{"id":"1075d25c-b343-4695-8785-9c9a4292dab1","arxiv_id":"1908.10205","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For real-valued objects, iterative phase retrieval can recover both the object and the missing intensity values when the missing fraction stays below 1 - 2/sigma^2 for diffraction patterns and 1 - 1/sigma^2 for holograms.","lead":"This paper shows that coherent diffraction patterns and holograms with many missing detector pixels can still be reconstructed using iterative phase retrieval. The tolerable missing fraction is set by the oversampling ratio, and the missing pixels must be spread out rather than clustered at the center.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Counting bound in Eq. 7 is presented as the maximum allowable missing fraction, but the paper's own unsymmetrized σ=8 runs fail at f=0.8, well below 0.969; sufficiency is assumed, not shown.","rationale":"The paper provides a clean, reproducible counting bound for when missing-pixel recovery is information-theoretically possible, and the simulations (especially with symmetrization) are coherent and support the usefulness of the approach. The stress-test concern is not that the counting is wrong as a necessary condition, but that the paper elevates it to 'should not exceed' and 'sufficient' without proof. The failure at f=0.8 in the unsymmetrized σ=8 case is the clearest evidence: it is below the claimed bound yet reconstruction fails, so the bound is not sufficient for the raw protocol. The symmetrization step is a legitimate use of prior knowledge for real-valued objects, but it must be part of the stated claim. The reader's CONDITIONAL verdict already captures this; our concrete test would determine whether the sufficiency gap is merely a matter of pipeline choice or a fundamental limitation, and would likely motivate a revised statement of the bound rather than a rejection.","tokens_in":14909,"tokens_out":13165,"duration_ms":129386,"concrete_test":"Repeat the σ=8 CDI experiment of Section 2.2.1 with f=0.8 exactly as in Appendix A but using a different real-valued object (e.g., an irregularly shaped sample rather than the 'man' image), both with and without the symmetrization preprocessing, and also with 1% Poisson noise added to the measured intensities. If the unsymmetrized reconstruction fails while the symmetrized one succeeds, Eq. 7 is not a sufficient condition on the raw missing fraction; the 'maximum allowable' claim should be restated as a necessary count and explicitly restricted to the symmetrized, noise-free protocol.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that f < 1 − 2/σ² (Eq. 7) or f < 1 − 1/σ² (Eq. 14) is the maximum allowable missing fraction rests on a counting argument (Section 2.1, Eq. 6): the number of measured intensity samples must exceed the number of object unknowns. That is a necessary condition, not a sufficient one. The paper's own data expose the gap: for σ=8, Eq. 7 allows f<0.969, yet the unsymmetrized protocol of Section 2.2.1 fails at f=0.8 and f=0.9 (Fig. 3, Table 2). Success at f=0.95–0.97 is achieved only after the symmetrization step of Section 2.2.2, which uses the centrosymmetry of the diffraction pattern to replace missing pixels with their symmetry partners and therefore changes the effective missing fraction. The abstract's headline example ('5% of measured values at σ=8') is thus only supported for the symmetrized pipeline, not for the raw incomplete data, and no proof or noise study establishes that the count bound is sufficient more generally. There is also an algebraic inconsistency in Section 2.1: N²/2 > N0² implies σ>√2, not the stated σ>2, so the relation between Eq. 2 and the established oversampling condition is unclear.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the recovery of missing intensity pixels in coherent diffraction imaging (CDI) and in-line holography using iterative phase retrieval. For a real-valued object, the author derives from a simple counting argument that the fraction f of missing pixels should not exceed 1 - 2/σ² for a diffraction pattern and 1 - 1/σ² for a hologram, where σ is the linear oversampling ratio. Simulations on a single real-valued amplitude object (the \"man\" image) with known tight support and no noise show that random missing pixels can be recovered up to high f when the diffraction pattern is centro-symmetrised, that missing pixels concentrated in the centre ruin reconstruction