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