Rubber band filters: optimal padding without edge artifacts

David Burghoff; Zhenyang Xiao

REVIEW 4 major objections 5 minor 18 references

Rubber band filters: optimal padding without edge artifacts

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that bandlimited filtering can be made free of boundary artifacts by recasting it as a weighted least-squares fit over an optimally de-weighted padded region, and demonstrates boundary errors roughly ten times lower than…

desk verdict A clean, useful least-squares bandlimited filtering method for spectroscopy, but the 'optimal' and 'artifact-free' claims outrun the oracle-assisted validation. read the letter →

arxiv 2507.07347 v1 pith:KY5XUNDF submitted 2025-07-10 physics.optics

Zhenyang Xiao , David Burghoff This is my paper · ORCID

classification physics.optics

keywords rubberbandfilterbandlimitedfilteringedgeartifactssymmetricpaddingweightedleastsquaresconjugategradientspectroscopyFouriertransform

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 introduces a "rubber band filter" for bandpass filtering of spectroscopic signals, aimed at removing the edge artifacts that conventional symmetric padding leaves at signal boundaries. It frames filtering as a weighted least-squares fit: the measured trace is symmetrically padded, the padded region is strongly de-weighted, and the best bandlimited reconstruction is found by solving a convolution system in the frequency domain. The authors show on terahertz time-domain, FTIR, and dual-comb spectroscopy data that the method preserves sharp edges with substantially lower error, around ten times lower in the THz example and about three times lower for dual-comb phase, while needing only a handful of extra Fourier transforms. If the method holds up, it gives spectroscopists a drop-in filtering step that does not distort the very edges where important signal features live.

What carries the argument

The central object is the weighted least-squares cost $J = \sum_n w(t_n) \left| s_f(t_n) - s(t_n) \right|^2$, with the filtered signal restricted to passband frequencies $\Omega$. Its stationary condition is the frequency-domain convolution equation $W * F = W * S$ restricted to $\Omega$, solved by conjugate gradient where each matrix-vector product $A F = W * F$ is evaluated with FFTs; the result is the optimal set of passband Fourier amplitudes $F(\omega_k)$. The work it does is letting the padded region absorb the boundary discontinuity smoothly while keeping the reconstruction strictly bandlimited, at $O(N \log N)$ cost.

What would settle it

If, on a synthetic signal with a known bandlimit, no single small weight (or simple weight rule) keeps the boundary error consistently below symmetric padding across noise and cut-off choices, then the claim that the method recovers the bandlimited signal without edge artifacts fails.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that edge-free bandlimited filtering can be posed as a convex least-squares problem over the passband Fourier amplitudes, with the padded region included in the fit but heavily down-weighted. The stationary condition reduces to the convolution equation $W * F = W * S$, where $S$ is the Fourier transform of the padded signal and $W$ is the transform of the weight function; solving only for frequencies in the passband $\Omega$ and setting the rest to zero yields a bandlimited trace that is allowed to relax in the padding instead of being forced into the discontinuity that symmetric padding creates. The paper validates this on three experimental records by comparing against a true filtered signal obtained from the full unpadded measurement, reporting consistently smaller boundary error, roughly ten times lower in the terahertz example and about three times lower for dual-comb phase, with a few FFT-based conjugate-gradient iterations.

Load-bearing premise

The load-bearing premise is that the weighted least-squares fit, with the padded region's weight chosen by hand (roughly 0.03 to 0.06 in the demonstrations), reproduces the true bandlimited signal; since the validation reference is the full measurement that practical filtering never has, the method's success depends on that weight being close to optimal for unseen data.

Editorial extensions

If this is right

For stored spectroscopic traces that are already windowed, rubber band filtering can replace symmetric padding as the default preprocessing step, preserving sharp features at both ends instead of rounding them off.
Because the output is itself a padded, bandlimited signal whose padding has been reshaped by the weights, downstream operations like phase extraction or derivative estimation see a smooth extension rather than a jump.
The same weighted-fit idea extends directly to any bandpass shape, not just ideal filters, by changing which Fourier amplitudes $\Omega$ are free parameters.
The error scaling in the three examples suggests that the advantage grows when the signal has steep slopes at the window edges, which is exactly where symmetric padding is most distorted.

Reading between the lines

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

A natural next test is to use the same rubber-band formulation for interpolation or denoising where the weights encode measurement confidence rather than padding, since the convex least-squares core only needs a weight function and a passband.
The method's dependence on a hand-tuned weight (0.03 vs 0.06 in the demonstrations) suggests a practical autotuning rule: choose the weight by cross-validating the residual outside the padding, which the paper does not propose.
For imaging data, the same weighted-least-squares padding could suppress ringing at image borders, although the paper only demonstrates one-dimensional spectroscopy traces.
Because convergence is fast only when the solution resembles the naively filtered spectrum, pathological passbands or extremely steep signal edges may require more iterations; a fixed iteration budget could be risky in real-time instruments.

