Non-linear Local Polynomial Regression Multiresolution Methods Using \(\ell ^1\)-norm Minimization with Application to Signal Processing
Harten’s Multiresolution has been developed and used for different applications such as fast algorithms for solving linear equations or compression, denoising and inpainting signals. These schemes are based on two principal operators: decimation and prediction. The goal of this paper is to construct an accurate prediction operator that approximates the real values of the signal by a polynomial and estimates the error using \(\ell ^1\)-norm in each point. The result is a non-linear multiresolution method. The order of the operator is calculated. The stability of the schemes is ensured by using a special error control technique. Some numerical tests are performed comparing the new method with known linear and non-linear methods.
KeywordsStatistical Multiresolution Multiscale decomposition Signal processing Generalized wavelets
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