The 16-Point Paradigm

In central vision, we need only 4 points for a homography. In non-central vision (multi-camera rigs), estimating the homography from a planar scene requires a more complex solver.

The Minimal Solver

The movement of a generalized camera over a plane can be solved using 16 point correspondences. This "minimal" set allows us to recover the relative 6-DOF pose ($R, \mathbf{t}$) and the plane parameters ($\mathbf{n}, d$).

\mathbf{q}_i \cdot (\mathbf{R} \mathbf{u}_i' \times \mathbf{n}) + \mathbf{u}_i \cdot (\mathbf{t} \times \mathbf{R} \mathbf{u}_i' + \mathbf{R} \mathbf{q}_i') = 0

This is the Generalized Epipolar Constraint (GEC) specialized for planar homography.

Linearization

By defining 16 intermediate variables, the problem can be linearized into a matrix $A$ of size $N \times 16$. Just like DLT, we solve this via SVD. This is the heart of the 16-Point RANSAC framework.

Complexity Alert 16 points is a "large" minimal set for RANSAC. This makes the iteration count $N$ much higher compared to the 4-point central case, requiring highly optimized sampling strategies.

Rig Position (X)

Rig Rotation (Y)

Observe how the 16 rays from different camera centers "anchor" the plane intersection.