{"id":"2b541158-ff0e-47b5-afb5-22b445d10416","arxiv_id":"2506.03944","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Compactly supported and bandlimited signals are unique up to global phase from certain OLCT and short-time OLCT magnitude measurements, extending prior Fourier and linear canonical transform phase retrieval theorems.","lead":"This paper proves when a signal can be recovered, up to a global phase, from the magnitude of its offset linear canonical transform, a five-parameter generalization of the Fourier and fractional Fourier transforms. The results are theoretical uniqueness guarantees for continuous, discrete, and windowed measurements, relevant to optics and signal processing.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4 is not well-defined as printed, and its proof omits the τ=0 slice needed for the final equality of ambiguity functions.","rationale":"The reader's weakest assumption already identifies the malformed statement of Theorem 3.4 and the dependence on a positive-measure set of ratios. I agree with that assessment. My stress-test adds a second, more specific proof gap: the Paley-Wiener continuation is applied for every τ, but for τ=0 the set Λτ is not positive measure, so the proof as written does not establish equality of A_f and A_g on the slice needed for (15). This is a genuine but repairable gap, so it does not change the reader's conditional verdict. I did not treat the STOLCT issues as the single load-bearing concern for the central claim, because the strongest claim identified by the reader is Theorem 3.4, and that theorem is independent of the STOLCT half. The central mathematical strategy of Theorem 3.4 is credible: compact support makes the ambiguity function entire in the second variable, and equality on positive-measure sets of ratios forces equality of the full ambiguity functions. The verdict should remain conditional pending the statement rewrite, the τ=0 repair, and the already-noted corrections in the STOLCT sections.","tokens_in":14349,"tokens_out":34870,"duration_ms":360218,"concrete_test":"Restrict Theorem 3.4 to the slice τ=0 and check whether the written hypotheses imply A_f(0,σ)=A_g(0,σ) for all σ. The printed proof derives equality only at σ=0 on this slice because Λ_0={0}. Supply an explicit argument: for a sequence τ_n→0, the positive-measure sets Λ_{τ_n} give equality of the entire functions σ↦A_f(τ_n,σ) and σ↦A_g(τ_n,σ) for every n; if the ambiguity functions are uniformly continuous in (τ,σ), taking n→∞ yields the missing slice. If this limiting argument cannot be completed, the conclusion of Theorem 3.4 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central uniqueness claim rests on Theorem 3.4, but the statement is internally inconsistent: Λ is first declared to be a subset of R with positive measure and then defined as a set of OLCT parameter matrices. Under the literal wording, the hypothesis \"|OA f|=|OA g| for all A∈Λ\" is undefined. Reading the intended hypothesis charitably as \"the ratios a/b of the matrices in the family Λ form a positive-measure set,\" the surrounding argument is essentially sound: (17) gives equality of the ambiguity functions along the line σ=-(a/b)τ, and for each fixed τ≠0 the collection Λτ={-a/b τ : A∈Λ} has positive measure, so the Paley-Wiener/identity-theorem step is valid. The gap is that the proof asserts this for every τ, including τ=0. For τ=0, Λτ={0}, a zero-measure set, so the Paley-Wiener step gives no information; yet the final invocation of (15) requires A_f=A_g on the full ambiguity domain, including the slice τ=0. A continuity-in-τ limiting argument would likely repair this, but it is not supplied. Thus the theorem is not established as written, even though the underlying mathematical idea appears sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":14580,"tokens_out":10873,"duration_ms":107314,"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":[{"comment":"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.","section":"Section 3.2, Theorem 3.4 statement"},{"comment":"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.","section":"Section 3.2, Theorem 3.4 proof, following Eq. (17)"},{"comment":"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.","section":"Section 5, Lemma 5.1 proof, Eqs. (20)–(22)"},{"comment":"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.","section":"Section 5.2, Theorem 5.5 proof, Eqs. (26)–(27)"}],"minor_comments":[{"comment":"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'.","section":"Abstract and text"},{"comment":"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.","section":"Section 5.1, definition of separability"},{"comment":"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.","section":"Sections 3.2 and 5.1, black-box citations"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central results are plausible extensions of known Fourier/LCT phase retrieval facts, and the exact identities (5), (6), and (10) provide a sound reduction strategy. However, the printed proof of Lemma 5.1 contains a demonstrably false equality, and Theorem 3.4 is both malformed and missing a continuity argument for the τ=0 slice. These issues are load-bearing and need to be corrected before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine but sloppy extension of known uniqueness results to OLCT phase retrieval. The core ideas are right, and the paper fills a real gap. The continuous and discrete OLCT ambiguity characterizations reduce cleanly to known Fourier/LCT results via identities (5) and (6). Theorem 3.4, modulo the τ=0 issue, gives a plausible multiple-OLCT uniqueness statement for compactly supported signals. The STOLCT results are not verifiable as printed.\n\nWhat's new: the OLCT and STOLCT uniqueness claims are new relative to the cited literature, which only covers FT, FrFT, LCT, and STFT/STLCT. Theorems 3.2 and 4.2 characterize nontrivial ambiguities via convolution, and the proofs honestly reduce to the known Fourier-domain results. That is a fair contribution. Theorem 3.4's idea is sound, and the STOLCT results extend [32] and [34] in a natural way.