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REVIEW 3 major objections 4 minor 65 references

Optimized Sampling for Non-Line-of-Sight Imaging Using Modified Fast Fourier Transforms

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Existing NLOS imaging setups oversample the relay surface, letting reconstruction work with roughly 100x fewer samples on irregular grids while keeping FFT-level speed.

desk verdict 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. read the letter →

arxiv 2501.05244 v1 pith:4SEIDU3F submitted 2025-01-09 eess.IV cs.CVeess.SPphysics.optics

Talha Sultan , Alex Bocchieri , Chaoying Gu , Xiaochun Liu , Pavel Polynkin , Andreas Velten This is my paper · ORCID
classification eess.IVcs.CVeess.SPphysics.optics
keywords non-line-of-sightimagingphasorfieldnon-uniformfastFouriertransformscaledRayleigh-SommerfelddiffractionsparsesamplingSPADarraycomputational
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 4 minor

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.

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 (3)
  1. [Supplement A.3, Eqs. (54)-(57)] 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.
  2. [Section 5 and Supplement A.3] 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.
  3. [Section 7.2.1 and Fig. 9] 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.
minor comments (4)
  1. [Section 5, step 3] 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.
  2. [Section 8.1, Fig. 13] 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.
  3. [Eq. (2)] The integrals over domega and dvec{x}_p are written without explicit limits; adding the integration ranges would improve reproducibility.
  4. [General notation] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the oversampling criterion is a parameter-free physical derivation, and the NUFFT/SFFT algorithms are validated against external baselines and public datasets.

full rationale

The paper's load-bearing claim—that NLOS relay surfaces are typically oversampled—is derived in Supplement A.3 from a stated point-source model and Fresnel/paraxial approximation (Eqs. 45-57), not fitted to the paper's own reconstructions. The ratio λsx/λsz = 2|zr−z0|/|xr−x0| (Eq. 57) is a mathematical consequence of the assumed phase model; it does not presuppose the conclusion. The NUFFT and SFFT algorithms are standard external tools, and NURSD/SRSD are compared against independent RSD and FBP reconstructions on public datasets. Citations to the authors' prior phasor-field work [12-14] supply the physical model and are not invoked as a uniqueness theorem to forbid alternatives. The paper itself flags the Poisson-noise limitation (Sec. 7.2.1), which affects applicability but not circularity. A potential factor-of-two concern in the instantaneous-frequency step of Eq. 55 is a correctness/mathematical issue, not a case of the result being equivalent to its inputs by construction. Overall, the derivation chain is self-contained.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivations rest on the phasor-field wave model and several user-set processing parameters (alpha, lambda_c, D). The oversampling claim additionally assumes paraxial propagation, bandlimited point sources, and noise-free measurements. These assumptions are stated in the paper but are not independently validated by external benchmarks beyond the empirical demonstrations.

free parameters (3)
  • alpha (and beta) lateral scaling factors = not reported; values between 0 and 1
    Chosen per reconstruction in SRSD to set depth-dependent voxel growth in Eqs. 17-18; controls volume and output grid but is not fitted to data.
  • lambda_c phasor field center wavelength = for example 4 cm; varied over a range in experiments
    Sets the bandpass filter in Eq. 7, the baseline sampling interval lambda_s = lambda_c/2, and the oversampling/downsampling ratio. User-selected and trades resolution against noise.
  • D downsampling ratio = D = 2, 4, 5 in the examples
    User-selected compression factor in Eq. 25/57 and in the data-savings calculation; not derived from measured data.
assumptions (6)
  • domain assumption The NLOS transient measurement is modeled with the phasor-field framework and reconstructed by the Rayleigh-Sommerfeld diffraction integral (Eq. 2).
    Adopted from the authors' prior phasor-field line of work [12-14]; all reconstructions in the paper inherit this forward model.
  • domain assumption The hidden scene can be represented as a sum of bandlimited point sources, P(k)=0 for k >= |k*|, each radiating e^{ikr} (Supplement A.3).
    Used to derive the oversampling criterion Eq. 57. Real scenes are not ideal bandlimited point sources, so this is an idealization.
  • domain assumption The paraxial/Fresnel approximation is valid, r approx (z-z0)+(x-x0)^2/[2(z-z0)] (Supplement Eq. 52).
    The ratio lambda_sx/lambda_sz = 2|z_r-z_0|/|x_r-x_0| follows from this approximation and fails for steep angles and very close relay geometry.
  • ad hoc to paper Photon (Poisson) noise can be ignored for the sampling-rate derivation.
    The compression and interpolation claims rely on clean measurements. The paper's own Section 7.2.1 shows naive interpolation degrades sharply under low exposure, so this assumption is load-bearing for the general oversampling claim.
  • ad hoc to paper NUFFT approximation error is negligible because the imaging system oversamples the relay surface.
    Invoked in Section 6 ('Accuracy') to justify using a generic MATLAB nufftn without custom error control. If oversampling is marginal, interpolation errors could corrupt reconstructions.
  • standard math Helmholtz reciprocity and separability of RSD allow swapping illumination and detection and plane-to-plane propagation.
    Standard wave-optics results cited to reciprocity references [48,49] and Born and Wolf [50]; these are background results, not new claims.

