Force curve segmentation by piecewise polynomial approximation: mathematical formulation and complete structure of the algorithm

In this technical report, we give a detailed formulation of the force curve segmentation problem based on the fitting of the curve by a piecewise polynomial. The segmentation algorithm is a discrete search in which the unknowns are the positions Dj, j=1,...,k of the discontinuity points. We first formulate the problem as the minimization of a least-square cost function with respect to D=[D1,...,Dk]\in R^k. Then, we describe the structure of the proposed optimization algorithm including the update of the list of discontinuity positions D when their number k increases.