\n\nWhere it's soft: the statement of Theorem 3.4 first says Λ ⊂ R and then defines Λ as a set of parameter matrices. That can't be right. If the intended hypothesis is that the set of ratios a/b has positive measure, the proof still has a gap: it asserts Λτ has positive measure for every τ, but for τ=0 this set is just {0}. The Paley-Wiener step gives equality of ambiguity functions only for τ≠0; the τ=0 slice needs a continuity-in-τ argument that isn't supplied. I think this is repairable, but it's a real missing step.\n\nLemma 5.1 is more serious. The proof sets f♯v(t)=fv(−t) and claims F(fv)=F(f♯v). That's false; you get F(fv)(−u) at best. The final formula (19) may still be correct, but the printed proof doesn't establish it, and the ambiguity-function conventions in that section are inconsistent. Since Theorems 5.3–5.6 all lean on Lemma 5.1, the STOLCT branch is unverified as written. This is a presentation-and-proof problem, not necessarily a wrong result, but it needs to be fixed.\n\nBottom line: for a reader who works on phase retrieval in generalized transform domains, this paper is worth reading after the authors clean it up. It doesn't provide algorithms, stability, or numerics, so its value is purely theoretical. It deserves a serious referee, but I'd ask for corrections to Theorem 3.4 and Lemma 5.1 before accepting.","headline":"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.","tokens_in":15117,"tokens_out":5018,"would_cite":false,"duration_ms":48545,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","42C40","94A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["phase retrieval","offset linear canonical transform","ambiguity function","short-time offset linear canonical transform","bandlimited signals","uniqueness","convolution ambiguities","Paley-Wiener theorem"],"falsifier":"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.","tokens_in":14142,"feed_emoji":"📡","tokens_out":14425,"duration_ms":151399,"temperature":0.7,"pith_summary":"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.","feed_headline":"OLCT magnitudes fix signals up to a global phase","feed_subtitle":"Uniqueness up to global phase extends Fourier and fractional phase retrieval to the five-parameter OLCT family.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the theorem (equation (15)) that equality of ambiguity functions forces two L2 signals to agree up to a unit-modulus constant.","marker":"[28]"},{"why":"Its proof of the convolution representation for linear canonical transform phase retrieval is the template for Theorem 3.2's OLCT ambiguity characterization.","marker":"[26]"},{"why":"Provides the Paley-Wiener theorem used to extend equality of ambiguity functions from a positive-measure set of slices to the whole function in Theorem 3.4.","marker":"[44]"},{"why":"Supplies Lemma 2.4 on the support and continuity of ambiguity functions of bandlimited signals, and the proof template for the sampled STOLCT theorems.","marker":"[32]"},{"why":"Provides the STLCT phase retrieval uniqueness theorems whose proof strategies the STOLCT results in Theorems 5.4 and 5.6 follow.","marker":"[34]"},{"why":"Supplies the discrete Fourier phase retrieval ambiguity theorem whose proof pattern is used for the discrete OLCT convolution characterization in Theorem 4.2.","marker":"[45]"},{"why":"Supplies Proposition 5.2, the nonseparability criterion that converts equality of magnitudes into equality up to a sign in the STOLCT theorems.","marker":"[46]"}],"fun_headline_variants":["OLCT magnitudes determine signals up to global phase","Phase retrieval unique for OLCT magnitudes","OLCT phase retrieval: only global phase ambiguity","Magnitudes alone recover OLCT signals up to phase","Offset linear canonical transform: phase from magnitudes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["OLCT magnitudes determine signals up to global phase","Phase retrieval unique for OLCT magnitudes","OLCT phase retrieval: only global phase ambiguity","Magnitudes alone recover OLCT signals up to phase","Offset linear canonical transform: phase from magnitudes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2093,"prompt_tokens":1063,"completion_tokens":1030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":961}},"tokens_in":679,"tokens_out":1030,"duration_ms":11420,"temperature":1.0,"reasoning_tokens":961,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:56:47.310540+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theorem (equation (15)) that equality of ambiguity functions forces two L2 signals to agree up to a unit-modulus constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Its proof of the convolution representation for linear canonical transform phase retrieval is the template for Theorem 3.2's OLCT ambiguity characterization."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Paley-Wiener theorem used to extend equality of ambiguity functions from a positive-measure set of slices to the whole function in Theorem 3.4."},{"cited_title":"Alaifari, M","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 2.4 on the support and continuity of ambiguity functions of bandlimited signals, and the proof template for the sampled STOLCT theorems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the STLCT phase retrieval uniqueness theorems whose proof strategies the STOLCT results in Theorems 5.4 and 5.6 follow."},{"cited_title":"Beinert, G","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete Fourier phase retrieval ambiguity theorem whose proof pattern is used for the discrete OLCT convolution characterization in Theorem 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 5.2, the nonseparability criterion that converts equality of magnitudes into equality up to a sign in the STOLCT theorems."}],"review_version":1}