26-EFT.WP.STG.Lensing v1.0 | Chapter 6 — Feature Lenses and Saliency (Bands / Morphology / Path Sparsity)

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Chapter 6 — Feature Lenses and Saliency (Bands / Morphology / Path Sparsity)

One-sentence goal: Construct feature lenses on graph spectra to extract salient structure by unifying three mechanisms—band filtering, morphological operators, and path sparsity—under parallel spectral–variational formulations, with auditable saliency measures and selection.

I. Scope & Objects

1. Inputs
   • Graph & operators: G = (V, E, w), L or L^vis / L_ani (see Chapters 4/5), with unit(L) = 1, dim(L) = [1].
   • Signal: x_in ∈ R^{|V|} (scalar, or per-channel independently).
   • Band design: band set B = { b }, per-band kernel g_b(λ) and weights a_b ≥ 0.
   • Morphological structuring element: S_r (geodesic ball of radius r on the graph).
   • Candidate path set: Γ = { gamma_k(ell) }, or spline candidates from shortest paths / manifold tangents.
   • Runtime parameters: mode ∈ { offline, streaming }, window win = { Δt_win, Δt_slide }.
2. Outputs
   • Band responses x_b = g_b(L_*) x_in, morphological response x_morph, selected sparse paths Γ_sel ⊆ Γ with coefficients α.
   • Saliency map Sal ∈ R^{|V|}_{≥0}, mask Mask ∈ {0,1}^{|V|}.
   • Dual-form gap delta_form_lens (spectral kernels vs. variational forms).
3. Constraints & boundaries
   • unit(x_b) = unit(x_in); morphological residuals share the same units; path measures are explicit as ( ∫_{gamma_k(ell)} • d ell ).
   • Anisotropy/visibility are optional; unify with L_* ∈ { L, L^vis, L_ani }.

II. Terms & Variables

• Band kernels & energy: x_b = U g_b(Λ) U^T x_in; per-band energy E_b = || x_b ||_2^2.
• Graph-domain morphology: dilate(x; r) = sup_{j∈S_r(i)} x_j, erode(x; r) = inf_{j∈S_r(i)} x_j.
• Morphological gradient & top-hat: grad_m = dilate − erode, top_hat = x_in − open(x_in; r).
• Path sparsity: edge–path incidence B_P, coefficients α, x_path = D_P α (path dictionary D_P).
• Saliency aggregation: Sal = w_spec*Sal_spec + w_morph*Sal_morph + w_path*Sal_path, with w_* ≥ 0.
• Dual-form gap: delta_form_lens = || x'_{spec} − x'_{var} ||_2 (per-band or overall).

III. Postulates P716-*

• P716-1 (Parallel spectral–variational): For every g_b and for morphology/sparsity forms, provide spectral realizations and equivalent variational forms in parallel; record delta_form_lens.
• P716-2 (Band regularity): Bands collectively cover spec(L_*), with controlled overlap between adjacent bands (see C71-62), avoiding spectral holes or excessive redundancy.
• P716-3 (Monotone structuring elements): If r1 ≤ r2 then S_{r1}(i) ⊆ S_{r2}(i); grad_m is non-decreasing with r.
• P716-4 (Explicit path measure): Any path operator must use ( ∫_{gamma_k(ell)} • d ell ) with gamma_k: [0, L_k] → R^d and traceable L_k.
• P716-5 (Units & dimensions): check_dim( Sal ) = "[1]"; unit(Mask) = 1; threshold τ_s is dimensionless.
• P716-6 (Temporal consistency): In win, time averages and peaks of saliency must declare weights and measures.

IV. Minimal Equations S716-*

1. S716-1 (Band feature lenses)
   • Band-pass kernels: g_b(λ) = exp( − τ_{b,lo} λ ) − exp( − τ_{b,hi} λ ) or g_b(λ) = (1+β_{b,lo}λ)^{−1} − (1+β_{b,hi}λ)^{−1}, with τ_{b,hi} > τ_{b,lo}.
   • Spectral realization: x_b = U g_b(Λ) U^T x_in;
   • Variational equivalence (quadratic case):
     x_b = argmin_x ( (1/2)||x − x_in||_2^2 + (β_{b,lo}/2) x^T L_* x ) − argmin_x ( (1/2)||x − x_in||_2^2 + (β_{b,hi}/2) x^T L_* x ).
2. S716-2 (Morphological saliency)
   • grad_m(i; r) = dilate(x_in; r)_i − erode(x_in; r)_i;
   • Top-hat: top_hat(i; r) = x_in(i) − open(x_in; r)_i;
   • Normalization: Sal_morph = norm( grad_m ⊙ mask_vis ), where mask_vis(i) = 1_{ vis_i > 0 }.
3. S716-3 (Path sparsity & selection)
   • Sparse coding: α = argmin_{α} ( (1/2)|| x_in − D_P α ||_2^2 + λ ||α||_1 );
   • Path-wise saliency: Sal_path(i) = ∑_{k: i∈gamma_k} |α_k| / L_k;
   • Variational equivalence (TV–path): x' = argmin_x ( (1/2)||x − x_in||_2^2 + λ || B_P x ||_1 ).
4. S716-4 (Bandwise saliency aggregation)
   • Spectral energy density: Sal_spec(i) = ∑_{b∈B} a_b * ( x_b(i) )^2;
   • Normalized decision: Mask(i) = 1_{ Sal(i) ≥ τ_s }, with Sal = w_spec*Sal_spec + w_morph*Sal_morph + w_path*Sal_path.
5. S716-5 (Dual-form gap & stability)
   • Per-band gap: delta_b = || x_{b,spec} − x_{b,var} ||_2;
   • Total gap: delta_form_lens = ( ∑_b delta_b^2 )^{1/2 }; require ρ( g_b(L_*) ) ≤ 1 + ε_b to keep energy bounded.

