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REVIEW 2 major objections 1 minor 33 references

Differential intensity measurements recover phase by solving PDEs.

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2026-06-26 03:38 UTC pith:3RPBO5SY

load-bearing objection The paper derives explicit PDEs for phase from differential Fourier intensity measurements by generalizing the transport of intensity equation, but leaves the first-order approximation without error bounds or a clear validity range. the 2 major comments →

arxiv 2606.27176 v1 pith:3RPBO5SY submitted 2026-06-25 math.NA cs.NA

On Fourier Phase Retrieval from Differential Intensity Measurements with Applications to Wavefront Sensing

Simon Hubmer , Lukas Weissinger , Ronny Ramlau , Otmar Scherzer This is my paper
classification math.NA cs.NA
keywords Fourier phase retrievaldifferential intensity measurementstransport of intensity equationwavefront sensingpyramid wavefront sensoradaptive optics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that Fourier phase retrieval can be achieved from intensity measurements that differ only by small modulations in the Fourier domain, such as those induced by prisms or phase plates. Generalizing the transport of intensity equation, it derives that the unknown phase satisfies certain partial differential or integro-differential equations. These equations then support the design of direct reconstruction algorithms for Fourier-type wavefront sensors. A reader would care because the method replaces iterative searches with the solution of linear differential equations, offering a faster route for applications like adaptive optics.

Core claim

Given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics.

What carries the argument

First-order differential approximation of intensity differences under small Fourier-domain modulations, generalizing the transport of intensity equation.

Load-bearing premise

The modulations induced by the optical elements are sufficiently small that a first-order differential approximation remains valid.

What would settle it

Measure the discrepancy between the phase solved from the derived equations and the true phase when the modulation amplitudes are increased until the first-order approximation visibly fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Efficient reconstruction algorithms become available for the pyramid wavefront sensor.
  • The same framework applies to other Fourier-type wavefront sensors used in adaptive optics.
  • Numerical experiments confirm practical performance on simulated and real data.

Where Pith is reading between the lines

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

  • The PDE approach could be hybridized with conventional iterative phase retrieval to handle cases where the small-modulation assumption is only approximately true.
  • Similar differential measurements might be engineered in non-optical domains that admit Fourier-domain modulation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper considers Fourier phase retrieval from differential intensity measurements obtained via slight modulations in the Fourier domain (induced by prisms, phase plates, or SLMs). Generalizing the transport of intensity equation, it derives that the phase satisfies certain partial differential or integro-differential equations, which are then used to design efficient reconstruction algorithms for Fourier-type wavefront sensors such as the pyramid sensor. Numerical experiments illustrate the approach.

Significance. If the first-order approximation is placed on a rigorous footing with explicit error control, the framework could supply direct, non-iterative solvers for phase recovery in adaptive-optics wavefront sensing, offering computational efficiency relative to iterative phase-retrieval methods. The reported numerical experiments provide initial evidence of practical utility.

major comments (2)
  1. [Derivation of the governing equations (abstract and §2–3)] The central derivation relies on a first-order Taylor expansion in the modulation parameter to obtain the governing PDEs/integro-differential equations. No quantitative remainder estimate (e.g., in terms of modulation amplitude and Sobolev norms of the field) or explicit validity regime is supplied, rendering the claim that “the phase can be determined as the solution” conditional on an uncharacterized small-modulation hypothesis. This is load-bearing for the main result.
  2. [Application to pyramid wavefront sensor and numerical experiments] When the general framework is specialized to the pyramid wavefront sensor, the manuscript does not verify that typical operating modulation amplitudes lie inside the regime where the first-order approximation remains accurate, nor does it compare reconstruction error against the size of the neglected higher-order terms.
minor comments (1)
  1. [Introduction and notation] Notation for the modulation operator and the resulting differential intensity should be introduced with a single consistent symbol set to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the first-order approximation. We address each major point below.

read point-by-point responses

  1. Referee: [Derivation of the governing equations (abstract and §2–3)] The central derivation relies on a first-order Taylor expansion in the modulation parameter to obtain the governing PDEs/integro-differential equations. No quantitative remainder estimate (e.g., in terms of modulation amplitude and Sobolev norms of the field) or explicit validity regime is supplied, rendering the claim that “the phase can be determined as the solution” conditional on an uncharacterized small-modulation hypothesis. This is load-bearing for the main result.

