Source localization by matched-field processing (MFP) generally involves solving a number of computationally intensive partial differential equations. This paper introduces a technique that mitigates this computational workload by “compressing” these computations. Drawing on key concepts from the recently developed field of compressed sensing, it shows how a low-dimensional proxy for the Green’s function can be constructed by backpropagating a small set of random receiver vectors. Then the source can be located by performing a number of “short” correlations between this proxy and the projection of the recorded acoustic data in the compressed space. Numerical experiments in a Pekeris ocean waveguide are presented that demonstrate that this compressed version of MFP is as effective as traditional MFP even when the compression is significant. The results are particularly promising in the broadband regime where using as few as two random backpropagations per frequency performs almost as well as the traditiona...
In this paper, we effectively demonstrate how classical localization procedures under a least-squares framework such as matched-field processing (MFP) may be solved in a reduced-dimensional space even without a-priori knowledge of the best'' dimension-reducing transform. This property has been shown in similar forms in the mainstream canon of Compressed Sensing (CS) literature. have described a number of useful variations on the theme of CS @cite_8 including a matched filtering detector. They have also described the smashed filter'' that is designed primarily for classification between a finite number of sets, but could easily be extended to parametric estimation. Wakin has also established some rigorous results on parameter estimation that relate the recovery properties of a general compressive estimation problem to the properties of the manifold that these parameters induce. This work could be used to analyze this problem via its manifold parameters @cite_14 .
@cite_14 @cite_8
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We study the notion of compressed sensing (CS) as put forward by Donoho, Candes, Tao and others. The notion proposes a signal or image, unknown but supposed to be compressible by a known transform, (e.g. wavelet or Fourier), can be subjected to fewer measurements than the nominal number of data points, and yet be accurately reconstructed. The samples are nonadaptive and measure 'random' linear combinations of the transform coefficients. Approximate reconstruction is obtained by solving for the transform coefficients consistent with measured data and having the smallest possible l1 norm.We present initial 'proof-of-concept' examples in the favorable case where the vast majority of the transform coefficients are zero. We continue with a series of numerical experiments, for the setting of lp-sparsity, in which the object has all coefficients nonzero, but the coefficients obey an lp bound, for some p ∈ (0, 1]. The reconstruction errors obey the inequalities paralleling the theory, seemingly with well-behaved constants.We report that several workable families of 'random' linear combinations all behave equivalently, including random spherical, random signs, partial Fourier and partial Hadamard.We next consider how these ideas can be used to model problems in spectroscopy and image processing, and in synthetic examples see that the reconstructions from CS are often visually "noisy". To suppress this noise we postprocess using translation-invariant denoising, and find the visual appearance considerably improved.We also consider a multiscale deployment of compressed sensing, in which various scales are segregated and CS applied separately to each; this gives much better quality reconstructions than a literal deployment of the CS methodology.These results show that, when appropriately deployed in a favorable setting, the CS framework is able to save significantly over traditional sampling, and there are many useful extensions of the basic idea. Compressive Sensing (CS) combines sampling and compression into a single sub-Nyquist linear measurement process for sparse and compressible signals. In this paper, we extend the theory of CS to include signals that are concisely represented in terms of a graphical model. In particular, we use Markov Random Fields (MRFs) to represent sparse signals whose nonzero coefficients are clustered. Our new model-based recovery algorithm, dubbed Lattice Matching Pursuit (LaMP), stably recovers MRF-modeled signals using many fewer measurements and computations than the current state-of-the-art algorithms. The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible image or signal from a small set of linear, non-adaptive (even random) projections. However, in many applications, including object and target recognition, we are ultimately interested in making a decision about an image rather than computing a reconstruction. We propose here a framework for compressive classification that operates directly on the compressive measurements without first reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the compressive domain; we find that the number of measurements required for a given classification performance level does not depend on the sparsity or compressibility of the images but only on the noise level. The second part of the theory applies the generalized maximum likelihood method to deal with unknown transformations such as the translation, scale, or viewing angle of a target object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear manifold in the high-dimensional image space. We find that the number of measurements required for a given classification performance level grows linearly in the dimensionality of the manifold but only logarithmically in the number of pixels samples and image classes. Using both simulations and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness of the smashed filter for target classification using very few measurements. Compressed sensing (CS) utilizes the sparsity of magnetic resonance (MR) images to enable accurate reconstruction from undersampled k-space data. Recent CS methods have employed analytical sparsifying transforms such as wavelets, curvelets, and finite differences. In this paper, we propose a novel framework for adaptively learning the sparsifying transform (dictionary), and reconstructing the image simultaneously from highly undersampled k-space data. The sparsity in this framework is enforced on overlapping image patches emphasizing local structure. Moreover, the dictionary is adapted to the particular image instance thereby favoring better sparsities and consequently much higher undersampling rates. The proposed alternating reconstruction algorithm learns the sparsifying dictionary, and uses it to remove aliasing and noise in one step, and subsequently restores and fills-in the k-space data in the other step. Numerical experiments are conducted on MR images and on real MR data of several anatomies with a variety of sampling schemes. The results demonstrate dramatic improvements on the order of 4-18 dB in reconstruction error and doubling of the acceptable undersampling factor using the proposed adaptive dictionary as compared to previous CS methods. These improvements persist over a wide range of practical data signal-to-noise ratios, without any parameter tuning.