Detection of irregular points by regularization in numerical differentiation and application to edge detection

Numerical differentiation is a typical ill-posed problem which can be treated by the Tikhonov regularization. In this paper, we prove that the L2-norms of the second-order derivatives of the regularized solutions blow up in any small interval I where the exact solution is not in H2(I). This generalizes the previous results by Wang, Jia and Cheng (2002 Inverse Problems 18 1461–76) where the interval I is assumed to be the whole interval. One application for image edge detection is presented.