The intersection of two epipolar lines al and a2 is the epip

The intersection of two epipolar lines al and a2 is the epipole p in image 1 of a stereo pair. Suppose we are given the image coordinates of two non-planar pairs of points (Xi\' X_1\') and (X_2X_2) and their respective homographies X_1\' =H pi_1X_1 and X_2\'=H_n^2X_2. Derive the expressions of epipole p and fundamental matrix F in terms of these points and their homographies.

Solution

The fundamental matrix F encapsulates this intrinsic geometry. It is a 3 × 3 matrix of rank 2. If a point in 3-space X is imaged as x in the first view, and x\' in the second, then the image points satisfy the relation x\'^TFx = 0.We begin with a geometric derivation of the fundamental matrix. The mapping from a point in one image to a corresponding epipolar line in the other image may be decomposed into two steps. In the first step, the point x is mapped to some.A point x in one image is transferred via the plane to a matching point x\' in the second image. The epipolar line through x\' is obtained by joining x\' to the epipole e\' . In symbols one may write x\' = Hx and l\' = [e\']×x\' = [e\' ]×Hx = Fx where F = [e\' ]×H is the fundamental matrix.

The fundamental matrix satisfies the condition that for any pair of corresponding points x x\' in the two images x\'^TFx = 0.This is true, because if points x and x0 correspond, then x0 lies on the epipolar line l 0 = Fx corresponding to the point x. In other words 0 = x0>l 0 = x0>Fx. Conversely, if image points satisfy the relation x0>Fx = 0 then the rays defined by these points are coplanar..

I would like to give an algebraic derivation for the same:

The ray back-projected from x by P is obtained by solving PX = x. The oneparameter family of solutions is of the form given as X() = P⁺x + C

where P⁺ is the pseudo-inverse of P, i.e. PP⁺ = I, and C its null-vector, namely the camera centre, defined by PC = 0. The ray is parametrized by the scalar . In particular two points on the ray are P⁺x (at = 0), and the first camera centre C (at = ). These two points are imaged by the second camera P\' at P\'P⁺x and P\' C respectively in the second view. The epipolar line is the line joining these two projected points, namely l \' = (P\'C) × (P\' P⁺x). The point P\' C is the epipole in the second image, namely the projection of the first camera centre, and may be denoted by e\'. Thus, l \' = [e\']×(P\' P⁺)x = Fx, where F is the matrix F = [e\']×P\'P⁺. Note that according to the question ,

H = P\' P⁺