From template matching to optimal approximation by piecewise smooth curves

Smoothing spline preserving discontinuities, are defined by standard energy minimization problems except that discontinuities are allowed at some fixed breakpoints. A natural approach to locate discontinuities is to further minimize the spline energy also with respect to the breakpoints. Such approaches have been much studied in computer vision (Blake and Zisserman, 1987). We show that, in the case ofn equally spaced data points and large n, such a highly nonconvex minimization problem has strong connections with the usual template matching techniques, and that it can be exactly solved by 0(n) direct algorithms (even for not equally spaced abscissae) provided all the breakpoints are distant enough compared to the smoothing scale. Otherwise a few (Gaus-Seidel type) iterations, based on the previous algorithms, are sufficient in many cases.