even at small f, and that holograms tolerate somewhat larger missing fractions. The paper concludes that even 5% of measured intensity values at σ = 8 suffice for CDI and 6% at σ = 4 for holography.","tokens_in":15197,"tokens_out":5961,"duration_ms":58285,"significance":"If the claimed bounds were established as sufficient conditions, they would provide simple quantitative guidance for experiments with dead pixels, beamstops, or saturated regions, which is practically valuable. The counting derivation is transparent and parameter-free, and the paper makes a useful, well-illustrated observation that missing low-frequency information in the centre of a diffraction pattern is far more damaging than uniformly distributed missing pixels. The error-metric comparison in Section 2.3 is also a useful practical note. However, the paper does not prove that the counting condition is sufficient; its own unsymmetrised simulations at σ = 8 fail at f = 0.8–0.9, well below the claimed limit of 0.969. The headline success at f = 0.95–0.97 relies on a symmetrisation step that changes the effective missing fraction, and all simulations are noise-free, use one object type, and assume a known tight support. The central claim therefore outruns the evidence, though the underlying phenomenon is real and the paper can be revised to state the necessary-condition status accurately.","major_comments":[{"comment":"There is an algebraic inconsistency between Eq. (2) and Eq. (4). From Eq. (2), N²/2 > N0², and with σ = N/N0, one obtains σ > √2, not σ > 2. If the intended statement is the standard, more conservative linear oversampling condition σ > 2, then Eq. (2) needs to be changed (for example, to N²/4 > N0² if one counts independent equations after accounting for both the real-valuedness and the centrosymmetry of the intensity). As written, the derivation of Eq. (7) inherits this inconsistency, and the factor of 2 inside Eq. (7) deserves a clear justification.","section":"Section 2.1, Eqs. (2)–(4)"},{"comment":"The claim that f < 1 - 2/σ² is the maximum allowable missing fraction (Abstract, Section 2.1, Conclusions) is not supported by the paper's own unsymmetrised results. For σ = 8, Eq. (7) allows f < 0.969, yet the unsymmetrised protocol of Section 2.2.1 fails at f = 0.8 and f = 0.9, with errors of 9.97×10⁻³ and 1.11×10⁻³ in Table 2 and stagnation in Fig. 3u. For σ = 4, the unsymmetrised protocol fails at f = 0.6 (Table 1), although Eq. (7) gives f < 0.875. The successes at f = 0.95–0.97 (Fig. 5, Table 4) are obtained only after the symmetrisation step in Section 2.2.2, which the paper itself notes changes the effective number of missing pixels. The counting argument in Eq. (6) is a necessary condition for the existence of a solution, not a proof that iterative phase retrieval will succeed; the manuscript should either rephrase the headline claim as a necessary condition, or explicitly and prominently qualify that the sufficiency assertion holds only for the symmetrised, noise-free, known-support examples actually simulated.","section":"Section 2.2.1 and Abstract; Eq. (7)"},{"comment":"The holography bound f < 1 - 1/σ² in Eq. (14) is likewise derived solely from a counting argument. The supporting simulation is a single noise-free in-line hologram of a real-valued amplitude object with a known tight support and a fixed reconstruction pipeline. The text says that reconstructions are \"identical\" up to f = 0.9 but only \"still resemble\" the original at f = 0.95, which is above the claimed limit of 0.938; this is not a demonstration that the bound is tight or sufficient. As with the CDI case, the paper should state clearly that Eq. (14) is a necessary condition, and that the empirical evidence for its sufficiency is limited to one idealised example. The current wording, including the Abstract's \"should not exceed\", overstates the strength of the result.","section":"Section 3.1 (Eq. 14) and Section 3.2"}],"minor_comments":[{"comment":"In the paragraph on error metrics, the sentence \"the reconstructions with the lowest error as defined by Eq. 7\" should read \"Eq. 8\", since Eq. 7 is the missing-fraction bound, not an error metric.","section":"Section 2.3"},{"comment":"There is a typo in the sentence \"a single inverse FT delivers a poorer reconstruction that the that obtained by iterative phase retrieval\"; it should be \"than that obtained\".","section":"Section 5 (Conclusions)"},{"comment":"The