Tropical Geometry Based Edge Detection Using Min-Plus and Max-Plus Algebra

TL;DR

This paper introduces a novel edge detection framework based on tropical geometry, reformulating classical operators with min-plus and max-plus algebra to improve edge sharpness and continuity, especially in challenging regions.

Contribution

It presents a new tropical algebra-based approach for edge detection, integrating multi-scale processing and classical operators to enhance boundary detection in low-contrast and textured images.

Findings

Improved boundary detection in low-contrast regions.

Enhanced edge continuity and sharpness.

Favorable results in quantitative edge metrics.

Abstract

This paper proposes a tropical geometry-based edge detection framework that reformulates convolution and gradient computations using min-plus and max-plus algebra. The tropical formulation emphasizes dominant intensity variations, contributing to sharper and more continuous edge representations. Three variants are explored: an adaptive threshold-based method, a multi-kernel min-plus method, and a max-plus method emphasizing structural continuity. The framework integrates multi-scale processing, Hessian filtering, and wavelet shrinkage to enhance edge transitions while maintaining computational efficiency. Experiments on MATLAB built-in grayscale and color images suggest that tropical formulations integrated with classical operators, such as Canny and LoG, can improve boundary detection in low-contrast and textured regions. Quantitative evaluation using standard edge metrics indicates favorable edge clarity and structural coherence. These results highlight the potential of tropical algebra as a scalable and noise-aware formulation for edge detection in practical image analysis tasks.