Detecting Critical Configurations for Euclidean 3D Reconstruction by Analyzing the Scaled Measurement Matrix

3D reconstruction is ambiguous under so-called critical motions or critical surfaces. This paper proposes an algorithm to detect a few critical configurations where Euclidean reconstruction degenerates. Assuming that the focal lengths are the only unknown intrinsic parameters, the following critical configurations are detected: (1) coplanar 3D points, (2) pure rotation; (3) rotation around two camera centers; (4) presence of excessive noise and outliers in the measurements. The configurations in Cases (1), (2) and (4) will affect the rank of the scaled measurement matrix (SMM). The number of camera centers in Case (3) will affect the number of independent rows of the SMM. By examining the rankness and the number of independent rows of the SMM, we are able to detect the above-mentioned critical configurations. Experimental results on both synthetic and real data demonstrate the effectiveness of the proposed algorithm on detecting the critical situations for factorization-based 3D reconstruction.