figure caption lists subplot labels (a)–(t) for f = 0 to f = 0.9 but the text and Table 1 omit f = 0.6 and f = 0.7 in the caption listing; please check the caption for consistency.","section":"Section 2.2.1, Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper's main quantitative claim is presented as a proven maximum, but the derivation yields only a necessary condition and the authors' own unsymmetrised simulations violate the claimed bound. The revision should reframe the central claim as a necessary condition plus an empirically observed sufficient condition for the symmetrised, noise-free, known-support case. The algebraic slip in Eqs. (2)–(4) should be corrected before publication. The paper relies on several of the author's own prior works ([12], [18], [19]); this is not disqualifying, but the editor may wish to confirm that the present contribution is clearly distinguished from [12], which already described iterative recovery of missing diffraction-pattern pixels."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives a clean counting argument for how many randomly missing pixels can be tolerated in CDI and in-line holography: for real-valued objects, f < 1 − 2/σ² in diffraction patterns and f < 1 − 1/σ² in holograms, where σ is the linear oversampling ratio. The simulations are extensive and the comparison of random versus central missing pixels is genuinely useful. The explicit inequalities and the systematic demonstration that random missing pixels are far more tolerable than a central gap are new relative to the author's prior work, which only noted that recovery is possible. If you work in coherent imaging, this is a handy rule of thumb for dead pixels and beamstop gaps.\n\nThe soft spot is real and load-bearing. Equations (6) and (7) are derived from a counting argument, which is a necessary condition for uniqueness, not a proof that iterative phase retrieval will find the solution. The paper's own data show the gap: for σ = 8, Eq. (7) allows f < 0.969, yet the unsymmetrized reconstructions in Fig. 3 fail at f = 0.8 and 0.9. The success at f = 0.95–0.97 is only achieved after the symmetrization step of Section 2.2.2, which uses the centrosymmetry of the diffraction pattern to fill missing pixels from their opposite-side partners, thereby lowering the effective missing fraction. The paper does admit this in Section 2.2.2, but the abstract and conclusions present the bound as a general maximum without that qualification. So the headline example ('even 5% of measured values at σ = 8') is supported only for the symmetrized pipeline, not for raw incomplete data.\n\nThere is also a minor algebra slip in Section 2.1: N²/2 > N₀² gives σ > √2, not σ > 2. The standard oversampling condition σ > 2 may still be the right practical requirement, but as written the derivation is inconsistent. The simulations are noise-free, use a single real-valued 'man' object with known tight support, and report the best of ten reconstructions selected by an error metric that uses the known original; that makes the quantitative success metrics optimistic. None of this invalidates the practical value, but it does mean the 'should not exceed' language is a conjecture supported by examples, not a proven theorem.\n\nWho is this for? Experimentalists in CDI and holography who want a simple guide to how many dead pixels or beamstop gaps they can tolerate, and how to distribute them. It deserves a serious referee; the issues are addressable with a better-qualified abstract, a statement that the bound is necessary but not sufficient, and a clearer separation of the symmetrized and unsymmetrized regimes. I'd send it out, with the expectation of major revision.","headline":"A useful counting bound for missing-pixel tolerance in CDI and holography, but the paper sells a necessary condition as sufficient and only the symmetrized pipeline supports the headline example.","tokens_in":15729,"tokens_out":3062,"would_cite":false,"duration_ms":29276,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that an object can be reconstructed from a diffraction pattern or hologram even when most measured intensity samples are missing, provided the experiment is oversampled and the missing pixels are randomly scattered.","keywords":["coherent diffraction imaging","in-line holography","iterative phase retrieval","missing intensity values","oversampling ratio","hybrid input-output algorithm","dead pixels","Nyquist-Shannon