Galaxy Morphological Classification with Zernike Moments and Machine Learning Approaches

Hamed Ghaderi; Hossein Safari; Nasibe Alipour

REVIEW 3 major objections 4 minor 1 cited by

Galaxy Morphological Classification with Zernike Moments and Machine Learning Approaches

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

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Pith's one-line read Zernike moments classify galaxy morphology with accuracy above 0.90 on Galaxy Zoo 2 samples.

desk verdict A useful, genuinely new application of Zernike moments to optical galaxy classification, but the headline TSS numbers are inflated by test-set model selection and the abstract overstates the 1D-CNN binary result. read the letter →

arxiv 2501.09816 v1 pith:M7E5QN5G submitted 2025-01-16 astro-ph.IM astro-ph.GA

Hamed Ghaderi , Nasibe Alipour , Hossein Safari This is my paper · ORCID

classification astro-ph.IMastro-ph.GA

keywords galaxymorphologyZernikemomentsmachinelearningclassificationsupportvectorconvolutionalneuralnetworkZoo2rotationinvarianceimbalanced

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 show that a compact vector of Zernike moments—coefficients of a galaxy image projected onto an orthogonal polynomial basis on the unit disc—carries enough morphological information to classify galaxies by machine learning. Using Galaxy Zoo 2 images, the authors build a galaxy-versus-non-galaxy classifier and a three-class spiral/elliptical/odd classifier, and they compare Zernike-feature models (SVM and one-dimensional CNN) against image-based deep networks (2D-CNN, ResNet50, VGG16 with a vision transformer). Their central claim is that the Zernike-based models are high-performance: TSS above 0.86 for the oversampled binary SVM, and accuracy above 0.90 with TSS above 0.86 for the SVM, 1D-CNN, and VGG16 models in the three-class task. If true, 1081 rotation-invariant numerical features per clean galaxy image can replace much heavier image-processing pipelines at lower computational cost, which would be useful for upcoming large surveys.

What carries the argument

The central object is the Zernike moment: the coefficient $Z^q_p = \frac{p+1}{\pi}\int I(r,\theta) ZP^{q*}_p \, r\,dr\,d\theta$ obtained by projecting the image intensity onto the complete orthogonal Zernike polynomials $ZP^q_p(r,\theta)=R^q_p(r)e^{iq\theta}$ on the unit disc. Because of the Fourier factor $e^{iq\theta}$, the magnitudes of the ZMs are invariant under image rotation, and orthogonality and completeness guarantee that the coefficient list is unique and can reconstruct the image. The paper uses the first 1081 moments ($p_{\max}=45$) per grayscale galaxy as the feature vector for an RBF-kernel SVM and a 1D-CNN, and contrasts this with activation-map features from 2D-CNN, ResNet50, and VGG16 models. The machinery also includes watershed segmentation of galaxy images to a centered 200x200 sub-image (giving translation and scale normalization) and oversampling of the minority non-galaxy class with RGB-channel ZMs.

What would settle it

Re-run the SVM and 1D-CNN pipelines with $p_{\max}$ chosen by cross-validation on training data only, then evaluate on a held-out test set untouched during model selection; if the held-out TSS for the binary classifier falls below 0.86, or the three-class accuracy falls below 0.90, the paper's high-performance claim as stated is not supported.

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Extended reading notes

Core claim

The authors' discovery claim is that Zernike moments (ZMs), taken to maximum order $p_{\max}=45$, give 1081 unique, rotation-invariant descriptors that are directly usable for morphological classification. For the binary task they use 545 non-galaxies and 11,735 galaxies selected by Galaxy Zoo 2 fraction thresholds, oversample the non-galaxy minority by adding the ZMs of the red, green, and blue channels (normalized by the total-intensity moment $Z^0_0$) to reach an imbalance ratio of about 0.2, and report SVM accuracy of 0.90 with TSS of 0.86 and AUC 0.93. For the three-class task, after segmenting images to 200x200 galaxy sub-images, they report weighted TSS of 0.88 for SVM with ZMs, 0.86 for 1D-CNN with ZMs, and 0.89 for VGG16 with a vision transformer, with accuracies 0.90, 0.90, and 0.93, matching or approaching the deep image models while using far fewer features.

