Method and Algorithms for Computing Fuzzy Fréchet and Hausdorff Distance

Abstract

Accurate image similarity assessment is a key problem in computer vision, particularly in segmentation and classification problems. Classical Hausdorff and Fréchet metrics provide pointwise distance values and do not allow similarity to be evaluated in the form of intervals, which limits their applicability in problems where uncertainty plays a significant role. In this study, a combined approach to computing distances between images based on fuzzy Fréchet and Hausdorff metrics is developed. Two theorems are proved demonstrating that, for convex polygonal contours, the fuzzy Hausdorff distance coincides with the fuzzy Fréchet distance. This result makes it possible to replace the computation of the fuzzy Hausdorff metric with the simpler fuzzy discrete Fréchet metric. A method and algorithms for determining the fuzzy discrete Fréchet distance and a combined distance between convex polygons are proposed; their computational complexity is evaluated, and an application example is provided. The results show that the combined fuzzy metric reduces computation time by at least a factor of two compared to the direct computation of the fuzzy Hausdorff metric, while preserving similarity assessment accuracy. The proposed approach can be applied to shape analysis, segmentation evaluation, and similarity modeling in image classification systems. Future research directions include extending the method to non-convex polygons and arbitrary geometric objects.

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