{"id":"c4789e0b-0a55-4a4c-bc79-94d710b97857","arxiv_id":"2501.05244","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Scaled and non-uniform FFT variants of the Rayleigh-Sommerfeld diffraction algorithm reconstruct NLOS scenes with flexible sampling grids and depth-dependent voxel sizes at FFT-level complexity.","lead":"This paper gives non-line-of-sight imaging algorithms that use modified fast Fourier transforms to reconstruct hidden scenes from unevenly spaced sensor points and to enlarge the reconstruction volume without extra memory. It also argues that current setups oversample the relay wall, which explains why heavy data compression works with little quality loss.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 57's oversampling ratio is too optimistic: Supplement A.3 uses the local phase slope k'' as the spatial frequency, but the instantaneous frequency of the Fresnel phase is 2k'', and the aperture-maximum frequency is ignored; the 'typically oversample / D=5 / 100x' premise is not established.","rationale":"Read in good faith: the paper's core algorithms (SRSD, NURSD) are demonstrated on real data, and that part stands independently. The load-bearing theoretical premise is the oversampling criterion because it justifies the 100x compression and motivates NUFFT-based sampling. Re-deriving Supplement A.3 shows a concrete algebraic slip: the quadratic Fresnel phase has local frequency dphi/dx = k(x_r-x_0)/L, not k''=k(x_r-x_0)/(2L), and the bandlimit over a finite aperture is set by the maximum of |x_r-x_0| across the aperture. This makes Eq. 57's ratio optimistic by a factor of 2 and its D-admissible table too permissive. The empirical downsampling experiments are suggestive but they only test two scenes at high SNR, with the reference being the same method's full-data output and without ground truth, so they cannot validate the broad quantitative claim. The concern does not invalidate the algorithms; it narrows the claimed theoretical justification. Hence the reader's CONDITIONAL verdict stands (unchanged), with the condition sharpened to 'correct the sampling-bound derivation and re-state the compression claims accordingly'.","tokens_in":26541,"tokens_out":13019,"duration_ms":129751,"concrete_test":"Re-derive the transverse sampling bound for a point source at (x0,L) over a relay aperture of half-width W: the maximum instantaneous frequency is max_{x_r}|x_r-x_0|/(lambda L), so the bound is delta_x <= lambda L/[2 max_{x_r}|x_r-x_0|]. Then re-evaluate Table 1 and the Sec. 7.2 downsampling experiment with the corrected bound for the 1.8 m x 1.3 m wall and L in [1,3] m, lambda_c=4 cm. If the corrected max admissible D is below 5 for typical hidden-scene offsets (including off-axis points), the 100x/'typically oversample' claim needs revision; if it still exceeds 5 for the measured scenes with margin, the concern is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section A.3 derives Eq. 57 by writing the Fresnel phase of a point source as exp[ik''(x_r-x_0)] with k''=k(x_r-x_0)/(2L), then uses k'' as the transverse spatial frequency. But the phase is quadratic: phi(x_r)=k(x_r-x_0)^2/(2L), so its instantaneous frequency is dphi/dx_r=k(x_r-x_0)/L = 2k''. The Nyquist spacing is therefore lambda_sx <= pi/(2k'') = lambda L/[2|x_r-x_0|], a factor of 2 smaller than Eq. 55. In addition, over a finite relay aperture the frequency varies across the aperture; the relevant bound uses max_{x_r}|x_r-x_0|, not a single local x_r. With the corrected bound the oversampling ratio becomes L/max|delta_x| instead of 2L/|delta_x|, so Table 1's admissible D values shrink by roughly 2x. For the paper's own geometry (1.8 m x 1.3 m wall, lambda_c=4 cm, depths 1-3 m), D=5 and the stated ~100x data reduction are not supported for scene points at moderate lateral offsets. The empirical reconstructions remain useful evidence, but the mathematical justification for 'typically oversample' is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes three families of modifications to the standard Rayleigh-Sommerfeld Diffraction (RSD) algorithm for non-line-of-sight (NLOS) imaging: Scaled RSD (SRSD), which changes the lateral voxel size with depth using a scaled Fourier transform; Non-Uniform RSD (NURSD-1/2/3), which uses the NUFFT to allow non-uniform sampling of the relay surface and of the reconstruction grid; and 3D NURSD for non-planar relay surfaces. The central theoretical claim is that existing NLOS relay-surface measurements are spatially oversampled, formalized as lambda_{sx}/lambda_{sz} = 2|z_r-z_0|/|x_r-x_0| > D (Eq. 