One-dimensional Edge-preserving Spline Smoothing for Estimation of Piecewise Smooth Functions

Splines are piecewise polynomials and widely used for interpolation and smoothing of observed data, due to their flexibility and optimality in the sense of certain variational problems for one-dimensional (1D) data. However, spline interpolation and smoothing are applicable only to the estimation of continuous functions, and not suitable for that of piecewise smooth functions. In this paper, we propose a novel spline smoothing technique for the estimation of 1D piecewise smooth functions. We newly define the set of breaking splines, which are permitted to have several discontinuous knots. Then, we estimate a piecewise smooth function as a breaking spline minimizing the sum of the data fidelity term, the roughness penalty term, and the number of the discontinuous knots. Numerical experiments show the effectiveness of the breaking splines compared to the conventional splines and the state-of-the-art total generalized variation (TGV) denoising.