L1-Based decomposition and reconstruction algorithms and W-matrices

Mallatʹs decomposition and reconstruction algorithms are very important in the field of wavelet theory and its application to signal processing. Wavelet theory is based on L2(R) space and the classical mean square error is employed naturally in many relevant applications. In the recent years, it is understood that the L2 space is not always the best one for all applications. Therefore, wavelet theory and its approximation properties were also studied in L1(R) by many researchers. The orthogonality was also developed in L1 space in our previous work. In this paper, Based on our previous work on L1 orthogonality, two novel decomposition and reconstruction algorithms, called MAE and ETO algorithms, are discussed in detail. The exact reconstruction algorithms are also established by extending the concept of W-matrices. Experiments are conducted to support these new algorithms.