Restoration of Non-rigidly Distorted Underwater Images using a Combination of Compressive Sensing and Local Polynomial Image Representations

Ajit Rajwade; Jerin Geo James; Pranay Agrawal

REVIEW 4 major objections 5 minor 37 references

Restoration of Non-rigidly Distorted Underwater Images using a Combination of Compressive Sensing and Local Polynomial Image Representations

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that wavy-water video distortion can be undone by recovering the full motion field from a few tracked points using compressed sensing with a 3D Fourier sparsity basis.

desk verdict A genuinely new CS-based motion interpolation method with reproducible wins over baselines, but the center-of-trajectory assumption is a real, untested limitation. read the letter →

arxiv 1908.01940 v1 pith:MKAGTBYM submitted 2019-08-06 cs.CV

Jerin Geo James , Pranay Agrawal , Ajit Rajwade This is my paper · ORCID

classification cs.CV

keywords underwaterimagerestorationnon-rigiddistortioncompressedsensingmotionvectorfield3DdiscreteFouriertransformopticalflowpolynomialexpansiontrajectorytracking

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

The paper tries to establish that a static scene filmed through wavy water can be restored by exploiting the spatio-temporal smoothness and periodicity of water waves. It tracks a small number of feature points, converts their trajectories into displacement measurements, and uses compressed sensing with a 3D discrete Fourier basis to estimate the dense motion field at every pixel. A simple optical-flow refinement based on local polynomial expansion is then applied, sometimes alone and sometimes after the compressed-sensing step. The authors claim these methods restore both synthetic and real underwater videos with lower error and better structural preservation than existing restoration techniques, while needing no training data, no distortion-free template, and no special hardware. A sympathetic reader would care because this offers a physically grounded, data-free route to a practical vision problem that interferes with monitoring, recognition, and tracking in underwater scenes.

What carries the argument

The central object is the complex-valued motion vector field $d = d_x + i d_y$, a 3D array over image space and time. Its key property is sparsity in the 3D DFT basis, justified by the spatial smoothness and temporal periodicity of water waves. The measurement model is $e = \Phi F \theta + \eta$, where $\Phi$ is a row-subsampled identity selecting tracked points and $F$ is the 3D DFT matrix; because $\Phi$ and $F$ are maximally incoherent, a LASSO solver can reconstruct the field from very few tracks. The second ingredient is PEOF, which fits local second-degree polynomials to image patches and solves for the displacement between each frame and the mean image, iterating over a pyramid for large motions.

What would settle it

Record a static target through wavy water with a known steady current or asymmetric wave train, track feature points, and compare each trajectory's mean against its known true position; if the mean offset exceeds a small threshold, the compressed-sensing reconstruction is fed biased measurements and the restored-image error should rise, whereas an algorithm that explicitly estimates drift would not degrade.

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Extended reading notes

Core claim

The central claim is that the full 3D motion vector field generated by a wavy water surface is compressible in the 3D discrete Fourier basis, so it can be recovered from a small set of tracked point trajectories by solving an L1-regularized least-squares problem, the LASSO. The recovered field is then used to warp every video frame back toward a common undistorted coordinate system. The paper further claims that applying polynomial-expansion optical flow (PEOF) after the compressed-sensing step preserves straight lines, text, and grid structure better than the compressed-sensing step alone, and that both stages outperform the compared existing methods on synthetic and real underwater videos in terms of RMSE, SSIM, and normalized mutual information. The two-stage CS+PEOF pipeline is presented as the best overall restoration approach, at computational cost far below the spline-plus-RPCA baseline.

Load-bearing premise

The load-bearing premise is that the time-average of each tracked point's positions equals its true undistorted position, which requires the water surface motion at every point to be locally symmetric and zero-mean over the captured frames.

Editorial extensions

If this is right

Dense non-rigid motion estimation for wavy water needs only a sparse set of tracked points: the paper reports reconstructions from fewer than 3% of pixel trajectories, with performance saturating as more tracks are added.
The compressed-sensing stage alone removes more than 90% of the motion in most test videos, indicating that the 3D-DFT sparsity model captures the physical distortion well.
A simple local-polynomial optical-flow stage, used alone, already beats the compared baselines on most numerical metrics, so elaborate models are not required for this task.
The two-stage CS+PEOF combination gives the best structural preservation of lines, text, and grids, as well as the best numerical quality in most comparisons.
The method needs no training data, no distortion-free template, and no special illumination or multiple viewpoints; only a short video on the order of 100 frames is sufficient.

