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REVIEW 4 major objections 7 minor 59 references

Denoising 3D images: robustness of persistent homology measures

T0 review · 4 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Vectorized and distance-based persistent-homology measures stay reliable under noise and denoising of 3D porous images; raw generator counts and average lifespans do not.

desk verdict Useful controlled ranking of PH summaries under matched noising/denoising on synthetic porous volumes; the hierarchy is partly baked into how the summaries treat short intervals. read the letter →

arxiv 2607.24579 v1 pith:PF7ZC26U submitted 2026-07-27 cs.CG cs.CVmath.AT

Ebru Dagdelen , Aakash Karlekar , Manav Arora , Matthew Illingsworth , Jonathan Jaquette , Linda J. Cummings , Lou Kondic This is my paper · ORCID
classification cs.CGcs.CVmath.AT MSC 55N3168U1062H35
keywords persistenthomologyimagedenoisingporousmediabottleneckdistanceWassersteinpersistencelandscapesimagestopologicaldataanalysis
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

Noise in 3D grayscale images of porous media floods persistent homology with millions of short-lived generators, making computation expensive and any analysis that uses generator counts unreliable. This paper tests how well common topological summaries survive a controlled noising-then-denoising cycle on three families of synthetic porous volumes. After adding spatially uncorrelated Gaussian noise, the authors smooth with either a Gaussian filter or a trained 3-D U-Net and compare the cleaned images with the known originals. They find that normalized bottleneck and Wasserstein distances, persistence landscapes, and persistence images recover a clear minimum near a modest smoothing strength and stay close to the original topology across a useful range of parameters; normalized generator counts and average lifespans swing wildly and are dataset-dependent. The practical message is that practitioners who must denoise before computing persistent homology should trust the vectorized and diagram-distance summaries and treat raw persistence statistics with caution.

What carries the argument

A controlled original-noisy-denoised pipeline on three synthetic porous datasets, evaluated by six normalized topological measures (generator count, average lifespan, bottleneck, Wasserstein-2, persistence-landscape L2, persistence-image L2).

What would settle it

Repeat the same measure comparison on real µCT volumes whose noise is known to be spatially correlated or intensity-dependent; if generator-count and lifespan measures suddenly become as stable as the vectorized distances, the claimed ranking fails.

Watch

Extended reading notes

Core claim

On synthetic 3-D porous-media volumes, L2-based vectorizations (persistence landscapes and images) and diagram distances (bottleneck, Wasserstein-2) are consistently more robust indicators of noising/denoising quality than scalar persistence statistics (normalized generator count and average lifespan), under both Gaussian convolution and U-Net denoising.

Load-bearing premise

Real experimental noise is adequately captured by spatially uncorrelated Gaussian noise of fixed strength, so the robustness ranking found on these synthetic trials transfers to actual scans.

Editorial extensions

If this is right

  • Denoising pipelines for sub/super-levelset PH should monitor landscape or image L2 distance (or bottleneck/Wasserstein) rather than raw Betti numbers.
  • A modest Gaussian bandwidth near the noise standard deviation simultaneously minimizes most robust measures across homology dimensions.
  • U-Net denoising preserves landscape and image structure more reliably than generator counts or average lifespans across heterogeneous pore geometries.
  • Persistence statistics that count or average short intervals should be discounted or heavily weighted when noise is present.
  • The same robustness hierarchy appears under both classical smoothing and learned denoising, suggesting it is intrinsic to the measures rather than the denoiser.

Reading between the lines

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

  • If the ranking holds for real correlated noise, software libraries could expose a default “robust PH score” based on landscapes or images instead of Betti curves.
  • The observation that restricting to the first 100 landscapes already acts as a soft denoiser suggests a cheap pre-filter before full barcode computation.
  • Edge-preserving or topology-aware denoisers could be scored by the same pipeline to decide whether they improve on simple Gaussian smoothing for PH fidelity.
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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

4 major / 7 minor

Summary. The manuscript studies how six topological summaries of 3D grayscale porous-media images respond to a controlled noising/denoising pipeline. Three synthetic dataset families (Fourier-series, overlapping-sphere PuMA, Worley/cellular) are generated with known ground truth, corrupted with i.i.d. Gaussian voxel noise (σ=5.76, calibrated from blank regions of a coquina µCT scan), and denoised either by Gaussian convolution with varying bandwidth σ or by a 3D U-Net trained on 450 volumes per morphology (288/72/90 split). Six normalized measures are compared between original and denoised images: generator-count difference ΔN_i (Eq. 1), average-lifespan difference ΔL_i (Eq. 2), normalized bottleneck and Wasserstein-2 distances, and normalized L2 distances between persistence landscapes (top 100 envelopes) and persistence images. The authors find a consistent optimal Gaussian bandwidth near σ≈0.5, which coincides with the minimizer of ∥f_o−f_d∥_∞ (Fig. 7), and conclude that L2/distance-based measures (bottleneck, Wasserstein, landscapes, and to a lesser degree images) are more robust indicators of denoising quality than the persistence statistics ΔN and ΔL, under both denoising paradigms.