sampling"],"falsifier":"Take an object different from the one used here and add Poisson noise to its diffraction pattern, delete a random fraction $f$ just below $1 - 2/\\sigma^2$ at $\\sigma = 8$, and run the same HIO protocol with a loose support. If reconstructions fail or stagnate at error levels well above the no-missing baseline, the counting bound is necessary but not sufficient.","tokens_in":14691,"feed_emoji":"🔬","tokens_out":7101,"duration_ms":67653,"temperature":0.7,"pith_summary":"This paper asks how much of a measured diffraction pattern or hologram can be missing before the object can no longer be recovered. Its answer is a pair of simple counting bounds: for a real-valued object, the missing fraction $f$ should stay below $1 - 2/\\sigma^2$ in coherent diffraction imaging and below $1 - 1/\\sigma^2$ in in-line holography, where $\\sigma$ is the linear oversampling ratio. At $\\sigma = 8$ that means as few as 5% of the diffraction-pattern intensities can be enough; at $\\sigma = 4$, 6% of hologram intensities suffice. The same iterative phase retrieval routine then recovers both the object and the values of the missing pixels. The practical catch is that the missing pixels must be randomly distributed, not concentrated at the center as a beamstop would produce.","feed_headline":"Missing 90% of a diffraction pattern still yields the image","feed_subtitle":"A 5% sample of the diffraction pattern can reconstruct both the object and the missing pixels.","key_machinery":"The load-bearing device is a counting inequality: the number of measured intensity values must exceed the number of unknown object pixels. Written with the linear oversampling ratio $\\sigma = N/N_0$, it becomes the missing-pixel thresholds $f < 1 - 2/\\sigma^2$ for diffraction patterns and $f < 1 - 1/\\sigma^2$ for holograms. The algorithm that carries the recovery is the hybrid input-output (HIO) iterative phase retrieval routine with a tight object support; at each iteration the missing Fourier or hologram amplitudes are replaced by the current estimate, so the support constraint effectively fills gaps in the data.","core_discovery":"The paper's central claim is that the information content of an oversampled diffraction measurement is far larger than the object itself, and that excess information can be spent on missing pixels. Counting equations against unknowns gives $f < 1 - 2/\\sigma^2$ for a real-valued object in coherent diffraction imaging and $f < 1 - 1/\\sigma^2$ for an amplitude object in holography. Simulations with the hybrid input-output algorithm, replacing each missing intensity by the current iterate, confirm the counting picture: at high oversampling ratios the object is recovered even when most pixels are missing, and symmetrization of the centrosymmetric diffraction pattern improves convergence and extends recovery toward the theoretical limit. The missing values themselves are reconstructed along with the object, so no separate inpainting step is needed.","pith_inferences":["In experiments with noise and imperfect support, the safe missing fraction will probably sit below the counting bound; a calibration curve of $f$ versus reconstruction error would be a direct test.","The same counting logic could be applied to complex-valued or three-dimensional objects, giving thresholds of the same form; whether iterative phase retrieval reaches them is an open question.","The center-missing failure is a clue that low spatial frequencies carry the object's background; combining phase retrieval with extrapolation or a prior on low frequencies might rescue beamstop data.","If these thresholds hold under noise, they suggest a detector-design trade-off: oversample more in exchange for accepting more defective pixels."],"forward_implications":["Detectors with random dead or saturated pixels can tolerate high pixel-loss fractions when the measurement is oversampled; no separate inpainting step is needed.","At linear oversampling ratio 8, retaining only about 5% of the diffraction-pattern intensities can be enough to recover both the object and the missing values for a real-valued object.","In-line holography inherits the same resilience, with the reference-wave extent playing the role of oversampling and a threshold of $f < 1 - 1/\\sigma^2$.","Centrally missing values, such as those hidden by a beamstop or a saturated central spot, are a different problem: even 1% missing in the center defeats