    Authors: We agree that the derivation rests on a first-order Taylor expansion and that the manuscript supplies no explicit remainder bound in Sobolev norms. Deriving such a bound would require a separate, technically involved analysis that lies outside the paper’s primary focus on deriving the governing equations and illustrating their use for wavefront sensors. In the revised manuscript we will insert a short subsection (or paragraph) in §2 that states the small-modulation hypothesis explicitly, supplies a heuristic remainder estimate based on the second-order term, and indicates the parameter regime in which the approximation is expected to be accurate. revision: partial

  2. Referee: [Application to pyramid wavefront sensor and numerical experiments] When the general framework is specialized to the pyramid wavefront sensor, the manuscript does not verify that typical operating modulation amplitudes lie inside the regime where the first-order approximation remains accurate, nor does it compare reconstruction error against the size of the neglected higher-order terms.

    Authors: The observation is correct. The current numerical section does not quantify how the chosen modulation amplitudes compare with the size of the neglected higher-order contributions. In the revised version we will add a targeted numerical test for the pyramid sensor: we will vary the modulation amplitude over a representative range, compute both the first-order reconstruction and a direct (non-linearized) intensity simulation, and report the discrepancy as a function of modulation strength. This will provide concrete evidence that typical operating values remain inside the regime where the first-order model is reliable. revision: yes

Circularity Check

0 steps flagged

No circularity: direct generalization of TIE via first-order expansion

full rationale

The derivation starts from differential intensity measurements and applies a first-order Taylor expansion in the modulation parameter to obtain PDEs or integro-differential equations, generalizing the known transport-of-intensity equation. No parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness result, and the target equations are not defined in terms of themselves. The small-modulation hypothesis is an explicit modeling assumption rather than a hidden tautology, so the claimed reconstruction equations are independent of the input data by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the derivation is presented as a direct generalization of the transport of intensity equation without additional fitted constants or new postulated objects.

pith-pipeline@v0.9.1-grok · 5660 in / 959 out tokens · 36399 ms · 2026-06-26T03:38:39.048331+00:00 · methodology

0 comments
read the original abstract

In this paper, we consider Fourier phase retrieval from differential intensity measurements, i.e., the problem of determining the phase of a complex-valued function from a series of intensity measurements differing only by slight modulations in Fourier domain. These modulations may be induced by optical elements such as prisms or phase plates, or via spatial-light modulators. Generalizing the principles behind the transport of intensity equation, we show that given such differential intensity measurements, the phase can be determined as the solution of certain partial differential or integro-differential equations. This is then used to design efficient reconstruction algorithms for a number of Fourier-type wavefront sensors, such as the pyramid wavefront sensor, commonly used in adaptive optics. Numerical experiments illustrate the usefulness of our proposed approach.

Figures

Figures reproduced from arXiv: 2606.27176 by Lukas Weissinger, Otmar Scherzer, Ronny Ramlau, Simon Hubmer.

Figure 4.1
Figure 4.1. Figure 4.1: Schematic depiction of a Fourier-type WFS measurement system [10, 12]. [PITH_FULL_IMAGE:figures/full_fig_p020_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Illustration of shape functions Ψ(ξ), with ξ = (ξ1, ξ2), as defined in [PITH_FULL_IMAGE:figures/full_fig_p022_4_2.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Datasets for quadratic modulation experiment: ground truth phase [PITH_FULL_IMAGE:figures/full_fig_p029_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Phase reconstructions in the quadratic modulation experiment: recon [PITH_FULL_IMAGE:figures/full_fig_p030_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Dataset for Fourier-type WFS experiments: ground truth wavefront phase [PITH_FULL_IMAGE:figures/full_fig_p031_5_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: depicts the wavefronts [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Fourier-type WFS experiments: wavefront reconstructions obtained via the [PITH_FULL_IMAGE:figures/full_fig_p032_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Fourier-type WFS experiments: reconstruction errors obtained via the [PITH_FULL_IMAGE:figures/full_fig_p033_5_5.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

