Epipolar geometry and fundamental matrix: Is the product equal to the distance from the epipolar lines?

Question

Given the fundamental matrix of a stereo camera setup $\boldsymbol{F}$ and two potentially corresponding image points from a left image $(u_l | v_l)$ and a right image $(u_r | v_r)$:
The product $$ \pmatrix{u_l & v_l & 1} \boldsymbol{F} \pmatrix{u_r \\ v_r \\ 1} $$ will equal $0$ if the image points meet the epipolar constraint. Am I right in assuming that if the product doesn't equal $0$ - and therefore the points do not meet the epipolar constraint - it will equal the distance of the points to their corresponding epipolar line in pixels?

Answer 1

You cannot generally express distance in this manner.

However, if $\mathbf{x}_r$ is a point in homogeneous coordinates and $\mathbf{F}$ a the fundamental matrix, then $$\mathbf{l}_l=\mathbf{F}\mathbf{x}_r$$ is a line in homogeneous coordinates. It can be represented as

$$\mathbf{l}_l=(n_{x},n_{y},-d)^{\top} $$

then you want to scale your vector representation such that

$$1=\sqrt{n_{x}^{2}+n_{y}^{2}}$$

and you will get the actual (signed) distance to the line by multiplication with any point

$$\text{signed distance} = \mathbf{l}_l^\top (u_l,v_l,1)^\top $$

Note the necessary use of the square root. Distance is not a projective quantity.