Gene Golub 1932-2007

Gene Golub has passed away ( http://genehgolub.blogspot.com )

Compressed Sensing: SpaRSA and Image Registration using Random Projections

Mário Figueiredo, Robert D. Nowak and Stephen Wright the authors of GPSR are coming up with a faster method for reconstructing solutions from Compressed Sensing measurements. The algorithm is presented in this preprint: Sparse Reconstruction by Separable Approximation the abstract reads:

Finding sparse approximate solutions to large underdetermined linear systems of equations is a common problem in signal/image processing and statistics. Basis pursuit, the least absolute shrinkage and selection operator (LASSO), wavelet-based deconvolution and reconstruction, and compressed sensing (CS) are a few well-known areas in which problems of this type appear. One standard approach is to minimize an objective function that includes a quadratic (ℓ2) error term added to a sparsity-inducing (usually ℓ1) regularizer. We present an algorithmic framework for the more general problem of minimizing the sum of a smooth convex function and a nonsmooth, possibly nonconvex, sparsity-inducing function. We propose iterative methods in which each step is an optimization subproblem involving a separable quadratic term (diagonal Hessian) plus the original sparsity-inducing term. Our approach is suitable for cases in which this subproblem can be solved much more rapidly than the original problem. In addition to solving the standard ℓ2 − ℓ1 case, our approach handles other problems, e.g., ℓp regularizers with p less than 1, or group-separable (GS) regularizers. Experiments with CS problems show that our approach provides state-of-the-art speed for the standard ℓ2 − ℓ1 problem, and is also efficient on problems with GS regularizers

The algorithm seems to promise a faster capability than GPSR_BB. Let us note the potential use of non-convex terms in the regularization term. Thank you Jort.



Another paper caught my attention: Fast Global Image Registration Using Random Projections by Dennis Healy, Jr. and Gustavo Rohd. The abstract reads:

We describe a new method for efficiently computing the global optimum of least squares registration problems based on the recently developed theory of signal processing using random projections. The method is especially attractive for large scale registration problems where the goal is to register many images to a standard template. We test our new algorithm using real images of cells’ nuclei and show our method can outperform more traditional dimension reduction methods such as projections onto lower dimensional B-spline function spaces.

This is a very nice application where image registration uses the work of Mike Wakin and Richard Baraniuk on Random Projection of Smooth manifolds. Nice.

On a totally unrelated note, on Monday, the Monday Morning Algorithm series will feature the algorithm presented here.

Compressed Sensing: Another deterministic code, a LASSO-CS root finder and a tree based greedy algorithm

It looks like I overlooked a fifth deterministic code for measurement in Compressed Sensing. It is featured in Annihilating filter-based decoding in the compressed sensing framework by Ali Hormati and Martin Vetterli
the abstract reads:

Recent results in compressed sensing or compressive sampling suggest that a relatively small set of measurements taken as the inner product with universal random measurement vectors can well represent a source that is sparse in some fixed basis. By adapting a deterministic, non-universal and structured sensing device, this paper presents results on using the annihilating filter to decode the information taken in this new compressed sensing environment. The information is the minimum amount of nonadaptive knowledge that makes it possible to go back to the original object. We will show that for a $k$-sparse signal of dimension $n$, the proposed decoder needs $2k$ measurements and its complexity is of $O(k^2)$ whereas for the decoding based on the $\ell_1$ minimization, the number of measurements needs to be of $O(k\log(n))$ and the complexity is of $O(n^3)$. In the case of noisy measurements, we first denoise the signal using an iterative algorithm that finds the closest rank $k$ and Toeplitz matrix to the measurements matrix (in Frobenius norm) before applying the annihilating filter method. Furthermore, for a $k$-sparse vector with known equal coefficients, we propose an algebraic decoder which needs only $k$ measurements for the signal reconstruction. Finally, we provide simulation results that demonstrate the performance of our algorithm.

