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REVIEW 4 major objections 8 minor 23 references

A Fast Unsupervised Scheme for Polygonal Approximation

T0 review · 4 major / 8 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Fast unsupervised scheme rivals top polygon-fit quality

desk verdict A plausible incremental algorithm whose main empirical claim is undercut by an unvalidated Rosin oracle from the author's own prior work. read the letter →

arxiv 2506.04664 v2 pith:2XOGDVKZ submitted 2025-06-05 cs.GR cs.CGcs.CV

Bimal Kumar Ray This is my paper
classification cs.GRcs.CGcs.CV MSC 68U0568T10
keywords polygonalapproximationunsuperviseddigitalcurvevertexinsertionmergingadjustmentRosin'smeasureMPEG7
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 claims a fast, unsupervised algorithm that approximates a closed digital curve by a polygon without requiring the user to set a vertex count or error threshold. The scheme first finds sharp corners by scanning the curve and watching for the scanning chord to shrink, then greedily inserts vertices where error is largest, merges away weak vertices, and adjusts the remaining vertices to reduce squared error. On the MPEG7 shape dataset the approximations look almost identical to those of a leading unsupervised method, run faster in every reported test, and are competitive under Rosin's fidelity-efficiency measure. If the claim holds, polygonal approximation becomes a cheaper preprocessing step for shape representation, recognition, and compression.

What carries the argument

The load-bearing mechanism is the sharp-turn detector: while a scanning segment endpoint moves along the curve, if the Euclidean distance from the fixed start to the endpoint decreases instead of increasing, the curve point just passed is marked as a probable vertex. This cheap test, run clockwise and counterclockwise, generates the initial segmentation in O(n). The rest of the pipeline, iterative vertex insertion using a stopping heuristic of vertex count plus squared error, weakest-vertex merging with a self-intersection guard, and strongest-vertex adjustment, refines this set while keeping the total complexity O(m'n).

What would settle it

Recompute Rosin's measure for the MPEG7 curves using an exact optimal polygonal approximation, such as dynamic programming for the min-# or min-ε problem, for the vertex counts and errors produced by both schemes; if the proposed scheme's merit, or its ranking against the comparison scheme, changes materially, the competitiveness claim fails.

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Extended reading notes

Core claim

The paper's central claim is that high-quality polygonal approximation can be obtained by combining an O(n) scan for sharp turns with iterative vertex insertion, iterative merging, and vertex adjustment, at overall complexity O(m'n). The paper reports that on the MPEG7 curves this scheme is faster than the comparison unsupervised scheme, is competitive under Rosin's measure, which is the geometric mean of fidelity and efficiency relative to an optimal polygon, and is more robust to rotation and scaling as measured by the coefficient of variation of compactness. The supporting evidence is the visual similarity of the overlaid polygons, the Rosin merit plot, and the execution-time plot.

Load-bearing premise

The competitiveness evidence depends on the approximate optimal scheme used to compute Rosin's measure being accurate enough; if that approximation is biased, the reported Rosin comparisons may not reflect true fidelity and efficiency.

Editorial extensions

If this is right

  • Users get a scale-free approximation: no vertex-count or error threshold needs to be supplied.
  • Because the initial scan is O(n) and the total cost is O(m'n), the scheme is practical for long digital boundaries.
  • The reported average runtime gap of 1.22 units on a 10Log10 scale places the scheme faster than the comparison method across the MPEG7 test set.
  • Sharp corners are retained faithfully, which benefits object recognition, character recognition, and signature verification.
  • Stable compactness under rotation and scaling suggests the approximation is suitable for invariant shape matching.

Reading between the lines

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

  • The scanning-chord contraction test is a discrete proxy for curvature extrema and could be adapted to open curves or to multi-scale analysis by varying the chord-length threshold.
  • The stopping heuristic, number of vertices plus squared error, is essentially a penalized-error criterion; formalizing the penalty could yield a principled model-selection rule.
  • Because Rosin's measure in the paper uses an approximate optimal scheme from the author's earlier work, an exact optimal comparison on small curves would test whether the competitiveness claim is robust.
  • The self-intersection guard in the merging phase suggests the scheme may extend to noisy or self-approaching curves if the error threshold is estimated adaptively.
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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 / 8 minor

Summary. The paper proposes an unsupervised polygonal approximation scheme for closed digital curves, consisting of initial segmentation by bidirectional scanning, iterative vertex insertion, iterative merging, and vertex adjustment. The central claims are that the scheme is faster than state-of-the-art methods and competitive with the Madrid-Cuevas et al. scheme in terms of Rosin's measure and aesthetic quality, with an overall computational complexity of O(m'n). Validation is performed on MPEG7 shapes using visual overlays, Rosin's measure, execution time, and compactness-based robustness under rotation and scaling.

