PCOD: Polar Coordinate-Based Outlier Detection

• Polar Coordinate-Based Outlier Detection is a method that converts sub-pixel edge points into polar coordinates to identify local anomalies in radial signals.
• It computes robust global deviation using the median of local standard deviations from sliding window analysis on sorted angles, ensuring sensitivity to noise.
• PCOD acts as a pre-filter for circle fitting, significantly improving measurement accuracy and stability in industrial metrology applications.

Searching arXiv for the cited papers to ground the article in current literature. Polar Coordinate-Based Outlier Detection (PCOD) is an outlier detection and removal algorithm introduced for circle fitting in noisy point sets, especially in machine vision workflows for industrial metrology. In the formulation reported in “Outlier Detection Algorithm for Circle Fitting” (Poyraz, 28 Jul 2025), the method transforms sub-pixel edge points into polar coordinates relative to a reference center, computes local and global standard deviations on the resulting radial signal, and identifies outliers by comparing local mean values with a robust global deviation estimate. The reported application focuses on high-precision diameter measurement of industrial washer parts, where contamination, imperfect edge detection, and background artifacts can otherwise bias subsequent circle fitting (Poyraz, 28 Jul 2025).

1. Problem formulation and geometric premise

PCOD operates on a set of 2D edge points extracted at sub-pixel precision from images of washer-like parts, $P = \{(x_i, y_i)\}$ for $i = 1, \ldots, N$, with the objective of removing outliers before circle fitting so that the estimated diameter and circle parameters are accurate and stable in industrial settings (Poyraz, 28 Jul 2025).

The method assumes that most points lie close to a true circle, typically with angular coverage spanning a full contour, while outliers arise from dust, oil vapor particles, burrs, dark spots on the glass, imperfect edge detection, and background artifacts. This geometric setting motivates the use of polar coordinates: for a perfect circle, the radius as a function of angle is approximately constant, so anomalous points appear as local deviations in the radial signal. The paper summarizes the core idea as follows: “a perfect circle appears as a constant radius signal $r(\theta)$,” and outliers “manifest as local deviations in $r$” (Poyraz, 28 Jul 2025).

A reference center is required for the polar transform. In the reported implementation, this center is estimated as the centroid of the edge points, $x_c = (1/N)\sum_{i=1}^N x_i$, $y_c = (1/N)\sum_{i=1}^N y_i$. The paper notes that if $(x_c, y_c)$ deviates from the true circle center, the radial values $r_i$ acquire an angle-dependent bias. PCOD mitigates moderate center error through local mean radii in small angular neighborhoods and a robust global threshold based on the median of local standard deviations, but excessive center error degrades the locality assumption and can increase false positives or false negatives (Poyraz, 28 Jul 2025).

2. Polar transformation and statistical construction

With respect to the centroid-based reference center, PCOD computes polar coordinates $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$ and $$\theta_i = \operatorname{atan2}(y_i - y_c, x_i - x_c)$$, then sorts the angles in ascending order to preserve contour continuity (Poyraz, 28 Jul 2025).

The method distinguishes between reference global statistics and the robust statistics actually used for detection. Classical global mean and standard deviation of the radii can be written as $i = 1, \ldots, N$0 and $i = 1, \ldots, N$1, using the “biased” population form with denominator $i = 1, \ldots, N$2. However, the paper explicitly states that PCOD does not threshold using $i = 1, \ldots, N$3 or $i = 1, \ldots, N$4; instead, it constructs a robust global deviation from local windows (Poyraz, 28 Jul 2025).

Local neighborhoods are defined by fixed-width sliding windows on the sorted angle sequence. The window is index-based rather than angle-based: contiguous blocks of $i = 1, \ldots, N$5 points advance by a stride $i = 1, \ldots, N$6. For a window starting at index $i = 1, \ldots, N$7 and covering points $i = 1, \ldots, N$8, PCOD computes the local mean

$i = 1, \ldots, N$9

and local standard deviation $r(\theta)$0. The robust global deviation is then defined as $r(\theta)$1. The use of the median of local standard deviations is intended to make the global threshold robust to clustered outliers that could inflate mean-based estimates (Poyraz, 28 Jul 2025).

