De-bursting

Most signals are smooth, but running medians assume they have no curvature. An alternate expression of this assumption is that the signal has minimal curvature $0 \approx h_{i+1} -2 h_{i} + h_{i-1}$;in other words, $\bold 0 \approx \nabla^2 \bold h$.Thus we propose to create the cleaned-up data $\bold h$from the observed data $\bold d$ with the fitting problem

$$\begin{array} {lll} 0 &\approx & \bold W (\bold h - \bold d) \\ 0 &\approx & \epsilon\ \nabla^2 \bold h \end{array}$$ (18)

where $\bold W$ is a diagonal matrix with weights sprinkled along the diagonal, and where $\nabla^2$ is a matrix with a roughener like (1,-2,1) distributed along the diagonal. This is shown in Figure 4 with $\epsilon = 1$.Experience showed similar performances for $0 \approx \nabla \bold h$ and $0 \approx \nabla^2 \bold h$.Better results, however, will be found later in Figure 6, where the $\nabla^2$ operator is replaced by an operator designed to predict this very predictable signal.