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arxiv: 2604.05199 · v2 · pith:MXGVUZECnew · submitted 2026-04-06 · 🌌 astro-ph.HE · astro-ph.IM· gr-qc

Approximating the Fourier Transform of Ring-Like Images: the Focal Expansion

Filip Niewi\'nski , Michael D. Johnson This is my paper

Pith reviewed 2026-05-10 18:35 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.IMgr-qc
keywords Fourier transform approximationHankel transformring-like imagesphoton ringinterferometryblack hole imagingfocal expansion
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The pith

A single focal expansion term approximates the Fourier transforms of ring-like images accurately across small and large spatial frequencies.

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

The paper develops a method to approximate the 2D Fourier transform of functions that are concentrated in rings, such as images of black hole photon rings. It reduces the transform to Hankel transforms of angular modes and approximates those with a focal expansion. This single approximation stays reliable across all frequencies, unlike previous methods that work only in limited regimes. The leading term alone works well for many ring-like shapes. This helps model interferometric signals for experiments like the Black Hole Explorer.

Core claim

The authors derive a focal expansion that approximates the Hankel transforms arising in the Fourier transforms of radially symmetric or ring-like 2D functions. The leading term of this expansion provides a global approximation that is accurate at both small and large spatial frequencies for a wide class of such functions, as demonstrated through examples and application to toy models of photon rings.

What carries the argument

the focal expansion, a series approximation applied to the Hankel transforms of each angular mode that captures the transform behavior from low to high spatial frequencies in one expression

If this is right

  • The method yields a single formula for interferometric visibilities of ring-like sources that works at both low and high baselines.
  • It removes the need to switch between moment expansions at small frequencies and asymptotic forms at large frequencies.
  • For photon-ring models it supplies the interferometric response directly, aiding analysis for missions targeting that feature.
  • The approach applies to any 2D radially concentrated image without requiring separate low-frequency and high-frequency treatments.

Where Pith is reading between the lines

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

  • The same style of expansion could be tested on 3D Fourier transforms or other oscillatory integrals that appear in wave propagation problems.
  • Direct comparison against calibrated visibility data from existing or future arrays would show whether the approximation supports quantitative ring-radius measurements.
  • Hybrid schemes that add the focal term to standard numerical integrators might extend the range to mildly non-ring images.

Load-bearing premise

The input images must be radially concentrated ring-like functions, and the leading focal term by itself must deliver sufficient accuracy without higher-order terms or per-case adjustments.

What would settle it

Numerically compute the exact 2D Fourier transform of a Gaussian ring profile or similar test function, then compare it to the leading focal term output over a dense grid of spatial frequencies from near zero to large values; persistent low relative error across the full range would support the claim.