Significance. If the claims are properly supported, the paper offers a simple, unsupervised heuristic with modest complexity and visually plausible output, and the large MPEG7 visual comparison is a useful asset. However, the evidence for the two headline claims is currently thin: only one baseline is used, the Rosin measure is computed with an unvalidated approximate oracle from the author's own prior work, execution-time results are reported without statistical support, and the timing summary is ambiguous. The robustness analysis is a positive addition. The contribution is potentially publishable after the evaluation is substantially strengthened.

major comments (4)
  1. [Section V, Figure 5 and Section III] The central quality claim, 'competitive with Rosin's measure,' rests entirely on Figure 5, but the Rosin values are computed using an approximate version of Perez and Vidal's optimal scheme taken from the author's prior work [23]. This approximate oracle is not described, no error bound is given, and it is not validated against the exact optimal algorithm. Since Rosin's measure compares each scheme against the optimal polygon for the same vertex count, a biased surrogate could systematically favor one scheme; the paper itself notes that Rosin's measure favors larger vertex counts and that Madrid-Cuevas et al. generally produce more vertices. The authors should compute Rosin's measure with the exact Perez-Vidal algorithm on at least a representative subset of curves, or validate the approximation against exact optimal values and report the maximum and mean absolute error. Without this, Figure 5 does not support the competitiveness claim.
  2. [Section V, Figure 6] The execution-time comparison uses a single baseline and reports that the Madrid-Cuevas et al. scheme is 'on average slower than the proposed scheme by a factor of 1.22 in 10Log10 scale.' This sentence is ambiguous: if 1.22 is the average difference in 10*log10(time), the geometric-mean ratio is about 1.32, whereas if it is a difference in log10(time) the ratio is implausibly large. No standard deviations, confidence intervals, or per-curve statistics are given, and the line diagram alone cannot establish a general speed advantage. The authors should define the scale precisely, report the distribution of timings (e.g., median and interquartile range over multiple runs), and include a statistical test such as the Wilcoxon signed-rank test over the MPEG7 set.
  3. [Section V, Table I and Figures 5-7] The abstract and conclusion claim superiority or competitiveness with 'state-of-the-art' approaches, but the paper compares only with Madrid-Cuevas et al. [5]. A single baseline cannot support a state-of-the-art claim. At least one or two additional recent unsupervised schemes, such as Carmona-Poyato et al. [15], Fernandez-Garcia et al. [18], or Parvez [20], should be included. In addition, Rosin's measure is the only quantitative quality metric; the authors should also report other measures such as the figure of merit or the compression ratio versus approximation error, and perform a statistical comparison across the dataset.
  4. [Section IV, iterative vertex insertion] The stopping criterion for vertex insertion is the heuristic m + E2, and the text states that using the second or third power of m caused over-smoothing. This makes the exponent an empirically selected free parameter, and no sensitivity analysis is provided. Because this heuristic determines the final vertex count and therefore affects all downstream quality comparisons, the authors should either justify the choice analytically or report how the results vary with the exponent (e.g., m, m^2, m^3) on a few representative curves. Without this, the 'unsupervised' claim is tied to a hand-fitted criterion rather than to a principled stopping rule.
minor comments (8)
  1. [Algorithm, Step 5] Step 5 says 'Repeat Step 4 and 5 until the approximation is stabilized,' which is self-referential; it should refer to Steps 3 and 4 (merging and vertex adjustment).
  2. [Abstract and Section IV] The text says the scheme comprises 'three phases' but then lists four: initial segmentation, iterative vertex insertion, iterative merging, and vertex adjustment. Please correct the number or the list.
  3. [Section II and References] The name 'Pervez and Mauhmud [19]' should be 'Parvez and Mahmoud'; also, 'Perez' in reference [7] is a different author, so the spelling should be checked throughout.
  4. [Section III] The definition of Rosin's measure is garbled: the sentence beginning 'Fidelity is defined by the ratio...' mixes fidelity and efficiency and should be rewritten for clarity.
  5. [Table I] The table lists 'Cup' twice, and the row header 'Horsesho e' contains a line-break error; also, several rows appear to be missing or misaligned in the text version.
  6. [Figures 5-7] Figures 5, 6, and 7 are referenced but not actually embedded in the manuscript text provided; the published version must include legible figures with axis labels, legends, and a clear statement of the curve ordering on the horizontal axis.
  7. [Section V, robustness] The 'coefficient of variation (scaled)' in Figure 7 is not defined; please state the scaling and how compactness is computed for polygons with different vertex counts.
  8. [Section IV, complexity] The overall complexity O(m'n) does not account for the outer loop that repeats merging and vertex adjustment until stabilization; either bound the number of iterations by a constant or include it in the complexity expression.

