26-EFT.WP.STG.Lensing v1.0 | Chapter 7 — De-Lensing and Inversion (Deconvolution / Regularization / Sparsity)

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Chapter 7 — De-Lensing and Inversion (Deconvolution / Regularization / Sparsity)

One-sentence goal: Provide robust inversion for blur/occlusion effects induced by lens operators through parallel spectral–variational formulations—combining deconvolution, regularization, and sparsity—with executable guidance for parameter selection, numerical stability, and contract auditing.

I. Scope & Objects

1. Inputs
   • Graph & operators: G = (V, E, w); L_* ∈ { L, L^vis, L_ani } (see Chapters 4/5), with unit(L_*) = 1, dim(L_*) = [1].
   • Lens kernel: K = g(L_*), or parameterized g(λ; θ) (focus/defocus/anisotropy, see Chapter 5).
   • Observations: y = K x_true + v (default v ~ N(0, σ^2 I); switchable to Poisson/Laplace).
   • Priors & operators: analysis operators Ψ ∈ { B, L_*^{p/2}, W_wavelet, D_P^T }; synthesis dictionary D (incl. path dictionaries; see Chapter 6).
2. Outputs
   • Inversion estimate: x_hat; optionally kernel parameters θ_hat (semi-blind/blind inversion).
   • Dual-form gap: delta_form_inv = || x_spec_inv − x_var ||_2.
   • Quality & uncertainty: res = || K x_hat − y ||_2, u_c(x_hat), cond(K).
3. Boundary
   unit(x_hat) = unit(y); if visibility/anisotropy is enabled, define on L_* and persist hash(*).

II. Terms & Variables

1. Spectral inverse kernel: g_inv(λ; μ, r) = g(λ) / ( g(λ)^2 + μ r(λ) ), with r(λ) ∈ { 1, λ^p }.
2. Variational objective: J(x) = (1/2) || Kx − y ||_2^2 + μ R(x), with
   R(x) ∈ { (1/2) || L_*^{p/2} x ||_2^2, || Ψ x ||_1, ElasticNet }.
3. Sparsity formulations
   • Analysis: min_x (1/2) || Kx − y ||_2^2 + λ || Ψ x ||_1.
   • Synthesis: min_α (1/2) || K D α − y ||_2^2 + λ || α ||_1, with x = D α.
4. Noise-robust losses: ρ_δ(•) is Huber, or D_KL(y || Kx) for Poisson.
5. Algorithms: CG/PCG (quadratic), FISTA/ADMM/PDHG (non-smooth); Chebyshev approximations for matrix–vector products with K, K^T.

III. Postulates P717-*

• P717-1 (Parallel dual forms): Every inversion must produce both the spectral form x_spec_inv and the variational form x_var, logging delta_form_inv.
• P717-2 (Stable regularization): g_inv(λ; μ, r) must be bounded and monotone in μ; recommend μ > 0 and r(λ) ≥ 0.
• P717-3 (Operator traceability): For K = g(L_*), persist kind / θ / hash / λ_max / order; record visibility/anisotropy choices.
• P717-4 (Noise matching): The data-fidelity loss must match the noise model, with declared unit(y) and dim(y).
• P717-5 (Safe step sizes): First-order methods satisfy σ τ ≤ 1 / ||A||^2, where A is the KKT linear operator.
• P717-6 (Explicit path measure): When using path sparsity or TV, write ( ∫_{gamma(ell)} • d ell ) with explicit L_gamma.

IV. Minimal Equations S717-*

1. S717-1 (Spectral de-lensing: unified Wiener/Tikhonov)
   • Spectral form: x_spec_inv = ( U g_inv(Λ; μ, r) U^T ) y, with L_* = U Λ U^T.
   • Typical choices: r(λ) = 1 (zeroth-order), r(λ) = λ^p (p ∈ {1, 2}) for smoothness/curvature.
2. S717-2 (Quadratic variational equivalence)
   x_var = argmin_x ( (1/2) || Kx − y ||_2^2 + (μ/2) x^T L_*^p x ), yielding
   x_var = ( K^T K + μ L_*^p )^{−1} K^T y, equivalent to x_spec_inv when co-diagonalizable.
3. S717-3 (ℓ1-sparse inversion: analysis/synthesis)
   • Analysis: x_var = argmin_x ( (1/2) || Kx − y ||_2^2 + λ || Ψ x ||_1 );
     prox_{λ||Ψ•||_1}(z) realized via soft-thresholding in the Ψ domain.
   • Synthesis: α_hat = argmin_α ( (1/2) || K D α − y ||_2^2 + λ || α ||_1 ), x_var = D α_hat.
     Equivalence condition: if D is complete/orthogonal and Ψ = D^T, the two are equivalent.
4. S717-4 (Robust/Poisson data terms)
   • Huber: min_x ∑_i ρ_δ( (Kx − y)_i ) + μ R(x);
   • Poisson: min_x D_KL(y || Kx) + μ R(x), D_KL(a||b) = ∑ ( a log(a/b) − a + b ).
5. S717-5 (Semi-blind kernel estimation)
   • min_{x, θ∈Θ} (1/2) || U g(Λ; θ) U^T x − y ||_2^2 + μ R(x) + γ R_θ(θ);
   • First-order condition: ∂/∂θ J = (∂g/∂θ)(Λ)^T ⊙ diag( U^T x, U^T r ) + γ ∂R_θ, with r = Kx − y.
6. S717-6 (Dual-form gap & conservation)
   • delta_form_inv = || x_spec_inv − x_var ||_2;
   • Energy control: E_out = || x_hat ||_2^2 ≤ (1 + ε) || y ||_2^2 (deconvolution overshoot suppression).

