0.7 Compressed sensing  (Page 4/5)

Recovery via greedy pursuit

At the expense of slightly more measurements, iterative greedy algorithms such as Orthogonal Matching Pursuit(OMP) [link] , Matching Pursuit (MP) [link] , and Tree Matching Pursuit (TMP) [link] , [link] have also been proposed to recover the signal $x$ from the measurements $y$ (see Nonlinear Approximation from Approximation ). In CS applications, OMP requires $c\approx 2ln\left(N\right)$ [link] to succeed with high probability. OMP is also guaranteed to converge within $M$ iterations. We note that Tropp and Gilbert require the OMP algorithm to succeed in the first $K$ iterations [link] ; however, in our simulations, we allow the algorithm to run up to the maximum of $M$ possible iterations. The choice of an appropriate practical stopping criterion (likely somewhere between $K$ and $M$ iterations) is a subject of current research in the CS community.

Impact and applications

CS appears to be promising for a number of applications in signal acquisition and compression. Instead of sampling a $K$ -sparse signal $N$ times, only $cK$ incoherent measurements suffice, where $K$ can be orders of magnitude less than $N$ . Therefore, a sensor can transmit far fewer measurements to a receiver, which can reconstruct the signal and then process itin any manner. Moreover, the $cK$ measurements need not be manipulated in any way before being transmitted, except possiblyfor some quantization. Finally, independent and identically distributed (i.i.d.) Gaussian or Bernoulli/Rademacher (random $\pm 1$ ) vectors provide a useful universal basis that is incoherent with all others. Hence, when using a random basis, CSis universal in the sense that the sensor can apply the same measurement mechanism no matter what basis the signal is sparse in(and thus the coding algorithm is independent of the sparsity-inducing basis) [link] , [link] , [link] .

These features of CS make it particularly intriguing for applications in remote sensing environments that might involvelow-cost battery operated wireless sensors, which have limited computational and communication capabilities. Indeed, in many suchenvironments one may be interested in sensing a collection of signals using a network of low-cost signals.

Other possible application areas of CS include imaging  [link] , medical imaging  [link] , [link] , and RF environments (where high-bandwidth signals may containlow-dimensional structures such as radar chirps)  [link] . As research continues into practical methods for signal recovery (see [link] ), additional work has focused on developing physical devices foracquiring random projections. Our group has developed, for example, a prototype digital CS camera based on a digitalmicromirror design  [link] . Additional work suggests that standard components such as filters (with randomized impulseresponses) could be useful in CS hardware devices  [link] .

The geometry of compressed sensing

It is important to note that the core theory of CS draws from a number of deep geometric arguments. For example, when viewedtogether, the CS encoding/decoding process can be interpreted as a linear projection $\Phi :{\mathbb{R}}^N\mapsto {\mathbb{R}}^M$ followed by a nonlinear mapping $\Delta :{\mathbb{R}}^M\mapsto {\mathbb{R}}^N$ . In a very general sense, one may naturally ask for a given classof signals $\mathcal{F}\in {\mathbb{R}}^N$ (such as the set of $K$ -sparse signals or the set of signals with coefficients ${\parallel \alpha \parallel }_{{\ell }_p}\le 1$ ), what encoder/decoder pair $\Phi ,\Delta$ will ensure the best reconstruction (minimax distortion) of all signals in $\mathcal{F}$ . This best-case performance is proportional to what is known as the Gluskin $n$ -width  [link] , [link] of $\mathcal{F}$ (in our setting $n=M$ ), which in turn has a geometric interpretation. Roughly speaking, the Gluskin $n$ -width seeks the $(N-n)$ -dimensional slice through $\mathcal{F}$ that yields signals of greatest energy. This $n$ -width bounds the best-case performance of CS on classes of compressible signals, and one of the hallmarks of CS is that,given a sufficient number of measurements this optimal performance is achieved (to within a constant)  [link] , [link] .