Reading between the lines

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

If the zero-mean water-motion assumption fails because of a steady current or wave drift, the center-of-trajectory estimate becomes biased; a natural extension would be to estimate and subtract a global or slowly varying drift term before compressed-sensing reconstruction.
The same 3D-DFT sparsity idea could be tested on other temporally periodic non-rigid distortions, such as short-exposure atmospheric turbulence, where similar band-limited motion fields arise.
The success of this parameter-free pipeline suggests that deep single-frame restoration methods lose information by ignoring temporal redundancy; training them on multiple frames, or combining them with a compressed-sensing motion-reduction front end, could close the performance gap.
Because the row-subsampled identity and the DFT are maximally incoherent, the reconstruction guarantees are near-optimal; replacing the DFT with another basis would likely require more tracked points for the same accuracy.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

What you should know about this paper: it's a genuine new idea -- using compressed sensing with a global 3D DFT sparsity prior to interpolate a dense non-rigid motion field from a few tracked trajectories -- and it ships code and data, with consistent wins over three published baselines on a 14-video test bed. The central claim holds up on their data; the biggest caveat is a load-bearing assumption that doesn't get stress-tested.

What's actually new: the CS-based MVF interpolation is, as far as I know, not in the prior literature. The prior sparsity-based optical flow works use local patches and two-frame brightness constancy; this works globally in space-time and uses displacement trajectories as measurements. That's a real contribution. The PEOF result is also useful: a simple, fast optical flow method outperforms more expensive SOTA on this task, and the two-stage CS+PEOF is better than either alone. The experiments are extensive: synthetic and real videos, motion reduction numbers, parameter sensitivity, and compute time. The paper is clearly written.

Where the soft spots are:

1. The COT assumption: Sec 2.3.2 equates the time-average of each trajectory with the undistorted coordinate, justified by 'local symmetry of water motion.' If there's a net current, Stokes drift, or an asymmetric wave train, every displacement measurement inherits a spatially varying bias, and the CS model is fed systematically wrong data. Their trajectory filtering rule (COT over first vs second half differing by >3px) won't catch a constant offset. Their real data comes from a mechanically-paddled tank, where waves are near-zero-mean, so this is untested. This is the paper's main limitation. It's not fatal, because zero-mean symmetric wave motion is a plausible and explicitly stated assumption, but it deserves a stress test (simulate drift, or find real data with current).

2. The Fourier sparsity 'verification' in Sec 3.3 is circular: they show the MVF estimated by their CS method (which assumes DFT sparsity) has sparse DFT coefficients. That doesn't corroborate the model independently. They could run PEOF or any other dense flow method and check its DFT spectrum.

3. No error bars on any metric. The gains are consistent across many videos, so this is minor, but repeated synthetic trials with random wave parameters would help.

4. The comparison to [15] is done across papers, using different sequences (their 'Middle/Small/Tiny' vs [15]'s 'Large/Medium/Small'). That's not a valid head-to-head. Either run the code or drop the claim.

Bottom line: the paper is a solid contribution, worth a serious referee. The COT issue is a revision-level concern, not a reject-level one. If the authors address the stress test and the circular verification, I'd be happy to see it in a good venue.

Recommendation: engage with it.

Referee Report

4 major / 5 minor

Summary. The paper addresses restoration of a static scene imaged through a wavy water surface, where each video frame is a non-rigidly distorted view of the same underlying image. The authors propose a compressed sensing (CS) pipeline: salient points are tracked across frames, their displacement trajectories are used as sparse space-time measurements of the dense motion vector field (MVF), and the MVF is recovered by solving a LASSO problem under the assumption that the MVF is sparse in the 3D DFT basis. The restored image is then obtained by warping each frame by the estimated MVF and taking the mean. The paper also evaluates a simpler optical flow method based on local polynomial expansion (PEOF), and a two-stage CS+PEOF variant. Experiments on synthetic videos, a mechanically-paddled real tank dataset, and the Real2 dataset of Tian and Narasimhan report numerical improvements over SBR-RPCA, LWB, and a deep-learning baseline in terms of RMSE, SSIM, and NMI, with substantially lower computation time than SBR-RPCA and LWB.