Load-bearing premise

The reported scores come from a test set that was also used to choose the maximum Zernike order ($p_{\max}=45$), so the headline accuracy and TSS may be optimistic for images the model has not been tuned on.

Editorial extensions

If this is right

A single 1081-component Zernike vector is enough to separate galaxy from non-galaxy images and spiral, elliptical, and odd galaxies at the accuracy levels reported, without training a deep network on the pixels.
Rotation invariance removes the need to align or rotationally augment galaxy images before classification, simplifying pipelines for surveys where orientation is arbitrary.
Because the 1D-CNN and SVM operate on one-dimensional moment vectors, the trained classifiers run on CPU resources, cutting the GPU cost of ResNet50- and VGG16-style pipelines.
For the three-class problem the ZM-based models are competitive with a VGG16 plus vision transformer on clean Galaxy Zoo 2 samples, suggesting ZMs are a viable low-cost feature set for large morphological samples.
Performance is class-dependent: odd objects have the lowest recall (0.84 for SVM with ZMs), so the practical use needs to weigh the odd-class error against the computational savings.

Reading between the lines

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

Editorial inference: the same 1081-moment representation could be tested for continuous structural regressions, since the paper shows that the $q=0$ moments are directly tied to the axisymmetric S\'ersic-like brightness profile; if the link holds, ZMs might estimate structural parameters rather than only discrete classes.
Editorial note: the abstract says TSS exceeds 0.86 for both binary ZM models, but the body table lists TSS=0.70 for the 1D-CNN binary model; the three-class TSS values quoted in the abstract and table agree, so readers comparing models should use the table values.
Editorial inference: transfer to other surveys is untested; retraining on GZ2 and scoring DECaLS, DES, or Euclid images would show whether the rotation invariance and compactness survive differences in seeing, depth, and bandpass.
Editorial inference: hybridizing Zernike moments with a small CNN could combine the interpretable, low-cost geometric description with deep texture features, possibly improving the odd-object class where the reported recall is lowest.

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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

The paper is worth reading, and the core idea is sound: Zernike moments are a compact, rotation-invariant description of galaxy images, and feeding 1081 of them to an SVM or a 1D-CNN gives accuracy above 0.90 and TSS above 0.86 on clean Galaxy Zoo 2 three-class samples, competitive with VGG16 with a vision transformer. That result, if confirmed, is cheap and practical for big surveys. The analytic section linking the Sersic profile to vanishing even-order Zernike moments is a genuinely nice touch, and the PDF decision-boundary analysis is a simple way to show the features carry class information without any learning model.

The problems are real but addressable. First, the abstract says TSS greater than 0.86 for both SVM and 1D-CNN in the oversampled galaxy/non-galaxy binary task, but Table 3 in the body reports 1D-CNN TSS = 0.70. The three-class claims for Models I, II, and V are consistent with the abstract, but the binary 1D-CNN claim is not. Second, and more important, the maximum Zernike order p_max was chosen by scanning 5 to 46 and keeping the value with highest accuracy on the same test split later used for the reported metrics. That selection-on-the-test-set inflates the generalization estimate. This is not a fatal flaw, because the descriptive power of Zernike moments is not in question, but it means the honest headline is "works well on this data set," not "generalizes at TSS > 0.86." A quick fix is to hold out a validation set before the p_max scan and report the test metrics only for the chosen p_max. Finally, the GitHub links in the abstract and text contain spaces and are malformed; that is minor but should be fixed.

Who benefits? Anyone working on fast morphological classification for large surveys, and people who want a concrete worked example of Zernike moments applied to a modern data set. The paper does not need to be a breakthrough to be useful, and the methods section is clear enough to reproduce once the links are fixed. My recommendation: send it to peer review, but require the authors to (a) correct the abstract/body inconsistency, (b) rerun the p_max selection with a proper validation split, and (c) fix the code links. Then the claimed numbers will be trustworthy and the contribution will stand.

