Improved Total Variation-Type Regularization Using Higher Order Edge Detectors

We present a novel deconvolution approach to accurately restore piecewise smooth signals from blurred data. The first stage uses Higher Order Total Variation 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 Total Variation restoration. The method replaces the first order derivative approximation used in standard Total Variation by a variable order derivative operator. Smooth segments as well as jump discontinuities are restored while the staircase effect typical for standard first order Total Variation regularization is avoided. As compared to first order Total Variation, 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 Total Variation problem and is, consequently, readily incorporated in existing Total Variation restoration codes.