Figures

Figures reproduced from arXiv: 2604.05199 by Filip Niewi\'nski, Michael D. Johnson.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Two possible extensions of a radial function to the [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Example radial profiles for an annulus ring, with polynomial modulation given by eq. (III.13) and for several different [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The Fourier transforms of several angularly symmetric annulus rings, described via [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Log-log plots showing the absolute error of the focal expansion applied to an annulus ring, for several approximation [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of the method from [13], both the full and Taylor-expanded versions described by the authors, to the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The density plot of an annulus ring with a rudimen [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The radial profiles of the non-vanishing angular [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison of the exact Fourier transform of the fig. (10) fuzzy ellipse to the LO and SLO focal expansion, at three [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Log-log plots showing the absolute error of the focal expansion applied to the fig. (10) fuzzy ellipse for several angles in [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Radial profiles of the first two angular modes of a Gaussian ring at three representative values of the standard deviation [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Density plot of an example Gaussian ring intensity [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. The comparison of the tails-on and tails-off versions of the focal expansion to the exact Fourier transform of a [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The comparison of the tails-on focal expansion at LO and SLO to the exact results for Gaussian rings with [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Examples of the behavior of the focal expansion when a non-optimal shift parameter is chosen – in this case [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The data from the first plot of the first row of [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Example of a Gaussian ring with non-trivial angular [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. The radial profiles of several examples of logarithmic-divergence rings, with both the exact profile and the approximate [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. The Fourier transform of several symmetric log-divergence rings calculated numerically and via the focal expansion. [PITH_FULL_IMAGE:figures/full_fig_p033_21.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. The [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. The comparison of the numeric result (dashed red) to the focal expansion (yellow, green) for the integral [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. A log-log plot of the absolute errors between the integral [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Comparison of the numerical evaluation (dashed red) of [PITH_FULL_IMAGE:figures/full_fig_p035_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: FIG. 27. Comparison of the numerics of integral [PITH_FULL_IMAGE:figures/full_fig_p036_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: FIG. 28. Several examples of even and odd log-divergence profiles modeled using the [PITH_FULL_IMAGE:figures/full_fig_p037_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Comparison of the numeric (dashed red) and focal (yellow, green) results for [PITH_FULL_IMAGE:figures/full_fig_p038_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. Comparison of the numeric results for [PITH_FULL_IMAGE:figures/full_fig_p039_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: FIG. 31. Comparison of numerical results (dashed red) and [PITH_FULL_IMAGE:figures/full_fig_p039_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32. An example of an asymmetric logarithmic ring with [PITH_FULL_IMAGE:figures/full_fig_p039_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: FIG. 33. The comparison of the function sin [PITH_FULL_IMAGE:figures/full_fig_p040_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: FIG. 34. Comparison of the positive- [PITH_FULL_IMAGE:figures/full_fig_p040_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: FIG. 35. Comparison of the numerical results (dashed red) for the integral [PITH_FULL_IMAGE:figures/full_fig_p041_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. Comparison of the residual of the numerically calculated integral [PITH_FULL_IMAGE:figures/full_fig_p042_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37. Several examples of lima¸cons with different values of [PITH_FULL_IMAGE:figures/full_fig_p044_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: FIG. 38. Comparisons of the numerical evaluations of the integrals [PITH_FULL_IMAGE:figures/full_fig_p045_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: FIG. 39. Comparison of the numerically calculated Fourier transform of a lima¸con (dashed red) with [PITH_FULL_IMAGE:figures/full_fig_p046_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: FIG. 40. Comparison of the numerically calculated Fourier transform of a lima¸con with [PITH_FULL_IMAGE:figures/full_fig_p047_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: FIG. 41. Comparison of the numerically calculated Fourier transform of a lima¸con with [PITH_FULL_IMAGE:figures/full_fig_p048_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: FIG. 42. A 4 [PITH_FULL_IMAGE:figures/full_fig_p052_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: FIG. 43. The comparison of the exact Cauchy kernel integral [PITH_FULL_IMAGE:figures/full_fig_p056_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: FIG. 44. Comparison of the real and imaginary parts of the [PITH_FULL_IMAGE:figures/full_fig_p056_44.png] view at source ↗
read the original abstract

We derive and showcase a novel approach to approximating Fourier transforms in higher dimensions, focusing specifically on the case of 2D radially concentrated ('ring-like') functions. We first reduce the problem to that of evaluating the Hankel transforms of each angular mode of the image and then use our focal expansion to approximate these remaining Hankel transforms. Our method provides a single approximation that remains accurate from small to large spatial frequencies, bridging regimes where moment-based or large-frequency asymptotic expansions are individually reliable. We explore a series of examples, showing that the leading focal term provides an accurate global approximation for a broad range of functions. We demonstrate the utility of this approximation by examining the interferometric response for toy models of a black hole's "photon ring," highlighting the application to experiments designed to measure this feature such as the Black Hole Explorer.

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

2 major / 2 minor

Summary. The paper derives a focal expansion to approximate the 2D Fourier transform of radially concentrated ('ring-like') functions. It reduces the problem to Hankel transforms of angular modes and approximates those transforms via the leading term of the focal expansion, claiming this single approximation remains accurate from small to large spatial frequencies and bridges moment-based and large-frequency asymptotic regimes. The utility is demonstrated on toy models of black hole photon rings for interferometric applications such as the Black Hole Explorer.

