compressive Sensing

Introduction

Faithfully and efficiently constructing a signal from a coded signal is an age old scientific problem. The Shannon/Nyquist sampling theorem states that in order to reconstruct a signal from a captured signal it should be sampled at least twice as fast as the maximum signal bandwidth. However, for many practical applications, like imaging systems, the Nyquist sampling rate is very high, paving the way for alternative methods. To avoid a high sampling rate, signal compression is one such approach. In this paper summary, a method of compressive sensing is outlined. Compressive sensing is a method of capturing and representing compressible signals at a rate significantly below the Nyquist rate. This method employs nonadaptive linear projections that maintain the structure of the signal, which is then reconstructed using the optimization techniques outlined in <ref name="R1"> C. Dick, F. Harris, and M. Rice, “Synchronization in software defined radios—Carrier and timing recovery using FPGAs,” in Proc. IEEE Symp. Field-Programmable Custom Computing Machines, NapaValley, CA, pp. 195–204, Apr. 2000.</ref> <ref name="R2">J. Valls, T. Sansaloni, A. Perez-Pascual, V. Torres, and V. Almenar, “The use of CORDIC in software defined radios: A tutorial,” IEEE Commun. Mag., vol. 44, no. 9, pp. 46–50, Sept. 2006.</ref>.

In summary, in this paper we will learn the definition of compressible signal, how to construct a compressible signal and then how to reconstruct the compressed signal.

Compressible Signals

Let us define a discrete time signal [math]\displaystyle{ \ x }[/math] such that it has following properties:

• Real valued
• Finite length
• One dimensional

Then [math]\displaystyle{ \ x }[/math] is a [math]\displaystyle{ N \times 1 }[/math] column vector in [math]\displaystyle{ \mathbb{R}^n }[/math] with elements [math]\displaystyle{ x[n], \; n=1,2,....N }[/math]. To define a signal in [math]\displaystyle{ \mathbb{R}^n }[/math] we define the [math]\displaystyle{ N \times 1 }[/math] basis vectors [math]\displaystyle{ \ \psi^{N}_{i} }[/math]. To keep it simple, assume that the basis is orthonormal. Using a [math]\displaystyle{ N \times N }[/math] basis matrix [math]\displaystyle{ \,\psi = [\psi_1|\psi_2|......|\psi_N] }[/math], the signal [math]\displaystyle{ \ x }[/math] can be expressed as
[math]\displaystyle{ x=\sum_{i=1}^N s_{i} \psi_{i} }[/math] or [math]\displaystyle{ \ x=\psi s }[/math],
where [math]\displaystyle{ \ s }[/math] is a [math]\displaystyle{ N \times 1 }[/math] column vector of weighting coefficients [math]\displaystyle{ s_i=\langle x,\psi_i \rangle=\psi^{T}_i x }[/math].
The signal is [math]\displaystyle{ \ K }[/math]-sparse if it is a linear combination of only [math]\displaystyle{ \ K }[/math] basis vectors. The signal will be compressible if the above representation has just a few large coefficients and many small coefficients. Please note that the above definition has the following inefficiencies:

• The initial number of samples may be very large even if the desired [math]\displaystyle{ \ K }[/math] is small.
• The set of all [math]\displaystyle{ \ N }[/math] transform coefficients, [math]\displaystyle{ \ s_i }[/math], must be calculated even though all but [math]\displaystyle{ \ K }[/math] of them will be discarded.
• Localization of the large coefficients must be encoded, thus introducing overhead.