From the same lab, but the document is private: Efficient and Stable Acoustic Tomography Using Sparse Reconstruction Methods by Ivana Jovanovic, Ali Hormati, Luciano Sbaiz and Martin Vetterli. The abstract reads:

We study an acoustic tomography problem and propose a new inversion technique based on sparsity. Acoustic tomography observes the parameters of the medium that influence the speed of sound propagation. In the human body, the parameters that mostly influence the sound speed are temperature and density, in the ocean - temperature and current, in the atmosphere - temperature and wind. In this study, we focus on estimating temperature in the atmosphere using the information on the average sound speed along the propagation path. The latter is practically obtained from travel time measurements. We propose a reconstruction algorithm that exploits the concept of sparsity. Namely, the temperature is assumed to be a linear combination of some functions (e.g. bases or set of different bases) where many of the coefficients are known to be zero. The goal is to find the non-zero coefficients. To this end, we apply an algorithm based on linear programming that under some constrains finds the solution with minimum l0 norm. This is actually equivalent to the fact that many of the unknown coefficients are zeros. Finally, we perform numerical simulations to assess the effectiveness of our approach. The simulation results confirm the applicability of the method and demonstrate high reconstruction quality and robustness to noise.

I also found the presentation of this paper from Ewout van den Berg and Michael Friedlander entitled " In pursuit of a root". The presentation is much clearer about what they are talking about ( I am a visual person). In essence they try to solve a Basis Pursuit problem by a Lasso and rely on the fact that the correspondence between the two problems is continuous and differentiable. Very nice.


Similarly, this presentation-talk by Minh Do entitled "Practical Compressed Sensing" puts a new light on the possible extension of compressed sensing. Namely the fact that not only most signals are sparse in a good basis of representation but most coefficients of importance lie in a tree across different scales



Minh Do has already published a scheme based on this observation called: Tree-Based Orthogonal Matching Pursuit Algorithm For Signal Reconstruction with Chinh La. I am curious on how the implementation will fare in comparison to other reconstruction techniques.

Photo Credit: JAXA, Japanese probe Kayuga transmitted this HDTV of an Earth rise above the Moon at 2:52 p.m. on November 7, 2007 (JST).The "Earth-rise" movie is here, and the "Earth-set" movie is here.

Sparsity: What is it good for ?

The solution of an underdetermined system of linear equations is generally too large (i.e. it generally is an entire subspace as opposed to a unique solution). When you have too many unknowns and not enough equations, experience tells you to wait for additional equations or constraints to obtain a potentially unique solution. In Compressed Sensing, sparsity of the solution is the element that provides an ability to put additional constraints in order to find a unique solution.

But sparsity can be also used for other purpose as shown by this excellent study by Jerome Bobin, Jean-Luc Starck, Jalal Fadili and Yassir Moudden in Sparsity and Morphological Diversity in Blind Source Separation where their approach "takes advantage of this “morphological diversity” to differentiate between the sources with accuracy". Evidently, the reconstruction techniques parallels the ones found in Compressed Sensing. The abstract reads:

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some new and essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper introduces a new BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.

The GMCALab matlab code for the implementation of this algorithm can be found here.

Monday Morning Algorithm Part deux: Reweighted Lp for Non-convex Compressed Sensing Reconstruction

Today's algorithm implements the algorithm found in Iteratively Reweighted Algorithms for Compressive Sensing by Rick Chartrand and Wotao Yin as mentioned in an earlier post. The cons for the algorithm is that it solves a non-convex problem and therefore there is no provable way of showing you achieved an optimal solution. The pros are:

• you don't need semi-definite programming
• you do better than the reweighted L1 technique and
• you do it faster
• all operations rely on simple linear algebra which can then be optimized for the problem at hand.
  