Significance. If the reported results hold, the CS-based MVF interpolation is a genuinely new use of compressed sensing for refractive distortion correction, and the finding that a simple PEOF optical flow method is competitive with or better than dedicated state-of-the-art restoration methods is practically valuable. The paper is refreshingly clear about its assumptions, provides synthetic and real data with ground truth, and makes code and datasets available, which strengthens reproducibility. The central claims, however, depend on two assumptions that are not independently verified: the zero-mean local symmetry of water motion that justifies the center-of-trajectory approximation, and the empirical sparsity of real MVFs in the DFT basis. The current evidence for the latter is circular, and the former is only tested in a wave-tank regime where waves are generated by mechanical paddles and are likely near-zero-mean. The CS-theoretic justification also contains a coherence error.

major comments (4)

[Sec. 3.3, 'Verification of Fourier Sparsity' and Fig. 1] The empirical verification of Fourier sparsity is circular. The MVF whose DFT coefficients are plotted in Fig. 1 is the output of the CS/LASSO estimator in Eq. (2), which explicitly minimizes the ℓ1 norm of the DFT coefficients and therefore enforces sparsity in the DFT basis. Showing that this estimate has a concentrated spectrum is a tautology, not evidence that real MVFs are sparse in the DFT basis. The authors do note that 'the actual MVF can only be estimated,' but they then use the CS estimate itself to support the modeling assumption. Please verify sparsity using an independently estimated MVF (for example, from PEOF/optical flow, or from the known synthetic MVF in the Synthetic dataset), or explicitly reword the section to state that Fig. 1 illustrates the estimator's output rather than validating the premise on which the estimator is built.
[Sec. 2.3.2, center-of-trajectory (COT) approximation, with Sec. 2.3.1 filter] The COT approximation x_i0 ≈ mean_t x_it is load-bearing for the entire CS stage, because every displacement in Eq. (1) is computed relative to the COT. The justification by 'local symmetry of water motion' fails when there is a net current, Stokes drift, or an asymmetric wave train: each trajectory then has a bias b_i = COT_i - p_i^0, and all measurements e in Eq. (1) are systematically shifted by b_i. The trajectory-filtering rule in Sec. 2.3.1 (COT over the first vs. last T/2 frames differing by more than 3 pixels) cannot detect this bias, because a constant offset adds equally to both halves. All real data come from a mechanically-paddled tank, where waves are plausibly near-zero-mean, so the failure mode is untested. Please add a synthetic experiment with a nonzero-mean velocity component (a constant drift or a biased wave) and report the restoration metrics and the resulting motion reduction; if the method degrades, state the operating domain clearly, and if it does not, explain why the bias is not harmful in CS reconstruction.
[Sec. 2.3.3, coherence argument after Eq. (1)] The claim that the coherence between the row-subsampled identity sampling matrix Φ and the DFT basis F 'reaches its lower bound of 1' is numerically incorrect as stated. With the definition given in the paper, |Φ_i F_j| = 1/√(NxNyT) for every pair of rows/columns, so the defined coherence is approximately 1/√(NxNyT), not 1; under the standard sensing-matrix coherence used in CS theory, the value depends on the chosen row set and for a random partial DFT is typically O(√(log(NxNyT))), not exactly 1. The lower bound of coherence is also not 1 in this setting. Please correct the coherence calculation and the resulting sample-complexity sentence; otherwise the CS-theoretic justification for the number of tracked points is unsupported. This is a local error in the theory section, but it is part of the key novelty claim, so it should be fixed.
[Table 1 and Sec. 3.3] The central empirical claim that CS, PEOF, and CS+PEOF outperform SBR-RPCA, LWB, and DL rests on single runs without error bars, repeated trials, or statistical tests. Several of the differences are small (e.g., Real1 Checker: CS+PEOF SSIM 0.892 vs. PEOF 0.890; Synthetic BrickWall: CS+PEOF RMSE 0.141 vs. PEOF 0.142), and all of the Synthetic videos are generated from 'randomly chosen parameters' with no indication of how many random realizations were used. Please report means and standard deviations over at least several random wave/tracking realizations for the Synthetic dataset, or otherwise characterize the run-to-run variability of the metrics. Without this, the superiority claims over the baselines are not yet supported at the level the paper asserts.