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Cite this review

Pith. "Pith review of Optimized Sampling for Non-Line-of-Sight Imaging Using Modified Fast Fourier Transforms." pith.science (2026). https://pith.science/paper/4SEIDU3F

@misc{pith2026250105244,
  author       = {Pith},
  title        = {Pith review of: Optimized Sampling for Non-Line-of-Sight Imaging Using Modified Fast Fourier Transforms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SEIDU3F}},
  note         = {Machine review of arXiv:2501.05244}
}
read the original abstract

Non-line-of-Sight (NLOS) imaging systems collect light at a diffuse relay surface and input this measurement into computational algorithms that output a 3D volumetric reconstruction. These algorithms utilize the Fast Fourier Transform (FFT) to accelerate the reconstruction process but require both input and output to be sampled spatially with uniform grids. However, the geometry of NLOS imaging inherently results in non-uniform sampling on the relay surface when using multi-pixel detector arrays, even though such arrays significantly reduce acquisition times. Furthermore, using these arrays increases the data rate required for sensor readout, posing challenges for real-world deployment. In this work, we utilize the phasor field framework to demonstrate that existing NLOS imaging setups typically oversample the relay surface spatially, explaining why the measurement can be compressed without significantly sacrificing reconstruction quality. This enables us to utilize the Non-Uniform Fast Fourier Transform (NUFFT) to reconstruct from sparse measurements acquired from irregularly sampled relay surfaces of arbitrary shapes. Furthermore, we utilize the NUFFT to reconstruct at arbitrary locations in the hidden volume, ensuring flexible sampling schemes for both the input and output. Finally, we utilize the Scaled Fast Fourier Transform (SFFT) to reconstruct larger volumes without increasing the number of samples stored in memory. All algorithms introduced in this paper preserve the computational complexity of FFT-based methods, ensuring scalability for practical NLOS imaging applications.

Figures

Figures reproduced from arXiv: 2501.05244 by the authors.

