Experimental and Theoretical Scrutiny of the Geometric Derivation of the Fundamental Matrix

Abstract

In this paper, we prove mathematically that the geometric derivation of the fundamental matrix F of the two-view reconstruction problem is flawed. Although the fundamental matrix approach is quite classic, it is still taught in universities around the world. Thus, analyzing the derivation of F now is a non-trivial subject. The geometric derivation of E is based on the cross product of vectors in R3. The cross product (or vector product) of two vectors is x × y where x = ⟨x1, x2, x3⟩ and y = ⟨y1, y2, y3⟩ in R3. The relationship between the skew-matrix of a vector t in R3 and the cross product is [t]×y = t × y for any vector y in R3. In the derivation of the essential matrix we have E = [t]×R which is the result of replacing t × R by [t]×R, the cross product of a vector t and a 3×3 matrix R. This is an undefined operation and therefore the essential matrix derivation is flawed. The derivation of F, is based on the assertion that the set of all points in the first image and their corresponding points in the second image are protectively equivalent and therefore there exists a homography H&pgr; between the two images. An assertion that does not hold for 3D non-planar scenes.