  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.