Figure 1
Figure 1. (A) An active imaging system acquires the NLOS measurement using a uniform grid on the relay surface. The standard RSD algorithm reconstructs the hidden scene plane by plane with low computational complexity but constrains both the input and output grids to be uniformly spaced Cartesian grids. We take a max filter along depth to display a 2D image of the 3D reconstruction (B) We develop the NURSD algorithm, which is… view at source ↗
Figure 2
Figure 2. These Modified Fast Fourier Transforms can convert between time t and frequency f with flexible sampling schemes without increasing computational complexity. ∆ is used for uniform spacing, while non-uniform samples are indexed by subscript ℓ. optics. In this work, we show that the NUFFT can be used to implement efficient reconstruction algorithms for datasets collected with these multi-pixel arrays. Moreover, we sho… view at source ↗
Figure 3
Figure 3. Showing the top view of the reconstruction volume when reconstructing with the RSD algorithm (left) and the SRSD algorithm (right). The dotted green lines demonstrate the increase in volume as we increase z, when the lateral field of view (FOV) is fixed. The RSD algorithm requires zero padding and increasing the side length, while the SRSD increases the lateral FOV without increasing the number of pixels. when xin =… view at source ↗
Figures from the paper (17 more)
Figure 4
Figure 4. Figure 4: (A) The relay surface is sparsely sampled at the green locations and interpolated to a dense grid denoted by the red locations. Reconstructing the hidden scene using the Standard RSD algorithm on the interpolated grid generates an output with minimal loss in reconstruc…
Figure 5
Figure 5. Figure 5: Column 1 presents the hidden scene as viewed from the relay surface. Columns 2, 3, and 4 display reconstructions using the standard RSD and the scaled RSD, where a max filter is applied along the depth to generate 2D images.The average reconstruction time in seconds is…
Figure 6
Figure 6. Figure 6: Column 1 shows the image of the hidden scene. Columns 2 and 3 display the standard RSD reconstruction and the scaled RSD reconstruction, respectively, after applying a max filter operation. Row 1: Two letter T’s of the same size are shown, with the T in the middle appe…
Figure 9
Figure 9. Figure 9: RSD reconstruction results after spatial downsampling and nearest neighbor interpolation to original dimensions for different exposure times for the office scene. The listed exposure time is for each spatial grid position. We note that the office scene uniquely has mor…
Figure 10
Figure 10. Figure 10: RSD reconstruction results after spatial downsampling and nearest neighbor interpolation to original dimensions for different wavelengths, λc, for the 5ms office scene. The number of frequency components used in the reconstructions, F, depends on the wavelength. SSIM …
Figure 11
Figure 11. Figure 11: RSD reconstruction results after spatially upsampling the original dataset using linear interpolation for different wavelengths, λc, for the 20ms office scene. This reveals previously missing details and slightly denoises the signal. The number of frequency components…
Figure 12
Figure 12. Figure 12: Comparison of reconstruction between the RSD and the NURSD-1 algorithms when the full grid is sampled on the relay surface. SSIM score is shown on the bottom right, and is computed with the RSD reconstruction as the reference [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Top row shows the relay wall mask applied to the dataset. The NURSD enables arbitrary spatial sampling of the relay surface. SSIM score is shown on the bottom right, and is computed with the reconstruction in the top row set to the reference. Blue SSIM score shown aft…
Figure 14
Figure 14. Figure 14: Comparison of reconstruction between the RSD and the NURSD-2 algorithms when the full voxel grid is sampled. SSIM score is shown on the bottom right, and is computed with the RSD reconstruction as the reference [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Top row shows the reconstruction mask for each plane in the reconstruction volume. The NURSD enables arbitrary spatial sampling of the voxel grid. Bottom row shows the corresponding reconstruction, along with computation time. allows similar flexibility. 8.4 Non-Plana…
Figure 16
Figure 16. Figure 16: Columns 1 and 2 show the spatial masks for the relay surface and the reconstruction plane respectively. Column 3 shows the reconstruction, along with the computation time. This demonstrates that the NURSD enables arbitrary sampling of both the relay surface and the vo…
Figure 17
Figure 17. Figure 17: We compare the performance of RSD, FBP, 3D RSD, and 3D NURSD in rows 2, 3, 4, and 5 respectively for datasets acquired on planar (Column 1) and non-planar (Column 2 ,3, 4) relay surfaces for the same hidden scene. The performance of 3D RSD and 3D NURSD is comparable, …
Figure 18
Figure 18. Figure 18: An image of our NLOS imaging system consisting of a pulsed laser, galvonometer, and SPAD array on the left. The image on the right displays how focusing the SPAD array to a large area (1.15 m x 0.7 m) on the planar 1.9 m x 1.9 m relay surface generates a non-uniform g…
Figure 19
Figure 19. Figure 19: Column 1 shows an image of the hidden scene and the Standard RSD reconstruction when the full 1.9 m x 1.9 m laser grid is used to acquire the dataset. Column 2 shows the standard RSD reconstruction when using a subset of the laser grid that is approximately equal to t…
Figure 20
Figure 20. Figure 20: We generate 4D videos of light transport in the hidden scene using the RSD (Row 1) and the Scaled RSD (Row 2).The mannequin and shelf towards the back of the hidden volume appear smaller in the scaled RSD (Row 2, Columns 3 and 4) due to perspective projection C.2 4D L…
Figure 21
Figure 21. Figure 21: In Row 1, we show the reconstructions for two different datasets using the Standard RSD (Columns 2 and 4) and Scaled RSD (Columns 3 and 5). In Row 2, we randomly subsample 10% of the relay surface (Column 1) for the same datasets and interpolate to a uniform grid. We …
Figure 22
Figure 22. Figure 22: We place the digit ”2” at various depths within the hidden scene. Column 1 indicates the average depth of the ”2” for each row. Column 2 shows the smartphone image of the hidden scene, with a yellow box marking the location of the ”2.” Column 3 presents the Standard R…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.