Incoprea research · leads O · photonics and imaging measurement · 3 November 2026

Two hundred leads on photonics and imaging measurement

This page continues the programme into instrumentation. The earlier parts dealt with derived catalogue columns, tail indices and material constants; the same failure appears in photonics under a different name. A resolution number is quoted without the PSF that produced it, a contrast is quoted without the number of coherence areas that were averaged, a detection limit is quoted without the noise model that turns a flux into a probability. In each case a model is doing part of the work and is not being reported, so the number cannot be checked by anyone who does not already have the model.

The three tests below were chosen because each isolates one of those hidden models. T1 asks how a resolution claim degrades when the forward model is slightly wrong, and finds that a 2% error in PSF width moves a 0.5σ separation by 26 to 38%. T2 asks what “the number of coherence areas” means, and finds that the familiar 1/√M law holds only for independent areas and that correlated areas impose a contrast floor. T3 asks what a 5σ detection limit is, and finds that the same nominal limit corresponds to a detection probability between 0.49 and 1.00 depending on how the threshold is computed. The proof states the condition under which a linear forward model determines its object at all, and why the answer does not depend on the estimator.

Part 1 · the leads

The four sections, and what the tags mean

test means the sentence could be settled by computation or by re-analysis of public data without new hardware. note means it needs a new measurement, instrument access, or a survey of practice. The tags are about how a claim could be settled, not about whether it is true; every lead on this page is untested.

AC. Resolution claims: what a point-spread function can and cannot support

Fifty leads on the gap between a PSF and a resolution claim: what the width, the wings, the sampling and the prior each contribute, and which of them a measurement actually fixes.

AD. Coherence, speckle and the statistics of a scattering measurement

Fifty leads on the statistics that a scattering measurement rests on: coherence areas, contrast, correlation and the assumptions that turn an intensity fluctuation into a number.

AE. Detectors: noise models, dynamic range and the limits of calibration

Fifty leads on the detector as a measuring instrument: which noise terms scale with what, where the dynamic range actually ends, and what a calibration can and cannot transfer.

AF. Computational imaging: priors, hallucination and the information actually recorded

Fifty leads on reconstruction: how much of an image is data, how much is prior, and how the difference could be measured rather than asserted.

Part 2 · tests run

Tests run

Three computations, all synthetic, each with a known generating truth. Code: incoprea_nov3/code/photonics_leads_compute.py. Output: incoprea_nov3/results/photonics_leads.json. Seed 20261103, numpy 2.5.4. Re-run with python3 photonics_leads_compute.py; the numbers reproduce exactly.

T1. How much of a super-resolution gain survives a slightly wrong forward model

Two equal point sources are imaged by a Gaussian PSF of width σ = 1 on a 321-point grid spanning ±4σ, with a separation s expressed in units of σ. The analysis fits two Gaussian sources with the width fixed at σ_m = (1+ε)σ and one free amplitude; the data are symmetrised so the equal-amplitude fit is exact. The reported resolution gain is G = σ/s, so a separation of 0.5σ is a claimed 2× gain. The question is what a small error in the width does to the recovered separation.

Read the sign of ε carefully. A model that is narrower than the truth (ε < 0) pushes the recovered separation up: at s = 0.4σ a 2% error gives 39.0%, and at 0.3σ the same error gives 63.9%. A model that is wider (ε > 0) pushes the separation down and, below about 0.7σ, eventually collapses the pair to a single source: the fit returns s = 0, which is honest (“unresolved”) but is a different claim from the one the analysis set out to make. At s = 1.0σ the same 2% error gives only 5.7% and -6.5%: model error matters more, in relative terms, the harder the resolution claim.

With noise, and the size of the effect relative to it

At s = 0.5σ with peak SNR 500 and 400 noise draws per case, the statistical error of the recovered separation is 0.0026σ, or 0.53%. The model error is far larger.

A 2% width error produces a bias of 25.8% or -38.3%, which is 49 and 73 times the photon-noise error. The claimed gain of 2× is reported as 1.59× or 3.24×: one is 80% of the claim, the other is 62% above it. A 5% width error either shrinks the reported gain to 1.30× from the true 2× or collapses the pair in 100% of draws.

A wrong functional form, not just a wrong width

The PSF need not be Gaussian. With a Lorentzian object of the same FWHM fitted by a Gaussian of fixed width, the recovered separation is driven upward by the mismatch in the wings:

At s = 0.4σ the fixed-width fit returns 1.26σ, a bias of 216%. Fitting the width as a free parameter halves the bias but does not remove it (107% at 0.4σ, 52% at 0.7σ). The reason is that a Gaussian cannot match a Lorentzian at both the core and the wings, and the fit spends the difference on separation. A model-error budget that considers only the width therefore understates the problem.

