Directional Projection of the Hessian: A Robust Method for Edge Detection

Abstract Edge detection remains a fundamental component in many computer vision and image processing applications, including image segmentation, object recognition, and scene understanding. Despite extensive research, achieving both high precision and robustness in the presence of noise and structural ambiguity remains challenging. This paper introduces a novel edge detection method based on Directional Hessian Projection (DHP), which exploits second-order differential structure by projecting the local Hessian matrix onto multiple orientations. The proposed approach constructs a directional edge map by averaging the absolute directional curvatures obtained from these projections. This mathematically grounded formulation enhances the detection of curvature-rich and fine-scale structures while improving robustness to noise and filtering artifacts. Extensive experiments on synthetic and natural images, including the BSDS500 benchmark and the MultiCue dataset, show that DHP consistently outperforms several classical edge detectors such as Sobel, Canny, Marr–Hildreth, and wavelet-based methods. Quantitatively, DHP achieves lower MSE, higher PSNR and accuracy, and an average precision (AP) of 0.77 on BSDS500. On the MultiCue dataset, it reaches an ODS of 0.901 and an OIS of 0.93, with precision of 0.85 and recall of 0.96. Compared with deep learning approaches, DHP does not require large annotated datasets while narrowing the performance gap with learning-based methods.