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

TL;DR

The paper addresses edge detection in challenging, low-contrast textures by introducing a tropical geometry-based reformulation that uses min-plus and max-plus algebra to emphasize dominant intensity transitions. Central to the approach are tropical convolution and gradient operators, e.g., $O_T(x,y) = min_{i,j}[K(i,j) + I(x - i, y - j)]$ and $delta_T I(x,y) = min(I(x+1,y) - I(x,y), I(x,y+1) - I(x,y))$, enabling a modular pipeline with multi-directional kernels, adaptive thresholding, Hessian filtering, and wavelet shrinkage. The work presents three tropical variants (adapt_thresh_min_plus, eight_kernels_min_plus, four_kernels_max_plus) and shows integration with classical detectors like Canny and LoG, achieving sharper edges and better structural coherence on MATLAB grayscale and color images. These findings suggest a scalable, noise-aware alternative for practical edge detection, with future directions including adaptive parameter tuning, real-time deployment, and integration with learning-based methods.

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.

Figures (4)

• Figure 1: Edge detection results using min-plus and max-plus algebraic operations
• Figure 2: Visual comparison of edge detection using classical operators and TG-integrated operators
• Figure 3: Edge detection results on MATLAB's tree image with varying thresholding levels
• Figure 4: Visual comparison of edge detection methods applied to the MATLAB Images using $TG^{\text{adapt tresh}}_{min-plus}$, $TG^{\text{8 kernels}}_{min-plus}$ and $TG^{\text{4 kernels}}_{max-plus}$ methods.

Theorems & Definitions (2)

• Example 1
• Example 2