This construction makes PCOD a localized radial-consistency test rather than a global residual filter. A plausible implication is that its behavior is closely tied to the angular ordering induced by the contour geometry, which differentiates it from generic point-cloud outlier detectors that do not exploit near-circular structure.

3. Detection rule, workflow, and computational profile

PCOD makes point-level rather than window-level decisions. For each point $r(\theta)$2 in window $r(\theta)$3, the algorithm flags the point as an outlier if $r(\theta)$4, where $r(\theta)$5 is the local mean for the window containing the point and $r(\theta)$6 is the median of the window-level standard deviations over the whole contour (Poyraz, 28 Jul 2025).

The paper denotes the threshold multiplier by $r(\theta)$7, equivalent to $r(\theta)$8, but does not report the exact numerical value used in the experiments. It provides only practical guidance: $r(\theta)$9 between 2 and 3 is typical for standard deviation-based filtering; larger $r$0 reduces false positives but may leave moderate outliers, whereas smaller $r$1 increases sensitivity but risks removing valid points under higher noise (Poyraz, 28 Jul 2025). Because the exact experimental value is not explicitly reported, parameter-specific performance attribution beyond that guidance would be unwarranted.

The reported pipeline is a single-pass procedure:

1. Acquire image and preprocess to remove small connected components while retaining the largest object.
2. Perform sub-pixel edge detection to obtain $r$2.
3. Estimate the center by centroid.
4. Convert to polar coordinates.
5. Sort points by $r$3.
6. Form sliding windows of width $r$4 and stride $r$5, and compute $r$6 and $r$7.
7. Compute $r$8.
8. Mark outliers using $r$9.
9. Remove outliers and convert the remaining points back to Cartesian coordinates, $x_c = (1/N)\sum_{i=1}^N x_i$0, $x_c = (1/N)\sum_{i=1}^N x_i$1. 10. Fit a circle to the filtered points (Poyraz, 28 Jul 2025).

The baseline PCOD described in the paper does not iterate center refinement after filtering. The source notes that iterative refinement can be added in practice by recomputing the centroid from filtered points and repeating the polar transformation and filtering until stable, but it also states that this refinement is not part of the reported experimental pipeline (Poyraz, 28 Jul 2025).

The stated computational complexity is $x_c = (1/N)\sum_{i=1}^N x_i$2 time and $x_c = (1/N)\sum_{i=1}^N x_i$3 memory, with angle sorting as the dominant step. Sliding-window statistics can be computed in $x_c = (1/N)\sum_{i=1}^N x_i$4 with prefix sums, whereas a naive implementation is $x_c = (1/N)\sum_{i=1}^N x_i$5. If angles are not sorted, complexity may be $x_c = (1/N)\sum_{i=1}^N x_i$6, but the paper sorts angles to maintain contour continuity (Poyraz, 28 Jul 2025).

4. Integration with circle fitting and comparison set

PCOD is designed as a pre-filter for circle fitting rather than as a standalone geometric estimator. After outlier removal, the filtered points are passed to a circle fitting algorithm. The standard circle model is $x_c = (1/N)\sum_{i=1}^N x_i$7, with geometric radial residuals $x_c = (1/N)\sum_{i=1}^N x_i$8, and geometric least-squares objective $x_c = (1/N)\sum_{i=1}^N x_i$9 (Poyraz, 28 Jul 2025).

The paper also states the algebraic circle form $y_c = (1/N)\sum_{i=1}^N y_i$0, with parameter recovery $y_c = (1/N)\sum_{i=1}^N y_i$1, $y_c = (1/N)\sum_{i=1}^N y_i$2, $y_c = (1/N)\sum_{i=1}^N y_i$3 (Poyraz, 28 Jul 2025).

Ten circle fitting methods were compared after outlier filtering: Least Squares Fitting (LSF), Pratt, Taubin, RANSAC, IRLS, HyperLS, M-Estimator, LMedS, TLS, and EDCircle. Their brief principles are given in the source, including algebraic least squares, generalized eigenproblem formulations, consensus-based fitting, iteratively reweighted least squares, median-based robust estimation, total least squares, and eigen-decomposition or SVD-based algebraic estimation (Poyraz, 28 Jul 2025).