Referee Report

3 major / 4 minor

Summary. The paper proposes using Zernike moments (ZMs) of galaxy images as compact, rotation-invariant features for morphological classification, and compares five models: SVM with ZMs, 1D-CNN with ZMs, and three image-based deep networks (2D-CNN, ResNet50, VGG16 with vision transformer). The data are drawn from GalaxyZoo 2. Two classification tasks are considered: galaxy vs. non-galaxy, and three-class galaxy morphology (spiral, elliptical, odd). The central claims are that ZM-based SVM and 1D-CNN achieve high performance (TSS > 0.86) on the galaxy/non-galaxy task, and that SVM with ZMs, 1D-CNN with ZMs, and VGG16 achieve accuracy > 0.90 and TSS > 0.86 on the three-class task. The authors also argue that the ZMs provide uniqueness, completeness, and invariances, and that the ZM-based methods have lower computational cost than deep image models.

Significance. If the reported performance estimates were unbiased, the paper would make a useful practical contribution: 1081 Zernike moments can match or approach deep image classifiers on a moderately clean Galaxy Zoo 2 sample while using far fewer parameters and less computational resources. The authors provide code and data on GitHub, which supports reproducibility. However, the headline performance numbers are currently undermined by a model-selection procedure that uses the test set to choose the maximum Zernike order p_max, and by an abstract that states TSS values inconsistent with the body. These issues are central to the paper's main claim and must be fixed before the results can be accepted as reliable.

major comments (3)

[Section 4.2 (and Section 4.3 if p_max is scanned there too)] The maximum Zernike order p_max was selected by evaluating accuracy on the same test set later used to report all performance metrics. The text states: 'Examining the ZMs for maximum order number (pmax) ranges from 5 to 46, we calculate the performance metric (e.g., accuracy) for both the SVM and classic 1D-CNN classifiers. We obtain the highest performance for pmax = 45.' With roughly 42 candidate values, choosing the best on the test set overstates the generalization performance of the final model, even if the training/test split is repeated ten times, because the selection criterion uses test-set labels. This directly affects the headline binary TSS values (SVM 0.86, 1D-CNN 0.70) and the three-class TSS values in Table 4. Please restructure the evaluation so that p_max is selected on an independent validation set (or via nested cross-validation) and the reported metrics are computed on a held-out test set that is not used at any stage of model selection.
[Abstract and Section 4.2] The abstract claims 'TSS greater than 0.86 for the SVM and 1D-CNN with ZMs for the oversampled galaxy-non-galaxy classifier,' but the body reports, for the oversampled binary task, SVM TSS = 0.86 +/- 0.01 and 1D-CNN TSS = 0.70 +/- 0.03. The 1D-CNN value is far below 0.86, and the SVM value is not strictly greater than 0.86 given the uncertainty. This is a factual discrepancy between the abstract and the results. Please correct the abstract and ensure all numeric claims match the tables and text.
[Section 4.3, Table 4] The text states that 'TSS is greater than 0.86 for Models I, II, and V' and the abstract says 'accuracy larger than 0.90 and TSS greater than 0.86' for SVM with ZMs, 1D-CNN with ZMs, and VGG16. Table 4 lists TSS = 0.88 +/- 0.003 (Model I), 0.86 +/- 0.01 (Model II), and 0.89 +/- 0.005 (Model V), and accuracy = 0.90 +/- 0.005, 0.90 +/- 0.01, and 0.93 +/- 0.003. For Model II, TSS is not strictly greater than 0.86, and for Models I and II accuracy is not strictly larger than 0.90. Please use phrasing consistent with the uncertainties, e.g., 'TSS at least 0.86' or report lower bounds from the confidence intervals.

minor comments (4)

