Improved total variation-type regularization using higher order edge detectors

We present a novel deconvolution approach for accurately restoring piecewise smooth signals from blurred data. There are two separate stages. The first stage uses higher order total variation (TV) restorations to obtain an estimate of the location of jump discontinuities from the blurred data. In the second stage the estimated jump locations are used to determine the local orders of a variable order TV restoration. The method replaces the first order derivative approximation used in standard TV by a variable order derivative operator. Smooth segments as well as jump discontinuities are restored, while the staircase effect typical for standard first order TV regularization is avoided. Compared to first order TV, signal restorations are more accurate representations of the true signal, as measured in a relative l2-norm. The method can also be used to obtain an accurate estimation of the locations and sizes of the true jump discontinuities. The approach is independent of the algorithm used for the standard TV problem and is, consequently, readily incorporated into existing TV restoration codes.