33 extracted references

  1. [1]

    R. M. Clare, B. E. Engler, S. Weddell, Iu. Shatokhina, A. Obereder, and M. Le Louarn. Numerical evaluation of pyramid type sensors for extreme Adaptive Optics for the European Extremely Large Telescope. InProceedings of the Fifth AO4ELT Conference, 2017

    2017

  2. [2]

    R. M. Clare, B. E. Engler, and S. J. Weddell. Wavefront reconstruction with the cone sensor. In35th International Conference on Image and Vision Computing New Zealand (IVCNZ), pages 1–6, 2020

    2020

  3. [3]

    B. L. Ellerbroeck and C. R. Vogel. Inverse problems in astronomical optics.Inverse Problems, 25(6):063001 (37pp), 2009. 33

    2009

  4. [4]

    H. W. Engl, M. Hanke, and A. Neubauer.Regularization of inverse problems.Dordrecht: Kluwer Academic Publishers, 1996

    1996

  5. [5]

    B. E. Engler, S. J. Weddell, and R. M. Clare. Wavefront sensing with prisms for astronomical imaging with Adaptive Optics. InInternational Conference on Image and Vision Computing New Zealand (IVCNZ), pages 1–7, 2017

    2017

  6. [6]

    L. C. Evans.Partial Differential Equations. Graduate studies in mathematics. American Mathe- matical Society, 1998

    1998

  7. [7]

    Fauvarque, B

    O. Fauvarque, B. Neichel, T. Fusco, J.-F. Sauvage, and O. Girault. General formalism for Fourier- based wave front sensing.Optica, 3(12):1440–1452, 2016

    2016

  8. [8]

    R. A. Gonsalves. Phase retrieval by differential intensity measurements.J. Opt. Soc. Am. A, 4(1), 1987

    1987

  9. [9]

    J. W. Goodman.Introduction to Fourier Optics. McGraw-Hill physical and quantum electronics series. W. H. Freeman, 2005

    2005

  10. [10]

    Hubmer, V

    S. Hubmer, V. Laidlaw, R. Ramlau, E. Sherina, and B. Stadler. On Phase Unwrapping via Digital Wavefront Sensors.SIAM Journal on Imaging Sciences, 18(2):1260–1283, 2025. Green OA

    2025

  11. [11]

    N. E. Hurt.Phase Retrieval and Zero Crossings. Dordrecht ; Boston : Kluwer Academic Pub- lishers, 1989”

    1989

  12. [12]

    Hutterer, A

    V. Hutterer, A. Neubauer, and J. Shatokhina. A mathematical framework for nonlinear wavefront reconstruction in adaptive optics systems with Fourier-type wavefront sensing.Inverse Problems, 39(3):035007, 2023

    2023

  13. [13]

    Janin-Potiron, V

    P. Janin-Potiron, V. Chambouleyron, L. Schatz, O. Fauvarque, C. Z. Bond, Y. Abautret, E. R. Muslimov, K. El Hadi, J.-F. Sauvage, K. Dohlen, B. Neichel, C. M. Correia, and T. Fusco. Adap- tive Optics with programmable Fourier-based wavefront sensors: a spatial light modulator ap- proach to the LAM/ONERA on-sky pyramid sensor testbed.Journal of Astronomical...

