Optimal Nonlinear Signal Approximations Based on Piecewise Constant Functions

We provide here an optimal method of approximating a signal by piecewise constant functions. To this end, we minimize over the signal subdomains a fidelity term between the signal and its corresponding piecewise approximations; subdomains being determined by the number of approximations samples used for. An optimal recursive relationship is then obtained and proven, which helps us to derive the proposed approximation algorithm. The complexity of the algorithm is O( $$\hbox {MN}^2$$ MN 2 ), where N is the number of samples of the processed signal and M is the number of piecewise constant approximation functions. There are different techniques to approximate a signal using piecewise constant functions, wavelet decomposition is one of them by means of a Haar wavelet. Our approach is then compared to linear and nonlinear wavelet-based approximations, and both qualitative and quantitative results are provided on various tested signals, showing the efficiency of the proposed approach.

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