We consider compressive imaging problems, where images are reconstructed from a reduced number of linear measurements. Our objective is to improve over existing compressive imaging algorithms in terms of both reconstruction error and runtime. To pursue our objective, we propose compressive imaging algorithms that employ the approximate message passing (AMP) framework. AMP is an iterative signal reconstruction algorithm that performs scalar denoising at each iteration; in order for AMP to reconstruct the original input signal well, a good denoiser must be used. We apply two wavelet-based image denoisers within AMP. The first denoiser is the "amplitude-scale-invariant Bayes estimator" (ABE), and the second is an adaptive Wiener filter; we call our AMP-based algorithms for compressive imaging AMP-ABE and AMP-Wiener. Numerical results show that both AMP-ABE and AMP-Wiener significantly improve over the state of the art in terms of runtime. In terms of reconstruction quality, AMP-Wiener offers lower mean-square error (MSE) than existing compressive imaging algorithms. In contrast, AMP-ABE has higher MSE, because ABE does not denoise as well as the adaptive Wiener filter.Index Terms-Approximate message passing, compressive imaging, image denoising, wavelet transform. 1053-587X

Compressed sensing typically deals with the estimation of a system input from its noise-corrupted linear measurements, where the number of measurements is smaller than the number of input components. The performance of the estimation process is usually quantified by some standard error metric such as squared error or support set error. In this correspondence, we consider a noisy compressed sensing problem with any additive error metric. Under the assumption that the relaxed belief propagation method matches Tanaka's fixed point equation, we propose a general algorithm that estimates the original signal by minimizing the additive error metric defined by the user. The algorithm is a pointwise estimation process, and thus simple and fast. We verify that our algorithm is asymptotically optimal, and we describe a general method to compute the fundamental information-theoretic performance limit for any additive error metric. We provide several example metrics, and give the theoretical performance limits for these cases. Experimental results show that our algorithm outperforms methods such as relaxed belief propagation (relaxed BP) and compressive sampling matching pursuit (CoSaMP), and reaches the suggested theoretical limits for our example metrics.

We consider a compressive hyperspectral imaging reconstruction problem, where three-dimensional spatio-spectral information about a scene is sensed by a coded aperture snapshot spectral imager (CASSI). The CASSI imaging process can be modeled as suppressing three-dimensional coded and shifted voxels and projecting these onto a two-dimensional plane, such that the number of acquired measurements is greatly reduced. On the other hand, because the measurements are highly compressive, the reconstruction process becomes challenging. We previously proposed a compressive imaging reconstruction algorithm that is applied to two-dimensional images based on the approximate message passing (AMP) framework. AMP is an iterative algorithm that can be used in signal and image reconstruction by performing denoising at each iteration. We employed an adaptive Wiener filter as the image denoiser, and called our algorithm "AMP-Wiener." In this paper, we extend AMP-Wiener to three-dimensional hyperspectral image reconstruction, and call it "AMP-3D-Wiener." Applying the AMP framework to the CASSI system is challenging, because the matrix that models the CASSI system is highly sparse, and such a matrix is not suitable to AMP and makes it difficult for AMP to converge. Therefore, we modify the adaptive Wiener filter and employ a technique called damping to solve for the divergence issue of AMP. Our approach is applied in nature, and the numerical experiments show that AMP-3D-Wiener outperforms existing widely-used algorithms such as gradient projection for sparse reconstruction (GPSR) and two-step iterative shrinkage/thresholding (TwIST) given a similar amount of runtime. Moreover, in contrast to GPSR and TwIST, AMP-3D-Wiener need not tune any parameters, which simplifies the reconstruction process.Comment: 13 pages, to appear in Journal of Selected Topics in Signal Processin