CS: Rice entries, recovery of sparse signals, inverse scattering using S-OMP, CS by inverse scale space and curvelet thresholding

Here some new papers from the Compressive Sensing Resources that I have not covered before:

A introductory lecture note by Richard Baraniuk entitled Compressive sensing and an introduction to compressive sampling by Emmanuel Candès and Michael Wakin that was featured in the IEEE magazine on compressive sensing.

The paper formerly known as "The Johnson-Lindenstrauss lemma meets compressed sensing" by Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin now has the title: A simple proof of the restricted isometry property for random matrices.

Emmanuel Candès, The restricted isometry property and its implications for compressed sensing. The abstract reads:

It is now well-known that one can reconstruct sparse or compressible signals accurately from a very limited number of measurements, possibly contaminated with noise. This technique known as "compressed sensing" or "compressive sampling" relies on properties of the sensing matrix such as the restricted isometry property. In this note, we establish new results about the accuracy of the reconstruction from undersampled measurements which improve on earlier estimates, and have the advantage of being more elegant.

T. Tony Cai, Guangwu Xu, and Jun Zhang, On recovery of sparse signals via ell-1 minimization.

This article considers constrained ℓ1 minimization methods for the recovery of high dimensional sparse signals in three settings: noiseless, bounded error and Gaussian noise. A unified and elementary treatment is given in these noise settings for two ℓ1 minimization methods: the Dantzig selector and ℓ1 minimization with an ℓ2 constraint. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg and Tao (2006) and Donoho, Elad, and Temlyakov (2006) are extended.

Here is an important paper for the solving of the Helmholtz equation by Jong Chul Ye and Su Yeon Lee entitled: Non-iterative exact inverse scattering using simultanous orthogonal matching pursuit (S-OMP). The abstract reads:

Even though recently proposed time-reversal MUSIC approach for inverse scattering problem is non-iterative and exact, the approach breaks down when there are more targets than sensors. The main contribution of this paper is a novel non-iterative exact inverse scattering algorithm that still guarantees the exact recovery of the extended targets under a very relaxed constraint on the number of source and receivers, where the conventional time-reversal MUSIC fails. Such breakthrough was possible from the observation that the induced currents on the unknown targets assume the same sparse support, which can be recovered accurately using the simultaneous orthogonal matching pursuit developed for multiple measurement vector problems. Simulation results demonstrate that perfect reconstruction can be quickly obtained from a very limited number of samples.

The conclusion states:

This paper derived a novel non-iterative exact inverse scattering algorithm using simultaneous OMP based on the observation that the induced current on the unknown targets assumes the same sparse support. Using the theoretical and numerical analysis, we showed that the maximum number of recoverable targets using the new method is upper bounded by the average of the number of transmitters and detectors. This is a significantly improvement over the conventional timereversal MUSIC especially when the number of source or detectors are significantly smaller than its counterpart. We believe that our results fill the missing gap toward the complete theory for the non-iterative exact inverse scattering theory.

Jianwei Ma, Compressed sensing by inverse scale space and curvelet thresholding.

Compressed sensing provides a new sampling theory for data acquisition, which says that compressible signals can be exactly reconstructed from highly incomplete sets of linear measurements. It is significant to many applications, e.g., medical imaging and remote sensing, especially for measurements limited by physical and physiological constraints, or extremely expensive. In this paper we proposed a combinatorial recovery algorithm from a view of reaction-diffusion equations, by applying curvelet thresholding in inverse scale space flows. Numerical experiments in medical CT and aerospace remote sensing show its good performances for recovery of detailed features from incomplete and inaccurate measurements, in comparison with some existing methods.

It may sound like a detail but I noted the comprehensive list of reconstruction techniques that gives some context into the newer method proposed in the paper:

On the other hand, a few recover algorithms have been proposed in last couple of years. Typically, e.g., LP [7], reweighed LP [11], GRSP [23], Greed/OMP [42], StOMP [20], and iterative thresholding (IT) [4, 17, 32{34, 37]. Here we concern the nonlinear recovery algorithm from a new way: partial differential equation (PDE) method

I am looking forward to seeing the code that implements this algorithm.

The papers I had already featured are listed below:

• Rayan Saab, Rick Chartrand, and Özgür Yulmaz, Stable sparse approximation via nonconvex optimization.
• Shuchin Aeron, Manqi Zhao, and Venkatesh Saligrama, Fundamental limits on sensing capacity for sensor networks and compressed sensing.
• Hong Jung, Kyunghyun Sung, Krishna S. Nayak, Eung Yeop Kim, and Jong Chul Ye, k-t FOCUSS: A general compressed sensing framework for high resolution dynamic MRI.

Credit: NASA/JPL-Caltech/University of Arizona/Texas A&M, Phoenix view on Sol 28.