Distância de Hausdorff morfológica discreta com elemento estruturante octogonal Minkowski-Chebyshev

Abstract

This work presents and evaluates a morphological variant of the Hausdorff distance based on an octagonal structuring element associated with a Minkowski–Chebyshev combination. The motivation stems from the use of the Hausdorff distance in shape comparison, image registration, segmentation assessment and linear feature analysis, while its application to high-resolution data requires a trade-off between geometric fidelity and computational cost. Simple discrete metrics, such as Chebyshev, favor locality and performance, but introduce anisotropy; on the other hand, approximations closer to the Euclidean metric tend to be computationally more expensive. In this context, the objective of this work was to implement and evaluate a two-dimensional morphological formulation using an octagonal Minkowski–Chebyshev metric, experimentally identified as RMC_p1, and compare it with Chebyshev, Chamfer 3–4 and Euclidean references. The methodology involved implementations in C++, CUDA and Java, the reproduction and extension of a two-dimensional benchmark based on Rodrigues’ work, and stress tests with 1000 synthetic pairs of binary masks. The main benchmark comprised 1520 executions, of which 1348 were successfully completed. The RMC_p1 metric achieved an average similarity of 91.72% with respect to the Euclidean reference, outperforming Chebyshev, which achieved 78.60%, while remaining below Chamfer 3–4, which achieved 96.79%. In the morphological CUDA versions, RMC_p1 achieved a geometric mean runtime close to Chebyshev, indicating a favorable trade-off between fidelity and performance. In the stress tests, the proposed metric maintained an average similarity of 93.08%, again outperforming Chebyshev, with 86.33%, and remaining below Chamfer 3–4, with 96.74%. The results indicate that the proposed octagonal element reduces the geometric error associated with the square Chebyshev element while preserving compatibility with local and parallelizable morphological implementations. It is concluded that RMC_p1 constitutes an intermediate alternative between computational cost and geometric fidelity in the morphological computation of the Hausdorff distance.