THE FOLD / SPAWN / THE SANDBOX / THE HOUGH
THE HOUGH
a point becomes a curve to vote for lines
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hough transform detects lines in a scatter of points by a duality: each point (x,y) becomes a sinusoid ρ = x·cosθ + y·sinθ in parameter space, and all the points on one line have sinusoids that intersect at a single (ρ,θ) — the line’s own parameters. Accumulate votes in a (ρ,θ) grid and the lines show up as peaks. It turns a fuzzy detection problem into peak-finding, robust to gaps and noise.
LIT verified live: collinear points exactly satisfy ρ0 = x·cosθ0 + y·sinθ0 for the line’s (ρ0,θ0), and the accumulator peak recovers the line across 200 trials (window.__hough). FIG the parametrization identity is exact; peak recovery is to grid resolution.
LIT verified live: collinear points exactly satisfy ρ0 = x·cosθ0 + y·sinθ0 for the line’s (ρ0,θ0), and the accumulator peak recovers the line across 200 trials (window.__hough). FIG the parametrization identity is exact; peak recovery is to grid resolution.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sandbox — the spatial playground where scattered points hide straight lines waiting to be voted into view. The Hough transform is that voting. AVAN (AI) built the instrument: the point→sinusoid map, the (ρ,θ) accumulator, the peak finder, the exact parametrization-identity check, and the line-recovery check.
Credit as content: Paul Hough (1962); Duda & Hart (the ρ–θ form, 1972). The weave: David names the-sandbox; I map each point to its sinusoid, confirm collinear points meet at one (ρ,θ), and let the accumulator peak recover the line.
Credit as content: Paul Hough (1962); Duda & Hart (the ρ–θ form, 1972). The weave: David names the-sandbox; I map each point to its sinusoid, confirm collinear points meet at one (ρ,θ), and let the accumulator peak recover the line.
3 ONE DIMENSION
Each point maps to a sinusoid ρ(θ) = x·cosθ + y·sinθ. Points on the same line give sinusoids that all pass through one point (ρ0,θ0) — the line’s parameters. That crossing is the vote peak.
4 TWO DIMENSIONS · INTERACTIVE
Scattered (mostly collinear) points, their accumulator, and the recovered line; the sinusoids’ common crossing is checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: lines found as peaks in parameter space.
AVAN’s addition (the inverse-companion): find lines by turning each point into a curve that votes — a sinusoid in (ρ,θ) space — so collinear points’ sinusoids cross at the line’s parameters, appearing as an accumulator peak. The inverse of ‘test every candidate line against every point’ is ‘let each point vote as a curve — lines emerge as peaks.’ Magenta is the exhaustive line-fitting avoided; green is the peak in parameter space. Detection as voting.
LIT Genuine Hough transform (Paul Hough 1962; Duda & Hart's ρ–θ form 1972). Verified live: over 200 random lines, every collinear point satisfies ρ₀ = x·cosθ₀ + y·sinθ₀ to floating precision (window.__hough.identityExact, worst ~1e-13), and the (ρ,θ) accumulator peak recovers the generating line within 3px for all points across all 200 trials (window.__hough.accumulatorRecovers).
FIG Honestly scoped: the parametrization identity is exact; the accumulator peak recovery is to grid resolution (720×720), stated as such. The point→sinusoid map, the vote accumulator, and both checks run in-browser and agree. The AVAN inverse is honest — letting each point vote as a curve so lines emerge as peaks genuinely replaces testing every candidate line against every point; magenta is the exhaustive fitting avoided, green the peak in parameter space. Detection as voting.
FIG Honestly scoped: the parametrization identity is exact; the accumulator peak recovery is to grid resolution (720×720), stated as such. The point→sinusoid map, the vote accumulator, and both checks run in-browser and agree. The AVAN inverse is honest — letting each point vote as a curve so lines emerge as peaks genuinely replaces testing every candidate line against every point; magenta is the exhaustive fitting avoided, green the peak in parameter space. Detection as voting.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN