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Uniqueness of phase retrieval from offset linear canonical transform

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Pith's one-line read Compactly supported signals are determined up to a global phase by the magnitudes of their offset linear canonical transforms over a positive-measure family of parameters, with analogous uniqueness for discrete, nonseparable, and…

desk verdict Genuine but sloppy OLCT phase retrieval results: the core ideas are right, but Theorem 3.4's statement is malformed and Lemma 5.1's proof has a false equality, so the STOLCT half is unverifiable until repaired. read the letter →

arxiv 2506.03944 v1 pith:DAXV7W55 submitted 2025-06-04 eess.SP cs.ITmath.IT

Jing Liu , Haiye Huo This is my paper
classification eess.SPcs.ITmath.IT MSC 42C1542C4094A12
keywords phaseretrievaloffsetlinearcanonicaltransformambiguityfunctionshort-timebandlimitedsignalsuniquenessconvolutionambiguitiesPaley-Wienertheorem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Phase retrieval is the problem of reconstructing a signal from the magnitude of a linear transform, which in optics and signal processing is often the only measurable quantity. The paper works with the offset linear canonical transform (OLCT), a five-parameter integral transform whose special cases include the Fourier, fractional Fourier, and linear canonical transforms. Its main result is that a compactly supported signal is uniquely determined up to a global phase by the magnitudes of its OLCTs over a positive-measure family of parameter choices, and that every other ambiguity between two compactly supported signals is exactly a chirp-modulated convolution pair. The same convolution description is established for discrete finite-support signals, and the paper further proves uniqueness (up to a sign or global phase) for nonseparable signals and for bandlimited signals from sampled short-time OLCT magnitudes. If these uniqueness results are correct, they tell practitioners in optics and signal processing that intensity-only OLCT measurements, under natural support or bandlimitedness priors, carry no hidden ambiguity beyond the unavoidable global phase.

What carries the argument

The load-bearing object is the ambiguity function $A_f(\tau,\eta)=\int f(t+\tau/2)\overline{f(t-\tau/2)}e^{-it\eta}\,dt$, the two-dimensional correlation of a signal with itself that jointly records time and frequency shifts. The paper's central identity is Lemma 3.3, which shows that applying the OLCT merely re-parameterizes and phase-modulates this function, so a one-dimensional intensity measurement $|O_A f|^2$ exposes a one-dimensional slice of $A_f$ through equation (17). A second identity, Lemma 5.1, writes the Fourier transform of the STOLCT spectrogram as a product of the signal's and window's ambiguity functions, $F(|V_\varphi^A f|^2)(u,-v)=e^{jy_0 v}A_{\tilde f}(bv,u)A_\varphi(bv,u)$; this is what lets the window's nonvanishing ambiguity slices be cancelled, leaving the signal's ambiguity function. Analytic continuation via the Paley-Wiener theorem then converts equality on positive-measure slices into equality of the full ambiguity function, which is equivalent to equality up to a global phase.

What would settle it

Compute, for a compactly supported test signal $f$ and a fixed OLCT parameter matrix $A$, both sides of equation (17): take the Fourier transform of $|O_A f|^2$ and compare it with the slice $e^{-j\eta(y_0+\frac12 ab\eta)}A_f(-b\eta,a\eta)$ of the ambiguity function. Any systematic disagreement would falsify Lemma 3.3 and collapse the proof of Theorem 3.4; conversely, agreement on a battery of random compactly supported signals would confirm the identity the argument rests on.

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Extended reading notes

Core claim

The central claim, made on the paper's own terms, is that phase retrieval in the OLCT domain is governed by the ambiguity function, not by the transform itself. Lemma 3.3 computes the ambiguity function of an OLCT-transformed signal: $A(O_A f, O_A g)(\tau,\eta) = e^{j[\tau\omega_0-\eta y_0+\frac12(d\tau-b\eta)(a\eta-c\tau)]} A(f,g)(d\tau-b\eta, a\eta-c\tau)$. Specializing to $f=g$ and $\tau=0$ gives $e^{-j\eta(y_0+\frac12 ab\eta)} A_f(-b\eta,a\eta) = \mathcal{F}(|O_A f|^2)(\eta)$, so the Fourier transform of the measured intensity is exactly one slanted slice of the signal's ambiguity function. When the ratios $a/b$ range over a positive-measure set, these slices fill a positive-measure set in the second variable for every first variable; compact support makes the ambiguity function analytic in that second variable, so the Paley-Wiener theorem extends equality of slices to equality of the whole ambiguity function. Because $A_f = A_g$ forces $g=\lambda f$ with $|\lambda|=1$, the signal is recovered up to global phase. The same slice-viewpoint, through the short-time identity in Lemma 5.1, yields the STOLCT uniqueness theorems.

Load-bearing premise

The main multiple-measurement theorem stands on the assumption that the measurement parameters cover a positive-measure set of ratios $a/b$, so the observed ambiguity-function slices have positive length; the theorem as printed describes its parameter set inconsistently, and if only a zero-measure set of ratios is intended, the analytic-continuation step collapses.

Editorial extensions

If this is right

  • Compactly supported signals can be recovered up to a global phase from OLCT magnitude measurements over a positive-measure family of parameters, so intensity-only OLCT data carry no hidden ambiguity beyond the global phase.
  • The convolution characterization (Theorem 3.2) says any two compactly supported signals with identical OLCT magnitudes differ by a chirp-modulated convolution of a common pair, with one factor reflected and shifted; this makes the ambiguity set explicit and testable.
  • Finite-support discrete signals satisfy the analogous convolution description, so discrete OLCT phase retrieval has the same ambiguity structure as one-dimensional Fourier phase retrieval.
  • Under a mild window condition, nonseparable real-valued signals are determined up to sign by STOLCT magnitudes, and complex-valued signals up to global phase when the window's ambiguity function vanishes nowhere.
  • FT-bandlimited and OLCT-bandlimited signals are determined up to a global phase by sampled STOLCT magnitudes at Shannon-type sampling rates, so sampled intensity data suffice for these signal classes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • This reader's inference: the same slice-of-ambiguity mechanism suggests a practical test for whether any finite set of OLCT parameters determines a given signal class, namely whether the corresponding slices of the ambiguity function cover, for each time lag, a set with an accumulation point in the frequency variable.
  • This reader's inference: the convolution ambiguity description could be read as saying the 'trivial' ambiguities (global phase, shift, conjugate reflection) are exactly the cases where one convolution factor is a chirp-modulated delta; making this explicit might simplify algorithm design, though the paper does not pursue it.
  • This reader's inference: since the STOLCT uniqueness proofs require the window's ambiguity function to be nonvanishing on specific slices, checking the condition for a candidate window reduces to computing one or two one-dimensional slices, a cheap numerical pre-check for experimental setups that the paper does not run.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. This paper studies uniqueness in phase retrieval from magnitude-only measurements of the offset linear canonical transform (OLCT). It proves that nontrivial ambiguities in continuous and discrete OLCT phase retrieval can be characterized via convolution factorizations, that compactly supported continuous signals are determined up to global phase by magnitude measurements over a positive-measure family of OLCT parameters, and that nonseparable or bandlimited signals can be uniquely recovered from short-time OLCT (STOLCT) magnitude measurements under conditions on the window's ambiguity function. The arguments proceed primarily by transferring known Fourier/LCT phase retrieval results through the exact identities (5), (6), and (10).

Significance. If the proofs are correct, the paper would provide a useful extension of known phase retrieval uniqueness results to the five-parameter OLCT framework, covering continuous, discrete, and short-time settings. The use of explicit transform identities to reduce OLCT problems to Fourier or LCT problems is natural and the claimed results are plausible. The paper also gives a concrete family of uniqueness conditions in terms of the window's ambiguity function. However, the printed proofs contain several load-bearing gaps and incorrect equalities, particularly in Theorem 3.4 and Lemma 5.1, so the significance of the contribution cannot be fully assessed until those issues are resolved.

major comments (4)
  1. [Section 3.2, Theorem 3.4 statement] The statement of Theorem 3.4 is internally inconsistent as printed. The first sentence declares 'Λ ⊂ R is a set with positive measure', while the next line defines Λ as a set of OLCT parameter matrices by 'Λ = {a/b ∈ Λ, b>0, ad−bc=1, ...}'. These two uses of Λ cannot both hold. Under the literal wording, the hypothesis '|OA f| = |OA g| holds for all A ∈ Λ' is undefined. The theorem should be restated unambiguously, for example by letting the family of matrices have ratios a/b ranging over a positive-measure subset of R.
  2. [Section 3.2, Theorem 3.4 proof, following Eq. (17)] The proof asserts that for every τ ∈ R the set Λτ = {−a/b τ : A ∈ Λ} has positive measure, and then applies a Paley—Wiener extension in the second variable. This is valid for τ ≠ 0, but for τ = 0 the set Λτ collapses to {0}, which has measure zero, so the extension argument gives no information on the slice τ = 0. Since the final invocation of (15) requires equality of the entire ambiguity functions, including on τ = 0, an additional continuity argument in τ is needed to bridge from τ ≠ 0 to τ = 0. This step is not supplied, and therefore Theorem 3.4 is not established as written.
  3. [Section 5, Lemma 5.1 proof, Eqs. (20)–(22)] The proof contains the false equality F(fv)(u) = F(fv^sharp)(u), where fv^sharp(t) = fv(−t). For a general complex-valued fv, one has F(fv^sharp)(u) = F(fv)(−u), which is not equal to F(fv)(u). Consequently the product identity in (22) and the first equality in (19) are not derived correctly. Since Theorems 5.3–5.6 all rely on Lemma 5.1, the STOLCT uniqueness results are not supported by the printed proof. The identity itself may be salvageable through a correct manipulation of conjugate-reversal, but the present argument is invalid.
  4. [Section 5.2, Theorem 5.5 proof, Eqs. (26)–(27)] In Eq. (26) the complex conjugate on the second Vφ term is missing, and the line 'F^{-1}(f^sharp) = F^{-1}f' with f^sharp(t) = f(−t) is also false in general. These errors affect the derivation of (27) and hence the claimed bandlimitedness of |V_A φ f(·,u)|^2. While the intended convolution formula can likely be repaired using the conjugate of the reversed function, the printed proof does not justify the conclusion.
minor comments (3)
  1. [Abstract and text] There are several typographical errors: 'demenstrate' in the abstract, 'resluts' in Section 5.2, 'lase step' after Eq. (22), and 'Form Proposition 2.3' in the proof of Theorem 5.6 should read 'From Proposition 2.3'.
  2. [Section 5.1, definition of separability] The definition of a separable function as f = f1 + f2 with f1 f2 = 0 relies on pointwise multiplication of L2 functions; it should be stated as equality almost everywhere or in terms of disjoint supports to be fully rigorous.
  3. [Sections 3.2 and 5.1, black-box citations] The proofs of Theorems 3.2 and 5.4 refer to 'following the proof process of [26, Theorem 3.2]' and 'a similar procedure for the proof of [34, Theorem 3.1]' without sketching the key steps. Given that these steps are central to the uniqueness claims, the paper would benefit from at least a brief indication of how the cited arguments apply to the OLCT setting.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation found; the results are proved from explicit OLCT/FT identities and external theorems, with only minor non-load-bearing definitional self-citations.