V. Metrology Pipeline M71-7 (Ready → Model → Solve → Verify → Persist)

1. Ready: fix K = g(L_*), noise model, prior R, and the pathway (analysis/synthesis); declare RefCond = { L_type, vis/ani, λ_max, order }.
2. Spectral design: select g_inv(λ; μ, r) based on R; for semi-blind, initialize θ and constraint set Θ.
3. Approximations & operators: implement matvec for K, K^T, g_inv(L_*) via Chebyshev/Lanczos; build Ψ / D.
4. Solve in dual forms:
   • Spectral: x_spec_inv ← g_inv(L_*) y;
   • Variational: use PCG for quadratic, FISTA/ADMM/PDHG for L1/TV, until res or relative change converges.
5. Parameter selection: choose μ / λ via discrepancy (target res ≈ κ σ √|V|), L-curve, or GCV; alternate (x, θ) for semi-blind.
6. Checks: evaluate delta_form_inv, cond(K^T K + μ L_*^p), E_out, robust-loss descent rate, and u_c(x_hat).
7. Persist:
   manifest.lens.inverse.* = { kernel.hash, kind, θ, μ, λ, prior, solver, iters, res, delta_form_inv, err_spec∞, contracts.*, signature }.

VI. Contracts & Assertions C71-7x (suggested thresholds)

• C71-71 (Dual-form gap): delta_form_inv ≤ 1e−3 || y ||_2 (quadratic); relax to 3e−3 for L1/TV.
• C71-72 (Stability): ρ( g_inv(L_*) K ) ≤ 1 + 0.02; cond( K^T K + μ L_*^p ) ≤ 1e6.
• C71-73 (Noise matching): for Gaussian noise, res_p50 ≈ σ √|V| ± 10%; for Poisson, χ²-normalized residual in [0.8, 1.2].
• C71-74 (Sparsity consistency): Jaccard similarity of supports between analysis/synthesis J ≥ 0.7 (when ||x||_0 is comparable).
• C71-75 (Parameter smoothness): semi-blind kernel drift || θ_t − θ_{t−1} || ≤ θ_guard; violations trigger rollback.
• C71-76 (Units & dimensions): check_dim( Kx − y ) = "[unit(y)]", check_dim(R) = "[1]".

VII. Implementation Bindings I71-7* (interfaces, I/O, invariants)

• invert_lens_spectral( L_*, g, μ, r, y ) -> x_spec_inv, report
  report: err_spec∞, matvec_calls, time_ms.
• invert_lens_variational( K, prior, params, y, solver ) -> x_var, report
  prior ∈ { "L2(p)", "L1-analysis(Ψ)", "L1-synthesis(D)", "ElasticNet", "TV_path(B_P)" }.
• select_reg_param( y, K, prior, rule ) -> { μ | λ }, diag (rule ∈ { discrepancy, Lcurve, GCV }).
• estimate_kernel_params( U, Λ, y, x_init, Θ, R_θ ) -> θ_hat, diag (semi-blind).
• compose_path_prior( G, Γ_spec ) -> { B_P | D_P }, meta (see Chapter 6).
• eval_delta_form( x_spec_inv, x_var ) -> delta_form_inv
• assert_inverse_contracts( ds, rules ) -> report
• emit_lens_manifest( results, policy ) -> manifest.lens
  Invariants: λ_max > 0; order ≥ 0; μ, λ ≥ 0; if anisotropy/visibility is enabled, record hash(L_*) and parameters.

VIII. Cross-References

• Lens kernels & spectral realizations: Chapter 5; feature lenses & path sparsity: Chapter 6.
• Graph operators & kernel approximations: EFT.WP.STG.Dynamics v1.0, Chapter 4; numerical stability & rigidity: Chapter 9.
• Path/topology priors: EFT.WP.Particle.TopologyAtlas v1.0, Chapters 7/9 (paths and spectral coordinates).
• Runtime & dashboard fields: EFT.WP.STG.Dynamics v1.0, Chapter 14.

IX. Quality & Risk Control

1. SLI/SLO: delta_form_inv_p99, residual_ratio, cond_p95, PSNR_gain (or SNR_out / SNR_in), latency_p95, matvec_per_iter.
2. Fallbacks:
   • Excessive gap → increase μ/λ or switch to L2;
   • Ill-conditioning → precondition / raise regularization order or narrow passband;
   • Semi-blind non-convergence → freeze θ and revert to non-blind;
   • Overshoot/ringing → add band limits or switch to TV_path.
3. Audit: persist samples of g(λ) and g_inv(λ), parameter trajectories θ_t, convergence curves, contract pass rates and outliers, and the manifest signature chain.

Summary

• Unifies de-lensing as spectral inversion + variational regularization + sparse priors under dual forms.
• Provides implementable formulas for L2 / L1 / TV / semi-blind, with parameter selection and stability bounds.
• For runtime, defines manifest.lens.inverse.*, delta_form_inv, and an auditable SLI/SLO and rollback pathway.