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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

Short version: this is a useful and clearly-written thing to know about if you filter spectroscopic time traces. The rubber band filter is basically a weighted least-squares projection onto a bandlimited subspace, with the weight deliberately made tiny in the symmetrically padded region. That is not a brand-new concept—it sits in the bandlimited extrapolation family (Papoulis-Gerchberg and relatives)—but the authors apply it cleanly to a practical spectroscopy problem, solve the normal equations with CG/FFTs, and demonstrate on THz, FTIR, and dual-comb phase data that it reduces edge error by roughly 3–10x versus symmetric padding. The derivation is correct, the convergence plot (Fig. 5b) is convincing, and the computational overhead is modest.

Where I part ways: the headline claims "optimal padding" and "artifact-free" are stronger than the evidence. The ground truth in Figs. 2–4 is the filtered version of the complete, untruncated measurement. That is a sensible oracle for a filter-evaluation paper, but it is not available in a real filtering job, and the paper tunes the weight constant (0.03 in the main examples; 0.06 "optimal" in Fig. 5a) against that same oracle. So the reported tenfold improvement could be partially an artifact of oracle-assisted weight selection. The stress-test note is right. What is missing is a blind test: pick the weight by a fixed rule or cross-validation, then compare against symmetric padding. Also missing: any comparison to established bandlimited extrapolation baselines, error bars or multiple realizations, and the promised code/data in the arXiv version. That last point is a real reproducibility gap.

Minor: "artifact-free" is contradicted by their own error plots, which still show elevated error at the edges. "Optimal padding" is also a misnomer—the padding itself is still symmetric; only the weight is optimized.

Verdict: the central method holds up as a useful engineering contribution, but the paper needs a softer title, a blind evaluation, and a baseline comparison before the strong quantitative claim is credible. It is still worth a serious referee—this kind of preprocessing is widely used in spectroscopy, and the method is clearly explained and likely to be adopted once its robustness is demonstrated. I would send it to review, but I would push for major revision.

Referee Report

4 major / 5 minor

Summary. The paper introduces a 'rubber band filter' for bandlimited filtering of truncated time-domain signals. The method symmetrically pads the signal, assigns weight 1 to the valid region and a small weight to the padded region, and solves a weighted least-squares problem with a bandlimited constraint via conjugate gradient, using FFTs for the required convolutions. The authors demonstrate the approach on terahertz time-domain spectroscopy, FTIR interferogram, and dual-comb spectroscopy data, reporting boundary errors roughly three to ten times lower than conventional symmetric padding. The central claim is that the method achieves 'optimal padding' and artifact-free bandlimited filtering with only a few extra FFTs over conventional approaches.

Significance. If the performance claims hold, the method would provide a practical and efficient alternative to symmetric padding for spectroscopic filtering, particularly where preserving sharp edge features is important. The derivation of the weighted least-squares formulation and the CG/FFT solution strategy is clear and technically sound, and the paper usefully identifies a real problem: symmetric padding introduces artifacts when the boundary slope is steep. The manuscript also makes a welcome effort to demonstrate the method on three distinct spectroscopic modalities. However, the evaluation is limited: errors are measured against an oracle reference (the full unpadded signal), the weight parameter is tuned per dataset, no error bars or statistical replicates are provided, and no comparison is made to existing bandlimited extrapolation or alternative padding methods. These limitations leave the 'optimal padding' claim stronger than the current evidence supports.

major comments (4)

[§2, Fig. 5a] The main examples in Figs. 2–4 use a padded-region weight of 0.03 times the Tukey window (α=0.2), but Fig. 5a reports that the optimal constant for the THz dataset is 0.06. The paper does not explain this discrepancy, leaving it unclear whether the reported 'approximately ten times lower' error is obtained at the chosen weight or at the tuned optimum, and whether the main examples are suboptimal.
[§2, Figs. 2–4 and Eq. (2)] The 'true filtered signal' in Figs. 2–4 is defined as the full, unpadded measurement filtered with the same passband, an oracle that is not available in practical filtering. The text states that the padding weight is 'initially chosen somewhat arbitrarily' and that it acts as a regularization constant, and Fig. 5a tunes that weight against the same oracle. Without a blind test or a principled a priori selection rule, the reported error reductions may reflect oracle-assisted parameter selection rather than a reproducible property of the algorithm, so the claim of 'optimal padding' is not yet supported by evidence available at the point of use.
[§2; §3] No comparison is made to established bandlimited extrapolation or alternative padding schemes, such as Papoulis–Gerchberg iteration, antireflective padding, or linear-prediction-based extension. The claim of 'overcoming longstanding challenges' and the assertion of superiority rest on a single baseline (symmetric padding), which is insufficient to establish the method's advantage relative to existing techniques that also aim to reduce boundary artifacts.
[§3, Fig. 5b] The convergence behavior is reported only as the residual against the direct solution A−1b of the CG iterate, with no information on the number of iterations or wall-clock time for the actual datasets, nor on how convergence depends on the weight parameter. Consequently, the claim that only 'a handful of iterations' and 'a few extra Fourier transforms' are needed cannot be quantitatively assessed from the presented results.