Outlier detection baselines included Z-Score, Median Absolute Deviation (MAD), DBSCAN, Local Outlier Factor (LOF), Percentile-based filtering, and “None” as a no-filter baseline, alongside the proposed PCOD (Poyraz, 28 Jul 2025). This comparison frame is important because the reported results evaluate PCOD as a preprocessing stage across multiple fitters rather than only against a single downstream estimator.

5. Experimental setting and reported results

The empirical study used a PSG-1500 Glass Dial Sorting Machine only for image acquisition, with a 4 MP industrial camera, telecentric lens with fixed working distance, and backlight illumination. Workpieces were fed via a vibration system to a rotating glass platform, and images were captured at the first station, where the black part appeared on a white background (Poyraz, 28 Jul 2025).

Preprocessing consisted of binarization and complement, followed by connected-component analysis to remove small components such as dust while retaining the largest object, then additional complement, filling, and re-complement operations to isolate the outer contour for diameter measurement. Sub-pixel edges were obtained using the Canny/Devernay sub-pixel edge detector via the IPOL implementation, with default parameters $y_c = (1/N)\sum_{i=1}^N y_i$4, high threshold $y_c = (1/N)\sum_{i=1}^N y_i$5, and low threshold $y_c = (1/N)\sum_{i=1}^N y_i$6 (Poyraz, 28 Jul 2025).

The dataset contained 45 disc-shaped parts described as motor injector setting discs, with tolerance $y_c = (1/N)\sum_{i=1}^N y_i$7 mm. Ground truth was obtained from a Coordinate Measurement Machine with $y_c = (1/N)\sum_{i=1}^N y_i$8m accuracy. Evaluation focused on the outer diameter; the inner diameter was not measured in order to avoid repeated preprocessing (Poyraz, 28 Jul 2025). A pixel-to-millimeter conversion factor was computed from the first workpiece and applied separately for each outlier removal method and fitting algorithm (Poyraz, 28 Jul 2025).

The reported evaluation metrics were Mean Absolute Error (MAE) in millimeters and the Standard Deviation of Absolute Error (SDAE) across the 45 parts (Poyraz, 28 Jul 2025).

Aspect	Reported result	Scope
MAE for Geometric LS, Pratt, Taubin, IRLS, HyperLS with PCOD	$y_c = (1/N)\sum_{i=1}^N y_i$9 mm	Compared with $(x_c, y_c)$0–$(x_c, y_c)$1 mm for Z-Score, MAD, DBSCAN, LOF, Percentile and $(x_c, y_c)$2 mm with None
RANSAC MAE	Reduced from $(x_c, y_c)$3 mm with None to $(x_c, y_c)$4 mm with Proposed	Better than Z-Score, MAD, DBSCAN, LOF at $(x_c, y_c)$5–$(x_c, y_c)$6 mm
M-Estimator and LMedS with PCOD	$(x_c, y_c)$7 mm and $(x_c, y_c)$8 mm	Competitive or better than other filters
TLS and EDCircle	TLS $(x_c, y_c)$9–$r_i$0 mm; EDCircle $r_i$1–$r_i$2 mm	PCOD did not improve these substantially
SDAE with PCOD	$r_i$3 mm for most methods	Lower than or comparable to baselines

The paper states that PCOD yields the lowest MAE across most circle fitting algorithms and improves both accuracy and stability over the baselines and over running the same fitters without outlier removal (Poyraz, 28 Jul 2025). It also notes that the proportion of points removed was not explicitly reported and that formal hypothesis tests were not reported; instead, visual and log-scale plots were used to demonstrate consistent improvements (Poyraz, 28 Jul 2025).

