Spline-based deconvolution

This paper proposes robust algorithms to perform deconvolution and inversion of the heat equation starting from 1D and 2D discrete noised data. The solutions are provided as splines that minimize some parameterized quadratic functionals. Parameters choice determines the trade-off between the regularity of the solution and the approximation of the initial data. The solutions are derived in an explicit form that uses aversion of harmonic analysis in splines spaces. The presented algorithms are easily implemented in a fast way while providing stable approximate solutions that restore the exact solutions with high accuracy even when the initial data are severely corrupted by noise. The approximate solutions have the same smoothness as the exact solution. The convergence of the approximate solutions to the exact solutions is proved when the sampling rate and the signals-to-noise ratios increase.