The result, stated plainly. At a claimed 2× gain, a 2% error in PSF width moves the recovered separation by +26% or −38%, against a photon-noise error of 0.5%. A 2× super-resolution gain does not survive a 2% model error in this model. What survives is the part of the gain that the model can support, and that part has to be measured by perturbing the model, not assumed.

T2. Speckle contrast and what “number of coherence areas” means

A detector that averages M independent coherence areas sees a contrast c = 1/√M. That is exact for exponential intensity statistics and it is the relation most measurements use in reverse to report a number of coherence areas. Simulating 200,000 pixels per case:

The law holds to the third decimal across two and a half decades (c√M between 0.997 and 1.003). What does not hold is the inversion: the estimator 1/c² has a spread that depends on the number of pixels used to estimate c. Median and 16–84% interval of the inferred M:

At M = 64 and 200 pixels the interval is roughly ±10%, so a reported “64 coherence areas” from a 200-pixel patch is not distinguishable from 58 or 71. At M = 4 the same 200 pixels give ±12%. The number of coherence areas is a property of the field; the precision of the reported number is a property of the estimator and the patch size, and the two are usually conflated.

Correlated areas: the contrast floor

The 1/√M law assumes independence. For a circular complex Gaussian field whose coherence matrix is equicorrelated, C = (1−ρ)I + ρJ, the averaged intensity has contrast² = tr(C²)/M², which tends to ρ as M grows. The floor is ρ, not zero. Simulated contrast (exact values agree to within 0.005 in every cell):

At ρ = 0.20 and M = 256 the measured contrast is 0.210, against 0.062 predicted by 1/√M — a factor 3.4. At ρ = 0.50 the contrast stalls at 0.50 no matter how many areas are averaged. Any inference of M from contrast in a correlated field therefore overestimates M, and the error grows with M: more averaging buys less than the law promises.

A finite pixel is not a number of coherence areas

The third ambiguity is geometric. A square pixel of side L over a speckle field with intensity correlation exp(−r²/ℓ²) has contrast² = L⁻⁴[∫(L−|u|)exp(−u²/ℓ²)du]², so for large L the contrast is √πℓ/L and the effective number of areas is (L/ℓ)²/π, not (L/ℓ)². A 2D FFT simulation with ℓ = 4 grid cells and 48 independent tiles:

The simulation matches the integral to about 1% at every L. The naive contrast (L/ℓ)⁻¹ is low by a factor that rises to √π ≈ 1.77 at large L, so the naive count (L/ℓ)² overestimates M_eff by π: at L/ℓ = 8 the naive count gives 64 areas, the exact count gives 20.4, and the measured contrast gives 23.6. The coefficient depends on the aperture and on the correlation shape, so “coherence areas per pixel” has no universal definition. What has a definition is the contrast, and it is the contrast that should be reported.

The result, stated plainly. c = 1/√M is exact and the inversion is not: 200 pixels determine M to about 10% only below M ≈ 60, and the relation fails altogether when the areas are correlated, where contrast stops at ρ. A “number of coherence areas” derived from contrast without stating independence, pixel size and correlation is not a measurement of the field.

T3. The detection limit versus integration time, and what “5σ” is a probability of

Let Φ be the source rate at the detector, t the exposure, D the dark or background rate per pixel, N the number of pixels and σ_r the read noise per pixel. With SNR = Φt/[Φt + Dt + Nσ_r²]^1/2, the flux that reaches SNR = k solves a quadratic:

Φmin(t) = [k² + √(k⁴ + 4k²(Dt + Nσr²))] / (2t)

Three configurations: A, read-noise only (σ_r = 1 e−, N = 1, D = 0); B, sCMOS-like (σ_r = 5 e−, N = 16, D = 0.01 e−/s/px); C, background-dominated (σ_r = 2 e−, N = 4, D = 1 e−/s/px, so read and background variance are equal at t = 16 s).

The local slope is −1 while the read-noise term dominates and approaches −1/2 once the background term does. Configuration C gives -0.875 over 1–100 s and -0.555 over 100–10,000 s. A has no background term at all and stays at −1; B is still read-limited at t = 10,000 s because its read/background crossover is 40000 s. There is no interior optimum. Φ_min falls monotonically with t, so the longest exposure the full well allows is always the best detection limit, and an “optimal integration time” quoted without a saturation or drift constraint is not optimal for this noise model.

