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.
Abstract (translated)
本文提出了一种基于热带几何的边缘检测框架，该框架使用极小-极大加法代数（min-plus和max-plus）重新表述卷积和梯度计算。这种热带公式化强调了主要强度变化，有助于获得更清晰、更连续的边缘表示。文中探讨了三种变体：自适应阈值方法、多核极小-极大加法方法以及着重于结构连续性的极大-加法方法。该框架集成了多尺度处理、赫斯（Hessian）滤波和小波收缩，以增强边缘过渡并保持计算效率。在MATLAB内置的灰度和彩色图像上进行的实验表明，与经典算子如Canny和LoG集成的传统边界的热带公式化方法可以在低对比度和纹理区域提高边界检测能力。使用标准边缘指标进行的定量评估显示了有利的边缘清晰度和结构连贯性。这些结果突显了热带代数作为边缘检测实用图像分析任务中可扩展且对噪声敏感的表述方案的潜力。
URL
https://arxiv.org/abs/2505.18625