minor comments (5)

[Sec. 2.3.3, Eq. (2)] The LASSO objective in Eq. (2) uses an unsquared data-fidelity term ||e−ΦFθ||_2, whereas the standard LASSO and the YALL1 solver typically use a squared ℓ2 term. Please clarify the exact optimization problem solved by YALL1 in this paper, and ensure the objective function matches the solver's formulation.
[Sec. 3.1 and Sec. 3.2] The description of the Synthetic dataset says the water surface is generated using a superposition of 2–6 sinusoidal waves with 'randomly chosen parameters,' but no details are given on the parameter ranges or the number of random trials. Adding this information would help reproducibility and would support the request for error bars in the major comments.
[Fig. 2] The caption of Fig. 2 lists the datasets used but does not identify which curve corresponds to which dataset or parameter; the graph itself is not visible in the text. Please add a legend or descriptive labels, and state the axes clearly.
[Sec. 3.3, 'Verification of Fourier Sparsity'] The motion-reduction measure MR is defined as a median over tracked points, but the denominator ||d_i||_2 is the norm of the original displacement trajectory; for points with very small displacement this ratio can be unstable. Please state whether any thresholding or regularization was applied, and report the distribution of the per-point ratios rather than only the median.
[Throughout] There are several small typographical and formatting issues, such as 'Y ALL1' with a space in Sec. 3.2, inconsistent time formatting in Table 1 (e.g., '29s' vs. '0m 23s'), and the phrase 'nonrigid transport object' in reference [19]. These do not affect the technical content but should be cleaned up.

Circularity Check

1 steps flagged · score 3.0 of 10

Fourier-sparsity 'verification' reuses the CS estimator's own output; central restoration claims are independently benchmarked.

self definitional [Section 3.3, 'Verification of Fourier Sparsity', Fig. 1]
"Verification of Fourier Sparsity: Here, we demonstrate the sparsity of the MVFs from real underwater sequences. This is shown in Fig. 1 for the 'Elephant' sequence ... We note that the actual MVF can only be estimated. However, we contend that the MVF estimated by our CS method is a good approximation to the actual one. ... scatter plot of frequencies which account for 99% of the squared magnitude of the estimated MVF using CS (right)."

The MVF d is estimated by solving Eq. (2), which minimizes λ||θ||1 + ||e - ΦFθ||2 with F the 3D DFT basis. This estimator is explicitly biased toward solutions with sparse DFT coefficients. The 'verification' then computes the DFT of that same estimated d and observes that few coefficients account for 99% of its energy. That concentration is imposed by the objective, not measured from an independent ground-truth MVF. The text even concedes 'the actual MVF can only be estimated' before using the estimate to validate the sparsity premise, so the demonstration is self-referential rather than independent confirmation.

full rationale

The paper's central derivation is not circular. It treats tracked displacement trajectories as CS measurements, assumes DFT sparsity as a physically motivated prior from fluid mechanics, and solves a standard LASSO problem; no fitted parameter is renamed as a prediction, and the restoration results are evaluated against external ground-truth still-water images and compared with SBR-RPCA, LWB, and deep-learning baselines. The center-of-trajectory approximation and the zero-mean wave assumption are genuine correctness risks, but they are assumptions, not circular reductions. The only circular step is the 'Verification of Fourier Sparsity' in Sec. 3.3, where the MVF estimated under the l1-DFT-sparsity objective is used to demonstrate the sparsity of real MVFs; this is self-consistent but does not independently confirm the premise. Because this circularity affects a supporting validation argument rather than the main restoration pipeline or its external benchmarks, a moderate score of 3 is appropriate.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method rests on three free parameters (lambda, DT downsampling factor, and a tracking filter threshold), four domain assumptions (DFT sparsity of the MVF, planar static scene, zero-mean symmetric water motion, bounded tracker noise), and one standard-math fact about coherence. No invented physical entities are introduced. The DFT sparsity axiom is the physically motivated core of the CS model; the COT assumption is the most fragile because it is checked only indirectly through final image quality.

