Edge Detection Filter based on Mumford-Shah Green Function

In this paper, we propose an edge detection algorithm based on the Green function associated with the Mumford–Shah segmentation model. This Green function has a singularity at its center. A regularization method is therefore proposed here to obtain an edge detection filter known here as the Bessel filter. This filter is robust in the presence of noise, and its implementation is simple. It is demonstrated here that this filter is scale invariant. A mathematical argument is also provided to prove that the gradient magnitude of the convolved image with this filter has local maxima in discontinuities of the original image. The Bessel filter enjoys better overall performance (the product of the detection performance and localization indices) in Canny-like criteria than the state-of-the-art filters in the literature. Quantitative and qualitative evaluations of the edge detection algorithms investigated in this paper on synthetic and real world benchmark images confirm the theoretical results presented here, indicating the scale invariant property of the Bessel filter. The numerical complexity of the algorithm proposed here is as low as any convolution-based edge detection algorithm.