[Equation (6)] The reconstruction formula I_R(r,theta) = sum_p sum_q Z_q^p ZP_q^p(r,theta) should explicitly include the complex conjugate of the Zernike polynomial or state that the real part is taken; as written, the sum over positive and negative q may not be manifestly real for a real image.
[Section 4.2, oversampling discussion] The oversampling procedure adds ZMs of the R, G, and B channels of the same non-galaxy images to the training set, but the test set contains only grayscale ZMs. Please clarify in the text that these augmented samples are deterministic transformations of the original minority samples and therefore do not add independent information; this affects how the uncertainty estimates and the improved TSS should be interpreted.
[Appendix/references] There are duplicate reference entries: Shamir 2009 appears twice, and Li et al. 2022 appears three times. Please consolidate the bibliography.
[Section 3.3 and captions] Minor wording issues: 'different architects' should be 'different architectures'; the GitHub URL in the text contains spaces and should be a proper hyperlink; and the phrase 'the task answer of GZ2' in Section 6 should be rephrased.

Circularity Check

1 steps flagged · score 6.0 of 10

Reported TSS is inflated by selecting p_max on the test set; the central performance claim is partially circular.

fitted input called prediction [Section 4.2 (Galaxy-non-galaxy classifier), with metrics in Table 3 and the Abstract]
"Examining the ZMs for maximum order number ( pmax) ranges from 5 to 46, we calculate the performance metric (e.g., accuracy) for both the SVM and classic 1D-CNN classifiers. We obtain the highest performance for pmax = 45 for both SVM and 1D-CNN models (see the analysis details in Section 4.3)."

The pmax hyperparameter is selected by maximizing accuracy on the same 25% test split that is later used to report the headline metrics. The paper states that 'Each model uses 75 percent of both classes in the training process, and the remaining 25 percent is applied to the test set.' The reported TSS=0.86 for SVM with ZMs (Abstract) and the Table 3 accuracy/TSS values are computed on that same test set after choosing pmax=45 as the best-performing value. Thus the reported 'prediction' is not an unbiased out-of-sample estimate; it is statistically forced by the test-set selection of pmax. This is the fitted-input-called-prediction pattern: a parameter (pmax) is fitted to the test set, then the test-set metric is presented as the model's performance.

full rationale

The Zernike-moment formalism itself is not circular: Equations (1)-(7) define the moments, derive rotation invariance from the exponential phase factor, and invoke standard orthogonality/completeness results cited to Teague (1980) and Khotanzad & Hong (1990). The reconstruction demonstration (Equation 6) is an identity/illustration, not a prediction. The use of the self-cited ZEMO package (Safari et al. 2023) is a tool citation and is not load-bearing for the central claim. The one genuine circular step is the selection of pmax=45 by evaluating accuracy on the test set, followed by reporting accuracy/TSS/AUC on that same test set as the headline performance. This makes the headline 'TSS greater than 0.86' an optimistically biased, test-set-conditioned quantity rather than an independent generalization estimate. Because the broader claim that Zernike moments are useful morphological descriptors still has substantial independent empirical content (PDF separations, reconstruction fidelity, comparisons with image-based models), the circularity is partial rather than total.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central contribution is an empirical feature-engineering comparison; the ledger entries above are the main tunable or assumed inputs on which the reported performance depends. No new physical entities are introduced, so invented_entities is empty.

free parameters (4)

p_max (maximum Zernike order) = 45
Chosen by scanning p_max from 5 to 46 and selecting the value with the highest test-set accuracy for SVM and 1D-CNN in Section 4.2. This is a performance-driven choice made on the same test set, so it inflates the reported metrics.
SVM penalty C = 1.5
Reported in Section 3.2 as the RBF kernel penalty; no search or justification is given, and results may depend on it.
Class fraction thresholds = 0.95 for spiral, 0.90 for elliptical and odd
Used to define clean class samples from GZ2 in Section 2. These cutoffs determine the dataset and are chosen by the authors, not derived from data.
Oversampling factor for non-galaxy minority = approximately 4, from grayscale plus R, G, B channels
Minority class is augmented by adding ZMs from the R, G, and B channels to the training set, raising the imbalance rate from 0.05 to about 0.2 in Section 4.2. This is a hand-chosen augmentation that affects the reported TSS.

