Estimation of F-Matrix and image rectification by double quaternion

Fundamental Matrix, or F-Matrix, is one of the most important and elemental tools in the field of computer vision. In conventional methods for estimating the F-Matrix, an eight-point algorithm is adopted. First, an approximate F-Matrix is calculated by a linear solver using at least eight corresponding pairs. Since this linear optimization method excludes an essential property, the rank 2 constraint, a method based on a singular value decomposition (SVD) is applied to impose the constraint. This last step with SVD, however, provides additional noise in the F-Matrix. Several methods introduce parameterizations taking into account the rank 2 constraint and optimized nonlinearly without SVD. In this paper, we propose a novel parameterization for the nonlinear optimization which includes this constraint. We adopt double quaternion (DQ) and a scalar as the parameter set. Experimental results show that the nonlinear optimization with our parameterization is competitive with other parameterization methods. Moreover, through the proposed parameterization, we can obtain two transformations for the two input images. These transformations lead to a novel method to estimate epipolar lines and to rectify the image pairs. This rectification method can deal with any image pairs in the same manner whether the epipoles are inside or outside the images.