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Essential Matrix Constraint Check

MediumPose 📚 CV: Structure from Motion and SLAM

Verify if a matrix satisfies the essential matrix constraint.

The essential matrix E encodes the relative pose between two calibrated cameras. It must satisfy:

Determinant constraint: $\det(E) = 0$
Singular value constraint: Two singular values are equal, one is zero

For this problem, we check only the determinant constraint. A valid essential matrix is rank-2 (determinant = 0) because it represents a degenerate transformation (all epipolar lines pass through a single point - the epipole).

The 3×3 determinant is computed using the rule of Sarrus or cofactor expansion:

$\det(E) = e_{00}(e_{11}e_{22} - e_{12}e_{21}) - e_{01}(e_{10}e_{22} - e_{12}e_{20}) + e_{02}(e_{10}e_{21} - e_{11}e_{20})$

Example:

Input:

is_essential([[0,0,0],[0,0,-1],[0,1,0]], 0.01)

Output:

True

Reasoning:

Computing determinant of the given matrix:

det = 0×(0×0 - (-1)×1) - 0×(0×0 - (-1)×0) + 0×(0×1 - 0×0)
- = 0×1 - 0×0 + 0×0
- = 0 Since |0| < 0.01, the matrix satisfies the essential matrix constraint.

Constraints:

E: 3x3 matrix
tolerance: threshold for considering det ≈ 0
Return True if |det(E)| < tolerance

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The task is to check whether a given $3\times 3$ matrix $E$ could be an essential matrix by verifying that its determinant is zero (or numerically, “close enough” to zero).

1. Background Knowledge

In epipolar geometry, the essential matrix $E$ relates corresponding points $x$ and $x'$ in two calibrated camera views via

x'^{\top} E x = 0.

It encodes the relative rotation and translation direction between the two cameras (pose), assuming camera intrinsics are known and removed.

An essential matrix is a special type of $3\times 3$ matrix with rank 2: its determinant is zero, and its singular values have the pattern $\sigma,$ \sigma $, 0$ . The rank-2 property reflects that all epipolar lines in each image intersect at a single point (the epipole), which corresponds to the camera center of the other view. In this problem, you ignore singular values and only check the determinant constraint:

\det(E) = 0.

For a $3\times 3$ matrix $E = [e_{ij}]$ , the determinant can be expanded as:

\det(E) = e_{00}(e_{11}e_{22} - e_{12}e_{21}) - e_{01}(e_{10}e_{22} - e_{12}e_{20}) + e_{02}(e_{10}e_{21} - e_{11}e_{20}).

2. Algorithm / General Approach

You can treat this as a direct formula evaluation problem:

Extract the 9 entries of the matrix.
Compute the determinant using the given $3\times 3$ formula.
Compare the determinant with zero (possibly using a small tolerance if using floating-point).
Return whether it satisfies the constraint.

This is a fixed-size computation with no loops or complex data structures.

3. Step-by-Step Strategy

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solution.py

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