What 5σ means

The same Φ at SNR = 5 is a different detection probability depending on how the threshold is set. With a blank-sky threshold k[N(Dt+σ_r²)]^1/2 the source shot noise is not in the threshold; with a total-noise threshold it is. At t = 1 s:

The nominal 5σ flux in the read-noise-limited case A is detected with probability 1.00 against a blank-sky threshold — the threshold is 5 e− while the flux is 26.0 e− — and with probability 0.49 against a total-noise threshold. In case B the two conventions give 0.72 and 0.51. Neither is 0.95. The flux needed for 95% completeness is 1.22× or 1.38× the nominal limit in case B, and 1.74× in case A. The two conventions bracket the familiar Gaussian rule: in the shot-noise limit 95% completeness needs (1 + 1.645/5)² = 1.766×, close to the measured 1.74×.

The probability curve makes the point without the table. For case B at t = 1 s, the simulated detection probability rises from 0.02 at half the nominal flux to 0.96 at 1.25× and 0.999 at 1.5×. A claim that a source is “detected at 5σ” is a claim about a threshold crossing, and the probability of that crossing at the flux the formula calls 5σ depends on a convention that is almost never stated.

The result, stated plainly. Φ_min ∝ t⁻¹ while read noise dominates and t^−1/2 once the background does, with no interior optimum; and the nominal 5σ flux is a 49% or 100% detection depending on the threshold convention. The detection limit is a curve indexed by a probability, not a single number.

Part 3 · proof

P1. What a forward model can and cannot invert

Status: proved. The propositions below are analytic. The table at the end is a numerical check of the rank and conditioning statements, computed in the same script as the tests.

Assumptions

1. A1. The measurement is linear in the object: y = Ax + e, with A a known real m×n matrix and x ∈ Rⁿ.
2. A2. For identifiability the noise is zero. For the perturbation statement e is zero-mean with covariance σ²I and A has full column rank.
3. A3. No prior: x ranges over all of Rⁿ, not over a sparse, parametric or bounded subset. A prior changes the answer, and that is the point of stating the assumption.
4. A4. The estimator is deterministic. A randomised estimator is still a function of y, so it does not evade Proposition 2.

Proposition 1 (identifiability)

With e = 0, the map x ↦ Ax is injective if and only if rank(A) = n. If rank(A) = r < n, then for every y in the range of A the solution set is an affine subspace x_p + ker(A) of dimension n − r, and no measurement can determine the component of x in ker(A). The linear functionals of x that are determined are exactly the rows of A, a space of dimension r.

Proof. If rank(A) = n then Ax = Ay implies A(x−y) = 0, and a full-column-rank A has trivial kernel, so x = y. If rank(A) < n, choose any nonzero v ∈ ker(A); then x and x + v are distinct and produce the same y, so the map is not injective. The solution set for a consistent y is the coset x_p + ker(A), whose dimension is n − r by rank–nullity. A functional φ^Tx is determined for all x exactly when φ annihilates ker(A), i.e. when φ lies in the row space. ∎

Corollary. The minimum-norm least-squares solution x̂ = A⁺y is one representative of that coset. It is the solution closest to the origin, which is a choice of prior (a zero-mean Gaussian prior with infinite variance in the row space and zero variance in the kernel), not a consequence of the data.

Proposition 2 (bandlimited imaging)

For a shift-invariant system on an n-point grid, A is circulant and is diagonalised by the discrete Fourier transform, so rank(A) = #{k : H(k) ≠ 0}, where H is the optical transfer function. If H(k) = 0 for |k| > k_c, then nullity = n − (2k_c + 1) in one dimension, and the null space is spanned by the complex exponentials outside the passband. Two objects that differ only outside the passband produce identical data; any deterministic estimator returns the same value for both, so at most one of them can be correct. Every spatial frequency in a reconstruction above the cutoff was supplied by the prior.

Proof. A circulant matrix has eigenvalues H(k) with the Fourier modes as eigenvectors, so its rank is the number of nonzero eigenvalues. The kernel is the span of the eigenvectors with H(k) = 0, which is exactly the set of frequencies outside the passband. The last sentence follows because the data for the two objects are equal, so no function of the data can distinguish them. ∎

Remark. This is why no algorithm, linear or not, can recover the missing band from noise-free data: the missing band is not weakly determined, it is absent. A real OTF has no exact zeros, so the kernel is trivial in exact arithmetic; the proposition then applies to the numerical kernel, the directions whose singular values are below the noise.

