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REVIEW 3 major objections 5 minor 28 references

Automated Rib Fracture Detection of Postmortem Computed Tomography Images Using Machine Learning Techniques

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims the HOMFLY knot invariant can separate fractured from intact rib CT images at 0.60 precision, but cannot yet label new images.

desk verdict Novel idea, honest write-up, but the reported HOMFLY 'classifier' is a descriptive list-overlap statistic, not a predictor. read the letter →

arxiv 1908.05467 v1 pith:FMVRWM2X submitted 2019-08-15 eess.IV cs.CV

Samuel Gunz , Svenja Erne , Eric J. Rawdon , Garyfalia Ampanozi , Till Sieberth , Raffael Affolter , Lars C. Ebert , Akos Dobay This is my paper · ORCID
classification eess.IVcs.CV
keywords HOMFLYpolynomialknottheoryribfracturedetectionpostmortemcomputedtomographytopologicalinvariantconvolutionalneuralnetworkunfoldingimageclassification
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

This paper tries to establish that a topological invariant, the HOMFLY polynomial knot invariant, can serve as the classifier in a machine-learning pipeline for detecting rib fractures in postmortem computed tomography images. The authors unfold the rib cage into a 2D projection, convert the grayscale image to pseudo-color, apply hand-picked convolution kernels, and pool the feature maps until they obtain 52 coordinates that are joined into a closed curve in RGB color space; each image then receives a HOMFLY polynomial as its class fingerprint. On 195 cases, the method separates fracture images from no-fracture images with class-exclusivity precision of 0.60 for the fracture class and 0.52 for the no-fracture class, whereas two convolutional neural networks reach F1 scores of 0.72 and 0.73. The paper states that the pipeline cannot yet be used to predict unlabelled images, so the contribution is a demonstration of a topology-based class separation rather than a deployable diagnostic tool. A sympathetic reader should care because the approach offers an alternative to statistically trained networks, one with potentially exact classification if the knot-stability assumption can be made to hold.

What carries the argument

The load-bearing object is the HOMFLY polynomial, a two-variable knot invariant computed from the closed piecewise-linear curve formed by joining 52 coordinates in RGB color space. The coordinates are produced by taking a pseudo-colored, unfolded rib CT image, convolving each RGB channel separately with 13 hand-picked $3\times3$ kernels, and softmax-pooling repeatedly until a $2\times2$ feature map per channel supplies four coordinate values; 13 kernels times four coordinates gives the 52 vertices of the curve. The HOMFLY polynomial is then computed from this curve with a program supplied by reference [20], and each image is assigned the corresponding knot type. The mechanism works if geometrically similar images yield topologically equivalent curves, and the paper measures class separation by counting which knot types appear in only one of the two class lists.

What would settle it

Take a fresh cohort of rib CT images, run the same 13-kernel, 52-coordinate pipeline, and check whether each image's HOMFLY polynomial falls exclusively into the fracture or no-fracture list; if a substantial share of new images has a knot type present in both lists or in neither, the claimed classification breaks, and the paper's own statement that unlabelled images cannot yet be predicted already points toward this outcome.

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Extended reading notes

Core claim

The central discovery, stated by the authors as a demonstration rather than a finished tool, is that rib CT images can be mapped to knot types whose HOMFLY polynomials separate, with measurable overlap, into a fracture list and a no-fracture list: precision is 0.60 for images with rib fractures and 0.52 for images without fractures, computed as the proportion of knot types not shared between classes. This separation is achieved without training any weights, using only the 52-coordinate curves derived from 13 convolution kernels on the pseudo-colored unfolded rib cage. The authors compare against two convolutional neural network baselines, which reach F1 scores of 0.72 and 0.73, and they report that the topology method cannot predict labels for unseen images because the current dataset does not cover the full catalogue of polynomials. On the paper's own terms, the result is an existence proof: a knot invariant can be inserted into an image-classification pipeline and yield a real, if modest, class signal.

