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.
In recent years there has been an increasing interest in the application of CS in various imaging modalities including but not limited to holography @cite_18 , coded aperture spectral imaging @cite_3 , fluorescent microscopy @cite_16 , and sub-wavelength imaging @cite_14 . The CS-based imaging systems are particularly interesting in applications where the measurements are time-consuming or expensive. Furthermore, by exploiting the sparsity of the scene image, the CS imaging methods can operate at SNR regimes where conventional imaging methods may perform poorly. A survey of practical advantages and challenges of various CS imaging systems can be found in @cite_5 . The first CS imaging system was introduced as the single-pixel camera'' in @cite_2 where a single sensor integrates the randomly masked versions of the scene image for a few different masks. Effectively, the single-pixel camera measures the inner product of the scene image and the randomly generated masks. Using the fact that natural images are often (nearly) sparse in some basis, it is shown in @cite_2 that CS allows accurate image reconstruction in this single-pixel architecture.
@cite_18 @cite_14 @cite_3 @cite_2 @cite_5 @cite_16
2114213376 2079319869 2047042135 2056283746
The compressed sensing (CS) synthetic aperture radar (SAR) imaging scheme can use random undersampled data to reconstruct images of sparse or compressible targets. However, compared to Nyquist sampling, the cost of the CS imaging scheme is the long reconstruction time, particularly for the conventional reconstruction strategy, which reconstructs the whole scene in one process. It also needs a large memory to access the sensing matrix used for reconstruction. In this paper, a segmented reconstruction strategy for the CS SAR imaging scheme is proposed. The whole scene is split into a set of small subscenes, so that the reconstruction time can be reduced significantly. The proposed method also needs much less memory for computation than the conventional method. In this proposed method, the range profiles are reconstructed first, and then, the range profiles can be split into subpatches. Subscenes can be reconstructed by using the subpatch data, and the whole scene can be obtained by combining the reconstructed subscenes. Simulation and experimental results are shown to demonstrate the validity of the proposed method. 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 recent years, compressed sensing (CS) has been applied in the field of synthetic aperture radar (SAR) imaging and shows great potential. The existing models are, however, based on application of the sensing matrix acquired by the exact observation functions. As a result, the corresponding reconstruction algorithms are much more time consuming than traditional matched filter (MF)-based focusing methods, especially in high resolution and wide swath systems. In this paper, we formulate a new CS-SAR imaging model based on the use of the approximated SAR observation deducted from the inverse of focusing procedures. We incorporate CS and MF within an sparse regularization framework that is then solved by a fast iterative thresholding algorithm. The proposed model forms a new CS-SAR imaging method that can be applied to high-quality and high-resolution imaging under sub-Nyquist rate sampling, while saving the computational cost substantially both in time and memory. Simulations and real SAR data applications support that the proposed method can perform SAR imaging effectively and efficiently under Nyquist rate, especially for large scale applications. Abstract Compressive sensing (CS) theory makes it possible – under certain assumptions – to recover a signal or an image sampled below the Nyquist sampling limit. In medical ultrasound imaging, CS could allow lowering the amount of acquired data needed to reconstruct the echographic image. CS thus offers the perspective of speeding up echographic acquisitions and could have many applications, e.g. triplex acquisitions for CFM B-mode Doppler imaging, high-frame-rate echocardiography, 3D imaging using matrix probes, etc. The objective of this paper is to study the feasibility of CS for the reconstruction of channel RF data, i.e. the 2D set of raw RF lines gathered at the receive elements. Successful application of CS implies selecting a representation basis where the data to be reconstructed have a sparse expansion. Because they consist mainly in warped oscillatory patterns, channel RF data do not easily lend themselves to a sparse representation and thus represent a specific challenge. Within this perspective, we propose to perform and assess CS reconstruction of channel RF data using the recently introduced wave atoms [1] representation, which exhibit advantageous properties for sparsely representing such oscillatory patterns. Reconstructions obtained using wave atoms are compared with the reconstruction performed with two conventional representation bases, namely Fourier and Daubechies wavelets. The first experiment was conducted on simulated channel RF data acquired from a numerical cyst phantom. The quality of the reconstructions was quantified through the mean absolute error at varying subsampling rates by removing 50–90 of the original samples. The results obtained for channel RF data reconstruction yield error ranges of [0.6–3.0] × 10−2, [0.2–2.6] × 10−2, [0.1–1.5] × 10−2, for wavelets, Fourier and wave atoms respectively. The error ranges observed for the associated beamformed log-envelope images are [2.4–20.6] dB, [1.1–12.2] dB, and [0.5–8.8 dB] using wavelets, Fourier, and wave atoms, respectively. These results thus show the superiority of the wave atom representation and the feasibility of CS for the reconstruction of US RF data. The second experiment aimed at showing the experimental feasibility of the method proposed using a data set acquired by imaging a general-purpose phantom (CIRS Model 054GS) using an Ultrasonix MDP scanner. The reconstruction was performed by removing 80 of the initial samples and using wave atoms. The reconstructed image was found to reliably preserve the speckle structure and was associated with an error of 5.5 dB.