Wedge Visualization (Interactive)

Wedge Visualization (Interactive)

Sep 21, 2025

GRG — Wedge Visualization: Interactive Uniform vs Adaptive Discretization

This mini-demo provides concrete, reproducible numbers for wedge-stratified GRG. Adjust the controls to compare a uniform wedge count against an adaptive profile on the outer annulus, and see how certified corridor half-widths contract roughly like 1/\sqrt{W_k} for comparable ring budgets.

How the Demo Works (plain language)

  • Dyadic edges. Rings follow turns-first dyadic refinement with b∈ℕ≥0, N=2^b, and quarter sampling Δu=2^{-b-2}.
  • Corridor half-width. Each sector’s estimate has a certified interval [E^- , E^+]. The half-width is (E^+−E^-)/2; decisions stop when this width ≤ tolerance τ.
  • Field model. We synthesize an anisotropic field with mixed angular orders up to fourth order to stress the wedges; values are computed in-page so the numbers are concrete and repeatable.
$$ F(\rho,\theta)=A(\rho)\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$
$$ \textbf{Stop when}\quad \widehat{E}_b^{+}-\widehat{E}_b^{-}\le \tau,\qquad \text{with}\ \ \Delta u=2^{-b-2}. $$

Controls

Results — Per-Sector Estimates on the Outer Annulus

Hover or tap a bar to read exact values (estimate and corridor half-width). The polar plots show the directional pattern and corridor bands.

Uniform W — outer annulus sectors (bars = estimate, whiskers = corridor half-width).
Adaptive outer Wk — tighter corridors where anisotropy is richer.
Polar — Uniform W.
Polar — Adaptive Wk.

Concrete Numbers (auto-computed)

The table aggregates sector values (outer annulus) for the current settings. Replace later with CSVs from your scripts; the structure is the same.

Notes & Practical Guidance

  • When to prefer GRG. As angular richness grows or is localized, adaptive W_k exposes structure with linear O(K·W) cost and certified decisions via τ.
  • When low-order bases suffice. Near-radial scenes with mild 2-fold variation are well captured by a short angular basis; choose GRG when auditability (corridors) matters.
  • Next steps. Publish side-by-side benchmarks (Zernike, B-splines, Fourier–Bessel) with the same corridor-aware stopping rule and public scripts/datasets.
GRG — Wedge Visualization

GRG — Wedge Visualization

Sep 21, 2025

Wedge-Stratified GRG — Visualizing Per-Sector Results

This section visualizes the anisotropic, wedge-stratified example discussed earlier. We compare a uniform W=8 discretization against an adaptive scheme on the outer annulus \rho\in[0.7,1), where angular content is richer. The plots show per-sector estimates with certified corridor half-widths.

Setup Recap

Field with mixed angular orders up to fourth order:

$$ F(\rho,\theta)=A(\rho)\,\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$

Rings and wedges:

$$ \text{Rings }k=1..K,\ \text{wedges }w=1..W_k,\quad I_{k,w}=\int_{\Theta_w}\!\int_{u_{k-1}}^{u_k} F(\rho,\theta)\,\rho\,d\rho\,d\theta. $$
  • Rings: K=32 (dyadic edges, quarter sampling).
  • Uniform: W=8 for all rings.
  • Adaptive: inner W_k=8, mid W_k=12, outer W_k=16 (used for plots below).

Per-Sector Results on the Outer Annulus

The numbers are the same illustrative values used in the write-up; replace them with your measured outputs when available. Error bars display corridor half-widths.

Left: uniform W=8. Right: adaptive outer ring with W_k=16. Hover/tap a bar to read exact values.

Polar View (Directional Pattern)

A simple polar plot places each sector’s estimate radially. Shaded rings visualize the corridor range.

Polar plot of per-sector estimates. Left: W=8. Right: W_k=16.

Reading the Plots

  • Corridor effect: with higher angular resolution, corridor half-widths shrink (≈ 1/\sqrt{W_k} for comparable ring budgets), revealing higher-order structure.
  • Decision making: choose the smallest dyadic level b such that the corridor width is below the task tolerance \tau.
$$ \textbf{Stop when}\ \ \widehat{E}^{+}_{b}-\widehat{E}^{-}_{b}\le\tau. $$