Load-bearing premise

The whole method rests on the assumption that images from the same class produce closed curves with the same knot type, so that a single HOMFLY polynomial identifies the class; the paper's data already undermine this, since more than one knot type appears within a single class.

Editorial extensions

If this is right

  • If the topology-based separation holds on larger cohorts, rib-fracture screening could run without neural network training, using only hand-crafted kernels and a knot invariant.
  • The reported precision values set a measurable baseline: a larger dataset is expected to improve them, because the paper attributes the current ceiling to an incomplete catalogue of knot types.
  • Because both CNN baselines plateau around F1 0.72-0.73, the topology pipeline's 0.60 precision gives a non-statistical reference point for future automated fracture detection.
  • The same image-to-knot recipe should transfer to other grayscale medical images where a pathology manifests as stable shape or texture changes, since nothing in the pipeline is rib-specific.

Reading between the lines

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

  • A fair reading is that what is established is a class-separation statistic, not a classifier: the list-overlap rule cannot label a new image whose knot type appears in both lists or in neither, which the paper concedes.
  • A natural extension is to treat the HOMFLY polynomial type as a categorical feature and fit a small probabilistic model over knot-type distributions, instead of requiring strict exclusivity between the two class lists.
  • Using a coarser invariant (for instance, only the unknot versus a small set of low-crossing types) could mitigate the instability from multiple knot types per class; this direction is untested but follows directly from the paper's own observation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a pipeline for automated detection of rib fractures in postmortem CT images by converting images into closed curves in RGB color space, computing HOMFLY polynomial knot invariants, and comparing the distribution of knot types between fracture and no-fracture classes. The authors report a 'precision' of 0.60 for the fracture class using this topological approach, alongside F1 scores of 0.72–0.73 for two convolutional neural network baselines (MobileNet and a custom Keras model). The paper claims in the abstract and conclusion that the method can classify CT images of rib fractures.

Significance. The application domain is relevant: automated rib fracture detection in forensic PMCT could reduce radiologist workload. The paper includes a legitimate comparison of two CNN baselines, with reported recall, precision, and F1 scores, which is a strength. However, the central novel claim—that HOMFLY polynomial knot invariants classify rib fracture images—is not supported by the experiments as presented. The reported 0.60 precision is not a predictive accuracy but a descriptive list-overlap statistic with no train/test split and no decision rule for new images. The paper's own Discussion (Section 4) explicitly states that the pipeline cannot be used for predicting unlabelled images. Consequently, the title, abstract, and conclusion overstate what was demonstrated. The work is at best a preliminary exploration of a topological descriptor, not a validated classifier.

major comments (3)
  1. [Eq. (2), Section 2.8, and Section 3] The quantity called 'precision' for the HOMFLY approach is defined as (K1 - K12)/K1, where K1 is the number of knot types in one class and K12 is the number shared between classes. This is a measure of list disjointness, not classification precision in the sense of TP/(TP+FP). It is computed over the entire dataset without any train/test split, and the paper provides no decision rule that would assign a label to a new image. Section 4 explicitly states: 'we could not use this pipeline for predicting unlabelled images.' Therefore, the abstract's claim of a 'precision of 0.60' and the conclusion's statement that the method 'can be applied to classify CT images of rib fractures' are not supported.
  2. [Section 4] The load-bearing invariance assumption stated in Section 4—that variations in size, color, and shape for the same object translate into small geometrical variations of the closed curve, so that distinct objects produce distinct curves with different topological properties—is contradicted by the authors' own finding: 'we ended up with more than one knot type for a given class.' This means the topological invariant is not stable within a class, and the method is reduced to comparing lists of knot types between two fixed collections of images. The claimed classification mechanism therefore does not hold, and the observed separation may be an artifact of the particular dataset rather than a generalizable property.
  3. [Sections 2.5 and 2.6] The hyperparameters of the topological pipeline—the number of convolution cycles (4), the subset of 13 out of 28 kernels, and the use of 52 vertices—were selected using the full dataset. Equation (1) defines the kernel-selection criterion D as a function of the centroids of the two classes' feature clusters, and Figure 3 shows the resulting values for all 28 kernels. No held-out validation or cross-validation is performed for the topological pipeline. As a result, even the descriptive list-overlap statistic is optimistically biased, and no reliable estimate of generalization performance is provided.
minor comments (5)
  1. [Abstract] 'To access the performance' should read 'To assess the performance'.
  2. [Section 2.7] There is a typographical error: 'HOMLFY polynomial' should be 'HOMFLY polynomial'.
  3. [Throughout] The word 'Euclidian' appears in Section 2.7 and elsewhere; the standard spelling is 'Euclidean'.
  4. [Section 2.5] The phrase 'we convolved each channel of the input images with a 3 × 3 kernel function ... followed by a downsampling using the softmax function' is unclear: softmax is not a downsampling operation. Please clarify the exact pooling or downsampling procedure used.
  5. [Section 3] The statement 'Using 52 vertices, we could classify all fractures' is ambiguous because the subsequent precision values indicate that knot types are shared between classes; please clarify what 'classify all fractures' means in this context.

