On the edge detection of an image by numerical differentiations for gray function

摘要

> The detection of image edges is of great importance in image processing. One of the efficient implementations for this image recovery problem is based on the identification of sharp jumps of the gray function of the image. Mathematically, this problem can be modeled by the numerical differentiation of the gray function with 2 variables. For this ill‐posed problem with nonsmooth solution, we investigate the regularization schemes with total variation and L 1 penalty term, respectively. We prove that the regularizing parameter under the Tikhonov regularization framework can be uniquely chosen in terms of the Morozov's discrepancy principle and then establish the convergence rate of the regularizing solutions in terms of the Bregman distance. The discrete schemes are performed by the lagged diffusivity fixed point iteration, with numerical implementations showing the validity of the proposed scheme.