minor comments (5)

[Introduction] The phrase 'effectively removing the redundancy information' is ungrammatical; it should be 'effectively removing redundant information' or 'removing the redundant signal components.'
[Fig. 5] The text says Fig. 5 highlights three aspects, including a 'padded region comparison between our method and symmetric padding method,' but the figure as described contains only panels (a) and (b) for weight dependence and convergence. Please either add the third panel or correct the caption and referencing text.
[§2, weight definition] The definition wpad(n) = 0.03 wTukey(n, α=0.2) is ambiguous about whether n denotes the number of padded samples or an index over the padded region; please clarify the notation.
[References] The reference list omits seminal work on bandlimited extrapolation, such as Papoulis (IEEE Trans. Circuits Syst., 1975) and Gerchberg (Optica Acta, 1974), which are directly relevant to the method's relation to prior art; adding these would help readers position the contribution.
[Data Availability] The data availability statement says data 'will be published prior to publication'; for a reproducibility-oriented methods letter, consider making the datasets and example code available at the time of submission.

Circularity Check

1 steps flagged · score 4.0 of 10

The method's least-squares core is self-contained, but the reported error reduction relies on a weight parameter tuned against the same full-signal oracle used for scoring.

fitted input called prediction [Section 2 (Results), Fig. 5a and accompanying text; weight w in Eqs. (2)-(3)]
"the weight assigned to the padded regions is initially chosen somewhat arbitrarily, but as we will show in Fig. 5a, minor tuning of these weight parameters can further reduce the overall error relative to the true filtered signal. It essentially takes the role of a regularization constant. ... This particular plot evaluates the performance using terahertz time-domain spectroscopy data and applies a Tukey window with α=0.2; under these conditions, the optimal weight for the constant in front of the Tukey window is found to be 0.06."

The weight w is the free regularization parameter in the cost function J (Eq. 2). The paper's quantitative evaluation defines error as deviation from the 'true filtered signal' (full-signal filtered output). Fig. 5a then chooses w by minimizing precisely that error on the THz dataset, reporting an optimal value (0.06) different from the value used in the main example (0.03). Any method with a tunable parameter will appear to improve when the parameter is selected against the same oracle used for scoring; the 'optimal weight' result is a fit, not a prediction. Since the full-signal true filtered signal is unavailable in a real filtering application, the reported error reductions (e.g., ~10x in Fig. 2b) are not shown to be reproducible under blind parameter selection.

full rationale

The mathematical core (Eqs. 1-5) is a self-contained derivation: minimizing a quadratic weighted least-squares cost over bandlimited amplitudes leads to the convolution system W*F = W*S, solved by conjugate gradients. Nothing in that derivation assumes the conclusion. The 'true filtered signal' is defined as the full-signal filtered output and used only as a benchmark; since the algorithm never receives the full signal, the benchmark is external and not circular. The paper's self-citations (refs. 10, 16-18) are data sources, not load-bearing for the method. The one genuine circular element is the weight parameter w: Eq. (2) makes w a free input, and the paper tunes w against the same oracle used to score performance (Fig. 5a, optimal 0.06 vs 0.03 used in the main THz example). Thus the specific quantitative claim that tuning 'can further reduce the overall error relative to the true filtered signal' is true by construction, and the factor-of-ten improvement is not shown to survive blind weight selection. This is a fitted-input-called-prediction issue in the evaluation, not a collapse of the derivation. Overall score 4.

