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

Abstract

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 L1 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.

A Novel Edge Detection Method Based on the Regularized Laplacian Operation

“…In order to calculate derivatives of the noisy data stably, some regularization methods should be introduced. There have been much work into this over the past years, such as the Tikhonov regularization [10], the Lavrentiev regularization [11], the Lanczos method [12], the mollification method [9] and the total variation method [13]. Some of the regularization methods for computing the first-order numerical differentiation have been applied to detecting image edges (see [10,13]).…”