26-EFT.WP.STG.Lensing v1.0 | Chapter 5 — Lens Kernels & Spectral Realizations (Focus / Defocus / Anisotropy)

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Chapter 5 — Lens Kernels & Spectral Realizations (Focus / Defocus / Anisotropy)

One-sentence goal: Present implementable spectral–variational formulations for focus/defocus/anisotropic lens kernels on graphs, together with stability and approximation-error bounds, and bind them to streaming execution.

I. Scope & Objects

1. Inputs
   • Graph & operators: G = (V, E, w), A, D, L or normalized L_n (unit(L) = 1, dim(L) = [1]).
   • Visibility & subgraph weights: W_vis, m_ij (see Chapter 4), optional anisotropy tensor C_ani(e).
   • Signals & observations: x_in ∈ R^{|V|}, y = H x_true + v.
   • Execution parameters: kind ∈ { focus, defocus, anisotropic }, params = { α, β, τ, p, order }, mode ∈ { offline, streaming }, win = { Δt_win, Δt_slide }.
2. Outputs
   • Spectral realization x'_spec = U g(Λ) U^T x_in and variational realization x'_var, with dual-form gap delta_form_lens.
   • Stability/approximation indicators: ρ(K), err_spec∞, cond, and anisotropy alignment align.
3. Boundary
   The kernel function g(λ) is bounded and approximable on λ ∈ [ λ_min, λ_max ]; unit(x') = unit(x_in).

II. Terms & Variables

• Spectral decomposition: L = U Λ U^T, Λ = diag(λ_k), 0 = λ_1 ≤ … ≤ λ_max.
• Kernels & operators: K = g(L) = U g(Λ) U^T; diffusion g_diff(λ) = exp(−τ λ); Tikhonov g_tik(λ) = (1 + β λ)^{−1}.
• Focus kernel: g_focus(λ) = 1 + α • h(λ) (high-pass boost), e.g., h(λ) = 1 − exp(−τ λ) or h(λ) = λ / (1 + β λ).
• Defocus kernel: g_defocus(λ) = exp(−τ λ) or g_defocus(λ) = (1 + β λ)^{−1}.
• Anisotropic Laplacian: L_ani = B^T C_ani B, with edge-wise C_ani(e) using c_∥, c_⊥ or a tensor spectrum.
• Approximation & scaling: Chebyshev order order, spectral scaling λ_scale ∈ [0, λ_max]; polynomials T_k.
• Dual-form gap: delta_form_lens = || x'_spec − x'_var ||_2.

III. Postulates P715-*

• P715-1 (Stability upper bound): If g(λ) ∈ [0, 1 + ε], then ρ(K) ≤ 1 + ε (default ε ≤ 0.02).
• P715-2 (Dual forms in parallel): Every kernel must provide both spectral and variational realizations computed in parallel; record delta_form_lens.
• P715-3 (Anisotropy PSD): C_ani(e) is positive semidefinite with c_∥ ≥ c_⊥ ≥ 0, ensuring L_ani is symmetric positive definite.
• P715-4 (Visibility consistency): If W_vis / m_ij is enabled, build kernels on L^vis or L_ani^vis; vis does not alter dimensions/units.
• P715-5 (Auditable approximation): Polynomial approximations must report err_spec∞ = max_{λ∈[0,λ_max]} | g(λ) − \tilde g(λ) | and order.

IV. Minimal Equations S715-*

1. S715-1 (Unified spectral form)
   x'_spec = ( U g(Λ) U^T ) x_in, or, in spectral-measure form, ( ∫_{λ∈spec(L)} g(λ) d μ_L(λ) ) acting on x_in.
2. S715-2 (Defocus: diffusion / Tikhonov)
   • Diffusion: x' = exp(−τ L) x_in = ( ∑_{k=0}^∞ (−τ)^k L^k / k! ) x_in.
   • Tikhonov: x' = ( I + β L )^{−1} x_in = argmin_x ( (1/2) || x − x_in ||_2^2 + (β/2) x^T L x ).
3. S715-3 (Focus: high-pass boost / regularized deconvolution)
   • High-pass boost (non-blind sharpening): x' = ( I + α ( I − exp(−τ L) ) ) x_in.
   • Deconvolution: given blur kernel K_blur = g_blur(L),
     x' = argmin_x ( (1/2) || K_blur x − x_in ||_2^2 + (μ/2) x^T L^p x ),
     with spectral response g_focus(λ) = 1 / ( g_blur(λ)^2 + μ λ^p ) * g_blur(λ) (p ∈ {1, 2}).
4. S715-4 (Anisotropic kernels)
   • Anisotropic diffusion: x' = exp( −τ L_ani ) x_in or x' = ( I + β L_ani )^{−1} x_in.
   • Directional alignment: align = ( ∑_{e∈E_Π} cos^2⟨ e, t_hat ⟩ / |E_Π| ) (with t_hat from path/texture orientation fields).
5. S715-5 (Chebyshev approximation & numerics)
   • With \tilde L = (2 / λ_max) L − I, apply g(L) x ≈ ∑_{k=0}^{order} c_k T_k( \tilde L ) x.
   • Error bound: err_spec∞ ≤ sup_{λ∈[0,λ_max]} | g(λ) − \tilde g(λ) |; runtime error || x'_spec − x'_{cheb} ||_2 ≤ err_spec∞ || x_in ||_2.
6. S715-6 (Dual-form consistency assertion)
   • In quadratic variational cases, x'_var = ( I + β L_* )^{−1} x_in matches the spectral kernel, with L_* ∈ { L, L^vis, L_ani }.
   • Record delta_form_lens = || x'_spec − x'_var ||_2 and compare against contractual thresholds.