The algorithm should be usable on Matlab and Octave. If you want to contribute a small algorithm next week or any week, please let me know. So here is the beast:

clear
% Algorithm implementing an approach suggested in
% "Iteratively Reweighted Algorithms for Compressive Sensing"
% by Rick Chartrand and Wotao Yin
%
% n is dimension of the ambient space (number of signal samples)
% m is the number of Compressed Sensing measurements
% p is the defining the order of the metric p=1 is l1 norm....
n= 100;
m = 40;
p = 0.9;
% Setting up the problem where one produces
% an input for the problem to be solved
phi = rand(m,n);
u_true = [1 10 120 90 950 4 100 1000 20 30 zeros(1,n-10)]';
u_noise = 10^(-4)* rand(n,1);
u_true = u_true + u_noise;
b = phi * u_true;
%
% solving for phi * u = b , where u is now the
%unknown to be found by the IRlp
%
epsilon = 1
% u_0 is the L_2 solution which would be exact if m = n,
% but here m is less than n
u_0 = phi\b;
u_old = u_0;
j=0
while epsilon > 10^(-8)
j = j + 1
w=(u_old.^(2) + epsilon).^(p/2-1);
v=1./w;
Q_n=diag(v,0);
tu=inv(phi*Q_n*phi');
u_new = Q_n * phi' * tu * b;
if lt(norm(u_new - u_old,2),epsilon^(1/2)/100)
epsilon = epsilon /10;
end
u_old = u_new;
end
% One plot showing the superposition of the least-square
% solution u_0 (symbol v), the
% reweighted Lp solution (symbol *) and the
% true solution (symbol o)
plot(u_0,'v')
title('Least-squares u_0 (v), reweighted Lp (*)...
, initial signal symbol (o)')
hold
plot(u_new,'*')
plot(u_true,'o')

As usual if you find a mistake or something interesting after playing with it, please let me know. Also, I you have a second and half, please answer these two questions, thanks in advance.








Credit Photo: NASA/JPL/Space Science Institute, Ring Tableau, Saturn's Moon: Mimas, Epimetheus, Daphnis

Compressed Sensing: Algorithms for Deterministic Compressed Sensing

Prompted by a commenter, I realized that there are now four, [update: strike that, make that five] different ways of building deterministic algorithms for the measurement matrices in Compressed Sensing. Terry Tao had previously introduced the subject. One can also read his take on DeVore's construction. There are currently at least three other lines of research yielding deterministic compressed sensing measurement matrices, that of Mark Iwen as mentioned before and Piotr Indyk as Muthu Muthukrishnan is pointing out in this entry. So to sumarize we have four constructions :

• Explicit Constructions for Compressed Sensing of Sparse Signals by Piotr Indyk.
• A Deterministic Sub-linear Time Sparse Fourier Algorithm via Non-adaptive Compressed Sensing Methods, by Mark Iwen.
• Deterministic Constructions of Compressed Sensing Matrices by Ron DeVore.
• Annihilating filter-based decoding in the compressed sensing framework by Ali Hormati and Martin Vetterli
• A Frame Construction and A Universal Distortion Bound for Sparse Representations by Mehmet Akçakaya and Vahid Tarokh.

For the last entry, Emmanuel Candès made a presentation at IMA ( Compressive Sampling and Frontiers in Signal Processing ) highlithing the connections of Compressed Sensing with information and coding theory.


Photo Credit: JAXA/NHK, First HDTV view of the moon from the Kayuga probe of the Japanese Space Agency this past week. The whole movie can be seen here.

Compressed Sensing: Sparse PCA and a Compressed Sensing Search Engine

Following up on their paper entitled Full Regularization Path for Sparse Principal Component Analysis, Alexandre d'Aspremont, Francis Bach, Laurent El Ghaoui have produced a new paper where they describe a greedy algorithm to perform their Sparse PCA analysis (as opposed to their previous approach using semidefinite programming). The new paper is: Optimal Solutions for Sparse Principal Component Analysis
the abstract reads:

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.