assumptions (5)

standard math Zernike polynomials form a complete and orthogonal basis on the unit disc
Invoked in Section 3.1 to claim uniqueness of ZM features and reconstruction via Equation 6.
standard math Magnitudes of Zernike moments are rotation invariant
Equation 7 shows rotation changes only the phase; used to assert robustness to inclination in Figure 9.
domain assumption The discrete approximation in Equation 5 faithfully represents the continuous ZM integral for galaxy images
The paper acknowledges tiny peaks from discretization in Figure 4; if discretization errors are large, the 1081 features could be noisy descriptors.
domain assumption Galaxy Zoo 2 vote fractions above the chosen thresholds provide correct class labels
All training and test labels come from threshold cuts on volunteer fractions in Section 2; label noise would degrade the reported metrics.
domain assumption The watershed segmentation isolates the central galaxy so that computed ZMs describe the galaxy, not the background
Applied in Section 4.3 before computing ZMs for the three-class classifier; poor segmentation would corrupt the features.

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Cite this review

Pith. "Pith review of Galaxy Morphological Classification with Zernike Moments and Machine Learning Approaches." pith.science (2026). https://pith.science/paper/M7E5QN5G

@misc{pith2026250109816,
  author       = {Pith},
  title        = {Pith review of: Galaxy Morphological Classification with Zernike Moments and Machine Learning Approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7E5QN5G}},
  note         = {Machine review of arXiv:2501.09816}
}

read the original abstract

Classifying galaxies is an essential step for studying their structures and dynamics. Using GalaxyZoo2 (GZ2) fractions thresholds, we collect 545 and 11,735 samples in non-galaxy and galaxy classes, respectively. We compute the Zernike moments (ZMs) for GZ2 images, extracting unique and independent characteristics of galaxies. The uniqueness due to the orthogonality and completeness of Zernike polynomials, reconstruction of the original images with minimum errors, invariances (rotation, translation, and scaling), different block structures, and discriminant decision boundaries of ZMs' probability density functions for different order numbers indicate the capability of ZMs in describing galaxy features. We classify the GZ2 samples, firstly into the galaxies and non-galaxies and secondly, galaxies into spiral, elliptical, and odd objects (e.g., ring, lens, disturbed, irregular, merger, and dust lane). The two models include the support vector machine (SVM) and one-dimensional convolutional neural network (1D-CNN), which use ZMs, compared with the other three classification models of 2D-CNN, ResNet50, and VGG16 that apply the features from original images. We find the true skill statistic (TSS) greater than 0.86 for the SVM and 1D-CNN with ZMs for the oversampled galaxy-non-galaxy classifier. The SVM with ZMs model has a high-performance classification for galaxy and non-galaxy datasets. We show that the SVM with ZMs, 1D-CNN with ZMs, and VGG16 with vision transformer are high-performance (accuracy larger than 0.90 and TSS greater than 0.86) models for classifying the galaxies into spiral, elliptical, and odd objects. We conclude that these machine-learning algorithms are helpful tools for classifying galaxy images.

Figures

Figures reproduced from arXiv: 2501.09816 by the authors.

**Figure 1.** The Zernike polynomials (ZPs) for different order numbers 𝑝 = 0, 1, 2, and 3 in polar coordinate. The ZPs in lower order numbers (𝑝 ⩽ 2) capture broad global features in an image. The order numbers (3 ⩽ 𝑝 ⩽ 5) identify finer details, such as asymmetries or specific distortions in an image. While the higher order numbers (𝑝 ⩾ 6) extract the tiny features like lines, sharp edges, and high-frequency variations due to t… view at source ↗