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

The method introduces no new physical entities. It does rely on a hand-tuned weight parameter and a modeling assumption that de-weighted padding does not bias the in-band solution. The paper treats these as design choices but does not provide a principled selection rule.

free parameters (3)

padding weight constant c = 0.03 (main figures), 0.06 (optimal for THz data)
Scales the Tukey window in the padded region; tuned per dataset to minimize error versus the true filtered signal (Fig. 5a).
Tukey window alpha = 0.2
Shape parameter of the Tukey window for the padding weight, chosen without stated justification; affects the results.
padded signal length = not specified explicitly
The number of padding points is not specified; it affects the filter solution and is chosen by the user.

assumptions (3)

domain assumption The measured signal is bandlimited with a known passband, so the true filtered signal is the projection onto that passband.
The method minimizes error within the chosen passband in Eq. (1) and (2); if the signal has out-of-band components that cannot be separated, the filter result may be biased.
ad hoc to paper De-weighting the padded region does not bias the in-band coefficients; the weighted least-squares solution is a good proxy for the true bandlimited signal.
This is the central modeling premise; the paper validates it empirically on three datasets but does not prove it generally. Introduced in Section 1, Algorithm 1.
domain assumption The full unfiltered signal can serve as ground truth for evaluating the filter.
Used to define the 'true filtered signal' in Figs. 2-4; valid only in the test setup where the full signal is available.

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Cite this review

Pith. "Pith review of Rubber band filters: optimal padding without edge artifacts." pith.science (2026). https://pith.science/paper/KY5XUNDF

@misc{pith2026250707347,
  author       = {Pith},
  title        = {Pith review of: Rubber band filters: optimal padding without edge artifacts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KY5XUNDF}},
  note         = {Machine review of arXiv:2507.07347}
}

read the original abstract

Bandpass filtering techniques are widely used in spectroscopy. However, conventional symmetric-padding filtering methods introduce boundary artifacts that distort the signal at the edges. We present a rubber band filter: a robust method for achieving band-limited filtering without these detrimental edge artifacts. The technique applies an optimal padding scheme during the filtering process, thereby overcoming longstanding challenges in achieving artifact-free filtering. Importantly, it is iterative and requires only a few extra Fourier transforms over conventional approaches. We demonstrate its superiority and versatility by applying it to three spectroscopic examples -- time-domain spectroscopy, Fourier-transform spectroscopy, and dual-comb spectroscopy.

Figures

Figures reproduced from arXiv: 2507.07347 by the authors.

**Figure 1.** Comparison between low-pass filtering without padding (left), with symmetric padding (middle), and with rubber band filtering (right). The shaded region shows the region where the signal is padded. The true signal is the signal without any padding. band-limited signal while eliminating edge artifacts by using optimal padding. An example is shown in [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** a. Comparison of low-pass signal filtering applied to a terahertz time-domain signal extracted from a complete measurement. The true filtered signal was generated by lowpass filtering the full, unpadded signal, serving as an ideal reference. The center normalized signal is unit-less, the shaded region shows a zoomed-in view of the signal at the edges. b. Comparison of the deviations from this true filtered signal u… view at source ↗

**Figure 4.** a. Comparison of low-pass signal filtering applied to a dual comb spectroscopy data after removing the linear phases. The center normalized signal is unit-less, the shaded region shows a zoomed-in view of the signal at the edges. b. Comparison of the deviations from this true filtered signal using different filtering methods. 2 4 Weight 10-1 10-2 100 a ×10-8 1 Error (a. u.) Iteration number 10-30 10-20 10-0 Error (… view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: a. Normalized error of the filtered signal as a function of the weight parameter, illustrating how increased weighting impacts the overall error. b. Normalized error plotted against iteration number, showing convergence behavior during the iterative Conjugate Gradient …

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Lu and B

G. Lu and B. Fei, J. biomedical optics 19, 010901 (2014)

work page 2014

[2] [2]

Fauvel, Y

M. Fauvel, Y . Tarabalka, J. A. Benediktsson,et al., Proc. IEEE 101, 652 (2012)

work page 2012

[3] [3]

Manolakis, D

D. Manolakis, D. Marden, G. A. Shaw et al., Linc. laboratory journal 14, 79 (2003)

work page 2003

[4] [4]

Chandel and G

R. Chandel and G. Gupta, Int. J. Adv. Res. Comput. Sci. Softw. Eng. 3 (2013)

work page 2013

[5] [5]

Burghoff, Y

D. Burghoff, Y . Y ang, and Q. Hu, Sci. advances 2, e1601227 (2016)

work page 2016

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Nazarian, I

S. Nazarian, I. Gkouzionis, M. Kawka, et al. , JAMA surgery 157, e223899 (2022)

work page 2022

[7] [7]

J. Li, B. Yu, W. Zhao, and W. Chen, Appl. Spectrosc. Rev. 49, 666 (2014)

work page 2014

[8] [8]

Lasch, Chemom

P . Lasch, Chemom. Intell. Lab. Syst. 117, 100 (2012)

work page 2012