Integrating Local Affine into Global Projective Images in the Joint Image Space

The fundamental matrix defines a nonlinear 3D variety in the joint image space of multiple projective (or “uncalibrated perspective”) images. We show that, in the case of two images, this variety is a 4D cone whose vertex is the joint epipole (namely the 4D point obtained by stacking the two epipoles in the two images). Affine (or “para-perspective”) projection approximates this nonlinear variety with a linear subspace, both in two views and in multiple views. We also show that the tangent to the projective joint image at any point on that image is obtained by using local affine projection approximations around the corresponding 3D point. We use these observations to develop a new approach for recovering multiview geometry by integrating multiple local affine joint images into the global projective joint image. Given multiple projective images, the tangents to the projective joint image are computed using local affine approximations for multiple image patches. The affine parameters from different patches are combined to obtain the epipolar geometry of pairs of projective images. We describe two algorithms for this purpose, including one that directly recovers the image epipoles without recovering the fundamental matrix as an intermediate step.