Summary: Variable Elimination for 3D from 2D.
Ji Zhanga, Mireille Boutinb and Daniel G. Aliagac
aSchool of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN, USA ;
b School of ECE, Purdue University, 465 Northwestern Av., West Lafayette, IN, USA;
cDept. of Computer Sc., Purdue University, 250 N. University St., West Lafayette, IN, USA.
Accurately reconstructing the 3D geometry of a scene or object observed on 2D images is a difficult problem: there
are many unknowns involved (camera pose, scene structure, depth factors) and solving for all these unknowns
simultaneously is computationally intensive and suffers from numerical instability. In this paper, we algebraically
decouple some of the unknowns so that they can be solved for independently. Decoupling the pose from the other
variables has been previously discussed in the literature. Unfortunately, pose estimation is an ill-conditioned
problem. In this paper, we algebraically eliminate all the camera pose parameters (i.e., position and orientation)
from the structure-from-motion equations for an internally calibrated camera. We then also fully eliminate the
structure coordinates from the equations. This yields a very simple set of homogeneous polynomial equations of
low degree involving only the depths of the observed points. When considering a small number of tracked points
and pictures (e.g., five points on two pictures), these equations can be solved using the sparse resultant method.
Keywords: Structure from Motion, pose-free structure from motion, Depth from motion.
Suppose a set of pictures of a 3D scene (or 3D object) is given. If the pictures were taken along a generic camera
path, it is possible to use them to reconstruct an approximation of the 3D shape of the scene. For example, this