M-RWTL: Learning Signal-Matched Rational Wavelet Transform in Lifting Framework

Transform learning is being extensively applied in several applications because of its ability to adapt to a class of the signals of interest. Often, a transform is learned using a large amount of training data, while only limited data may be available in many applications. Motivated with this, we propose wavelet transform learning in the lifting framework for a given signal. Significant contributions of this paper are: 1) the existing theory of lifting framework for the dyadic wavelet is extended to more generic rational wavelet design, where dyadic is a special case and 2) the proposed work allows to learn rational wavelet transform from a given signal and does not require large training data. Since it is a signal-matched design, the proposed methodology is called Signal-Matched Rational Wavelet Transform Learning in the Lifting Framework (M-RWTL). The proposed M-RWTL method inherits all the advantages of lifting, i.e., the learned rational wavelet transform is always invertible, method is modular, and the corresponding M-RWTL system can also incorporate non-linear filters, if required. This may enhance the use of RWT in applications, which is so far restricted. M-RWTL is observed to perform better compared with the standard wavelet transforms in the applications of compressed sensing-based signal reconstruction.