A Mathematical Survey of Image Deep Edge Detection Algorithms: From Convolution to Attention

Edge detection, a cornerstone of computer vision, identifies intensity discontinuities in images, enabling applications from object recognition to autonomous navigation. This survey presents a mathematically grounded analysis of edge detection’s evolution, spanning traditional gradient-based methods, convolutional neural networks (CNNs), attention-driven architectures, transformer-backbone models, and generative paradigms. Beginning with Sobel and Canny’s kernel-based approaches, we trace the shift to data-driven CNNs like Holistically Nested Edge Detection (HED) and Bidirectional Cascade Network (BDCN), which leverage multi-scale supervision and achieve ODS (Optimal Dataset Scale) scores 0.788 and 0.806, respectively. Attention mechanisms, as in EdgeNAT (ODS 0.860) and RankED (ODS 0.824), enhance global context, while generative models like GED (ODS 0.870) achieve state-of-the-art precision via diffusion and GAN frameworks. Evaluated on BSDS500 and NYUDv2, these methods highlight a trajectory toward accuracy and robustness, yet challenges in efficiency, generalization, and multi-modal integration persist. By synthesizing mathematical formulations, performance metrics, and future directions, this survey equips researchers with a comprehensive understanding of edge detection’s past, present, and potential, bridging theoretical insights with practical advancements.