Naushad Ansari

Naushad Ansari, Anubha Gupta

Transform learning is being extensively applied in several applications because of its ability to adapt to a class of 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 work are: 1) the existing theory of lifting framework of 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 nonlinear 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 to standard wavelet transforms in the applications of compressed sensing based signal reconstruction.

Naushad Ansari, Anubha Gupta

Motivated with the concept of transform learning and the utility of rational wavelet transform in audio and speech processing, this paper proposes Rational Wavelet Transform Learning in Statistical sense (RWLS) for natural images. The proposed RWLS design is carried out via lifting framework and is shown to have a closed form solution. The efficacy of the learned transform is demonstrated in the application of compressed sensing (CS) based reconstruction. The learned RWLS is observed to perform better than the existing standard dyadic wavelet transforms.

Naushad Ansari, Anubha Gupta

This paper proposes a joint framework wherein lifting-based, separable, image-matched wavelets are estimated from compressively sensed (CS) images and used for the reconstruction of the same. Matched wavelet can be easily designed if full image is available. Also matched wavelet may provide better reconstruction results in CS application compared to standard wavelet sparsifying basis. Since in CS application, we have compressively sensed image instead of full image, existing methods of designing matched wavelet cannot be used. Thus, we propose a joint framework that estimates matched wavelet from the compressively sensed images and also reconstructs full images. This paper has three significant contributions. First, lifting-based, image-matched separable wavelet is designed from compressively sensed images and is also used to reconstruct the same. Second, a simple sensing matrix is employed to sample data at sub-Nyquist rate such that sensing and reconstruction time is reduced considerably without any noticeable degradation in the reconstruction performance. Third, a new multi-level L-Pyramid wavelet decomposition strategy is provided for separable wavelet implementation on images that leads to improved reconstruction performance. Compared to CS-based reconstruction using standard wavelets with Gaussian sensing matrix and with existing wavelet decomposition strategy, the proposed methodology provides faster and better image reconstruction in compressive sensing application.

Naushad Ansari, Anubha Gupta, Rahul Duggal

Image-matched nonseparable wavelets can find potential use in many applications including image classification, segmen- tation, compressive sensing, etc. This paper proposes a novel design methodology that utilizes convolutional neural net- work (CNN) to design two-channel non-separable wavelet matched to a given image. The design is proposed on quin- cunx lattice. The loss function of the convolutional neural network is setup with total squared error between the given input image to CNN and the reconstructed image at the output of CNN, leading to perfect reconstruction at the end of train- ing. Simulation results have been shown on some standard images.