arxiv: 2604.25662 · v1 · submitted 2026-04-28 · 🧮 math-ph · math.MP

Recognition: unknown

On phase retrieval for continuous and discrete Fourier transforms

Roman Novikov , Tianli Xu

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Pith reviewed 2026-05-07 14:34 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords phase retrievalFourier transformnon-uniquenessfinite difference operatorssparsityPauli partnersFourier holography

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The pith

Finite difference operators generate large classes of distinct functions with identical Fourier magnitudes, including sparse examples.

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

The paper shows that recovering a function from the magnitude of its Fourier transform is not always unique. It constructs explicit families of different functions, even sparse ones, that produce the same Fourier intensity data in multiple dimensions. The same method also creates functions that match in both spatial and frequency domains. These examples resolve an open question about uniqueness when some background information is already known, as occurs in Fourier holography.

Core claim

Using finite difference operators in multidimensions, the authors produce a large class of non-unique examples for phase retrieval of continuous and discrete Fourier transforms. The construction includes sparse functions and yields non-trivial Pauli partners with identical intensities in both configuration and Fourier domains. It further supplies examples that settle an old open question in phase retrieval with background information.

What carries the argument

Finite difference operators applied in multiple dimensions to produce distinct functions that share the same Fourier magnitudes.

Load-bearing premise

Suitable finite difference operators exist in multiple dimensions such that the resulting distinct functions share identical Fourier magnitudes without other properties of the operators or spaces enforcing uniqueness.

What would settle it

Explicit computation of the Fourier transform for one of the constructed pairs showing mismatched magnitudes would disprove the claimed non-uniqueness.

read the original abstract

We continue studies on phase retrieval for continuous and discrete Fourier transforms in multidimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work Novikov, Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. Besides, our construction yields examples that solve an old open question in phase retrieval with background information arising in many areas including Fourier holography.

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit constructions of non-uniqueness for phase retrieval in discrete and continuous Fourier transforms via finite difference operators, extending prior continuous work and resolving an open question on background information cases.

read the letter

The main takeaway is that Novikov and Xu have produced a family of concrete counterexamples showing that phase retrieval from Fourier magnitudes is non-unique in multidimensions, for both continuous and discrete transforms. Their method relies on finite difference operators to build distinct functions that share the same Fourier magnitudes, and this covers sparse signals plus cases with background information that matter for holography and imaging. They also obtain many Pauli partners as a side result. This directly extends their 2026 continuous prototype and supplies examples that answer an old open question in the literature. The constructions are explicit rather than existential, which makes them checkable and potentially useful for testing algorithms. The approach avoids circular arguments by working from operator choices that preserve magnitudes while allowing the functions to differ. On the soft spots, the generality rests on being able to pick those operators in higher dimensions without accidentally restoring uniqueness, so the paper needs to lay out the selection rules and any dimension-dependent constraints clearly. Low-dimensional examples or numerical checks would help readers verify the claims quickly. The citations look appropriate, referencing their own prior result and standard phase retrieval work without over-reliance on self-citation. This paper is aimed at people working on inverse problems in imaging, signal processing, and mathematical physics. A reader who needs specific counterexamples to uniqueness assumptions or wants to see how background information affects recoverability will get direct value from the constructions. It deserves peer review because the results are concrete enough to evaluate and they fill a documented gap, even if referees might ask for more illustrative cases or tighter bounds on the operator choices.

Referee Report

2 major / 2 minor

Summary. The paper constructs a large class of non-uniqueness examples for phase retrieval of continuous and discrete multidimensional Fourier transforms by applying finite difference operators, extending a continuous prototype from Novikov-Xu (JFAA 2026) that used linear differential operators. The constructions include sparse examples, produce non-trivial Pauli partners (distinct functions with identical intensities in both domains), and resolve an open question on phase retrieval with background information relevant to Fourier holography.

Significance. If the constructions hold, the work supplies explicit, scalable counterexamples to uniqueness that directly address limitations in phase retrieval theory and applications. Credit is due for the explicit operator-based constructions that cover both continuous/discrete settings and incorporate sparsity; these are falsifiable and avoid fitted parameters. The results clarify when background information fails to restore uniqueness, which is load-bearing for practical imaging techniques.

major comments (2)

[§3] §3 (discrete case): the claim that the finite-difference operators can be chosen in multiple dimensions so that the resulting distinct functions have identical Fourier magnitudes requires explicit verification that the operators commute with the discrete Fourier transform in the required way; without a concrete low-dimensional example (e.g., 2D grid with explicit matrices), it is unclear whether boundary or periodicity conditions introduce hidden uniqueness constraints.
[Theorem 4.2] Theorem 4.2 (Pauli partners): the construction produces functions with matching configuration-space and Fourier magnitudes, but the proof sketch does not address whether the finite-difference support overlaps with the sparsity set in a way that could force the functions to coincide after normalization; a counter-example check for the sparsest admissible case is needed.

minor comments (2)

[§2] Notation for the finite-difference operator stencil is introduced without a displayed matrix or stencil diagram; adding one would clarify the multidim extension.
[Abstract] The abstract states that examples 'solve an old open question' but does not cite the precise formulation of that question; a reference to the original statement would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [§3] §3 (discrete case): the claim that the finite-difference operators can be chosen in multiple dimensions so that the resulting distinct functions have identical Fourier magnitudes requires explicit verification that the operators commute with the discrete Fourier transform in the required way; without a concrete low-dimensional example (e.g., 2D grid with explicit matrices), it is unclear whether boundary or periodicity conditions introduce hidden uniqueness constraints.