The code PathSPCA: A fast greedy algorithm for sparse PCA can be found here.

I have set up a search capability on the right side of this blog that uses the new capability for Google search to search through this page and ALL the sites I am linking to. Since I am making an effort at linking to all the researchers involved in Compressed Sensing, searching though this interface enables searching through most websites relevant to this subject.

Photo Credit: NASA, The Earth and the Moon as seen from Galileo en route to Jupiter.

Compressed Sensing: Hardware Implementation, Part IV, Hyperspectral Coded Aperture, Connection between the CS camera and the primary visual cortex ?

Ashwin Wagadarikar mentions a page dedicated to his work on hyperspectral imaging using coded/dispersive apertures and using compressed sensing reconstruction techniques to recover data. It adds to a previous entry on the explosive birth of coded aperture on designs using either a single or dual disperser. The paper on the single disperser is now available on that page or you can directly request a preprint from Ashwin. It is an excellent read and as mentioned before, they used the Gradient Projection for Sparse Reconstruction (GPSR) method to perform the reconstructions.


Marco Duarte answers by a comment in a previous entry on their multiscale random projection
where I mentioned that there was no description of the regularizing kernels:

"..For our paper on multiscale random projections, our regularization kernel relies on coarse pixelation, since we were applying our technique on the single pixel camera. This is to say that the random measurements are obtained at different resolutions to get the projections of the different regularizations of the target image..."

The ability to command the pixel on the DMD allows for some interesting convolutions. In other words, in order to deal with the non-differentiability of Image Appearance Manifolds, the CS camera has to do smoothing projections at different scales. This process is not unlike the parallel I had made on the current state of the art in compressed sensing and its potential use in the modeling of the primary visual cortex. The thesis of Thomas Serre makes it clear that random projections seem to be in the play (at least in his model that takes into account most of what we know about biological processes). He also shows that those projections are done at different scales as well.

Compressed Sensing: Extending Reed-Solomon codes.

Mehmet Akçakaya and Vahid Tarokh have written a series of papers where they use the theoretical framework of algebraic coding/decoding to do Compressive sampling. The measurement matrices are what they call Vandermonde frames that generalize Reed-Solomon codes using Vandermonde matrices. Reconstruction of the signal can be computed using many well known Reed-Solomon type decoding algorithms. In the noiseless case, they have better recovery results (in regards to sparsity) than previously.


Shannon Theoretic Limits on Noisy Compressive Sampling by Mehmet Akçakaya, Vahid Tarokh. The abstract reads:

In this paper, we study the number of measurements required to recover a sparse signal in ${\mathbb C}^M$ with $L$ non-zero coefficients from compressed samples in the presence of noise. For a number of different recovery criteria, we prove that $O(L)$ (an asymptotically linear multiple of $L$) measurements are necessary and sufficient if $L$ grows linearly as a function of $M$. This improves on the existing literature that is mostly focused on variants of a specific recovery algorithm based on convex programming, for which $O(L\log(M-L))$ measurements are required. We also show that $O(L\log(M-L))$ measurements are required in the sublinear regime ($L = o(M)$).

On Sparsity, Redundancy and Quality of Frame Representations by Mehmet Akçakaya, Vahid Tarokh. The abstract reads:

We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N2) operations, as long as the number of non-zero coefficients in the sparse representation of r is N for some 0    0.5, thus improving on a result of Candes and Tao [4]. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.

A Frame Construction and A Universal Distortion Bound for Sparse Representations by Mehmet Akçakaya, Vahid Tarokh.The abstract reads:

We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N^2) operations, as long as the number of non-zero coefficients in the sparse representation of r is N for some 0    0.5. It is known that   0.5 cannot be relaxed without violating uniqueness. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame.

Compressed Sensing: stopping criterion, random multiscale projections, manifold learning, sparse image reconstruction

The Rice Compressed Sensing Repository has some new preprints/articles:

Sparse signal reconstruction from noisy compressive measurements using cross validation by Petros Boufounos, Marco Duarte, and Richard Baraniuk.