The Problem Statement for the Compressive Sensing Problem

Compressive sensing directly acquires the compressed signal without going through the intermediate stage of acquiring [math]\displaystyle{ \ N }[/math] samples. In order to do this, a general linear measurement process is defined such that [math]\displaystyle{ \, M \lt N }[/math] inner products between [math]\displaystyle{ \ x }[/math] and a collection of vectors [math]\displaystyle{ \{\phi \}_{j=1}^M }[/math] are computed to give [math]\displaystyle{ y_{j}=\langle x, \phi_j \rangle }[/math].
Ordering the new measurement vectors [math]\displaystyle{ \ y_j }[/math] in a [math]\displaystyle{ M\times 1 }[/math] vector [math]\displaystyle{ \ y }[/math], and the measurement vectors [math]\displaystyle{ \phi_{j}^{T} }[/math] as rows in an [math]\displaystyle{ M \times N }[/math] matrix [math]\displaystyle{ \ \phi }[/math] we can use a previous definition to get
[math]\displaystyle{ \ y=\phi x=\phi \psi s=\theta s }[/math],
where [math]\displaystyle{ \ \theta=\phi \psi }[/math] is an [math]\displaystyle{ M \times N }[/math] matrix. This measurement process is nonadaptive, which implies that [math]\displaystyle{ \ \phi }[/math] is fixed and does not depend on the signal [math]\displaystyle{ \ x }[/math]. This leads to following conditions necessary for designing a compressive sensing method:

• Constructing a stable measurement matrix [math]\displaystyle{ \ \phi }[/math] which preserves the information while reducing the dimension from [math]\displaystyle{ \ N }[/math] to [math]\displaystyle{ \ M }[/math].
• Designing a reconstructing algorithm such that it recovers [math]\displaystyle{ \ x }[/math] from [math]\displaystyle{ \ y }[/math] by using only [math]\displaystyle{ M \approx K }[/math] measurements.

Constructing a Stable Measurement Matrix

The measurements matrix [math]\displaystyle{ \,\phi }[/math] must allow the reconstruction of the signal[math]\displaystyle{ \, x \lt \math\gt from \lt math\gt M \lt N }[/math]. If, [math]\displaystyle{ \,x }[/math] is [math]\displaystyle{ \,K }[/math]-sparse and the locations of the nonzero coefficients in [math]\displaystyle{ \,s }[/math] are known, then problem can be solved provided [math]\displaystyle{ M \geq K }[/math]. To keep this problem well conditioned we have necessary and sufficient condition as
[math]\displaystyle{ 1-\epsilon \leq \frac{\parallel \theta v\parallel_2}{\parallel v\parallel_2}\leq 1+\epsilon }[/math]
Where [math]\displaystyle{ \,v }[/math]is a vector sharing the same non zero entries as[math]\displaystyle{ \,s }[/math]and [math]\displaystyle{ \epsilon \geq 0 }[/math]. This means [math]\displaystyle{ \,\theta }[/math] must preserve the lengths of these particular K-sparse vectors. This condition is referred to as restricted isometry property (RIP)<ref name="R1"> C. Dick, F. Harris, and M. Rice, “Synchronization in software defined radios—Carrier and timing recovery using FPGAs,” in Proc. IEEE Symp. Field-Programmable Custom Computing Machines, NapaValley, CA, pp. 195–204, Apr. 2000.</ref> <ref name="R2">J. Valls, T. Sansaloni, A. Perez-Pascual, V. Torres, and V. Almenar, “The use of CORDIC in software defined radios: A tutorial,” IEEE Commun. Mag., vol. 44, no. 9, pp. 46–50, Sept. 2006.</ref>. A similar property called incoherence requires that the rows [math]\displaystyle{ \,\{\phi_j\} }[/math] of [math]\displaystyle{ \,\phi }[/math] cannot sparsely represent the columns [math]\displaystyle{ \,\{\psi_i\} }[/math] of [math]\displaystyle{ \,\psi }[/math]

The RIP and incoherence can be satisfied by choosing random matrix [math]\displaystyle{ ,\phi }[/math] most of the times. For example, let us construct a matrix [math]\displaystyle{ \,\phi_{j,i} }[/math] such that its matrix elements are independent and identically distributed (iid) random variables from a Gaussian probability distribution with mean zero and variance [math]\displaystyle{ \,1/N\lt \math\gt [1],[2],[4]. Therefore the measurements \lt math\gt \,y\lt \math\gt are merely \lt math\gt \,M\lt \math\gt different randomly weighted linear combinations of the elements of \lt math\gt \,x\lt \math\gt . The matrix \lt math\gt \,\phi }[/math] has following properties,

1. The incoherence property is satisfied by matrix [math]\displaystyle{ \,\phi }[/math].
2. The [math]\displaystyle{ \,\phi }[/math] is universal meaning that it will satisfy the RIP property with high probability.