full rationale

No specific circular reduction can be exhibited. The main results (Theorems 3.2, 3.4, 4.2, 5.3, 5.4, 5.5, 5.6) are obtained by chaining explicit OLCT-FT identities (Eqs. (5), (6), (10)), ambiguity-function identities (Lemmas 3.3 and 5.1), and external theorems cited from the literature: the Paley-Wiener theorem [44], Shannon's sampling theorem [43], Jaming's ambiguity-function uniqueness result [28], Beinert-Plonka's discrete Fourier phase-retrieval characterization [45], and the nonseparability criterion of Chen-Cheng [46]. For example, Theorem 3.4 reduces the hypothesis |O_A f| = |O_A g| to equality of ambiguity functions on the sets Lambda_tau via Eq. (17), then invokes Paley-Wiener analytic continuation and the external equivalence (15); neither step assumes the desired conclusion, and no parameter is fitted. The only self-citations, [38,39], support Definition 2.1 and the inverse OLCT formula; these are definitional facts and are not load-bearing for the uniqueness claims. There is a correctness gap in Theorem 3.4 as printed: Lambda is simultaneously declared to be a subset of R and a set of OLCT parameter matrices, and the proof does not explicitly justify the tau=0 slice needed to conclude A_f = A_g on the entire ambiguity domain. That is a rigor defect, not circularity. The score 2 reflects the presence of minor non-load-bearing definitional self-citations; the central claims retain independent mathematical content.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters or invented physical entities. The proof relies on a chain of established theorems: Fourier phase retrieval ambiguity characterization, ambiguity function uniqueness, Paley-Wiener and Shannon sampling, nonseparable magnitude uniqueness, and the support and continuity lemma for bandlimited ambiguity functions. The OLCT-to-FT identities (5) and (10) are derived in the paper.

assumptions (6)
  • standard math Fourier phase retrieval ambiguity characterization: any pair with equal Fourier magnitudes factors as f=h1*h2 and g=e^{j beta}(h1(.-t0)*h2(-.))
    Invoked in Theorem 3.2 via 'following the proof of [26, Theorem 3.2]' and in Theorem 4.2 via [45, Theorem 2.3]; not reproved in the paper.
  • standard math Ambiguity function uniqueness: Af=Ag implies g=lambda f with |lambda|=1, equation (15), cited to [28]
    Used to conclude uniqueness in Theorems 3.4 and 5.4; requires the Hermitian, conjugated, ambiguity function convention, which the printed Section 2.4 lacks.
  • standard math Paley-Wiener theorem and Shannon sampling for bandlimited functions, Proposition 2.3, references [43,44]
    Used to extend equality of ambiguity functions from a positive-measure set to all sigma in Theorem 3.4 and to reconstruct continuous |V_phi^A f|^2 from samples in Theorems 5.5 and 5.6.
  • standard math Nonseparable real-valued continuous functions are uniquely determined by their magnitudes up to sign, Proposition 5.2, reference [46]
    Used in Theorems 5.3 and 5.5 to pass from |f|=|g| to f=plus or minus g.
  • standard math Lemma 2.4 from [32]: ambiguity functions of bandlimited functions are uniformly continuous and supported in R times [-2 Omega, 2 Omega]
    Used to conclude equality of ambiguity functions at a set of points implies equality everywhere in Theorems 5.5 and 5.6.
  • domain assumption Restriction to b>0 for OLCT
    The paper assumes b>0 without loss of generality throughout because b=0 reduces OLCT to a chirp multiplication.

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Cite this review

Pith. "Pith review of Uniqueness of phase retrieval from offset linear canonical transform." pith.science (2026). https://pith.science/paper/DAXV7W55

@misc{pith2026250603944,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of phase retrieval from offset linear canonical transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAXV7W55}},
  note         = {Machine review of arXiv:2506.03944}
}
read the original abstract

The classical phase retrieval refers to the recovery of an unknown signal from its Fourier magnitudes, which is widely used in fields such as quantum mechanics, signal processing, optics, etc. The offset linear canonical transform (OLCT), which is a more general type of linear integral transform including Fourier transform (FT), fractional Fourier transform (FrFT), and linear canonical transform (LCT) as its special cases. Hence, in this paper, we focus on the uniqueness problem of phase retrieval in the framework of OLCT. First, we prove that all the nontrivial ambiguities in continuous OLCT phase retrieval can be represented by convolution operators, and demonstrate that a continuous compactly supported signal can be uniquely determined up to a global phase from its multiple magnitude-only OLCT measurements. Moreover, we investigate the nontrivial ambiguities in the discrete OLCT phase retrieval case. Furthermore, we demenstrate that a nonseparable function can be uniquely recovered from its magnitudes of short-time OLCT (STOLCT) up to a global phase. Finally, we show that signals which are bandlimited in FT or OLCT domain can be reconstructed from its sampled STOLCT magnitude measurements, up to a global phase, providing the ambiguity function of window function satisfies some mild conditions.

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