Geometric Interpretations of Compatibility for Fundamental Matrices

Abstract

In recent work, algebraic computational software was used to provide the exact algebraic conditions under which a sixtuple $\{F^{ij}\}$ of fundamental matrices, corresponding to $4$ images, will be compatible, i.e. there will exist cameras $\{P_i\}_{i=1}^4$ such that each pair $P_i,P_j$ has fundamental matrix $F^{ij}$; it has been further demonstrated that quadruplewise compatibility is sufficient for the problem of $n>4$ images. We expand on these prior results by proving equivalent geometric conditions for compatibility. We find that when the camera centers are in general position, compatibility can be characterized via the intersections of epipolar lines in each image. When the camera centers are coplanar, compatibility occurs when the prior condition holds and additionally any one camera center can be reconstructed via the other three.

Figures (1)

• Figure 1: Left: Generically, the three epipolar lines have no common intersection. Right: If the three epipolar lines share a common intersection $x_4$, then there exists a camera $C_4$ yielding the desired fundamental matrices.

Theorems & Definitions (30)

• Proposition 1.1: Hartley2004
• Definition 1.2
• Lemma 2.1: Hartley2004
• Proposition 2.2: braatelund2023compatibility
• Theorem 2.3: braatelund2023compatibility
• Theorem 3.1: Non-Collinear Case
• proof
• Theorem 3.2: Collinear Case
• proof
• Corollary 3.3