Circularity Check

2 steps flagged · score 6.0 of 10

The reported HOMFLY precision is a descriptive list-overlap statistic computed on the same images used to build the lists, not a predictive classifier result; the paper itself concedes the pipeline cannot predict unlabelled images.

  1. fitted input called prediction [Section 2.8 (Eq. 2) and Section 4 (Discussion)]
    "In case of the polynomial knot invariant the precision was modified to account for the relative proportion of knot types that were not shared between the two classes, i.e. (K1− K12)/K1 (2) where K1 is the total number of knot types in one class and K12 is the total number of knot types in both classes. ... Hence, although we can show this approach can classify rib fracture with a precision of 0.60, we could not use this pipeline for predicting unlabelled images."

    The claimed precision of 0.60 is defined by Eq. 2 as the fraction of knot types appearing in only one class list. Both lists are constructed from the full dataset of images, and the same images are then used to compute the overlap statistic. There is no train/test split and no decision rule that maps a new image to a class; the paper explicitly states that the pipeline cannot predict unlabelled images. Therefore the 'classification precision' is not a predictive performance measure but a restatement of how distinct the two label-conditioned lists of polynomials are. The conclusion that the method 'can be applied to classify CT images' is thus forced by the way precision was defined, not by any demonstrated predictive result.

  2. fitted input called prediction [Section 2.5 and Section 2.6 (Eq. 1)]
    "The number of convolution cycles was determined to be 4 for the current dataset. This value was obtained by maximizing the following quantity D = ||v1c− v2c|| / (Σ||v1i− v1c|| + Σ||v2j− v2c||) (1) ... A total of 13 kernel functions out of the 28 listed in Table 1 were chosen to generate a final set of 52 coordinates per image."

    The convolutional pipeline is tuned directly on the entire dataset: the number of convolution cycles is chosen by maximizing the cluster-separation statistic D, and the selection of 13 kernels is made on the same images. The subsequent HOMFLY polynomials and the overlap-based precision are then computed from these same tuned features on the same images. The reported 0.60 precision therefore measures how well the manually selected kernels and cycle count separate the already-seen data, rather than how the method would generalize to new images. This is a form of evaluating a fitted representation on its own training input, which makes the claimed classification performance optimistically biased and statistically dependent on the choices made from the data.