Significance. If the conclusions hold, the paper gives practitioners of TDA on µCT porous-media data useful, concrete guidance: which PH summaries can be trusted after denoising, and a practical observation that the optimal Gaussian bandwidth is well tracked by the (computationally cheap) sup-norm of the image difference. The experimental design has real strengths: known ground truth on three morphologically distinct synthetic families, averages over 10 realizations (Gaussian path) and 90 hold-out volumes (ML path), and an explicit numerical verification of the bottleneck stability bound (Fig. 7) that also yields the falsifiable observation that most measures are simultaneously minimized near the ∥·∥_∞ minimizer. The within-measure results — existence and location of an optimal denoising level, and the unreliability of ΔL at high σ — appear solid. The significance is tempered by the fact that the headline cross-measure robustness hierarchy is partly under-identified (Major Comment 1), and by the restriction to a single, spatially uncorrelated noise model, which limits transfer to real scans.

major comments (4)
  1. [Secs. 3.2, 5.1.5–5.1.6, 6 (Eqs. (1)–(2), Def. 4/Eq. (6))] The cross-measure robustness hierarchy — the paper's central comparative claim — is confounded by heterogeneous treatment of short-lived generators. The compared measures do not operate at a common level of feature weighting: (a) the landscape measure PL_i uses only the first 100 upper envelopes (Sec. 5.1.5), which the authors themselves note 'has a denoising effect' (Sec. 6); (b) the persistence image uses m=1, θ=L_max in Eq. (6), so every nonmaximal interval is linearly downweighted by lifespan; (c) the bottleneck distance is insensitive by its minimax definition to the multiplicity of near-diagonal points; whereas (d) ΔN_i (Eq. (1)) and ΔL_i (Eq. (2)) count every generator with equal weight. Since the added i.i.d. noise predominantly creates enormous numbers of short-lived generators (Table 1: e.g., Cellular N_1 grows from ~12k to ~1.3M), the compared measures include the perturbation
  2. [Sec. 5.1, Figs. 8–13] All curves are stated to be averages over 10 realizations (Sec. 2), but no measure of variability (error bars, shaded bands, standard deviation) is shown anywhere in the Gaussian-denoising results. This matters specifically where the paper draws conclusions from fine structure: the oscillatory behavior and secondary minima of ΔL for PuMA at σ≈5 and σ≈7 (Fig. 9b), the 'minor oscillatory behavior' acknowledged for PI (Fig. 13), and the precision of the optimal band σ∈[0.5,0.7]. With n=10, realization-to-realization spread could plausibly explain these features; if it does not, showing the spread would substantially strengthen the claims. Please add variability indicators, at least for Figs. 9 and 13.
  3. [Sec. 5.2 vs. Sec. 6] The ML-based path reports only four of the six measures, excluding bottleneck and Wasserstein distances 'due to prohibitive computation times' (Sec. 5.2). The Discussion nonetheless states that 'the measure robustness hierarchy identified under Gaussian denoising ... carries over to the ML setting.' That claim is only partially supported: two of the measures identified as most robust in Sec. 5.1 are exactly the ones missing, so the ML evidence bears only on PL/PI vs. ΔN/ΔL. Either temper the sentence to scope the carried-over hierarchy to the four measured quantities, or provide BD/W_2 on a subsample (e.g., 10 of the 90 test volumes, possibly at reduced resolution) to justify the stronger statement.
  4. [Sec. 4.1 (noise model), Table 1, Figs. 8c, 14c, 15c] The noising step clips voxel values to [0,255], but the Cellular dataset's histogram is strongly skewed toward dark values (Fig. 4c), so for a large fraction of its voxels the additive N(0,5.76) noise is one-sided after clipping — i.e., the effective noise on the dataset that drives most of the 'less robust' verdicts for ΔN/ΔL (broad ML distributions in Figs. 14c–15c, anomalous i=2 optimum σ≈2 in Fig. 8c, ~100× generator explosion in Table 1) is not the i.i.d. Gaussian the paper assumes. Please quantify the clipped fraction per dataset and discuss how much of Cellular's outlier status is attributable to clipping-induced noise asymmetry rather than to morphology per se. This also bears on the transferability of the σ≈0.5 regime and the rankings to real µCT noise, which may be spatially correlated or intensity-dependent; the Discussion's one-sentence limitation should be expanded according
minor comments (7)
  1. [Sec. 4.1, denoising normalization] Step (ii) wraps out-of-range values around modulo 256 (overflow 255→0, underflow 0→255). Since convolution with a normalized Gaussian kernel is a convex combination of its inputs, convolved values cannot leave [min,max] of the (already clipped) noisy image, so wrap-around should never trigger; if it does trigger in practice (e.g., due to the integer truncation in step (i)), it would create spurious extreme gradients (white voxels becoming black) with large topological consequences. Please clarify, or replace with clipping for consistency with the noising step.
  2. [Sec. 3.2, Def. 4 / Sec. 5.1.6] Implementation details of the persistence image are incomplete: the grid resolution M×N (Def. 5) is never stated, and the units of the kernel variance σ=0.5 (birth–lifespan grayscale units?) are ambiguous. These materially affect the measure; please report them. Similarly, the common mesh used for the landscape L2 computation (Sec. 5.1.5) is unspecified.
  3. [Eq. (6)] The weight function is written in terms of |x−y|, but the persistence surface is constructed on the transformed (birth, lifespan) coordinates where lifespan is the second coordinate; as written it is unclear whether |x−y| refers to pre- or post-transformation coordinates. Please make the notation consistent.
  4. [Secs. 5.2.2, Fig. 15] Unlike the other normalized measures, ΔL_i can exceed 1 (the Cellular distributions in Fig. 15c extend to ~3), so its values are not on the same [0,1] scale the Discussion implicitly assumes when comparing measure magnitudes. Worth one sentence when the measure is introduced (Eq. (2)).
  5. [Fig. 14 caption] Caption reads 'over 90×3 testing datasets' while Figs. 15–17 say 'over 90 testing datasets'; please make the captions consistent (presumably 90 volumes × 3 homological dimensions).
  6. [Sec. 2.1] The Fourier coefficient decay 1/(1+i^{1/2}+j^{1/2}+k^{1/2}) is unusually slow (exponent 1/2), giving substantial high-frequency content; a one-line motivation (or a note on how the choice affects the multiscale feature distribution) would help readers judge representativeness.
  7. [General] No code or data availability statement is given. Given that the generators (PuMA, Porespy/PyFastNoiseSIMD) and the PH code (Cubicle) are public, releasing the noising/denoising scripts and U-Net configuration would make the study reproducible and is standard for the venue.