6. Robustness, limitations, and practical usage

The reported behavior of PCOD depends on three principal factors: threshold selection, local window design, and center accuracy. For the threshold multiplier $r_i$4, the source warns that a value that is too low may prune valid inliers under moderate noise, while a value that is too high may leave outliers that continue to bias the fitted circle. For window width $r_i$5, larger values increase the stability of $r_i$6 and $r_i$7 but may smooth over localized defects; smaller values increase locality but raise the variance of $r_i$8. The stride $r_i$9 controls overlap and computation, and the source identifies $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$0 as a common compromise, though this is presented as practical guidance rather than as a fixed experimental setting (Poyraz, 28 Jul 2025).

The centroid-based center estimate is described as sufficiently accurate for the studied dataset, but large center errors can introduce angle-dependent biases that inflate $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$1 unevenly and harm detection. Similarly, the method uses fixed-count windows on sorted $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$2, which adapt to local sampling density, but highly non-uniform sampling may call for choosing $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$3 proportional to local density or using fixed-angle-width bins. The reported method itself uses fixed-count windows (Poyraz, 28 Jul 2025).

The paper also discusses partial arcs, occlusions, and clustered outliers. It states that PCOD remains effective on arcs because local statistics are computed along available segments, but if coverage is very limited or dominated by an outlier cluster, the median-based $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$4 can become too small or too large depending on window composition. It suggests, as practical mitigation, choosing $r_i = \sqrt{(x_i - x_c)^2 + (y_i - y_c)^2}$5 relative to arc length and considering a two-pass strategy that first prunes gross outliers such as with DBSCAN when arc coverage is very short. This should be read as implementation guidance rather than as part of the reported pipeline (Poyraz, 28 Jul 2025).

A separate limitation concerns the downstream estimator rather than the filter itself. TLS and EDCircle exhibited consistently high errors regardless of filtering, which the paper suggests may reflect sensitivity of purely algebraic homogeneous formulations to image noise and calibration. On the reported dataset, more stable fits included Pratt, Taubin, and IRLS after PCOD (Poyraz, 28 Jul 2025).

For practical deployment, the paper recommends clean parts and optics, connected-component filtering for obvious small contaminants, reliable sub-pixel edge detection with the reported Canny/Devernay settings, application of PCOD before circle fitting, and separate application to each contour when multiple circles such as inner and outer boundaries are present (Poyraz, 28 Jul 2025). This suggests that PCOD is best understood as one stage in a broader metrology pipeline rather than as a substitute for image preprocessing or robust fitting.

7. Terminological scope and relation to other polar-coordinate methods

Within the cited literature, the acronym “PCOD” refers to distinct polar-coordinate ideas, and the distinction is technically important. In “Outlier Detection Algorithm for Circle Fitting” (Poyraz, 28 Jul 2025), PCOD denotes Polar Coordinate-Based Outlier Detection for 2D edge-point filtering prior to circle fitting. By contrast, “Polar Depth for Potentially Heavy-Tailed Data” (Clemençon et al., 29 May 2026) develops a notion of polar depth for multivariate extremes, expressed through radial–angular decomposition, angular halfspace depth, and extreme-value asymptotics. That paper also presents a “PCOD step-by-step algorithm” for anomaly detection based on depth estimation in bulk or tail regimes (Clemençon et al., 29 May 2026).

The two usages share a polar-coordinate viewpoint but address different statistical objects. The circle-fitting PCOD of (Poyraz, 28 Jul 2025) operates on a near-circular contour where the relevant signal is the local constancy of radius as a function of angle. The polar-depth framework of (Clemençon et al., 29 May 2026) instead ranks multivariate observations by joint radial and directional extremeness under regular variation. Any direct equivalence between these methods would therefore be unwarranted. A plausible implication is that “PCOD” has become a broader label for polar-coordinate-based outlier or anomaly analysis, but in current arXiv usage the term is not unique and must be interpreted from context.

In the more specific sense established by (Poyraz, 28 Jul 2025), Polar Coordinate-Based Outlier Detection is a geometry-aware preprocessing method for circle fitting that leverages local radial regularity, robust aggregation via the median of local standard deviations, and lightweight computation. Its reported contribution lies not in replacing circle fitting, but in improving the quality of the point set supplied to circle fitters under realistic industrial contamination and imaging noise.