Fractional Dirac Operators for Edge Detection

The Dirac operator links harmonic analysis, physics and hypercomplex signal representations. However, most Dirac-based imaging methods remain integer order and lack spectral adaptability. In this paper, we propose a novel fractional Dirac framework for edge detection. Some fundamental properties are obtained, including square factorization, Liouville-type properties, and uncertainty principles with sharpened constants in a limiting case. Then, a numerically stable discrete realization is developed based on the quaternion Fourier transform. This realization yields an edge detector for both grayscale and RGB images. Experiments on benchmark datasets show that the proposed method produces coherent contours and competitive boundary-detection performance compared with classical gradient methods and recent transform-based detectors.