Circularity Check

2 steps flagged · score 1.0 of 10

No significant circularity: robustness ranking is an empirical comparison against known clean originals, not a result forced by definition or self-citation.

  1. other [Sec. 5.1.5 / Sec. 6 (persistence landscapes)]
    "To speed up the runtime of this computation (and avoid calculating millions of landscape functions), we only consider the first 100 upper envelopes as defined in Definition 3 when calculating this measure. ... First, recall that we used only the first 100 persistence landscapes due to computational constraints; with thousands to millions of persistence intervals, we did not compute all landscapes. This restriction to the 100 most prominent landscapes in itself has a denoising effect."

    Not circular derivation. Restricting to the top 100 envelopes is an explicit computational/design choice that the authors themselves flag as having a denoising effect. It makes PL less sensitive to the millions of short noise generators by construction of the summary, which weakens the force of the cross-measure 'robustness hierarchy' as an apples-to-apples comparison, but it does not make the reported PL recovery curves tautological predictions of their own inputs. The curves remain empirical distances to known clean originals.

  2. other [Sec. 3.2 Def. 4 / Sec. 6 (persistence images weighting)]
    "We employ the piecewise linear weight function ω_θ,m defined in (6) with m=1 and θ=L_max, where L_max denotes the maximum lifetime value in the PD. Consequently, features are weighted linearly according to their lifetimes, assigning greater importance to longer-lived topological features. ... We used a weighting of m=1, but we speculate that a larger power might help to diminish the effect of the small intervals, and make persistence images more robust to the noising/denoising process."

