Compressed sensing is an emerging approach for signal acquisition wherein theory has shown that a small number of linear, random projection of a signal contains enough information for reconstruction of the signal. Despite its potential to enable lightweight and inexpensive sensing hardware that simultaneously combines signal acquisition and dimensionality reduction, the compressed sensing of images and video still entails several challenges, in particular, a sensing-measurement operator which is difficult to apply in practice due to the heavy memory and computational burdens. Block-based random image sampling coupled with a projection-driven compressed-sensing recovery is proposed to address this challenge. For images, the block-based image acquisition is coupled with reconstruction driven by a directional transform that encourages spatial sparsity. Specifically, both contourlets as well as complex-valued dual-tree wavelets are considered for their highly directional representation, while bivariate shrinkage is adapted to their multiscale decomposition structure to provide the requisite sparsity constraint. Smoothing is achieved via a Wiener filter incorporated into iterative projected Landweber compressed-sensing recovery, yielding fast reconstruction. Also considered is an extension of the basic reconstruction algorithm that incorporates block-based measurements in the domain of a wavelet transform. The pro-posed image recovery algorithm and its extension yield images with quality that matches or exceeds that produced by a popular, yet computationally expensive, technique which minimizes total variation. Additionally, reconstruction quality is substantially superior to that from several prominent pursuits-based algorithms that do not include any smoothing. For video, motion estimation and compensation is utilized to promote temporal sparsity. Because video sequences have temporal redundancy in locations in which objects are moving while the background is still, a residual between the current frame and the previous frame compensated by object motion is shown to be more sparse than the orig-inal frame itself. By using residual reconstruction, information contained in the previous frame contributes to the reconstruction of the current frame. The proposed block-based compressed-sensing reconstruction for video outperforms a simple frame-byrame reconstruction as well as a 3D volumetric reconstruction in terms of visual quality. Finally, quantization of block-based compressed-sensing measurements is considered in order to generate a true bitstream from a compressed-sensing image acquisition. Specifically, a straightforward process of quantization via simple uniform scalar quantization applied in conjunction with differential pulse code modulation of the block-based compressed-sensing measurements is proposed. Experimental results demonstrate significant improvement in rate-distortion performance as compared scalar quantization used alone in several block-based compressed-sensing reconstruction algorithms. Additionally, rate-distortion performance superior to that of alternative quantized-compressed-sensing techniques relying on optimized quantization or reconstruction is observed.

The compressed sensing (CS) model of signal processing, while offering many unique advantages in terms of low-cost sensor design, poses interesting challenges for both signal acquisition and recovery, especially for signals of large size. In this work, we investigate how CS might be applied practically and efficiently in the context of natural video. We make use of a CS video acquisition approach in line with the popular single-pixel camera framework of blind, nonaptive, random sampling while proposing new approaches for the subsequent recovery of the video signal which leverage interrame redundancy to minimize recovery error. We introduce a method of approximation, which we term multihypothesis (MH) frame prediction, to create accurate frame predictions by comparing hypotheses drawn from the spatial domain of chosen reference frames to the non-overlapping, block-by-block CS measurements of subsequent frames. We accomplish this frame prediction via a novel distance-weighted Tikhonov regularization technique. We verify through our experiments that MH frame prediction via distance-weighted regularization provides state-of-the-art performance for the recovery of natural video sequences from blind CS measurements. The distance-weighted regularization we propose need not be limited to just frame prediction for CS video recovery, but may also be used in a variety of contexts where approximations must be generated from a set of hypotheses or training data. To show this, we apply our technique to supervised hyperspectral image (HSI) classification via a novel classifier we term the nearest regularized subspace (NRS) classifier. We show that the distance-weighted regularization used in the NRS method provides greater classification accuracy than state-of-the-art classifiers for supervised HSI classification tasks. We also propose two modifications to the core NRS classifier to improve its robustness to variation of input parameters and and to further increase its classification accuracy.