25/57). This is used to justify spatial subsampling and interpolation, leading to a claimed ~100x data reduction. The algorithms are validated on public datasets and on a custom 16x16 SPAD-array acquisition, with comparisons to standard RSD and filtered backprojection.","tokens_in":26849,"tokens_out":5956,"duration_ms":55852,"significance":"If the oversampling claim holds, the paper provides both a theoretical explanation for the empirically observed compressibility of NLOS measurements and practical tools to exploit it, while preserving FFT-level computational complexity. The SRSD additionally provides a memory-efficient, perspective-correct reconstruction volume. The manuscript gives detailed derivations of the SFFT and NUFFT variants, uses public datasets and external baselines, and demonstrates the algorithms on real non-planar relay surfaces and a real SPAD array. These strengths make the work potentially useful to the NLOS imaging community. The main caveat is that the central sampling-rate derivation is too optimistic by a factor of two and lacks a worst-case analysis for extended scenes, so the quantitative scope of the compression claim needs revision.","major_comments":[{"comment":"The derivation equates the local phase slope k'' = k(x_r-x_0)/(2(z_r-z_0)) with the transverse spatial frequency, but the instantaneous frequency of the quadratic Fresnel phase phi(x_r) = k(x_r-x_0)^2/(2(z_r-z_0)) is dphi/dx_r = k(x_r-x_0)/(z_r-z_0) = 2k''. For a finite relay aperture, the relevant Nyquist bound is set by the maximum of this frequency over the aperture, not by its value at a single local coordinate. The corrected bound is lambda_{sx} <= lambda*(z_r-z_0)/(2 max|x_r-x_0|), which makes the oversampling ratio in Eq. (25) and Eq. (57) too large by a factor of about two. With the paper's own geometry (1.8 m x 1.3 m wall, lambda_c = 4 cm, depths 1-3 m), the admissible D values in Table 1 shrink accordingly, and the '~100x' data-reduction claim in Sections 1.1 and 7.2.3 is not supported for scene points at moderate lateral offsets. The empirical reconstructions remain useful evidence, but the mathematical justification for 'typically oversample' is not established as stated.","section":"Supplement A.3, Eqs. (54)-(57)"},{"comment":"The sampling criterion is derived for a single paraxial point source. Real hidden scenes contain extended objects at multiple depths and lateral positions, so the required sampling rate is determined by the worst-case combination of maximum lateral offset and minimum depth over all points contributing to the measurement. The paper does not provide such a worst-case analysis, and the statement in Section 1.1 that 'most NLOS imaging setups currently oversample' is therefore not quantitatively justified for general scenes. The authors should either supply a worst-case derivation or explicitly restrict the claim to scenes satisfying specified geometric bounds.","section":"Section 5 and Supplement A.3"},{"comment":"The paper correctly states that the sampling derivation ignores photon noise and demonstrates experimentally that interpolation degrades sharply at low exposure times. However, the abstract and Section 1.1 present spatial oversampling and 'roughly 100x less data' as general properties without this caveat. The compression claim should be explicitly conditioned on sufficient photon flux and on the geometric conditions from the corrected sampling criterion, so that readers do not take the unqualified statement as the paper's central result.","section":"Section 7.2.1 and Fig. 9"}],"minor_comments":[{"comment":"The interpolation protocol says 'Reconstruct using fast RSD algorithm, with the wavelength set to 2 lambda_s', but the experiments in Section 7.2 use lambda_c = 4 cm with 1 cm sampling; the relationship between lambda_s and lambda_c in this step should be clarified.","section":"Section 5, step 3"},{"comment":"The text refers to a 'low pass filter' applied to compute the blue SSIM scores, but does not specify the filter parameters or implementation; please provide details.","section":"Section 8.1, Fig. 13"},{"comment":"The integrals over domega and dvec{x}_p are written without explicit limits; adding the integration ranges would improve reproducibility.","section":"Eq. (2)"},{"comment":"The symbol N is used both for the side length of the reconstruction cube and for the number of samples per dimension; defining these separately would avoid confusion in Sections 3.1, 4.2, and 6.