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arxiv: 2606.00968 · v1 · pith:3UOZRGLYnew · submitted 2026-05-31 · ⚛️ physics.optics · quant-ph

Fundamental Limit for One versus Two Point Sources Detection using Direct Imaging

Parth Hemant Darekar , Amit Kumar Jha , Michael R. Grace , Aqil Sajjad , Saikat Guha This is my paper

Pith reviewed 2026-06-28 16:55 UTC · model grok-4.3

classification ⚛️ physics.optics quant-ph
keywords direct imagingpoint source detectionBhattacharyya distancesub-Rayleigh regimeamplitude spread functionoptical resolutionweak incoherent sources
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The pith

The Bhattacharyya distance for one versus two point source detection via direct imaging has a leading term whose scaling with separation depends on whether the amplitude spread function has zeros.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper calculates the leading order term in the small-separation expansion of the Bhattacharyya distance between the intensity patterns produced by one and two weak incoherent point sources, using ideal direct imaging. Previous work missed the case distinction when the amplitude spread function vanishes at certain points. The calculation shows that the scaling with separation θ changes depending on the presence of zeros in the ASF, allowing accurate comparison to other measurement schemes and the quantum limit.

Core claim

For ideal direct imaging of weak incoherent sources, the leading term of the Bhattacharyya distance in the sub-Rayleigh regime scales as a specific power of θ that differs according to whether the amplitude spread function has zeros; this completes the analysis for arbitrary ASF and corrects earlier scaling reports.

What carries the argument

The Bhattacharyya distance computed from the Poisson statistics of photon counts on an ideal focal plane array, expanded to leading order in small source separation θ.

If this is right

  • The performance of direct imaging can now be compared precisely to other schemes for any given ASF.
  • Gaussian and Sinc ASFs show good agreement between the analytic leading term and numerical results.
  • The same ASF-dependent scaling appears in separation estimation and change detection tasks.
  • Results apply specifically in the sub-Rayleigh regime for weak sources.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar scaling distinctions may appear in other quantum optical detection tasks when the measurement is intensity-based.
  • Extending the analysis to non-ideal detectors or coherent sources could reveal further dependencies.
  • Testing the predicted scalings with actual optical setups would validate the small-θ approximation.

Load-bearing premise

The sources must be weak and mutually incoherent so that the intensity pattern is a simple sum and photon arrivals follow independent Poisson statistics.

What would settle it

Numerical computation of the Bhattacharyya distance for a specific ASF with a zero, at very small θ, should match the predicted leading power of θ rather than the power for a zero-free ASF.

Figures

Figures reproduced from arXiv: 2606.00968 by Amit Kumar Jha, Aqil Sajjad, Michael R. Grace, Parth Hemant Darekar, Saikat Guha.

Figure 1
Figure 1. Figure 1: Schematic of the setup. Under H1, we have a weak thermal source (in blue) at the origin. Under H2, we have two equally bright, weak thermal sources (in red) with an image plane separation of θ that is scaled to the PSF width. For small separation, the intensity distributions of light on the image plane under each hypothesis become difficult to distinguish. Here ψ(x) is the ASF. For ease of notation, we lab… view at source ↗
Figure 2
Figure 2. Figure 2: Bhattacharyya distance for ideal direct imaging for the Gaussian [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bhattacharyya distance for ideal direct imaging for the Sinc Amplitude [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

We consider the task of distinguishing between a single weak incoherent optical point source and two weak incoherent optical point sources located symmetrically about the first source. $\theta$ is the separation between the two point sources scaled to the Point Spread Function (PSF) width in the image plane. Using an ideal focal plane array of intensity detectors (ideal direct imaging), we quantify the performance using the Bhattacharyya distance and find the scaling of its leading order term in terms of $\theta$ in the sub-Rayleigh regime. A suite of previous analyses of this problem lacked a comprehensive analysis for when the amplitude spread function (ASF) of the imaging system has zeros and reported a scaling that we find to be incorrect. We complete this analysis by explicitly calculating the leading order term of the Bhattacharyya distance for ideal direct imaging with any ASF, for small $\theta$ and show the difference in scaling based on the presence or absence of zeros in the ASF. This is similar to the ASF dependent performance in the task of estimating the separation between the two point sources and the task of detecting a change to an object. We then apply our results to the specific example of a Gaussian and a Sinc ASF and show good agreement with numerical calculations. Our results allow the accurate comparison of other measurement schemes with ideal direct imaging, and to the quantum limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript analyzes the binary hypothesis test of distinguishing one versus two weak incoherent point sources separated by scaled distance θ using ideal direct imaging (focal-plane intensity detection). It derives the leading-order term in the small-θ expansion of the Bhattacharyya distance for arbitrary amplitude spread functions (ASFs), demonstrating that the scaling differs according to the presence or absence of zeros in the ASF. The general result is specialized to Gaussian (zero-free) and Sinc (with zeros) ASFs and shown to agree with numerical evaluation; the work positions the result as a benchmark against other schemes and the quantum limit.

Significance. If the central derivation holds, the paper supplies an explicit, ASF-general leading-order expression that corrects an incomplete scaling reported in prior work and parallels ASF-dependent behavior already known for separation estimation. The closed-form result plus numerical confirmation for two representative ASFs constitutes a concrete, usable benchmark for comparing direct imaging to quantum-optimal or other measurement strategies in the sub-Rayleigh regime.

minor comments (3)
  1. [Abstract] The abstract refers to 'a suite of previous analyses' that reported an incorrect scaling; adding the specific citations in the introduction would allow readers to locate the discrepancy immediately.
  2. In the derivation of the Bhattacharyya distance, state the precise order of the retained term (e.g., O(θ^4) or O(θ^2)) immediately after the general expression so that the claimed difference in scaling is unambiguous without consulting the appendix.
  3. Figure captions for the numerical comparisons should explicitly note the value of the source strength parameter and the range of θ over which the leading-order analytic curve is plotted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the central derivation supplies an explicit ASF-general leading-order expression and serves as a usable benchmark. We are pleased that the work is recommended for acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation is direct from Bhattacharyya definition and Taylor expansion

full rationale

The central result is an explicit leading-order calculation of the Bhattacharyya distance between the one-source and two-source hypotheses under the standard weak-incoherent Poisson imaging model. The paper applies the Bhattacharyya distance definition to the intensity pattern, performs a small-θ expansion, and obtains closed-form scalings that differ according to whether the ASF has zeros. This chain begins from the model equations and the distance definition; no step reduces by construction to a fitted parameter, a self-citation, or an ansatz imported from prior work by the same authors. The numerical checks for Gaussian and Sinc ASFs are independent verifications of the analytic expressions rather than the source of the result. The paper corrects earlier scalings in the literature rather than relying on them.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; full paper not available so ledger is minimal. No free parameters or invented entities mentioned. Relies on standard definition of Bhattacharyya distance.

axioms (1)
  • standard math Bhattacharyya distance quantifies distinguishability between two hypotheses for the observed intensity pattern
    Used as the performance metric for the detection task

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discussion (0)

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Reference graph

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