    Not circular. Lifespan weighting is part of the standard persistence-image construction the paper adopts; it intentionally down-weights short intervals. That choice (together with bottleneck's minimax definition) means PI/BD are less exposed to the noise-born generators that dominate ΔN and ΔL, so the robustness ranking partly reflects different feature weightings rather than a pure empirical discovery. This is a commensurability/identification issue for the hierarchy claim, not a case in which a 'prediction' reduces to a fitted input or a self-definition.

full rationale

The paper's central claim is a comparative empirical ranking of topological measures under a controlled noising/denoising pipeline on synthetic volumes whose ground-truth clean images are known a priori. Normalized differences (ΔN_i, ΔL_i, BD_i, WD_i, PL_i, PI_i) are computed between original and denoised PDs/vectorizations; the ranking (L2/vector and diagram distances more robust than raw persistence statistics) is read off the resulting curves and histograms (Secs. 5–6). Normalizations by original-to-empty or original-norm quantities are conventional scale-setting and do not force which measure recovers better. Optimal Gaussian σ ≈ 0.5 is observed to coincide with minimization of the independent pixel-space quantity ||f_o - f_d||_∞ (Fig. 7), which is a stability check rather than a tautology of the topological scores. U-Net results are likewise hold-out distributions against known originals. Built-in filtering (top-100 landscapes; lifespan weighting of persistence images; bottleneck's minimax pairing) is acknowledged by the authors and affects sensitivity, but that is a methodological design choice about what each summary measures, not circular derivation of the ranking from its inputs. No self-citation is load-bearing for the ranking; standard TDA citations supply definitions. Score 1 reflects only the minor, non-circular observation that some summaries intentionally discount short intervals while others do not.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central ranking rests on standard TDA stability and definitions, a domain noise model, synthetic morphology generators, and several hand-chosen numerical parameters (noise level, landscape truncation, PI weights, U-Net training setup). No new physical entities are postulated. The load-bearing modeling choice is i.i.d. Gaussian noise plus the claim that the three synthetic classes are representative enough for the robustness conclusions.

free parameters (6)
  • noise_std_sigma = 5.76 (grayscale units)
    Voxel-wise Gaussian noise std fixed at 5.76 from blank regions of one coquina µCT sample; all robustness curves depend on this single noise strength.
  • gaussian_denoise_sigma = optimal ~0.5–0.7 for most measures
    Smoothing bandwidth swept on [0,10]; reported ‘optimal’ near 0.5 is selected by minimizing errors vs ground truth, not by a parameter-free rule usable without clean data.
  • persistence_image_weight_m_and_kernel = m=1, kernel σ=0.5, θ=L_max
    PI construction uses Gaussian kernel variance 0.5 and piecewise weight ω with m=1, θ=L_max; authors note m≥d heuristics do not transfer to superlevel-set PH.
  • landscape_truncation_k_max = k_max=100
    Only first 100 upper envelopes used for PL L2 distances for runtime; this itself denoises the comparison.
  • Fourier_mode_cutoff_M = M=30
    Synthetic Fourier field truncated at M=30 with random N(0,1) coefficients and specific amplitude weights.
  • U-Net_training_hyperparameters
    Architecture, Huber loss, 64/16/20 split, Nv=128 volumes, 450 volumes per morphology class; exact optimizer schedule and width not fully pinned to a public artifact.
assumptions (5)
  • standard math Bottleneck stability: db(PD(f1),PD(f2)) ≤ ||f1−f2||_∞ for sub/superlevel-set filtrations.
    Invoked in Sec. 3.2 and checked numerically in Sec. 4.2/Fig. 7; standard Cohen-Steiner–Edelsbrunner–Harer theorem.
  • domain assumption Combined experimental noise sources may be modeled as i.i.d. Gaussian by the central limit theorem.
    Stated in Introduction and operationalized in Sec. 4.1; underpins the entire noising protocol.
  • domain assumption Superlevel-set cubical filtrations on 8-bit grayscale voxels with F=Z/2Z coefficients correctly capture the intended 0/1/2-dimensional porous-media topology.
    Sec. 3.1 setup used for all PH computations via Cubicle.
  • ad hoc to paper Fourier, overlapping-sphere (PuMA), and Worley/cellular synthetics are sufficiently representative of porous-media topology for robustness conclusions to be meaningful.
    Sec. 2 motivates each class by analogy to materials, but representativeness is an untested modeling choice relative to real µCT.
  • ad hoc to paper Normalized errors that limit to 1 under infinite smoothing are appropriate scales for comparing measure robustness.
    Sec. 3.2–5 normalizations (e.g. divide by distance to empty diagram or original vector norm) shape all plotted curves and the ‘robustness’ reading.

