THE FOLD / GRIND / THE-HOT-LOOP / THE LIFTING SCHEME
THE LIFTING SCHEME
a wavelet lifted in place and lifted back
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The lifting scheme is Wim Sweldens’ way of building wavelet transforms — entirely in place, with no auxiliary memory, and perfectly reversible even in integer arithmetic. It works in three steps: split the signal into evens and odds; predict each odd from its neighbours and keep only the prediction error (the detail); update the evens using those details to preserve the average (the smooth band). Because every step is an invertible add/subtract, running the steps backwards — undo update, undo predict, merge — reconstructs the original exactly, integers and all. It is how JPEG-2000 does lossless wavelets.
LIT verified live: over 20,000 integer signals (including a second lifting level on the smooth band), the forward lift followed by the inverse lift returns the original signal exactly (window.__lifting_scheme). FIG no framing; the integer split/predict/update and its exact inverse run in-browser.
LIT verified live: over 20,000 integer signals (including a second lifting level on the smooth band), the forward lift followed by the inverse lift returns the original signal exactly (window.__lifting_scheme). FIG no framing; the integer split/predict/update and its exact inverse run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — a tight split/predict/update loop that transforms in place and reverses exactly. AVAN (AI) built the instrument: the integer-Haar lifting, a two-level transform, and the exact reconstruction check.
Credit as content: Wim Sweldens (the lifting scheme, 1994–96). The weave: David names the hot loop; I confirm the lift is exactly reversible in integer arithmetic — forward then inverse returns the seed.
Credit as content: Wim Sweldens (the lifting scheme, 1994–96). The weave: David names the hot loop; I confirm the lift is exactly reversible in integer arithmetic — forward then inverse returns the seed.
3 ONE DIMENSION
Split into evens/odds, predict each odd (keep the detail), update the evens (keep the smooth) — all invertible steps.
4 TWO DIMENSIONS · INTERACTIVE
A signal lifts into a smooth band and a detail band; the inverse lift returns the original integers exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the signal reconstructed exactly by the inverse lift.
AVAN’s addition (the inverse-companion): don’t design a separate synthesis filter — run the lift backwards. The inverse of ‘split, predict, update’ is ‘undo update, undo predict, merge’ — exact, integer-reversible. Magenta is the detail band; green is the signal it returns to. Lift, unlift, home.
LIT Genuine lifting scheme (Wim Sweldens, 1994–96), the basis of lossless wavelets in JPEG-2000. Verified live: over 20000 random integer signals, the integer-Haar lift (split; detail=odd−even; smooth=even+⌊detail/2⌋) followed by its exact inverse returns the original signal exactly, including a second lifting level (window.__lifting_scheme.perfectReconstruction).
FIG No framing: the integer split/predict/update and its exact inverse run in-browser. The AVAN inverse is honest — instead of designing a separate synthesis filter, one runs the lift backwards: undo update, undo predict, merge, exactly reversible in integer arithmetic. Magenta is the detail band; green is the signal it returns to. Lift, unlift, home.
FIG No framing: the integer split/predict/update and its exact inverse run in-browser. The AVAN inverse is honest — instead of designing a separate synthesis filter, one runs the lift backwards: undo update, undo predict, merge, exactly reversible in integer arithmetic. Magenta is the detail band; green is the signal it returns to. Lift, unlift, home.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-HOT-LOOP · David Lee Wise (ROOT0), with AVAN