# 0.7 Compressed sensing  (Page 3/5)

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The second statement of the theorem differs from the first in the following respect: when $K , there will necessarily exist $K$ -sparse signals $x$ that cannot be uniquely recovered from the $M$ -dimensional measurement vector $y=\Phi x$ . However, these signals form a set of measure zero within the set of all $K$ -sparse signals and can safely be avoided if $\Phi$ is randomly generated independently of $x$ .

Unfortunately, as discussed in Nonlinear Approximation from Approximation , solving this ${\ell }_{0}$ optimization problem is prohibitively complex. Yet another challenge is robustness; in the setting ofTheorem "Recovery via ℓ 0 optimization" , the recovery may be very poorly conditioned. In fact, both of these considerations (computational complexity and robustness) can be addressed, but atthe expense of slightly more measurements.

## Recovery via convex optimization

The practical revelation that supports the new CS theory is that it is not necessary to solve the ${\ell }_{0}$ -minimization problem to recover $\alpha$ . In fact, a much easier problem yields an equivalent solution (thanks again to the incoherency of thebases); we need only solve for the ${\ell }_{1}$ -sparsest coefficients $\alpha$ that agree with the measurements $y$ [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link]

$\stackrel{^}{\alpha }=argmin{\parallel \alpha \parallel }_{1}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{s.t.}\phantom{\rule{4.pt}{0ex}}y=\Phi \Psi \alpha .$
As discussed in Nonlinear Approximation from Approximation , this optimization problem, also known as Basis Pursuit [link] , is significantly more approachable and can be solved with traditionallinear programming techniques whose computational complexities are polynomial in $N$ .

There is no free lunch, however; according to the theory, more than $K+1$ measurements are required in order to recover sparse signals via Basis Pursuit. Instead, one typically requires $M\ge cK$ measurements, where $c>1$ is an oversampling factor . As an example, we quote a result asymptotic in $N$ . For simplicity, we assume that the sparsity scales linearly with $N$ ; that is, $K=SN$ , where we call $S$ the sparsity rate .

Theorem

[link] , [link] , [link] Set $K=SN$ with $0 . Then there exists an oversampling factor $c\left(S\right)=O\left(log\left(1/S\right)\right)$ , $c\left(S\right)>1$ , such that, for a $K$ -sparse signal $x$ in the basis $\Psi$ , the following statements hold:

1. The probability of recovering $x$ via Basis Pursuit from $\left(c\left(S\right)+ϵ\right)K$ random projections, $ϵ>0$ , converges to one as $N\to \infty$ .
2. The probability of recovering $x$ via Basis Pursuit from $\left(c\left(S\right)-ϵ\right)K$ random projections, $ϵ>0$ , converges to zero as $N\to \infty$ .

In an illuminating series of recent papers, Donoho and Tanner [link] , [link] , [link] have characterized the oversampling factor $c\left(S\right)$ precisely (see also "The geometry of Compressed Sensing" ). With appropriate oversampling, reconstruction via Basis Pursuit is also provably robust tomeasurement noise and quantization error [link] .

We often use the abbreviated notation $c$ to describe the oversampling factor required in various settings even though $c\left(S\right)$ depends on the sparsity $K$ and signal length $N$ .

A CS recovery example on the Cameraman test image is shown in [link] . In this case, with $M=4K$ we achieve near-perfect recovery of the sparse measured image. Compressive sensing reconstruction of the nonlinear approximation Cameraman image from [link] (b). Using M = 16384 random measurements of the K -term nonlinear approximation image (where K = 4096 ), we solve an ℓ 1 -minimization problem to obtain the reconstruction shown above. The MSE with respect to the measured image is 0.08 , so the reconstruction is virtually perfect.

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