    2019

  14. [14]

    F. W. King.Hilbert Transforms. Cambridge University Press, 2009

    2009

  15. [15]

    Kirisits, K

    C. Kirisits, K. Raik, O. Scherzer, C. Strohmenger, and J. Yan.Using the transport of intensity and the transport of phase equation for phase retrieval, pages 53–68. De Gruyter, Berlin, Boston, 2025

    2025

  16. [16]

    Generalized non-linear reconstructor for all Fourier-type wavefront sensors

    Victoria Laidlaw. Generalized non-linear reconstructor for all Fourier-type wavefront sensors. Opt. Express, 33(15):31281–31294, 2025

    2025

  17. [17]

    W. C. H. McLean.Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Uni- versity Press, 2000

    2000

  18. [18]

    Pircher and R

    M. Pircher and R. J. Zawadzki. Review of adaptive optics OCT (AO-OCT): Principles and applications for retinal imaging.Biomed Opt Express, 8(5):2536–62, 2017

    2017

  19. [19]

    B. C. Platt and R. Shack. History and principles of shack-hartmann wavefront sensing.Journal of Refractive Surgery, 17(5):S573–S577, 2001

    2001

  20. [20]

    J. Primot. Theoretical description of shack–hartmann wave-front sensor.Optics Communications, 221(1):81–92, 2003

    2003

  21. [21]

    Ragazzoni

    R. Ragazzoni. Pupil plane wavefront sensing with an oscillating prism.J. of Modern Optics, 43(2):289–293, 1996

    1996

  22. [22]

    Ragazzoni, E

    R. Ragazzoni, E. Diolaiti, and E. Vernet. A pyramid wavefront sensor with no dynamic modula- tion.Optics Communications, 208:51–60, 2002. 34

    2002

  23. [23]

    F. Roddier. Curvature sensing and compensation: a new concept in adaptive optics.Applied Optics, 27(7):1223–1225, 1988

    1988

  24. [24]

    Roddier.Adaptive optics in astronomy

    F. Roddier.Adaptive optics in astronomy. Cambridge University Press, 1999

    1999

  25. [25]

    Rosensteiner

    M. Rosensteiner. Cumulative Reconstructor: fast wavefront reconstruction algorithm for Ex- tremely Large Telescopes.J. Opt. Soc. Am. A, 28(10):2132–2138, 2011

    2011

  26. [26]

    Rosensteiner

    M. Rosensteiner. Wavefront reconstruction for Extremely Large Telescopes via CuRe with Domain Decomposition. Technical Report E-TRE-AAO-528-0018, Austrian In-Kind Contribution - AO, 2011

    2011

  27. [27]

    Rouan, P

    D. Rouan, P. Riaud, A. Boccaletti, Y. Cl´ enet, and A. Labeyrie. The four-quadrant phase-mask coronagraph. i. principle.Publications of the Astronomical Society of the Pacific, 112(777):1479– 1486, November 2000

    2000

  28. [28]

    Shatokhina, V

    Iu. Shatokhina, V. Hutterer, and R. Ramlau. Review on methods for wavefront reconstruction from pyramid wavefront sensor data.Journal of Astronomical Telescopes, Instruments, and Sys- tems, 6(1):1 – 39, 2020

    2020

  29. [29]

    E. Vera, F. Guzman, C. Weinberger, N. Hernandez, and J. Tapia. Deep Optics for Wavefront Sensing: Beyond the Pyramid Wavefront Sensor. In7th International Conference on Adaptive Optics for Extremely Large Telescopes, AO4ELT 2023, Avignon, France, 2023

    2023

  30. [30]

    V´ erinaud

    C. V´ erinaud. On the nature of the measurements provided by a pyramid wave-front sensor.Optics Communications, 233:27–38, 2004

    2004

  31. [31]

    Zernike and F

    F. Zernike and F. J. M. Stratton. Diffraction theory of the Knife-Edge test and its improved form, the phase-contrast method.Monthly Notices of the Royal Astronomical Society, 94(5):377–384, 1934

    1934

  32. [32]

    Zhariy, A

    M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau. Cumulative wavefront reconstructor for the Shack-Hartmann sensor.Inverse Problems and Imaging, 5:893–913, 2011

    2011

  33. [33]

    C. Zuo, J. Li, J. Sun, Y. Fan, J. Zhang, L. Lu, R. Zhang, B. Wang, L. Huang, and Q. Chen. Transport of intensity equation: a tutorial.Optics and Lasers in Engineering, 135:106187, 2020. 35

    2020