**Figure 2.** The original image of galaxies recorded by SDSS (left column), the magnitude of Zernike moments (ZMs) for 𝑝max = 45 (middle column), and reconstructed (Equation 6) image (right column). The first, second, and third rows correspond to spiral, elliptical, and odd objects, respectively [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The visualization scheme for the convolutional neural network (CNN) layers includes a convolutional layer with a 3 × 3 kernel, a max-pooling layer with a 2 × 2 kernel, a flattening layer, and a fully connected layer, applied to a sample galaxy image. to use convolutional layers containing filters (kernels) for one-dimensional features or images. These filters contain the learning parameters applied for feature extra… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: To samples of elliptical galaxies (left column) and the corresponding ZMs with 𝑝max = 45 (right column) for Sersic index ´ 𝑛 = 2 (top row) and 𝑛 = 4 (bottom row). The 𝑝max = 45 returns 1081 ZMs with the set of orders and repetitions labeled by (𝑝, 𝑞) [PITH_FULL_IMAGE:…

**Figure 5.** Figure 5: The variation of ZMs 𝑍 0 0 (black line), 𝑍 0 2 (blue line), 𝑍 0 4 (orange line), 𝑍 0 6 (green line), 𝑍 0 8 (red line), and 𝑍 0 10 (purple line) versus ellipticity parameter of the elliptical galaxies with 𝑛 = 2 (left panel) and 𝑛 = 4 (right panel). 𝑝 = 2 (𝑍 −2 2 , 𝑍 0 …

**Figure 6.** Figure 6: (Left panel) the simulated spiral galaxies with the arm amplitude of 0.0 (a), 0.2 (b), and 0.8 (c), (right panel) the ZMs with 𝑝max = 4 for each simulated spiral galaxy. The other parameters for three simulated spiral galaxies including the Sersic index ´ 𝑛, effective …

**Figure 7.** Figure 7: The probability density function (PDF) of the root square ZMs of 𝑍3 for 4077 elliptical (solid blue line), 6139 spiral (dashed orange line), and 1519 odd objects (green dots) collected by GZ2. We illustrate the discriminant boundary between the PDF of elliptical with s…

**Figure 8.** Figure 8: The PDF of 𝑍3 for 545 non-galaxy images (dashed orange line) and 780 galaxy images (solid blue line). The vertical solid red line indicates the discriminant boundary of 𝑍3 for galaxy and non-galaxy samples. with ZMs), II (classic 1D-CNN with ZMs), III (classic 2D-CNN w…

**Figure 9.** Figure 9: (top panel) the simulated spiral galaxy profile with Sersic index ( ´ 𝑛 = 1), 𝑅𝑒 = 100, and 𝐼𝑒 = 18 for inclination angle 𝑖 = 0 ◦ (left) and 𝑖 = 60◦ (right), (middle panel) the ZMs with 𝑝max = 45 for two perspectives, and (bottom panel) the scatter plot of ZMs for spir…

**Figure 10.** Figure 10: A sample confusion matrix for Model I representing the predicted class vs. true class. True positive (TP) shows the number of galaxies that are correctly classified in galaxy class (positive class). In contrast, false positive (FP) indicates the number of non-galaxy i…

**Figure 11.** Figure 11: The receiver operating characteristic (ROC) curve for Model I (blue), Model II (orange), Model III (green), Model IV (red), and Model V (purple) of galaxy-non-galaxy classification and random (black dashed line). The area under the curve (AUC) is obtained to be 0.83±0…

**Figure 12.** Figure 12: The original image recorded by SDSS (left panel), segmented galaxy represented within a green rectangle (middle panel), galaxy sub-image (right panel) [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]

**Figure 13.** Figure 13: The schematic table of the confusion matrix represents the actual label vs predicted labels for a three-class model. 4.3. Galaxy classifier The galaxy classifier is designed to classify the galaxy images into three main groups: spiral, elliptical, and odd objects. We …

**Figure 14.** Figure 14: The receiver operating characteristic (ROC) curve for spiral (left panel), elliptical (middle panel), and odd objects (right panel) classes of Model I (blue), Model II (orange), Model III (green), Model IV (red), and Model V (purple), and random (black dashed line). T…

**Figure 15.** Figure 15: Variation of Accuracy (orange) and TSS (blue) versus absolute magnitudes of galaxies (AM). the galaxy images. Rotation invariant properties are essential concepts for galaxy classification and morphological prediction in the machine learning framework (e.g., Dieleman …

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