Circularity Check

1 steps flagged · score 4.0 of 10

The Rosin-measure comparison, the paper's main quantitative quality evidence, is computed with an unvalidated approximate optimal scheme from the author's own prior work [23], making the competitiveness claim partly self-referential.

  1. self citation load bearing [Section V, 'Experiments and Analysis', paragraph after Figure 5]
    "Perez and Vidal ’s [7] optimal scheme was used to compute Rosin’s measure. The comput ational complexity of Perez and Vidal is significantly high, which is why an approximate version of the scheme [23] is used here."

    Rosin's measure, defined in Section III, is calibrated against an optimal polygonal approximation. The paper substitutes an 'approximate version' drawn from [23], the author's own prior work, without describing it, bounding its error, or validating it against the exact Perez-Vidal scheme. Since Figure 5's Rosin values are the only quantitative support for the central claim of being 'competitive' with Madrid-Cuevas et al., the comparison is anchored to an oracle that is a self-citation rather than an independent optimum. If the approximate oracle is biased—for example, if it underestimates optimal error for the proposed scheme's vertex counts or overestimates it for the baseline's larger vertex counts—the conclusion is forced by the choice of surrogate.

full rationale

The proposed algorithm itself is self-contained: initial segmentation, vertex insertion, merging, and adjustment are described independently, with complexity stated. The speed comparison and robustness-to-transformation experiments do not depend on the author's prior work. However, the Rosin's-measure comparison—the primary quantitative evidence for the paper's headline claim—uses an approximate optimal scheme from the author's own [23] without validation. That makes the central quality claim partially self-referential or at least dependent on an unexamined surrogate. The hand-tuned stopping heuristic (sum of vertex count and squared error, with the paper reporting that second and third powers over-smooth) is a design choice rather than a circular derivation; it does not reduce the claimed result to its inputs. Overall circularity is moderate: the method is not equivalent to its input, but the main benchmark's oracle is a self-citation, so the competitiveness claim is not fully independent.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical or mathematical entities. It relies on several domain assumptions about detecting corners and measuring approximation quality, plus two hand-fitted criteria. The most significant burden is the use of a self-cited approximate optimal scheme to compute the quality metric.

free parameters (1)
  • exponent of vertex count in the insertion heuristic = 1 (yielding m + E2)
    Section IV, the author reports trying powers 2 and 3 and rejecting them due to over-smoothing; the exponent is chosen by hand to make the approximation look good.
assumptions (4)
  • domain assumption A sharp turn on a digital curve can be detected when the length of a scanning line segment decreases as it passes a point.
    Section IV, initial segmentation. This heuristic is the basis for detecting initial vertices, and the paper acknowledges it misses low-curvature vertices (Figure 3 discussion).
  • ad hoc to paper The approximate version of Perez and Vidal's optimal scheme from the author's prior work [23] computes Rosin's measure with sufficient accuracy.
    Section V, Rosin measure computation. The approximate optimal scheme is self-cited and not independently validated.
  • domain assumption Compactness, defined as area divided by the square of the perimeter, is a valid proxy for shape similarity under rotation and scaling.
    Section V, robustness evaluation. No justification is provided that compactness captures the fidelity of a polygonal approximation.
  • ad hoc to paper The sum of the number of vertices and the sum of squared errors (m + E2) is an appropriate heuristic for stopping vertex insertion.
    Section IV. The author selected this form after trying second and third powers and observing over-smoothing, so it is a hand-fitted criterion.

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Cite this review

Pith. "Pith review of A Fast Unsupervised Scheme for Polygonal Approximation." pith.science (2026). https://pith.science/paper/2XOGDVKZ

@misc{pith2026250604664,
  author       = {Pith},
  title        = {Pith review of: A Fast Unsupervised Scheme for Polygonal Approximation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XOGDVKZ}},
  note         = {Machine review of arXiv:2506.04664}
}
read the original abstract

This paper proposes a fast and unsupervised scheme for the polygonal approximation of a closed digital curve. It is demonstrated that the approximation scheme is faster than state-of-the-art approximation and is competitive with Rosin's measure and aesthetic aspects. The scheme comprises of three phases: initial segmentation, iterative vertex insertion, iterative merging, and vertex adjustment. The initial segmentation is used to detect sharp turns, that is, vertices that seemingly have high curvature. It is likely that some of the important vertices with low curvature might have been missed in the first phase; therefore, iterative vertex insertion is used to add vertices in a region where the curvature changes slowly but steadily. The initial phase may pick up some undesirable vertices, and thus merging is used to eliminate redundant vertices. Finally, vertex adjustment was used to enhance the aesthetic appearance of the approximation. The quality of the approximations was measured using the Rosin's method. The robustness of the proposed scheme with respect to geometric transformation was observed.

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Reference graph

Works this paper leans on

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