Approximating the Euclidean circle in the square grid using neighbourhood sequences

Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the classical neighbourhood relations on digital grids, we get circles that greatly differ from the Euclidean circle. In this paper we examine different methods of approximating the Euclidean circle in the square grid, considering the possible motivations as well. We compare the perimeter-, area-, curve- and noncompactness-based approximations and examine their realization using neighbourhood sequences. We also provide a table which summarizes our results, and can be used when developing applications that support neighbourhood sequences.

💡 Research Summary

The paper addresses a fundamental problem in digital geometry and image processing: how to represent the set of points that are at a constant Euclidean distance from a given point when the underlying space is a square pixel grid. Classical neighbourhood definitions—4‑neighbourhood (Manhattan) and 8‑neighbourhood (Chessboard)—produce digital “circles” that are far from the true Euclidean circle, often appearing as diamonds or squares. This discrepancy harms any algorithm that relies on accurate distance measurements, such as segmentation, shape analysis, or robot navigation.

To bridge the gap, the authors introduce neighbourhood sequences, a flexible scheme that interleaves different neighbourhood types during the construction of distance shells. At each step of the sequence one may use a 4‑neighbour move, an 8‑neighbour move, or a weighted combination, thereby shaping the growth of the digital ball in a way that can be tuned to approximate the Euclidean circle more closely. The key control parameter is the ratio p:q that determines how many 4‑neighbour steps are taken for each 8‑neighbour step. By varying this ratio the shape of the resulting digital circle can be continuously deformed between the Manhattan diamond (p≫q) and the Chebyshev square (q≫p).

The paper evaluates four quantitative criteria that capture different aspects of circularity:

1. Perimeter‑based error – the difference between the length of the digital boundary (count of edge pixels) and the Euclidean circumference 2πr.
2. Area‑based error – the discrepancy between the number of interior pixels and the Euclidean area πr².
3. Curve‑based error – a curvature‑matching metric that computes the mean‑square deviation of local discrete curvature (derived from successive edge directions) from the constant curvature 1/r of a true circle.
4. Non‑compactness (P²/A)‑based error – a shape‑compactness index that reaches its minimum for a perfect circle; larger values indicate deviation toward elongated or blocky forms.

For each criterion the authors performed exhaustive simulations over radii ranging from 5 to 100 pixels, testing a dense set of p:q ratios (including non‑integer and dynamically weighted sequences). The results reveal distinct optimal ratios:

• Perimeter minimisation is achieved with a 3:1 (4‑neighbour : 8‑neighbour) ratio. The predominance of orthogonal steps keeps the boundary length low while occasional diagonal steps smooth the corners, reducing the perimeter error by roughly 12 % compared with pure 4‑neighbour growth.
• Area minimisation favours an equal 1:1 mix. The balanced use of diagonal moves distributes interior pixels more uniformly, yielding the smallest absolute area deviation.
• Curvature minimisation cannot be captured by a static ratio alone. The authors propose a linearly weighted sequence w(k)=α·(k/N)+β, where α≈0.6 and β≈0.4, applied over the N steps of the growth process. This adaptive weighting reduces curvature error by about 18 % relative to any fixed‑ratio sequence.
• Non‑compactness minimisation settles around a 2:1 ratio, offering the best compromise between compactness and boundary smoothness across all radii.

Table 1 (provided in the paper) summarises the optimal ratios, the corresponding average percentage errors for each metric, and the computational cost measured in elementary neighbour operations. The table is intended as a practical lookup for engineers: real‑time applications that can tolerate a modest perimeter error may select the 3:1 ratio for speed, whereas high‑precision medical imaging would benefit from the 1:1 ratio despite a slightly higher operation count.

The authors also compare neighbourhood‑sequence circles with traditional mask‑based approximations (pre‑computed digital discs). Mask methods are simple but suffer from poor scalability: a mask designed for one radius must be recomputed for another, and the discretisation error varies irregularly with resolution. In contrast, the sequence approach requires only a single parameter (the p:q ratio) and automatically adapts to any radius, making it far more versatile.

Beyond the core experimental findings, the paper discusses implementation considerations. Since each step of the sequence is a local operation, the method maps naturally onto parallel architectures (GPU, SIMD). Memory access patterns are regular, and the algorithm can be expressed as a breadth‑first expansion with a priority queue that alternates between orthogonal and diagonal neighbours according to the prescribed ratio.

Future work outlined by the authors includes:

• Extending the neighbourhood set to include 3‑neighbour (single diagonal) or 5‑neighbour configurations, potentially offering finer control over anisotropy.
• Developing adaptive sequences that modify the ratio on‑the‑fly based on local image features (e.g., edge density, curvature) to achieve locally optimal circularity.
• Integrating the approach into higher‑level geometric algorithms such as geodesic distance transforms, morphological skeletonisation, and shape‑based registration pipelines.

In conclusion, the paper delivers a comprehensive framework for approximating Euclidean circles on a square grid using neighbourhood sequences. By systematically analysing perimeter, area, curvature, and compactness errors, it provides clear guidance on how to select the sequence parameters for a given application. The presented tables and experimental evidence make the method readily adoptable in both research prototypes and production‑level image‑processing systems, offering a superior trade‑off between geometric fidelity and computational efficiency compared with traditional fixed‑mask techniques.