lines or the point where they intersect? Be specific. Also there's an impossibility in your premise: all three angles cannot be 90, since [angles around a point should always add up to 360](http://www.mathsisfun.com/angle360.html). – legends2k Feb 06 '14 at 11:10
  • I tried to fill in some details. Basically I need a solution for the position and orientation of a 3d cartesian coordinate system from its projection into a 2d image plane. The picture shows the 2d projection, the original angles (in 3d) are 90DEG. – taalas Feb 06 '14 at 11:11
  • So you need both the lines and the point, since it's a frame. What you need is to undo the projection transform you applied to the 3d points. Since projection is really a deformation, the chances are bleak. Add the projection transform to the question. – legends2k Feb 06 '14 at 11:19

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Once a scale factor for the segments is chosen, the positions of the 3D points are actually known up to pose (unknown rotation and translation).

Assuming the 2D measurements are not exact but tainted with some noise, one can try to recover the unknown pose of the 3D points from the 2D projections by solving the corresponding non-linear least squares optimization problem (minimizing their reprojection error). For this to work, a good initial approximate solution is however needed. This paper proposes a globally convergent solution with a different approach. Unfortunately, none of these solutions are what I would call an "easy way" to recover the 3D positions.

user3146587
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