Find Epipole from Fundamental Matrix

The core idea is: the epipole in an image is the unique point where all epipolar lines from that camera intersect; algebraically it is a null-space vector of the fundamental matrix $F$ (right epipole) or $F^\top$ (left epipole). In homogeneous 2D coordinates, lines are 3D vectors and their intersection is a cross product.

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1. Background Knowledge

In stereo vision, two cameras view the same 3D scene. Any 3D point projects to a pair of image points $$\mathbf{x}$$ and $$\mathbf{x}$'$. These pairs are constrained by the fundamental matrix $F$, which encodes the epipolar geometry between the two views. The fundamental relation is:

$$\mathbf{x}'^{\top} F \mathbf{x} = 0$$

For a fixed point $$\mathbf{x}$$ in the first image, the set of all possible matches $$\mathbf{x}$'$ in the second image lie on a line

$$\mathbf{l}' = F \mathbf{x}$$

called the epipolar line.

The epipole is the projection of one camera center into the other camera’s image plane. All epipolar lines in an image pass through that epipole, so the epipole is the intersection point of all epipolar lines in that image. Algebraically, the right-image epipole $\mathbf{e}'$ satisfies:

$$F \mathbf{e}' = \mathbf{0}$$

and the left-image epipole $\mathbf{e}$ satisfies:

$$F^\top \mathbf{e} = \mathbf{0}$$

i.e., each epipole is a (right or left) null-space vector of $F$.

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2. Algorithm / Approach Pattern

Two equivalent patterns you should understand:

1. Null-space approach (most robust):

• The epipole is any non-zero vector in the null space of $F$ (or $F^\top$ depending on which epipole you want).
• Compute this using SVD or eigen-decomposition.

2. Line-intersection approach (geometric):

• Pick two (or more) points in one image.
• Map them to epipolar lines in the other image: $\mathbf{l}_i = F$\mathbf{x}$_i$ (or $\mathbf{l}_i' = F^\top$\mathbf{x}$_i'$).
• Compute the intersection of two epipolar lines:

$$\mathbf{e} = \mathbf{l}_1 \times \mathbf{l}_2$$

• With more than two lines, you can make the estimate more robust (e.g., least squares).

For coding problems like this, the simplest and numerically stable pattern is SVD-based null-space extraction, but understanding the line-intersection via cross product connects the algebra to the geometry.

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3. Step-by-Step Strategy

Assuming you are given a $3 \times 3$ fundamental matrix $F$ and asked for, say, the epipole in the right image:

A. Using null space (recommended)

1. Compute SVD of $F$:

• $F = U \Sigma V^\top$
• In most libraries: U, S, Vt = np.linalg.svd(F).

2. Get the right epipole $\mathbf{e}'$:

• The right null-space is the last column of $V$ (or last row of Vt).
• Let $\mathbf{e}' = V[:, -1]$ (or Vt[-1, :] transposed).