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Cite this review

Pith. "Pith review of Denoising 3D images: robustness of persistent homology measures." pith.science (2026). https://pith.science/paper/PF7ZC26U

@misc{pith2026260724579,
  author       = {Pith},
  title        = {Pith review of: Denoising 3D images: robustness of persistent homology measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PF7ZC26U}},
  note         = {Machine review of arXiv:2607.24579}
}
read the original abstract

When computing sub/super-level-set persistent homology (PH), the effect of noise may introduce millions of (short-lived) topological generators, presenting an obstacle to both the computation of PH of large 3D images, and any analysis of PH that incorporates the number of generators. As such, it is often necessary to denoise the data before computing its PH. We analyze the PH of synthetic 3D images of porous media in the presence of spatially uncorrelated noise, and perform a comparative analysis of various topological measures (e.g. bottleneck distance, Wasserstein distance, persistence statistics and persistence images) to assess their robustness to both noise and the denoising process (i.e. adding spatially uncorrelated Gaussian noise, and denoising by either a Gaussian convolution or a machine learning approach).

Figures

Figures reproduced from arXiv: 2607.24579 by the authors.

Figure 1
Figure 1. Flowchart of methodology. We start with the original synthetic dataset; add noise to produce a noisy dataset; then [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Fourier dataset discrete convolution on the image with a Gaussian kernel of standard deviation σ = 1.1 as described in Sec. 4.1, obtaining a multi-modal grayscale distribution that emulates what one might find in experimental samples. A 2D cross-section and 3D visualization of the resulting PuMA dataset is shown in Figs. 3a and 3b, and the distribution of grayscale values in Fig. 3c. The final PuMA dataset may be co… view at source ↗
Figure 3
Figure 3. PuMA dataset 2.3. Cellular Dataset The “cellular” dataset comes from the Porespy library — an open-source Python toolkit designed for the quantitative analysis of images of porous materials [41]. The Porespy library provides python functions for generating artificial porous media, calculating geometric and topological metrics, and extracting pore net￾works. Underlying the Porespy library is the PyFastNoiseSIMD packa… view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Cellular dataset 3. Definitions of key topological concepts and measures This section briefly introduces the key concepts from TDA that will be used throughout the rest of the paper. Section 3.1 below gives the key ideas in the context of this work, while Sec. 3.2 intr…
Figure 5
Figure 5. Figure 5: Three representations of the 0-dimension ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Persistence barcodes for the dimension 0, 1 and 2 PH ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Visualization of the Stability Theorem of Persistent Homology. Shown in black is [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Normalized difference of the number of generators measure ( [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Normalized difference of the average lifespan measure ( [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Normalized bottleneck distance (BDi, see Sec. 5.1.3) between the original and denoised datasets, for various denoising levels σ ∈ [0, 10]. 5.1.2. Average Lifespan of Generators When examining the average lifespan of generators measure (see Eq. (2)), we observe differi…
Figure 11
Figure 11. Figure 11: Normalized Wasserstein distance (W Di for i = 0, 1, 2, see Sec. 5.1.4) between the original and denoised datasets, for various denoising levels σ ∈ [0, 10]. (a) Fourier dataset (b) PuMA dataset (c) Cellular dataset [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Normalized L2 distance, P Li (see Sec. 5.1.5) between persistence landscapes of the original and denoised datasets, for various denoising levels σ ∈ [0, 10]. error here may be large. This similarity can be seen in the definition of the Wasserstein distance, which is s…
Figure 13
Figure 13. Figure 13: Normalized L2 distance P Ii (see Sec. 5.1.6) between persistence images of the original and denoised datasets, for various denoising levels σ ∈ [0, 5]. 5.1.6. Persistence Images The last measure we consider is the persistence image error measure, P Ii := ∥IPDo i − IPD…
Figure 14
Figure 14. Figure 14: Distribution of the normalized difference of the number of generators measure ( [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: Distribution of the normalized difference of the average lifespan measure ( [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 16
Figure 16. Figure 16: Distribution of the normalized L2 distance, P Li (see Sec. 5.1.5) over 90 testing datasets. 5.2.4. Persistence Images Results for the persistence image based measure P Ii , defined in Eq. (9), are presented in [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
Figure 17
Figure 17. Figure 17: Distribution of the normalized L2 distance P Ii (see Sec. 5.1.6) over 90 testing datasets. 6. Discussion Since our difference measures on the denoised and original datasets are normalized, we are able to compare their robustness to the noising/denoising process. We fi…

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Pith tools

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