We investigate the problem of reconstructing signals from a subsampled convolution of their modulated versions and a known filter. The problem is studied as applies to a specific imaging architecture that relies on spatial phase modulation by randomly coded “masks.” The diversity induced by the random masks is deemed to improve the conditioning of the deconvolution problem while maintaining sampling efficiency. We analyze a linear model of the imaging system, where the joint effect of the spatial modulation, blurring, and spatial subsampling is represented concisely by a measurement matrix. We provide a bound on the conditioning of this measurement matrix in terms of the number of masks @math , the dimension (i.e., the pixel count) of the scene image @math , and certain characteristics of the blurring kernel and subsampling operator. The derived bound shows that the stable deconvolution is possible with high probability even if the number of masks (i.e., @math ) is as small as @math , meaning that the total number of (scalar) measurements is within a logarithmic factor of the image size. Furthermore, beyond a critical number of masks determined by the extent of blurring and subsampling, use of every additional mask improves the conditioning of the measurement matrix. We also consider a more interesting scenario where the target image is known to be sparse. We show that under mild conditions on the blurring kernel, with high probability the measurement matrix is a restricted isometry when the number of masks is within a logarithmic factor of the sparsity of the scene image. Therefore, the scene image can be reconstructed using any of the well-known sparse recovery algorithms such as the basis pursuit. The bound on the required number of masks grows linearly in sparsity of the scene image but logarithmically in its ambient dimension. The bound provides a quantitative view of the effect of the blurring and subsampling on the required number of masks, which is critical for designing efficient imaging systems.
 We show that, in contrast to popular belief, sub-wavelength information can be recovered from the far-field of an optical image, thereby overcoming the loss of information embedded in decaying evanescent waves. The only requirement is that the image is known to be sparse, a specific but very general and wide-spread property of signals which occur almost everywhere in nature. The reconstruction method relies on newly-developed compressed sensing techniques, which we adapt to optical super-resolution and sub-wavelength imaging. Our approach exhibits robustness to noise and imperfections. We provide an experimental proof-of-principle by demonstrating image recovery at a spatial resolution 5-times higher than the finest resolution defined by a spatial filter. The technique is general, and can be extended beyond optical microscopy, for example, to atomic force microscopes, scanning-tunneling microscopes, and other imaging systems. Imaging spectroscopy involves the sensing of a large amount of spatial information across a multitude of wavelengths. Conventional approaches to hyperspectral sensing scan adjacent zones of the underlying spectral scene and merge the results to construct a spectral data cube. Push broom spectral imaging sensors, for instance, capture a spectral cube with one focal plane array (FPA) measurement per spatial line of the scene [1], [2]. Spectrometers based on optical bandpass filters sequentially scan the scene by tuning the bandpass filters in steps. The disadvantage of these techniques is that they require scanning a number of zones linearly in proportion to the desired spatial and spectral resolution. This article surveys compressive coded aperture spectral imagers, also known as coded aperture snapshot spectral imagers (CASSI) [1], [3], [4], which naturally embody the principles of compressive sensing (CS) [5], [6]. The remarkable advantage of CASSI is that the entire data cube is sensed with just a few FPA measurements and, in some cases, with as little as a single FPA shot. In this article, the authors present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a broader spectral range than conventional silicon-based cameras. The approach fuses a new camera architecture based on a digital micromirror device with the new mathematical theory and algorithms of compressive sampling. The emerging field of compressed sensing has potentially powerful implications for the design of optical imaging devices. In particular, compressed sensing theory suggests that one can recover a scene at a higher resolution than is dictated by the pitch of the focal plane array. This rather remarkable result comes with some important caveats however, especially when practical issues associated with physical implementation are taken into account. This tutorial discusses compressed sensing in the context of optical imaging devices, emphasizing the practical hurdles related to building such devices, and offering suggestions for overcoming these hurdles. Examples and analysis specifically related to infrared imaging highlight the challenges associated with large format focal plane arrays and how these challenges can be mitigated using compressed sensing ideas. The mathematical theory of compressed sensing (CS) asserts that one can acquire signals from measurements whose rate is much lower than the total bandwidth. Whereas the CS theory is now well developed, challenges concerning hardware implementations of CS-based acquisition devices—especially in optics—have only started being addressed. This paper presents an implementation of compressive sensing in fluorescence microscopy and its applications to biomedical imaging. Our CS microscope combines a dynamic structured wide-field illumination and a fast and sensitive single-point fluorescence detection to enable reconstructions of images of fluorescent beads, cells, and tissues with undersampling ratios (between the number of pixels and number of measurements) up to 32. We further demonstrate a hyperspectral mode and record images with 128 spectral channels and undersampling ratios up to 64, illustrating the potential benefits of CS acquisition for higher-dimensional signals, which typically exhibits extreme redundancy. Altogether, our results emphasize the interest of CS schemes for acquisition at a significantly reduced rate and point to some remaining challenges for CS fluorescence microscopy.