A Regularization Method for one Dimensional Edge-Dectection and Edge-Preserving Smoothing

We consider the problem of detecting discontinuities and estimating an unknown discontinuous function from noisy data. This is an ill-posed inverse problem, which needs to be regularized beyond the conventional dilemma between the fidelity to the data and the degree of the global smoothness which now doesn't exist. In this paper, we introduce a regularization functional having two items. The first is a measure of piecewise-smoothness of the function while the second is penalized by the local components: locations, sizes, and degrees of the discontinuities, and is also controlled by the global Components: the number of discontinuity points and the degree of piecewise-smoothness. We develop a methodology for the problem of edge-preserving smoothing and edge-detection. Two algorithms are proposed and the simulations were run for several one-dimensional synthetic images. We assess the results in the light of some performance criteria described by Canny (1986).