One-dimensional regularization with discontinuities

Regularization is equivalent to fitting smoothing splines to the data so that efficient and reliable numerical algorithms exist for finding solutions. however, the results exhibit poor performance along edges and boundaries. To cope with such anomalies, a more general class of smoothing splines that preserve corners and discontinuities is studied. Cubic splines are investigated in detail, since they are easy to implement and produce smooth curves near all data points except those marked as discontinuities or creases. A discrete regularization method is introduced to locate corners and discontinuities in the data points before the continuous regularization is applied