V. Metrology Pipeline M71-5 (Design → Approximate → Solve → Verify → Persist)

1. Kernel selection: choose among defocus / focus / anisotropic; set params and RefCond = { L_type, λ_max, scaling }.
2. Spectral design: construct g(λ) (pass/stop bands, roll-off); if using W_vis or L_ani, define on the corresponding operator.
3. Fast approximation: choose order and λ_scale; generate Chebyshev coefficients c_k or a Lanczos approximation.
4. Solve in dual forms:
   • Spectral: iterate polynomial–vector products to obtain x'_spec;
   • Variational: solve by primal–dual / CG to obtain x'_var (use CG/PCG for deconvolution).
5. Checks & stability: evaluate ρ(K), err_spec∞, delta_form_lens, align (if anisotropy enabled); compute u_c, U.
6. Persist & publish: write to manifest.lens: kernel.kind, g.hash, order, λ_max, err_spec∞, ρ(K), delta_form_lens, contracts.*, signature.

VI. Contracts & Assertions C71-5x (suggested thresholds)

• C71-51 (Spectral stability): ρ(K) ≤ 1 + ε (default ε ≤ 0.02).
• C71-52 (Approximation error): err_spec∞ ≤ 1e−3 or || x'_spec − x'_{approx} ||_2 ≤ 1e−3 || x_in ||_2.
• C71-53 (Dual-form gap): delta_form_lens ≤ 1e−3 || x_in ||_2.
• C71-54 (No overshoot in sharpening): overshoot = || x' − Π_{range(x_in)}(x') ||_2 / || x_in ||_2 ≤ 0.02 (when energy conservation is required).
• C71-55 (Anisotropy alignment): align ≥ 0.8; otherwise degrade to an isotropic kernel.
• C71-56 (Unit consistency): check_dim( x' − x_in ) = "[same]"; kernel unit(g) = 1.

VII. Implementation Bindings I71-5* (interfaces, I/O, invariants)

• design_spectral_kernel(kind, params, λ_max) -> { g, meta }
  out: callable/table; meta = { passband, rolloff, stability_bound }.
• chebyshev_apply( L|L^vis|L_ani, g, order, λ_max, x_in ) -> x'_spec, report
  report: err_spec∞ estimate, matvec_calls, time_ms.
• solve_variational( kind, L|L_ani, x_in, extras ) -> x'_var, report
  extras: K_blur (for deconvolution) or regularization order p.
• build_anisotropic_L( graph, orientation|paths, c_∥, c_⊥ ) -> L_ani, align, meta
  inv: c_∥ ≥ c_⊥ ≥ 0; returns align and condition-number estimate.
• compose_kernel_runtime( Φ_prev, K, mode, win ) -> Φ_new
• eval_delta_form_lens( x'_spec, x'_var ) -> delta_form_lens
• assert_kernel_contracts( ds, rules ) -> report
• emit_lens_manifest( results, policy ) -> manifest.lens
  Invariants: order ≥ 0; λ_max > 0; hash(*) is traceable; if visibility or anisotropy is enabled, record the corresponding hashes and parameters.

VIII. Cross-References

• Graph operators & kernel approximation: EFT.WP.STG.Dynamics v1.0, Chapter 4; numerical stability/rigidity: Chapter 9.
• View weights & occlusion gating: EFT.WP.STG.Lensing v1.0, Chapter 4.
• Path directions & anisotropic fields: EFT.WP.Particle.TopologyAtlas v1.0, Chapter 7.
• Runtime & dashboard metrics: EFT.WP.STG.Dynamics v1.0, Chapter 14.

IX. Quality & Risk Control

• SLI/SLO: track err_spec∞, delta_form_lens_p99, ρ(K), align_p50/p90, latency_p95, matvec_per_sample.
• Fallbacks: if ρ(K) violates bounds → shrink α/τ or increase β; if err_spec∞ is high → raise order or change spectral mapping; if align is low → degrade to isotropic.
• Audit: preserve sampled g(λ) curves with order/λ_max, evidence for err_spec∞, dual-form gap distributions and outliers, and runtime resource traces.

Summary

• Provides parallel spectral–variational realizations and stability criteria for focus/defocus/anisotropic lens kernels.
• Supplies Chebyshev fast approximation with auditable error paths.
• New manifest.lens keys: kernel.kind, g.hash, order, λ_max, ρ(K), err_spec∞, delta_form_lens, anisotropy.{ c_∥, c_⊥, align }, contracts.*, signature.