free parameters (3)

lambda (Eq. 2, LASSO regularization weight) = chosen by cross-validation; optimal value reported stable across datasets
L1 regularization weight in the CS reconstruction; cross-validation selects it from held-out trajectories, making the reconstruction data-dependent.
DT downsampling factor = 8
Tracked displacement trajectories are downsampled by a factor of 8 in X and Y before CS; hand-chosen to speed up YALL1 and align with the bandlimited assumption.
track filtering threshold = 3 pixels
Trajectories whose center over first half and second half differ by more than 3 pixels are discarded; hand-chosen heuristic that changes which measurements feed the CS solver.

assumptions (5)

domain assumption The complete motion vector field d is sparse or compressible in the 3D DFT basis.
Sec 2.2; the entire CS model in Eq. 1 rests on this. It is motivated by smoothness and periodicity of water waves and cited fluid mechanics, but real turbulent or non-periodic flows violate it.
domain assumption Static planar scene at depth h, orthographic camera looking straight down.
Sec 2.1; inherited from Murase (1992) and used by prior work; restricts applicability to flat scenes and distant or normal viewing geometry.
domain assumption Center of trajectory approximates the true undistorted location (zero-mean symmetric water motion).
Sec 2.3.2; needed to convert tracked pixel positions into displacement trajectories; a biased COT biases all CS measurements.
standard math The DFT basis and the row-subsampled identity sampling matrix have coherence 1, supporting CS recovery guarantees.
Sec 2.3.3; standard result, used only to motivate applicability of CS theory, not to derive numerical constants.
domain assumption Tracker output errors are bounded and enter Eq. 1 as small noise eta.
Sec 2.3.1 through 2.3.3; KLT and other trackers produce outliers; some are filtered, but residual errors must be small for LASSO to reconstruct accurately.

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Cite this review

Pith. "Pith review of Restoration of Non-rigidly Distorted Underwater Images using a Combination of Compressive Sensing and Local Polynomial Image Representations." pith.science (2026). https://pith.science/paper/MKAGTBYM

@misc{pith2026190801940,
  author       = {Pith},
  title        = {Pith review of: Restoration of Non-rigidly Distorted Underwater Images using a Combination of Compressive Sensing and Local Polynomial Image Representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MKAGTBYM}},
  note         = {Machine review of arXiv:1908.01940}
}

read the original abstract

Images of static scenes submerged beneath a wavy water surface exhibit severe non-rigid distortions. The physics of water flow suggests that water surfaces possess spatio-temporal smoothness and temporal periodicity. Hence they possess a sparse representation in the 3D discrete Fourier (DFT) basis. Motivated by this, we pose the task of restoration of such video sequences as a compressed sensing (CS) problem. We begin by tracking a few salient feature points across the frames of a video sequence of the submerged scene. Using these point trajectories, we show that the motion fields at all other (non-tracked) points can be effectively estimated using a typical CS solver. This by itself is a novel contribution in the field of non-rigid motion estimation. We show that this method outperforms state of the art algorithms for underwater image restoration. We further consider a simple optical flow algorithm based on local polynomial expansion of the image frames (PEOF). Surprisingly, we demonstrate that PEOF is more efficient and often outperforms all the state of the art methods in terms of numerical measures. Finally, we demonstrate that a two-stage approach consisting of the CS step followed by PEOF much more accurately preserves the image structure and improves the (visual as well as numerical) video quality as compared to just the PEOF stage.

Figures

Figures reproduced from arXiv: 1908.01940 by the authors.

**Figure 2.** Effect of increase in number of frames T (top) and number of salient points N (bottom) on restoration performance for CS method. Results shown on ‘Middle’ and ‘Small’ from Real2, and a few sequences from Real1/Synthetic. λˆ did not affect the performance much. Hence the time for cross-validation is not included in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Left to right, top to bottom order in each of the 3 groups [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

Halder, M

K. Halder, M. Paul, M. Tahtali, S. Anavatti, and M. Murshed. Correction of geometrically distorted underwater images us- ing shift map analysis. J. Opt. Soc. Am. A , 34(4):666–673, Apr 2017. 1, 5

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