recovery, so such gaps require other strategies."],"supporting_citations":[{"why":"Supplies the oversampling condition and the equation-count argument that the new thresholds extend.","marker":"[13]"},{"why":"Provides the hybrid input-output phase retrieval algorithm used for all reconstructions.","marker":"[15]"},{"why":"Gives the simulation and reconstruction protocol, including object padding and HIO parameters, on which the numerical examples are built.","marker":"[12]"},{"why":"Supplies the Shannon sampling argument underlying the oversampling requirement.","marker":"[14]"},{"why":"States the reference-wave-extent condition that motivates the holography oversampling ratio.","marker":"[5]"},{"why":"Provides the angular-spectrum method used to simulate and reconstruct in-line holograms.","marker":"[18]"},{"why":"Provides the twin-image elimination procedure used in the hologram reconstruction loop.","marker":"[19]"},{"why":"Offers the comparison point for recovery from incomplete frequency information via convex optimisation.","marker":"[22]"}],"fun_headline_variants":["5% diffraction samples reconstruct object and missing pixels","Oversampling lets you drop most diffraction data and still image","Random missing pixels in patterns? Iterative phase retrieval copes","Reconstruct objects from a mere sliver of hologram pixels","Lose most diffraction data, keep the image via oversampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that satisfying the counting inequality is enough for the iterative algorithm to converge to the true object; this is supported by noise-free simulations of one real-valued test object with known support, not by a proof.","fun_headline_variants_meta":{"raw":{"variants":["5% diffraction samples reconstruct object and missing pixels","Oversampling lets you drop most diffraction data and still image","Random missing pixels in patterns? Iterative phase retrieval copes","Reconstruct objects from a mere sliver of hologram pixels","Lose most diffraction data, keep the image via oversampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000293,"raw_usage":{"total_tokens":1659,"prompt_tokens":850,"completion_tokens":809,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":466,"completion_tokens_details":{"reasoning_tokens":726}},"tokens_in":466,"tokens_out":809,"duration_ms":7939,"temperature":1.0,"reasoning_tokens":726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:13:05.239115+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an object different from the one used here and add Poisson noise to its diffraction pattern, delete a random fraction $f$ just below $1 - 2/\\sigma^2$ at $\\sigma = 8$, and run the same HIO protocol with a loose support. If reconstructions fail or stagnate at error levels well above the no-missing baseline, the counting bound is necessary but not sufficient.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the oversampling condition and the equation-count argument that the new thresholds extend."},{"cited_title":"Fienup, Phase retrieval algorithms – a comparison, Appl","cited_arxiv_id":null,"evidence_quote":"Provides the hybrid input-output phase retrieval algorithm used for all reconstructions."},{"cited_title":"Latychevskaia, Iterative phase retrieval in coherent diffractive imaging: practical issues, Appl","cited_arxiv_id":null,"evidence_quote":"Gives the simulation and reconstruction protocol, including object padding and HIO parameters, on which the numerical examples are built."},{"cited_title":"Sayre, Some implications of a theorem due to Shannon, Acta Crystall., 5 (1952) 843–843","cited_arxiv_id":null,"evidence_quote":"Supplies the Shannon sampling argument underlying the oversampling requirement."},{"cited_title":"Gabor, Microscopy by reconstructed wave-fronts, Proc","cited_arxiv_id":null,"evidence_quote":"States the reference-wave-extent condition that motivates the holography oversampling ratio."},{"cited_title":"Latychevskaia, H.-W","cited_arxiv_id":null,"evidence_quote":"Provides the angular-spectrum method used to simulate and reconstruct in-line holograms."},{"cited_title":"Latychevskaia, H.-W","cited_arxiv_id":null,"evidence_quote":"Provides the twin-image elimination procedure used in the hologram reconstruction loop."},{"cited_title":"Candes, J","cited_arxiv_id":null,"evidence_quote":"Offers the comparison point for recovery from incomplete frequency information via convex optimisation."}],"review_version":1}