full rationale

The core topological claim reduces by construction: the paper defines its HOMFLY 'precision' as a list-overlap ratio (Eq. 2) computed from the knot-type lists of the training images themselves, and then explicitly concedes in Section 4 that the pipeline cannot predict unlabelled images. Because no held-out data, cross-validation, or classification rule for new images is involved, the reported 0.60 precision is a descriptive statistic about the two label-conditioned polynomial lists, not a measured classification performance. The kernel selection and convolution-cycle count are also optimized on the full dataset, compounding the dependence between the reported number and the data that produced it. This is not a self-citation or imported-uniqueness problem: the HOMFLY computation uses an external program by Ewing and Millett, and the CNN baselines are evaluated with standard train/validation splits and are independent. The circularity is instead that the paper's central 'HOMFLY classifier' claim is supported only by a metric that is, by Eq. 2, an overlap count on the training set, with the paper's own admission ruling out prediction. That warrants a score of 6: some 'predictions' reduce by construction, while the independent CNN comparison and the genuine mathematical computation of knot invariants keep the paper from being entirely definitionally circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new physical or mathematical entities are postulated; the closed curve in RGB space is a methodological construction, not an independent entity.

free parameters (3)
  • Number of convolution cycles = 4
    Determined by maximizing the cluster separation D in Eq. (1) on the dataset (Section 2.6).
  • Subset of kernels = 13 of 28
    Selected to reduce the number of crossings when projecting the curve (Section 2.5); data-dependent choice.
  • Image resolution = 950x950 for HOMFLY, 60x60 for Keras
    Chosen processing resolution for the two pipelines (Section 2.5).
assumptions (4)
  • standard math HOMFLY polynomial is a knot invariant that distinguishes knots
    Used in Section 2.7 to classify closed curves; the polynomial is assumed to separate distinct knot types.
  • domain assumption Small variations in image properties produce small geometrical variations in the closed curve, preserving knot type within a class
    Stated in Section 4 as the central premise linking images to topology; the paper notes this assumption fails as multiple knot types per class appear.
  • domain assumption Rib unfolding tool produces consistent, informative projections across cases
    The images are extracted with Syngo.via rib unfolding (Section 2.3); the method assumes these projections are comparable and encode fracture state.
  • ad hoc to paper The selected 13 kernels and 4 convolution cycles capture fracture-relevant features
    Chosen based on data-dependent criteria (Section 2.5-2.6), with no independent theoretical justification.

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Cite this review

Pith. "Pith review of Automated Rib Fracture Detection of Postmortem Computed Tomography Images Using Machine Learning Techniques." pith.science (2026). https://pith.science/paper/FMVRWM2X

@misc{pith2026190805467,
  author       = {Pith},
  title        = {Pith review of: Automated Rib Fracture Detection of Postmortem Computed Tomography Images Using Machine Learning Techniques},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMVRWM2X}},
  note         = {Machine review of arXiv:1908.05467}
}
abstract

Imaging techniques is widely used for medical diagnostics. This leads in some cases to a real bottleneck when there is a lack of medical practitioners and the images have to be manually processed. In such a situation there is a need to reduce the amount of manual work by automating part of the analysis. In this article, we investigate the potential of a machine learning algorithm for medical image processing by computing a topological invariant classifier. First, we select retrospectively from our database of postmortem computed tomography images of rib fractures. The images are prepared by applying a rib unfolding tool that flattens the rib cage to form a two-dimensional projection. We compare the results of our analysis with two independent convolutional neural network models. In the case of the neural network model, we obtain an $F_1$ Score of 0.73. To access the performance of our classifier, we compute the relative proportion of images that were not shared between the two classes. We obtain a precision of 0.60 for the images with rib fractures.

Figures

Figures reproduced from arXiv: 1908.05467 by the authors.

Figure 1
Figure 1. Flowchart depicting the calculation of a polynomial knot invariant starting from a CT image. The first step con [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Colormap conversion applied to transversal CT images of a human rib using the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Box plot of the optimal number of convolution steps for each kernel listed in Table 1. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Crossing convention for projecting a knot onto a 2D plane. (A) Regular projection of a knot. (B to D) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Rib unfolding. (A) Rib unfolding with no rib fractures and 12 ribs on either side of the vertebral column. [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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