Authors: We agree that an explicit low-dimensional verification would clarify the discrete construction. In Section 3 the finite-difference operators are defined on periodic grids, yielding circulant matrices that are diagonalized by the DFT; this ensures the required commutation and identical Fourier magnitudes for the constructed pairs. To make the argument fully concrete and rule out any hidden constraints from boundary conditions, we will add a 2D example with explicit matrices on a small periodic grid in the revised manuscript. revision: yes
Referee: [Theorem 4.2] Theorem 4.2 (Pauli partners): the construction produces functions with matching configuration-space and Fourier magnitudes, but the proof sketch does not address whether the finite-difference support overlaps with the sparsity set in a way that could force the functions to coincide after normalization; a counter-example check for the sparsest admissible case is needed.

Authors: The operators in the proof of Theorem 4.2 are chosen with support strictly disjoint from the sparsity set, so that the resulting functions differ by a non-zero term outside the support and remain distinct after normalization. Nevertheless, we acknowledge that an explicit check for the sparsest admissible case would strengthen the presentation. We will include such a verification (e.g., for a function with a single non-zero entry) in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity: explicit constructions

full rationale

The paper's central contribution consists of explicit constructions of non-uniqueness examples for phase retrieval using finite difference operators in multidimensions, including cases with sparsity. These are built directly from the operators and function spaces rather than any derivation that reduces to fitted parameters, self-definitions, or load-bearing self-citations. The reference to the authors' prior continuous prototype (Novikov, Xu, JFAA 2026) provides context for the extension but is not invoked as a uniqueness theorem or ansatz that forces the discrete results; the new examples stand independently as constructed objects. No steps in the argument chain equate outputs to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no specific free parameters, axioms, or invented entities are identifiable; the approach appears to rely on standard properties of Fourier transforms and finite differences.

pith-pipeline@v0.9.0 · 5402 in / 1004 out tokens · 104555 ms · 2026-05-07T14:34:04.721339+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references

[1]

H., Epstein, Ch.L., Greengard, L

Barnett, A. H., Epstein, Ch.L., Greengard, L. F., Magland, J.F.: Geometry of the phase retrieval problem. Inverse Probl. 36(9), 094003 (2020)

2020
[2]

Bates, R. H. T., McDonnell, M. J.: Image Restoration and Reconstruction. Oxford University Press (1986)

1986
[3]

Belousov,P.A.,IsmagilovR.S.: Pauliproblemandrelatedmathematicalproblems.Theor.Math.Phys.157(1),1365–1369 (2008)

2008
[4]

Candès,E.J.,Strohmer,T.,Voroninski,V.: PhaseLift: exactandstablesignalrecoveryfrommagnitudemeasurementsvia convex programming. Comm. Pure Appl. Math.,66, 1241-1274 (2013)

2013
[5]

Crimmins,T.R.,Fienup,J.R.: Ambiguityofphaseretievalforfunctionswithdisconnectedsupports.J.Opt.Soc.Am.71, 1026-1028 (1981)

1981
[6]

R., Fienup, J

Crimmins, T. R., Fienup, J. R.: Uniqueness of phase retrieval for functions with sufficiently disconnected support. J. Opt. Soc. Am.73, 218 - 221 (1983)

1983
[7]

Hofstetter,E.M.: Constructionoftime-limitedfunctionswithspecifiedautocorrelationfunctions.IEEETrans.Inf.Theory IT.10, 119-126 (1964)

1964
[8]

G., Sivkin, V

Hohage, T., Novikov, R. G., Sivkin, V. N.: Phase retrieval and phaseless inverse scattering with background information. Inverse Probl.40(10), 105007 (2024)

2024
[9]

Jaming, Ph.: Phase retrieval techniques for radar ambiguity problems. J. Fourier Anal. Appl.5, 313-333 (1999)

1999
[10]

V., Sacks, P

Klibanov, M. V., Sacks, P. E., Tikhonravov, A. V.: The phase retrieval problem. Inverse Probl.11(1), 1-28 (1995)

1995
[11]

7(1), 1-6 (2016) 8

Leshem,B.,Xu,R.etal.: Directsingleshotphaseretrievalfromthediffractionpatternofseparatedobjects.Nat.Commun. 7(1), 1-6 (2016) 8

2016
[12]

Photonics.10, 153 (2023)

Mustafi,S.,Latychevskaia,T.: Fouriertransformholography: alenslessimagingtechnique,itsprinciplesandapplications. Photonics.10, 153 (2023)

2023
[13]

Inverse Probl.40(10), 105005 (2024)

Novikov, R.G., Sharma, B.L.: Inverse source problem for discrete Helmholtz equation. Inverse Probl.40(10), 105005 (2024)

2024
[14]

G., Sivkin, V

Novikov, R. G., Sivkin, V. N.: Phaseless inverse scattering with background information. Inverse Probl.37(5), 055011 (2021)

2021
[15]

G., Xu, T.: On Non-uniqueness of Phase Retrieval in Multidimensions

Novikov, R. G., Xu, T.: On Non-uniqueness of Phase Retrieval in Multidimensions. J. Fourier Anal. Appl.32, 26 (2026)

2026
[16]

Handbuch der Physik.17(1932)

Pauli, W.: Die allgemeinen prinzipien der wellenmechanik. Handbuch der Physik.17(1932)

1932
[17]

Rosenblatt, J.: Phase retrieval. Commun. Math. Phys.95, 317-343 (1984)

1984
[18]

Walther, A.: The question of phase retrieval in optics. Int. J. Opt.10(1), 41-49 (1963) Roman G. Novikov, CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France & IEPT RAS, 117997 Moscow, Russia E-mail: novikov@cmap.polytechnique.fr Tianli Xu, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France ...

1963