Assuming that cameras satisfy pinhole model, we have the following geometry constraints from two perspective views.
Epipoles or epipolar points: e1,e2
Epipolar lines: l1,l2
Epipolar plane: the plane formed by optical centers O1,O2 and point P
Fundamental Matrix
Suppose that the transformation from frame O1 to frame O2 can be described by a rotation matrixR and a 3D translation vector t. Let P1 denote the position of point P in frame O1, and P2 the position of point P in frame O2. We then have the relation
P2=RP1+t To derive the beautiful and concise epipolar constraint, which reflects the fact that O1,O2 and P are co-planar, we first left-multiply both sides of (1) by[t] to obtain
[t]P2=[t]RP1+[t]t where[t] is the bracket notation (i.e. express cross product as a skew-symmetric matrix) of vector t. The term [t]t is always zero and can be crossed out. We then left-multiply both sides of (2) by P2T to obtain
P2T[t]P2=P2T[t]RP1 The termP2T[t]P2 is always zero, because P2 is perpendicular to[t]P2 and the dot product of the two is always zero. Finally, we obtain the epipolar constraint
P2T[t]RP1=0 Let F=[t]R be the Fundamental Matrix, we can rewrite the epipolar constraint in a concise form
P2TFP1=0 The epipolar constraint in this form relates the coordinates of the same point in two references frames.
Essential Matrix
Recall the pinhole camera model, we can project point P onto the image planes in two views by
Taking the vector product with t, followed by the dot product with p2 we obtain p2T[t]×Rp1=0.
E=t∧R,F=K−TEK−1,x2TEx1=p2TFp1=0 its normalized image coordinates (i.e. x1=K−1p1), and the