Proposition 3 (perturbation)

Let A be invertible, y ≠ 0, and suppose the reconstruction uses A + δA on data y + δy. If κ(A)||δA||/||A|| < 1, then

||x̂ − x|| / ||x|| ≤ [κ / (1 − κ||δA||/||A||)] · (||δA||/||A|| + ||δy||/||y||)

Proof. x̂ − x = (A+δA)⁻¹(y+δy) − A⁻¹y = −(A+δA)⁻¹δA x + (A+δA)⁻¹δy. Taking norms gives ||x̂−x|| ≤ ||(A+δA)⁻¹||(||δA||||x|| + ||δy||). The Neumann bound gives ||(A+δA)⁻¹|| ≤ ||A⁻¹||/(1 − ||A⁻¹δA||), and ||A⁻¹δA|| ≤ κ||δA||/||A||. Dividing by ||x|| and using 1/(||A||||x||) ≤ 1/||y|| yields the stated bound. ∎

Corollary (Gaussian PSF). For a Gaussian PSF of width σ pixels the OTF is H(k) = exp(−σ²k²/2), so κ = H(0)/H(π) = exp(π²σ²/2). At σ = 2 pixels, κ ≈ 3.6×10⁸; at σ = 3 pixels it exceeds 10¹⁹, beyond double precision. The highest spatial frequency is invertible only if the model is known to a relative accuracy of 1/κ, which is the honest form of the statement that super-resolution is prior-limited rather than photon-limited once the PSF is wider than a pixel or two.

Connection to T1. A two-source separation s is a near-null direction: the two columns of the fitting basis have condition number ≈ 2.83/s, listed in the third table below. Proposition 3 bounds the damage from a model error in that direction, and T1 measures it: a 2% error at s = 0.5σ produces 26 to 38%, of the same order as the linear bound with κ = 7.1 at s = 0.4σ, and the extra factor comes from the nonlinear way the fit trades separation against width.

Numerical check

Three forward models, all computed in the same script. First, an ideal low-pass OTF with 2k_c+1 passing bins on n = 129 samples; the rank and nullity are exact integers and the null space is the stated dimension:

Second, a Gaussian PSF, which has no exact spectral zeros and therefore full rank, but a condition number that grows exponentially with the square of the PSF width. The SVD column is blank once κ exceeds 10¹², where double precision can no longer resolve the smallest singular value:

Third, the two-source basis from T1, which is full rank at every separation but increasingly ill-conditioned as the separation falls:

The proof, stated in one line. A linear forward model determines its object if and only if its matrix has full column rank; otherwise only the row-space component is determined, the kernel component is supplied by the prior, and near-null directions amplify model error by the condition number.

Part 4 · limits

Limits

• Everything measured here is simulated. The generating truth is known because it was written, not because it was observed. The tests measure how estimators behave under stated models; they say nothing about the prevalence of those models in real instruments.
• T1 uses a symmetric, equal-amplitude, one-dimensional model with a fixed width. Real doublets have unequal amplitudes, background and pixellation. The collapse and inflation effects are real in this model; their numerical sizes are model-specific.
• T2 assumes stationary, fully developed speckle. The correlation models (equicorrelated complex Gaussian, Gaussian intensity correlation) were chosen because they are exactly solvable. A real scattering medium has a correlation function that is neither, and the coefficient √π is not universal.
• T3 ignores saturation, drift, cosmic rays and dead time. Those are exactly the effects that make the longest exposure not the best one in practice. The claim that there is no interior optimum is a claim about the stated noise model only.
• The three tests use one seed. Seed 20261103 is recorded and the numbers reproduce exactly on re-run, but the Monte Carlo standard errors are not quoted in every cell. Where a number is a single draw it carries sampling error of the size shown by the replicate spreads.
• The proof is exact but narrow. It assumes linearity, a known matrix and no prior. It does not say that priors are illegitimate; it says that what a prior supplies is not measured, and that the difference is computable from the null space. Nonlinear forward models need a local version of the same argument, which is not given here.
• The 200 leads are untested by construction. Several are probably known in the optics and metrology literature under other names; no literature search was run. Their value is as a falsifiable checklist, not as findings.
• No claim that any named instrument or paper is wrong. The failure modes discussed here are ones this programme can construct and measure in simulation, not accusations against particular measurements.

If one sentence survives: a resolution, a coherence area and a detection limit are all model-dependent numbers, and in each case the model error is larger than the photon noise at the point where the claim is most ambitious — 26 to 38% from a 2% PSF error, a contrast floor at ρ, and a nominal 5σ that is a 49% to 100% detection depending on a convention nobody states.

Incoprea Research · leads O · 3 November 2026. Two hundred leads, three simulations, one proof, no measured data. Code and JSON in incoprea_nov3/code/ and incoprea_nov3/results/. No novelty claimed; no systematic literature search.