The class of L1-regularized optimization problems has received much attention recently because of the introduction of “compressed sensing,” which allows images and signals to be reconstructed from small amounts of data. Despite this recent attention, many L1-regularized problems still remain difficult to solve, or require techniques that are very problem-specific. In this paper, we show that Bregman iteration can be used to solve a wide variety of constrained optimization problems. Using this technique, we propose a “split Bregman” method, which can solve a very broad class of L1-regularized problems. We apply this technique to the Rudin-Osher-Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging.

/pdf/the-split-bregman-method-for-l1-regularized-problems-3qiniu5mpj.pdf

The Split Bregman Method for L1-Regularized Problems

Wideband analog signals push contemporary analog-to-digital conversion (ADC) systems to their performance limits. In many applications, however, sampling at the Nyquist rate is inefficient because the signals of interest contain only a small number of significant frequencies relative to the band limit, although the locations of the frequencies may not be known a priori. For this type of sparse signal, other sampling strategies are possible. This paper describes a new type of data acquisition system, called a random demodulator, that is constructed from robust, readily available components. Let K denote the total number of frequencies in the signal, and let W denote its band limit in hertz. Simulations suggest that the random demodulator requires just O(K log(W/K)) samples per second to stably reconstruct the signal. This sampling rate is exponentially lower than the Nyquist rate of W hertz. In contrast to Nyquist sampling, one must use nonlinear methods, such as convex programming, to recover the signal from the samples taken by the random demodulator. This paper provides a detailed theoretical analysis of the system's performance that supports the empirical observations.

/pdf/beyond-nyquist-efficient-sampling-of-sparse-bandlimited-4l4hsdu9v4.pdf

Beyond Nyquist: Efficient Sampling of Sparse Bandlimited Signals

This paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(slog s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.

Compressed Sensing Off the Grid

A stylized compressed sensing radar is proposed in which the time-frequency plane is discretized into an N times N grid. Assuming the number of targets K is small (i.e., K Lt N2), then we can transmit a sufficiently ldquoincoherentrdquo pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel-compressed sensing approach offers great potential for better resolution over classical radar.

pdf/high-resolution-radar-via-compressed-sensing-j2yobgkpg3.pdf

High-Resolution Radar via Compressed Sensing

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

/pdf/sensitivity-to-basis-mismatch-in-compressed-sensing-4f52mixbp7.pdf

Sensitivity to Basis Mismatch in Compressed Sensing

We introduce a new approach to radar imaging based on the concept of compressive sensing (CS). In CS, a low-dimensional, nonadaptive, linear projection is used to acquire an efficient representation of a compressible signal directly using just a few measurements. The signal is then reconstructed by solving an inverse problem either through a linear program or a greedy pursuit. We demonstrate that CS has the potential to make two significant improvements to radar systems: (i) eliminating the need for the pulse compression matched filter at the receiver, and (ii) reducing the required receiver analog-to-digital conversion bandwidth so that it need operate only at the radar reflectivity's potentially low "information rate" rather than at its potentially high Nyquist rate. These ideas could enable the design of new, simplified radar systems, shifting the emphasis from expensive receiver hardware to smart signal recovery algorithms.

/pdf/compressive-radar-imaging-47z5fx3bs3.pdf

Compressive Radar Imaging

Sensing and imaging systems are under increasing pressure to accommodate ever-larger and higher-dimensional data sets; ever-faster capture, sampling, and processing rates; ever-lower power consumption; ever-smaller form factor; and new sensing modalities. These needs have motivated the development of new approaches to signal acquisition and processing. We provide an introduction to the field of compressive sensing (CS), which has stimulated a rethinking of sensor and signal processing system design. In CS, analog signals are digitized and processed not via uniform sampling but via measurements using more general, even random, test functions. In contrast to conventional wisdom, the new theory asserts that one can combine “sub-Nyquist rate sampling” with large-scale optimization for efficient and accurate signal acquisition when the signal has a sparse structure. Particular topics addressed include signal sparsity, randomized sampling, optimization-based signal recovery, and perspectives on applica...

Compressive sensing: A new approach to seismic data acquisition

Time‚ÄìFrequency Analysis of Seismic Reflection Data

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Compressive Radar Imaging

Compressive sensing: A new approach to seismic data acquisition

Time‚ÄìFrequency Analysis of Seismic Reflection Data