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The core algorithms (SRSD, NURSD variants) are well specified and validated against external baselines, which is a genuine strength. The main obstacle is the oversampling criterion: the factor-of-two error in Eq. (57) and the lack of a worst-case analysis mean the paper's headline compression claim is overstated. This is fixable within the manuscript's scope - the authors can correct the derivation and restate the claim with explicit geometric and photon-flux conditions - and the empirical results would still support a useful but more modest contribution. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the algorithmic work is genuinely useful: fusing the scaled FFT and the non-uniform FFT with the Rayleigh-Sommerfeld diffraction integral for NLOS imaging (SRSD, NURSD-1/2/3, 3D NURSD) is new, preserves FFT-level complexity, and is validated on public datasets and a real 16x16 SPAD array. Second, the paper's headline theoretical claim—that NLOS relay surfaces are typically oversampled, with Eq. 25/57 giving a D>1 criterion and 100x compression—has a real flaw. The supplement's derivation uses k'' = k(xr-x0)/(2L) as the spatial frequency, but the instantaneous frequency of the quadratic Fresnel phase is 2k''. Also, over a finite relay aperture you need to use the maximum |xr-x0|, not a single local value. So the correct ratio is about L/max|Δx|, roughly half what they claim, and D=5 is not supported for points at moderate lateral offsets in their geometry. That doesn't kill the empirical demonstrations—their downsampled reconstructions do hold up on the tested scenes—but the mathematical justification for \"typically oversample\" is not established as stated.\n\nThe paper does well in several respects. The SRSD's depth-dependent voxel growth is an elegant fix for the resolution-versus-depth tradeoff and produces perspective-correct projections. NURSD-1 on the SPAD array matches FBP quality while being orders of magnitude faster, and the non-planar 3D NURSD is a clean replacement for 3D RSD's interpolation stage. The complexity analysis is sound. The authors also honestly show the noise limitation in Section 7.2.1: their simple interpolation degrades badly at low exposure, so the compression claim is conditional on photon noise.\n\nThe main soft spots are the oversampling proof, the noise sensitivity, and the lack of released code. None of these is fatal for the algorithmic contributions, but they mean the paper oversells its theoretical foundation. This is a paper for computational imaging readers—especially people building SPAD-based NLOS systems—and for signal processing folks interested in non-uniform diffraction algorithms. It deserves a serious referee; the core methods are reproducible and novel. I'd recommend accepting it with major revision: fix or heavily qualify the sampling derivation, tighten the claims about 100x compression, and add explicit SNR/parameter tables.","headline":"Solid NUFFT/SFFT-based NLOS algorithms with real data, but the oversampling theorem has a factor-of-two error and the 100x compression claim is not justified.","tokens_in":27412,"tokens_out":5027,"would_cite":true,"duration_ms":44128,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Existing NLOS imaging setups oversample the relay surface, letting reconstruction work with roughly 100x fewer samples on irregular grids while keeping FFT-level speed.","keywords":["non-line-of-sight imaging","phasor field","non-uniform fast Fourier transform","scaled fast Fourier transform","Rayleigh-Sommerfeld diffraction","sparse sampling","SPAD array","computational imaging"],"falsifier":"Reconstruct a hidden point or structured object placed close to the relay wall with large lateral offset, so $2|z_r - z_0|/|x_r - x_0| < 1$, using the downsampled-and-interpolated pipeline at short exposure time. The paper's criterion predicts visible aliasing or speckle-driven degradation in exactly that geometry; a clean reconstruction would contradict the claimed oversampling condition.","tokens_in":26313,"feed_emoji":"📷","tokens_out":6287,"duration_ms":55844,"temperature":0.7,"pith_summary":"Non-line-of-sight imaging reconstructs hidden scenes from light that bounces off a visible relay surface. The fastest reconstruction algorithms use fast Fourier transforms, but those require evenly spaced measurement and voxel grids, which real multi-pixel sensors do not produce. This paper argues that standard NLOS setups oversample the relay surface laterally: a point source's wavefront changes slowly along the surface, so the Nyquist rate in the lateral direction is far lower than along depth. It turns that