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.
Many compressive imaging algorithms have been proposed in the literature. For example, Som and Schniter @cite_37 modeled the structure of the wavelet coefficients by a hidden Markov tree (HMT), and applied a turbo scheme that alternates between inference on the HMT structure with standard belief propagation and inference on the compressed sensing measurement structure with the generalized approximate message passing algorithm. He and Carin @cite_20 proposed a hierarchical Bayesian approach with Markov chain Monte Carlo (MCMC) for natural image reconstruction. Soni and Haupt @cite_24 exploited a hierarchical dictionary learning method @cite_34 and assumed that projecting images onto the learned dictionary will yield tree-sparsity, and therefore the nonzero supports of the dictionary can be identified and estimated accurately by setting an appropriate threshold.
@cite_24 @cite_37 @cite_34 @cite_20
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Breakthrough results in compressive sensing (CS) have shown that high dimensional signals (vectors) can often be accurately recovered from a relatively small number of non-adaptive linear projection observations, provided that they possess a sparse representation in some basis. Subsequent efforts have established that the reconstruction performance of CS can be improved by employing additional prior signal knowledge, such as dependency in the location of the non-zero signal coefficients (structured sparsity) or by collecting measurements sequentially and adaptively, in order to focus measurements into the proper subspace where the unknown signal resides. In this paper, we examine a powerful hybrid of adaptivity and structure. We identify a particular form of structured sparsity that is amenable to adaptive sensing, and using concepts from sparse hierarchical dictionary learning we demonstrate that sparsifying dictionaries exhibiting the appropriate form of structured sparsity can be learned from a collection of training data. The combination of these techniques (structured dictionary learning and adaptive sensing) results in an effective and efficient adaptive compressive acquisition approach which we refer to as LASeR (Learning Adaptive Sensing Representations.) We propose a novel algorithm for compressive imaging that exploits both the sparsity and persistence across scales found in the 2D wavelet transform coefficients of natural images. Like other recent works, we model wavelet structure using a hidden Markov tree (HMT) but, unlike other works, ours is based on loopy belief propagation (LBP). For LBP, we adopt a recently proposed “turbo” message passing schedule that alternates between exploitation of HMT structure and exploitation of compressive-measurement structure. For the latter, we leverage Donoho, Maleki, and Montanari's recently proposed approximate message passing (AMP) algorithm. Experiments with a large image database suggest that, relative to existing schemes, our turbo LBP approach yields state-of-the-art reconstruction performance with substantial reduction in complexity. We propose to combine two approaches for modeling data admitting sparse representations: on the one hand, dictionary learning has proven effective for various signal processing tasks. On the other hand, recent work on structured sparsity provides a natural framework for modeling dependencies between dictionary elements. We thus consider a tree-structured sparse regularization to learn dictionaries embedded in a hierarchy. The involved proximal operator is computable exactly via a primal-dual method, allowing the use of accelerated gradient techniques. Experiments show that for natural image patches, learned dictionary elements organize themselves in such a hierarchical structure, leading to an improved performance for restoration tasks. When applied to text documents, our method learns hierarchies of topics, thus providing a competitive alternative to probabilistic topic models. Bayesian compressive sensing (CS) is considered for signals and images that are sparse in a wavelet basis. The statistical structure of the wavelet coefficients is exploited explicitly in the proposed model, and, therefore, this framework goes beyond simply assuming that the data are compressible in a wavelet basis. The structure exploited within the wavelet coefficients is consistent with that used in wavelet-based compression algorithms. A hierarchical Bayesian model is constituted, with efficient inference via Markov chain Monte Carlo (MCMC) sampling. The algorithm is fully developed and demonstrated using several natural images, with performance comparisons to many state-of-the-art compressive-sensing inversion algorithms.