Image Reconstruction using Matched Wavelet Estimated from Data Sensed Compressively using Partial Canonical Identity Matrix

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.