Symmetric Boundary Alignment (MODWT)

This note documents the practical centering/orientation used for symmetric-boundary MODWT inverse (IMODWT) in VectorWave, based on an empirical sweep across Haar and DB4.

Overview

• Padding strategy: SYMMETRIC with centered, level-aware alignment.
• τ_j: computed from base filter length via τ_j = floor(((L0 − 1) · 2^(j − 1)) / 2).
• Orientation: branch-specific (approx/detail) to balance reflection phase.
• Filters: cascade IMODWT uses per‑level upsampled reconstruction filters with 1/√2 scaling (same as periodic cascade synthesis).

Heuristic (current)

• Detail branch (G): plus orientation always; level‑conditional Δτ_G:
  • j ≤ 2: Δτ_G = 0
  • j ≥ 3: Δτ_G = −1
• Approx branch (H): level‑ and family‑conditional
  • Haar (L0 = 2): plus orientation; Δτ_H = 0 for j = 1, and Δτ_H = −1 for j ≥ 2
  • DB4 and longer (L0 ≥ 8): minus orientation; Δτ_H = 0 for j = 1, and Δτ_H = −1 for j ≥ 2

Rationale and Evidence

• We introduced a sweep diagnostic (MultiLevelMODWTSymmetricAlignmentSweepTest) that tries:
  • Orientations: plus/minus per branch
  • Offsets: Δτ in {−1, 0, +1}
  • Wavelets: Haar, DB4
  • Sizes: N in {129, 257, 512}
  • Levels: up to maximum allowed
• Observed best configurations:
  • Haar: H:plus dH=−1, G:plus dG=0 (best RMSE ≈ 0.25–0.29)
  • DB4: H:minus dH in {0,−1}, G:plus dG=0 (best RMSE = 0.65–0.70)
• We codified these findings in an internal strategy class: SymmetricAlignmentStrategy.

Practical Accuracy

• Periodic: near machine precision (RMSE ≤ 1e−8) with Haar/DB4 across sizes/levels.
• Zero-padding: expected boundary loss; periodic remains the reference for strict invariants.
• Symmetric: RMSE < 0.70 on Haar/DB4 across 512 and levels up to max (test: MultiLevelMODWTSymmetricRMSETest).

Implementation

• Strategy class: com.morphiqlabs.wavelet.modwt.SymmetricAlignmentStrategy returns a per-level decision:
  • approxPlus (orientation), deltaApprox (Δτ_H), detailPlus, deltaDetail (Δτ_G).
• The inverse synthesis applies:
  • Approx (H): idx = sym(t ± l ∓ τ_H) depending on orientation
  • Detail (G): idx = sym(t + l − τ_G) (plus)
• τ_H, τ_G = computeTauJ(L0, j) + Δτ.

Diagnostics

• Periodic diagnostic: PeriodicCascadeInverseDiagnosticTest asserts RMSE ≤ 1e−8 (Haar/DB4) at small N.
• Symmetric sweep: MultiLevelMODWTSymmetricAlignmentSweepTest prints best RMSE/config.

Future Work

• If applications require stricter symmetric accuracy, consider:
  • Choosing Δτ_H = 0 for DB4 at specific levels/sizes (seen as near-equal in the sweep).
  • Family‑specific tables for additional orthogonal wavelets (DB6/DB8) with the same mechanism.