observation into FFT-compatible algorithms, the scaled RSD and several non-uniform RSD variants, that accept sparse or irregular input grids, reconstruct at arbitrary output locations, and enlarge the output volume with depth. If the oversampling bound holds in real scenes, the practical payoff is roughly two orders of magnitude less data, relaxed calibration, and acquisition times compatible with SPAD arrays.","feed_headline":"NLOS imaging drops 100x of samples and still reconstructs at FFT speed","feed_subtitle":"Mathematical proof that relay-surface measurements are oversampled lets sparse non-uniform grids use fast Fourier transforms.","key_machinery":"The load-bearing objects are the phasor field, a virtual coherent wavefront that represents the time-of-flight measurement at the relay surface; the Fresnel/paraxial approximation of a spherical phase $e^{ikr}$, which gives the slow transverse phase variation that makes lateral oversampling possible; and the sampling-ratio identity $\\lambda_{sx}/\\lambda_{sz} = 2|z_r - z_0|/|x_r - x_0|$ that quantifies the allowed downsampling $D$. Two modified Fourier transforms carry the algorithms: the non-uniform fast Fourier transform, which spreads irregular samples onto a dense grid, applies an FFT, and deconvolves the spreading kernel in $O(N\\log N)$ work, and the scaled (fractional) FFT, which evaluates the Fourier transform on a scaled output grid using a chirp convolution with three extra FFTs. These are fused with the Rayleigh-Sommerfeld diffraction kernel to produce NURSD-1/2/3, 3D NURSD, and SRSD.","core_discovery":"Using the phasor field formalism, the paper derives a spatial oversampling criterion for NLOS measurements. For a single point source at $(x_0,z_0)$ radiating a spherical phase wavefront, the Fresnel approximation turns the phase at relay-wall position $(x_r,z_r)$ into a quadratic term; the local transverse spatial frequency is $k(x_r-x_0)/(z_r-z_0)$, so the lateral sampling interval can be as large as $\\lambda^*(z_r-z_0)/(x_r-x_0)$. Comparing that with the depth sampling interval $\\lambda^*/2$ gives $\\lambda_{sx}/\\lambda_{sz} = 2|z_r-z_0|/|x_r-x_0| > D$, meaning the lateral axis can be downsampled by $D$ whenever the hidden object is far enough from the relay wall relative to its lateral offset. Experimental reconstructions with one out of five spatial samples retained, interpolated by nearest neighbors, give SSIM scores close to 1 against the full-data reconstruction, and the NUFFT versions reconstruct from randomly subsampled and non-uniform SPAD-array grids with quality comparable to filtered backprojection. The SRSD variant uses a scaled FFT to grow the lateral voxel size linearly with depth, matching the known resolution loss and producing perspective-correct views.","pith_inferences":["Because confocal acquisition effectively doubles the baseline sampling rate, the paper's argument implies confocal datasets lose more from aggressive downsampling than non-confocal ones, which the authors note matches prior observations.","The oversampling bound also suggests an adaptive-sampling rule: spend pixels on near-field, small-offset regions where $D$ is smaller, and sparsely sample regions far from the relay wall.","Replacing the naive interpolation step with photon-noise-aware denoising or learned reconstruction should extend the compression benefit to low-exposure acquisitions, where the paper's own results show the simple scheme fails.","If the Fresnel/paraxial assumption is relaxed, the same sampling-ratio derivation should produce a generalized $D$ that depends on angle, giving a testable prediction for non-paraxial relay geometries."],"forward_implications":["Relay-wall calibration no longer has to produce a uniform Cartesian grid; sparse, non-uniform, or non-planar sampling can feed the same FFT-complexity reconstruction.","Spatial compression by a factor of about 100 (with the paper's $D=5$ and temporal bandpass) cuts the memory and readout burden of large SPAD arrays.","Reconstruction voxels can be placed arbitrarily and coarsened with depth, so large hidden volumes can be stored and computed with about $O(Z)$ output samples instead of $O(Z^3)$ for a fixed field of view.","The same modified-FFT trick applies to other FFT-based NLOS reconstruction algorithms, not just RSD.","Perspective-correct reconstructions match the view from the relay wall, easing comparison with normal cameras."],"supporting_citations":[{"why":"Defines the standard RSD algorithm that the paper extends; provides the plane-to-plane convolution and complexity