V. Metrology Pipeline M71-6 (Design → Solve → Aggregate → Verify → Persist)

1. Design bands: choose B, g_b(λ), and weights a_b; choose L_* (visible/anisotropic).
2. Spectral approximation: generate Chebyshev or Lanczos approximations for each g_b; set order_b and λ_max.
3. Solve in dual forms:
   • Spectral: use polynomial–vector iterations to obtain each x_b;
   • Variational: solve the corresponding quadratic/TV problems via CG / primal–dual to get x_{b,var} and path-sparsity α.
4. Morphology: compute grad_m / top_hat for r ∈ R_set, and multiply by the vis mask to suppress occlusion artifacts.
5. Path sparsity: build D_P / B_P, solve for α and Sal_path; optionally perform non-maximum suppression along gamma_k.
6. Saliency aggregation: form Sal and Mask; choose adaptive threshold τ_s = median(Sal) + κ * mad(Sal).
7. Checks & publication: evaluate coverage_s, δ overlap, err_spec∞, delta_form_lens, sparsity_level; write to manifest.lens.

VI. Contracts & Assertions C71-6x (suggested thresholds)

• C71-61 (Stability): for any band b, ρ( g_b(L_*) ) ≤ 1 + 0.02.
• C71-62 (Band overlap control): overlap(B_b, B_{b+1}) ≤ 0.25 (via ||g_b g_{b+1}||_2 / ||g_b||_2||g_{b+1}||_2).
• C71-63 (Approximation error): per-band err_spec∞(b) ≤ 1e−3.
• C71-64 (Dual-form gap): delta_form_lens ≤ 1e−3 || x_in ||_2.
• C71-65 (Reasonable morphology scales): r ≤ diam(G)/4 and the p99 of grad_m does not exceed clip_max (prevent over-dilation).
• C71-66 (Sparsity level): sparsity = ||α||_0 / |Γ| ≤ 0.1, and ∑_k |α_k| is of the same order as || x_in ||_2.
• C71-67 (Coverage & false positives): coverage_s = |{ i | Mask(i) = 1 }| / |V| ∈ [0.05, 0.6] (task-tunable).
• C71-68 (Unit checks): check_dim( Sal ) = "[1]", check_dim( x_b − x_in ) = "[same]".

VII. Implementation Bindings I71-6* (interfaces, I/O, invariants)

• design_band_kernels(L_meta, bands_spec) -> { g_b }_B, meta
• apply_spectral_bands(L_*, { g_b }, order_b, x_in) -> { x_b }_B, report
• solve_variational_band(L_*, band_params, x_in) -> { x_{b,var} }_B, report
• graph_morphology(x_in, R_set, mode) -> { grad_m, top_hat }, meta
• build_path_dictionary(G, Γ_spec) -> { D_P, B_P, Γ }, meta
• sparse_path_select(x_in, D_P|B_P, λ) -> { α, Γ_sel, Sal_path }, report
• compose_saliency({ x_b }, grad_m, Sal_path, weights) -> { Sal, Mask }, meta
• eval_delta_form({ x_b }, { x_{b,var} }) -> delta_form_lens
• assert_feature_contracts(ds, rules) -> report
• emit_lens_manifest(results, policy) -> manifest.lens
  Invariants: order_b ≥ 0; λ_max > 0; R_set ⊂ N^+; Γ is traceable with L_k > 0; hash(*) fully recorded.

VIII. Cross-References

• Graph kernels & spectral approximation: EFT.WP.STG.Dynamics v1.0, Chapter 4.
• View & occlusion: EFT.WP.STG.Lensing v1.0, Chapter 4 (W_vis masking).
• Anisotropic orientation fields & path construction: EFT.WP.Particle.TopologyAtlas v1.0, Chapters 7/9.
• Runtime publication & dashboards: EFT.WP.STG.Dynamics v1.0, Chapter 14.

IX. Quality & Risk Control

• SLI/SLO: err_spec∞_p95, delta_form_lens_p99, overlap_p50, sparsity_level, coverage_s, latency_p95.
• Fallbacks: if approximation error exceeds bounds → raise order_b or narrow the bands; if delta_form_lens is high → unify on the variational form and lower β/τ; if sparsity fails → increase λ or simplify Γ; if morphological artifacts appear → reduce r and enable the vis mask.
• Audit: retain sampled g_b(λ), order_b / λ_max, Sal distributions, Γ_sel and histograms of α, contract pass rates, and anomalous samples.

Summary

• This chapter unifies band filtering, morphological gradients, and path sparsity into a single feature-lens framework, ensuring consistent and auditable implementations via spectral–variational dual forms.
• Key manifest.lens.* fields:
  bands.*, g_b.hash, order_b, λ_max, R_set, Γ.hash, λ_sparsity, Sal.hash, Mask.ratio, err_spec∞, delta_form_lens, contracts.*, signature.