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

ABSTRACT This paper presents a tropical geometry‐based edge detection framework that reformulates convolution and gradient operations using min‐plus and max‐plus algebra. By emphasizing dominant intensity variations, the tropical formulation yields sharper and more continuous edge representations. Three core variants are explored: an adaptive threshold‐based method, a multi‐kernel min‐plus approach, and a max‐plus formulation that promotes structural continuity. The framework integrates multi‐scale filtering, Hessian‐based refinement, and wavelet shrinkage to enhance edge transitions while maintaining computational efficiency. Experiments conducted on MATLAB built‐in grayscale and color images demonstrate that tropical formulations, when integrated with classical operators such as Canny and LoG, improve boundary localization in low‐contrast and textured regions. Quantitative evaluation using standard edge quality metrics confirms favorable clarity, noise resilience, and structural coherence. These findings underscore the potential of tropical algebra as a scalable, robust, and mathematically grounded alternative for practical edge detection tasks.