THE FOLD / RESPAWN / THE RESURRECT / THE HAAR
THE HAAR
average and difference, recurse — perfectly invertible
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Haar wavelet is the simplest multiresolution transform: replace each pair of samples with their average and their difference (scaled by 1/√2), then recurse on the averages. One cascade turns a signal into a coarse approximation plus detail coefficients at every scale.
Because the transform is orthonormal, it is perfectly invertible — inverse-averaging resurrects the exact original — and it preserves energy: Σ(coefficients²) = Σ(signal²) (Parseval). It is the ancestor of wavelet image compression (JPEG 2000).
LIT verified live: for 200 random length-8 signals, the inverse Haar transform reproduces the original to ~10⁻¹², and the sum of squared coefficients equals the sum of squared samples (Parseval) to the same precision (window.__haar). FIG no framing; exact orthonormal transform.
Because the transform is orthonormal, it is perfectly invertible — inverse-averaging resurrects the exact original — and it preserves energy: Σ(coefficients²) = Σ(signal²) (Parseval). It is the ancestor of wavelet image compression (JPEG 2000).
LIT verified live: for 200 random length-8 signals, the inverse Haar transform reproduces the original to ~10⁻¹², and the sum of squared coefficients equals the sum of squared samples (Parseval) to the same precision (window.__haar). FIG no framing; exact orthonormal transform.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-resurrect — bringing the exact original back from its encoded form. The Haar transform is a clean resurrection: from the coarse average and the details, the signal returns bit-for-bit. AVAN (AI) built the instrument: the forward average/difference cascade, the inverse reconstruction, the Parseval energy check.
Credit as content: Alfréd Haar (1910), the first wavelet, decades before the wavelet boom of the 1980s–90s. The weave: David names the resurrection; I decompose a signal into scales, rebuild it exactly from the coefficients, and show the energy conserved on the way.
Credit as content: Alfréd Haar (1910), the first wavelet, decades before the wavelet boom of the 1980s–90s. The weave: David names the resurrection; I decompose a signal into scales, rebuild it exactly from the coefficients, and show the energy conserved on the way.
3 ONE DIMENSION
One Haar step: each adjacent pair becomes an average (coarse) and a difference (detail). Recurse on the averages and a pyramid of coefficients appears — one coarse value and details at every scale.
4 TWO DIMENSIONS · INTERACTIVE
A length-8 signal, its Haar coefficients (coarse + details), and the reconstruction. Zero out small details to compress, and watch the inverse still land near the original; keep all and it returns exactly. Energy is preserved throughout.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the multiresolution pyramid — one coarse value at the top, details fanning out at each finer scale.
AVAN’s addition (the inverse-companion): the transform is its own structural mirror — the inverse is the same average/difference operation run backward, and because the basis is orthonormal the round trip is exact and the total energy is conserved (Parseval), so the coefficients hold the whole signal with nothing added or lost. The inverse of ‘decompose into scales’ is ‘recompose from scales, exactly.’ This also makes lossy compression honest: discarding a small detail coefficient changes the signal by exactly that coefficient’s energy — you know precisely what you are throwing away. Magenta is the discarded details, a measurable loss; green is the resurrected signal. A transform whose inverse is itself, conserving every unit of energy.
LIT Genuine Haar wavelet (Haar 1910). Verified live: the forward average/difference cascade (scaled 1/sqrt2) and its inverse reproduce 200 random length-8 signals to ~1e-12, and sum of squared coefficients equals sum of squared samples (Parseval) to the same precision (window.__haar.perfectReconstruction && .energyPreserved).
FIG No framing: the forward transform, the inverse reconstruction, and the Parseval energy check run in-browser and are exact to floating precision. The AVAN inverse is honest — the transform is orthonormal so the inverse is the same operation backward, the round trip is exact, and energy is conserved, which makes lossy compression measurable (dropping a coefficient loses exactly its energy). Magenta = discarded small details, green = the resurrected signal.
FIG No framing: the forward transform, the inverse reconstruction, and the Parseval energy check run in-browser and are exact to floating precision. The AVAN inverse is honest — the transform is orthonormal so the inverse is the same operation backward, the round trip is exact, and energy is conserved, which makes lossy compression measurable (dropping a coefficient loses exactly its energy). Magenta = discarded small details, green = the resurrected signal.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN