2 Arandjelović , Zisserman : Computing the F Affine from Two Ellipses

The af ne fundamental matrix FA represents the epipolar geometry in the case that the two camera projections are af ne [2]. The problem we address is estimating FA given a pair of corresponding ellipses. This is a minimal con guration for the problem, in a similar manner to the 4 point case. We will start with a counting argument and then derive the solution. FA has four degrees of freedom (dof) corresponding to the ratio of the ve non-zero entries (see the form of (1)). In the case of points, each 3D point has 3 dof and each correspondence provides 4 measurements (from the position of the point in each image). Thus for n points there are 4n constraints, and 3n+ 4 unknowns, and for n = 4 the number of constraints equals the number of unknowns. In this minimal case there is a unique solution. In the case of conics, each 3D conic has 8 dof (3 specifying the plane of the conic and 5 for the conic on that plane) and each correspondence provides 10 measurements (5 from each image conic). Thus for n conics there are 10n constraints and 8n+ 4 unknowns, and n= 2 is minimal. As will be seen, in this case there are 4 possible solutions. As this is a minimal solution all constraints are exactly satis ed. Consequently the following necessary conditions must be satis ed: the centres of corresponding ellipses satisfy the epipolar constraint, and for each epipolar line tangent to an ellipse in one image there is a corresponding epipolar line tangent to the corresponding ellipse in the other image. There are 4 steps: (i) the images are rst transformed by an af ne transformation to simplify the computation; (ii) given the image measurements in the transformed frame, the necessary conditions are used to obtain a quartic in a scalar variable; (iii) each solution of the quartic speci es a possible FA which satis es all the necessary conditions (there may be 0, 2 or 4 real solutions); nally, (iv) the FA are transformed back to the original images.