Significance. If the leading focal term can be shown to deliver controlled accuracy for general ring-like profiles, the method would offer a practical, unified tool for computing Fourier transforms in astrophysical imaging, especially for photon-ring modeling where efficient evaluation across frequency regimes is valuable. The reduction to Hankel transforms and the concrete interferometry examples are clear strengths; the absence of a general error bound currently limits the result's immediate applicability beyond the presented cases.

major comments (2)
  1. [focal expansion derivation and error analysis] The central claim that the leading focal term alone suffices for accurate global approximation of the Hankel transforms (and thus the 2D FT) for a broad range of radially concentrated functions lacks a derivation or bound on the truncation error. Without this, it is impossible to assess whether the approximation degrades for broader, multi-peaked, or otherwise non-ideal radial profiles outside the specific toy-model widths and contrasts shown.
  2. [examples and validation] Validation in the examples relies on a handful of photon-ring toy models with fixed radial parameters and frequency ranges. No systematic study (e.g., error metrics versus radial width, contrast, or number of peaks) is provided to support the assertion that the leading term works across a 'broad range' without case-by-case tuning or higher-order corrections.
minor comments (2)
  1. [derivation sections] Ensure all equations for the focal expansion, its relation to the Hankel transform, and the angular-mode decomposition are explicitly stated with consistent notation.
  2. [figures and examples] Add a brief comparison table or plot quantifying the approximation error against standard moment expansions and asymptotic forms for the same test functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments correctly identify that the manuscript relies primarily on examples rather than a general error bound, and that the validation could be more systematic. We address each point below and will incorporate revisions to strengthen the presentation of the focal expansion's accuracy and applicability.

read point-by-point responses

  1. Referee: The central claim that the leading focal term alone suffices for accurate global approximation of the Hankel transforms (and thus the 2D FT) for a broad range of radially concentrated functions lacks a derivation or bound on the truncation error. Without this, it is impossible to assess whether the approximation degrades for broader, multi-peaked, or otherwise non-ideal radial profiles outside the specific toy-model widths and contrasts shown.

    Authors: Section 2 derives the focal expansion and isolates the leading term for the Hankel transform of each angular mode. The manuscript does not supply a general truncation-error bound that holds for arbitrary radial profiles. We agree this omission makes it difficult to judge performance outside the demonstrated cases. In revision we will add a brief discussion of the leading-term error scaling with radial concentration (drawing on the expansion's asymptotic properties) together with numerical error estimates for the existing toy models and one additional broader-profile case. revision: partial

  2. Referee: Validation in the examples relies on a handful of photon-ring toy models with fixed radial parameters and frequency ranges. No systematic study (e.g., error metrics versus radial width, contrast, or number of peaks) is provided to support the assertion that the leading term works across a 'broad range' without case-by-case tuning or higher-order corrections.

    Authors: The examples in Section 3 use several photon-ring toy models with different radial widths and contrasts to illustrate global accuracy. We acknowledge that these do not constitute a systematic parameter survey. We will expand the validation to include quantitative error metrics (relative L2 and pointwise errors) plotted against radial width, contrast, and a two-peak configuration, all evaluated over the same frequency range used for the interferometric examples. revision: yes

Circularity Check

0 steps flagged

No circularity: focal expansion derived independently from Hankel reduction

full rationale

The paper reduces the 2D Fourier transform of radially concentrated functions to a set of Hankel transforms over angular modes, then introduces the focal expansion as a direct approximation for those transforms. This chain is presented as a mathematical derivation without any self-definition (the expansion is not defined using the target FT result), without fitting parameters to data and relabeling them as predictions, and without load-bearing self-citations or imported uniqueness theorems. Validation occurs via explicit examples on toy models rather than by construction, and the leading-term claim is tested rather than assumed tautologically. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

The abstract does not enumerate free parameters, background axioms, or additional invented entities beyond the focal expansion itself, which is the core new contribution.

invented entities (1)
  • focal expansion no independent evidence
    purpose: to approximate Hankel transforms of angular modes for ring-like functions across all frequencies
    Presented as the central new technique; no independent falsifiable evidence is supplied in the abstract.

pith-pipeline@v0.9.0 · 5446 in / 1195 out tokens · 29694 ms · 2026-05-10T18:35:39.416702+00:00 · methodology

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