Computer Vision

A 3-D point at depth $Z$ projects to the image plane by similar triangles: $x = fX/Z$, $y = fY/Z$.

In pixels this is packaged as $\tilde{x} = K [R \mid t] X$, where $K$ holds focal length and principal point, and $[R \mid t]$ places the point in the camera frame.

$Z = \dfrac{f B}{d}$

Closer points have larger disparity; depth resolution degrades quadratically with distance since $Z$ is inversely proportional to $d$.

For a correspondence $(x, x')$ between two views: $x'^\top F x = 0$.

This says $x'$ must lie on the epipolar line $l' = Fx$ — it collapses a 2-D search for correspondences into a 1-D search along a line. $E = K'^\top F K$ relates the calibrated and uncalibrated forms.

Repeatedly: sample a minimal set of correspondences, fit a model ($F$, $E$, or $H$), count inliers within a threshold, keep the best-scoring model.

The number of trials needed grows sharply as the outlier fraction rises, which is why a good initial match filter (e.g. descriptor ratio test) matters before RANSAC runs.

$x' \simeq H x$ holds when points lie on a common plane, or when the camera only rotates between views (no baseline, no parallax).

General 3-D scenes with camera translation need the epipolar constraint instead — a single $H$ can't explain parallax.

Given two calibrated views and a matched pixel pair, the 3-D point is the intersection of the two back-projected rays.

In practice the rays rarely intersect exactly due to noise, so triangulation finds the point minimizing reprojection error in both images (linear least squares or iterative refinement).