Compressive sensing is a new data acquisition technique that aims to measure sparse and compressible signals at close to their intrinsic information rate rather than their Nyquist rate. Recent results in compressive sensing show that a sparse or compressible signal can be reconstructed from very few incoherent measurements. Although the sampling and reconstruction process is robust to measurement noise, all current reconstruction methods assume some knowledge of the noise power or the acquired signal to noise ratio. This knowledge is necessary to set algorithmic parameters and stopping conditions. If these parameters are set incorrectly, then the reconstruction algorithms either do not fully reconstruct the acquired signal (underfitting) or try to explain a significant portion of the noise by distorting the reconstructed signal (overfitting). This paper explores this behavior and examines the use of cross validation to determine the stopping conditions for the optimization algorithms. We demonstrate that by designating a small set of measurements as a validation set it is possible to optimize these algorithms and reduce the reconstruction error. Furthermore we explore the trade-off between using the additional measurements for cross validation instead of reconstruction.

This paper sets up a cross validation procedure that tries to remove the heuristics of the residual stopping criterion used in either Basis Pursuit or Greedy algorithms.


Multiscale random projections for compressive classification by Marco Duarte, Mark Davenport, Michael Wakin, Jason Laska, Dharmpal Takhar, Kevin Kelly, and Richard Baraniuk

We propose a framework for exploiting dimension-reducing random projections in detection and classification problems. Our approach is based on the generalized likelihood ratio test;
in the case of image classification, it exploits the fact that a set of images of a fixed scene under varying articulation parameters forms a low-dimensional, nonlinear manifold. Exploiting
recent results showing that random projections stably embed a smooth manifold in a lower-dimensional space, we develop the multiscale smashed filter as a compressive analog of the familiar matched filter classifier. In a practical target classification problem using a single-pixel camera that directly acquires compressive image projections, we achieve high classification rates using many fewer measurements than the dimensionality
of the images.

It looks to me like a very nice outgrowth of the smashed filter introductory paper mentioned here. In this paper though, some thoughts is given to the fact that target identification must be done at different scales because images appearance manifolds are not smooth manifolds as initially shown by Donoho and Grimes. In this paper, they use regularizing kernels of different supports to provide the ability to smooth out image manifolds and then use nearest neighbor approaches to compare unknown objects and pose to libraries of known objects and pose. No description of these kernels is given though.

Random projections for manifold learning [Technical Report] by Chinmay Hegde, Michael Wakin, and Richard Baraniuk.

We propose a novel method for linear dimensionality reduction of manifold modeled data. First, we show that with a small number M of random projections of sample points in RN belonging to an unknown K-dimensional Euclidean manifold, the intrinsic dimension (ID) of the sample set can be estimated to high accuracy. Second, we rigorously prove that using only this set of random projections, we can estimate the structure of the underlying manifold. In both cases, the number of random projections required is linear in K and logarithmic in N, meaning that K <>

This paper tries to answer a central problem is dimensionality reduction. What is the intrinsic dimension of a manifold using random projections ? This is a useful tool.

Multichannel image estimation via simultaneous orthogonal matching pursuit, Ray Maleh and Anna Gilbert,
The abstract reads:

In modern imaging systems, it is possible to collect information about an image on multiple channels. The simplest example is that of a color image which consists of three channels (i.e. red, green, and blue). However, there are more complicated situations such as those that arise in hyperspectral imaging. Furthermore, most of these images are sparse or highly compressible. We need not measure thoroughly on all the channels in order to reconstruct information about the image. As a result, there is a great need for efficient algorithms that can simultaneously process a few measurements on all channels. In this paper, we discuss how the Simultaneous Orthogonal Matching Pursuit (SOMP) algorithm can reconstruct multichannel images from partial Fourier measurements, while providing more robustness to noise than multiple passes of ordinary Orthogonal Matching Pursuit (OMP) on every channel. In addition, we discuss the use of SOMP in extracting edges from images that are sparse in the total variational sense and extend the ideas presented in this paper to outline how sparse-gradient multichannel images can be recovered by this powerful algorithm.