baseline.","marker":"[3]"},{"why":"Introduces phasor-field virtual wave optics and supplies the experimental NLOS datasets used in the subsampling experiments.","marker":"[2]"},{"why":"Gives the Huygens-like phasor field wave model whose spherical phase term is the starting point for the sampling-ratio derivation.","marker":"[13]"},{"why":"Supplies the NUFFT spreading/deblurring scheme and complexity analysis used in the NURSD algorithms.","marker":"[53]"},{"why":"Provides the fractional/scaled Fourier transform that underlies the SRSD's chirp-convolution scaling.","marker":"[11]"},{"why":"Introduces the 3D RSD algorithm with interpolation onto a uniform grid, which the 3D NURSD replaces by a 3D NUFFT.","marker":"[38]"},{"why":"Describes the fast-gated 16x16 SPAD array hardware whose non-uniform detection grid motivates NURSD.","marker":"[6]"},{"why":"Demonstrates low-latency SPAD-array acquisition at 5 frames per second, the speed context that makes data-rate compression important.","marker":"[4]"}],"fun_headline_variants":["NLOS imaging goes sparse with modified FFTs","Sparse non-uniform NLOS sampling solved by NUFFT and SFFT","Oversampling proof cuts NLOS data, keeps FFT speed","Sparse NLOS imaging via modified FFTs keeps quality","NUFFT and SFFT make NLOS sampling flexible and fast"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The oversampling bound is derived for a single point source under the Fresnel/paraxial approximation with no photon noise; real scenes contain extended objects, steep angles, and shot noise, so the conclusion that the $D>1$ regime holds depends on those idealizations carrying over.","fun_headline_variants_meta":{"raw":{"variants":["NLOS imaging goes sparse with modified FFTs","Sparse non-uniform NLOS sampling solved by NUFFT and SFFT","Oversampling proof cuts NLOS data, keeps FFT speed","Sparse NLOS imaging via modified FFTs keeps quality","NUFFT and SFFT make NLOS sampling flexible and fast"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001046,"raw_usage":{"total_tokens":4461,"prompt_tokens":1076,"completion_tokens":3385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":3294}},"tokens_in":692,"tokens_out":3385,"duration_ms":22133,"temperature":1.0,"reasoning_tokens":3294,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:13:13.151627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Reconstruct a hidden point or structured object placed close to the relay wall with large lateral offset, so $2|z_r - z_0|/|x_r - x_0| < 1$, using the downsampled-and-interpolated pipeline at short exposure time. The paper's criterion predicts visible aliasing or speckle-driven degradation in exactly that geometry; a clean reconstruction would contradict the claimed oversampling condition.","supporting_citations":[{"cited_title":"Phasor field diffraction based reconstruction for fast non-line-of-sight imaging systems,","cited_arxiv_id":null,"evidence_quote":"Defines the standard RSD algorithm that the paper extends; provides the plane-to-plane convolution and complexity baseline."},{"cited_title":"Non-line-of-sight imaging using phasor-field virtual wave optics,","cited_arxiv_id":null,"evidence_quote":"Introduces phasor-field virtual wave optics and supplies the experimental NLOS datasets used in the subsampling experiments."},{"cited_title":"Phasor field waves: A Huygens-like light transport model for non-line-of-sight imaging applications,","cited_arxiv_id":null,"evidence_quote":"Gives the Huygens-like phasor field wave model whose spherical phase term is the starting point for the sampling-ratio derivation."},{"cited_title":"The Fractional Fourier Transform and Applications,","cited_arxiv_id":null,"evidence_quote":"Provides the fractional/scaled Fourier transform that underlies the SRSD's chirp-convolution scaling."},{"cited_title":"Fast Non-line-of-sight Imaging with Non- planar Relay Surfaces,","cited_arxiv_id":null,"evidence_quote":"Introduces the 3D RSD algorithm with interpolation onto a uniform grid, which the 3D NURSD replaces by a 3D NUFFT."},{"cited_title":"Fast-Gated 16 16 SPAD Array with 16 on-chip 6 ps Time-to-Digital Converters for Non-Line-of-Sight Imaging,","cited_arxiv_id":null,"evidence_quote":"Describes the fast-gated 16x16 SPAD array hardware whose non-uniform detection grid motivates NURSD."},{"cited_title":"Low-latency time-of-flight non-line-of-sight imaging at 5 frames per second,","cited_arxiv_id":null,"evidence_quote":"Demonstrates low-latency SPAD-array acquisition at 5 frames per second, the speed context that makes data-rate compression important."}],"review_version":1}