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 $l^2$-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.

The concentration edge -detection and Gegenbauer image-reconstruction methods were previously shown to improve the quality of segmentation in magnetic resonance imaging. In this study, these methods are utilized as a pre-processing step to the Weibull E-SD field segmentation. It is demonstrated that the combination of the concentration edge detection and Gegenbauer reconstruction method improves the accuracy of segmentation for the simulated test data and real magnetic resonance images used in this study.

Gibbs ringing is a well known artifact that effects reconstruction of images having discontinuities. This is a problem in the reconstruction of magnetic resonance imaging (MRI) data due to the many different tissues normally present in each scan. The Gibbs ringing artifact manifests itself at the boundaries of the tissues, making it difficult to determine the structure of the brain tissue. The Gegenbauer reconstruction method has been shown to effectively eliminate the effects of Gibbs ringing in other applications. This paper presents the application of the Gegenbauer reconstruction method to neuro-imaging.