knowledge base/practices

Three points and their apparent sizes predict a fourth point of a tilted plane far better than a parallelogram

added 2026-09-28·view source

geometrycomputer-visionimage-processing

Fact

A plane seen through a camera maps to the image as p = N(u, v) / w(u, v), with N and w both affine in plane coordinates. w is proportional to depth, and a feature's apparent size is roughly proportional to 1/w. So three reference points with known plane coordinates and measured sizes s_i fix the perspective. Weight each point's image position by 1/s_i, take the affine combination the target has in plane coordinates, do the same with the weights, and divide. The parallelogram rule is the special case of equal sizes.

Why it matters

With only three fiducials (a QR code's finder patterns, three marker corners) the usual guess for the fourth corner, B + C - A, ignores foreshortening. Under a modest tilt it misses by several module widths. That is enough to put the search for the real feature in the wrong place, or to sample a grid that cannot be decoded.

How to apply

// Target at A + alpha (B - A) + beta (C - A) in plane coordinates, where
// A, B, C are image points with apparent sizes sA, sB, sC. y likewise.
const [wA, wB, wC] = [1 / sA, 1 / sB, 1 / sC];
const d = wA + alpha * (wB - wA) + beta * (wC - wA);
const x = (A.x * wA + alpha * (B.x * wB - A.x * wA) + beta * (C.x * wC - A.x * wA)) / d;
  • Measure sizes along the plane's own axes. A size taken along image rows is inflated by rotation.
  • Use it to aim a search, then refine with a real correspondence. The size-to-depth link is only approximate.

Sources