用热带代数重构卷积与梯度，让边缘更清晰连续。
Tropical Geometry Based Edge Detection Using Min-Plus and Max-Plus Algebra
- 用极小-加和极大-加代数重写卷积与梯度计算
- 在低对比度和纹理区域提升边界检测能力
- 适合需要抗噪和结构连续性的图像分析任务
本文提出一种基于热带几何的边缘检测框架，将卷积与梯度计算重新表述为极小-加和极大-加代数形式。该方法突出显著的强度变化，有助于生成更锐利、更连续的边缘表示。研究探索了三种变体：自适应阈值法、多核极小-加法和强调结构连续性的极大-加法。框架融合多尺度处理、Hessian滤波与小波去噪，增强边缘过渡同时保持计算效率。在MATLAB内置灰度与彩色图像上的实验表明，结合传统算子（如Canny、LoG）的热带形式可有效提升低对比度与纹理区域的边界检测性能。标准边缘指标的定量评估显示其具备优异的边缘清晰度与结构一致性。结果表明，热带代数在实际图像分析中具有可扩展且抗噪的边缘检测潜力。
原文摘要 · Abstract (English)
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.
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