Essential Matrix Estimation

Implement a method to estimate the essential matrix between two camera views using the 8-point algorithm. This task is crucial in image alignment and stitching as it relates corresponding points in two views through the equation $\mathbf{p}_2^T E \mathbf{p}_1 = 0$ . The essential matrix $E$ encodes the rotation and translation between views, up to scale, and is a fundamental concept in computer vision.

To estimate $E$ , the process involves several steps:

Normalizing points to improve numerical stability
Building a constraint matrix $A$ from correspondences
Solving $Af = 0$ via Singular Value Decomposition (SVD)
Enforcing a rank-2 constraint on $E$

E = \begin{bmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \end{bmatrix}

This technique is widely used in structure from motion and stereo vision applications.

Example:

Input:

pts1 = [8 or more corresponding points]
pts2 = [corresponding points in second view]

Output:

3×3 essential matrix

Reasoning:

Each correspondence gives one constraint. 8 constraints → 8 equations for 9 unknowns in E. SVD finds the null space solution. Enforce rank-2 by zeroing smallest singular value.

Constraints:

pts1, pts2: Corresponding normalized points (N, 2)
Return: Essential matrix E (3×3) with rank 2

Example:

Constraints:

Test Results