Sparse gradient image reconstruction done faster by Ray Maleh, Anna Gilbert, and Martin Strauss, The abstract reads:

In a wide variety of imaging applications (especially medical imaging), we obtain a partial set or subset of the Fourier transform of an image. From these Fourier measurements, we want to reconstruct the entire original image. Convex optimization is a powerful, recent solution to this problem. Unfortunately, convex optimization in its myriad of implementations is computationally expensive and may be impractical for large images or for multiple images. Furthermore, some of these techniques assume that the image has a sparse gradient (i.e., that the gradient of the image consists of a few nonzero pixel values) or that the gradient is highly compressible. In this paper, we demonstrate that we can recover such images with GRADIENTOMP, an efficient algorithm based upon Orthogonal Matching Pursuit (OMP), more effectively than with convex optimization. We compare both the qualitative and quantitative performance of this algorithm to the optimization techniques.

Again convexity is a good way of doing business because there are currently mathematically provable bounds. But it also looks like there is some heuristics showing that a non-convex problem can provide faster solution. I am going to have to come back to that at some point because it looks like there are many methods that seem to trade convexity for faster, non-convex and heuristically optimal solution techniques.

Compressed sensing image reconstruction via recursive spatially adaptive filtering by
Karen Egiazarian, Alessandro Foi, and Vladimir Katkovnik, the abstract reads:

We introduce a new approach to image reconstruction from highly incomplete data. The available data are assumed to be a small collection of spectral coefficients of an arbitrary linear transform. This reconstruction problem is the subject of intensive study in the recent field of “compressed sensing” (also known as “compressive sampling”). Our approach is based on a quite specific recursive filtering procedure. At every iteration the algorithm is excited by injection of random noise in the unobserved portion of the spectrum and a spatially adaptive image denoising Þlter, working in the image domain, is exploited to attenuate the noise and reveal new features and details out of the incomplete and degraded observations. This recursive algorithm can be interpreted as a special type of the Robbins-Monro stochastic approximation procedure with regularization enabled by a spatially adaptive filter. Overall, we replace the conventional parametric modeling used in CS by a nonparametric one. We illustrate the effectiveness of the proposed approach for two important inverse problems from computerized tomography: Radon inversion from sparse projections and limited-angle tomography. In particular we show that the algorithm allows to achieve exact reconstruction of synthetic phantom data even from a very small number projections. The accuracy of our reconstruction is in line with the best results in the compressed sensing field.

In light of this entry, two other extended previews grabbed my interest:

• Mark A. Davenport, Chinmay Hegde, Michael B. Wakin and Richard G. Baraniuk, Manifold-Based Approaches for Improved Classification, To appear in NIPS Workshop on Topology Learning, December 2007.
• Chinmay Hegde, Mark A. Davenport, Michael B. Wakin and Richard G. Baraniuk, Efficient Machine Learning Using Random Projections, To appear in NIPS Workshop on Efficient Machine Learning, December 2007

Not found on the Rice page yet are the following:

Sparsity in Time–Frequency Representations , by Gotz Pfander and Holger Rauhut, the abstract reads:

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an S-sparse Gabor representation in Cn with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S  Cn/ log(n). Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.

Note on sparsity in signal recovery and in matrix identification by Gotz Pfander, the abstract reads:

We describe a connection between the identification problem for matrices with sparse representations in given matrix dictionaries and the problem of sparse signal recovery. This allows the application of novel compressed sensing techniques to operator identification problems such as the channel measurement problem in communications engineering.

Credit Photo: NASA, the International Space Station as seen from the Shuttle (STS-120) shortly after 5:00 AM (CST) yesterday.