Reconstruction Algorithm

Now we are left with the task to design reconstruction algorithm.The reconstruction algorithm must take into account the M measurements in the vector y, the random measurement matrix [math]\displaystyle{ \phi }[/math] and the basis [math]\displaystyle{ \psi }[/math]. Then reconstruct the length N signal x or its sparse coefficient vector s. Since M<N then we have s^' which satisfies,
[math]\displaystyle{ \theta s_{'}=y }[/math]
The justification for this is that [math]\displaystyle{ \theta s=y }[/math] then [math]\displaystyle{ \theta (s+r)=y }[/math] for any vector r in the null space [math]\displaystyle{ N(\theta) }[/math] of [math]\displaystyle{ \theta }[/math]. This leads to finding signal's sparse coefficient vector in the (N-M) dimensional translated null space [math]\displaystyle{ H=N(\theta)+s }[/math], in-order to construct the signal. The reconstruction process can be attempted in the following ways,

• Minimum [math]\displaystyle{ l_2 }[/math] norm reconstruction

Recall that [math]\displaystyle{ \parallel l\parallel_p=sum_{i=1}^{N} |s_i|^p }[/math]. In-order to solve this type of problems the classical approach is to find the vector in the transformed null space with the smallest [math]\displaystyle{ l_2 }[/math] norm also called energy norm by solving.
[math]\displaystyle{ \hat{s}=argmin\parallel s_{'}\parallel_2 }[/math] such that [math]\displaystyle{ \theta s_{'}=y }[/math]
The closed form solution is given by [math]\displaystyle{ \hat{s}=\theta_{T}(\theta \theta_{t})^{-1}y }[/math]. However the minimization returns non-sparse [math]\displaystyle{ \hat{s} }[/math] with many non zero elements in other words it almost never able to reconstruct the actual signal.

• Minimum [math]\displaystyle{ l_0 }[/math] norm reconstruction

Let us now evaluate the [math]\displaystyle{ l_0 }[/math] norm which counts the number of non-zero entries in s.
[math]\displaystyle{ \hat{s}=argmin\parallel s_{'}\parallel_0 }[/math] such that [math]\displaystyle{ \theta s_{'}=y }[/math]
At first glance it looks attractive as it recovers the K-sparse signal exactly with high probability using only M=K+1 iid Gaussian measurements. However, solving the equation is numerically unstable and NP-complete which requires exhaustive search of all [math]\displaystyle{ (_K^{N}) }[/math] possible localization of the non-zero entries in s.

• Minimum [math]\displaystyle{ l_1 }[/math] norm reconstruction

Now, if perform minimization on [math]\displaystyle{ l_1 }[/math] norm
is able to recover K-sparse signals and closely approximate compressible signal with high probability using only [math]\displaystyle{ M \geq cKlog(\frac{N}{K}) }[/math] iid Gaussian measurements.
[math]\displaystyle{ \hat{s}=argmin\parallel s_{'}\parallel_1 }[/math] such that [math]\displaystyle{ \theta s_{'}=y }[/math]
This optimization is convex optimization problem which reduces to linear program basis pursuit<ref name="R1"> C. Dick, F. Harris, and M. Rice, “Synchronization in software defined radios—Carrier and timing recovery using FPGAs,” in Proc. IEEE Symp. Field-Programmable Custom Computing Machines, NapaValley, CA, pp. 195–204, Apr. 2000.</ref> <ref name="R2">J. Valls, T. Sansaloni, A. Perez-Pascual, V. Torres, and V. Almenar, “The use of CORDIC in software defined radios: A tutorial,” IEEE Commun. Mag., vol. 44, no. 9, pp. 46–50, Sept. 2006.</ref>, having computational complexity[math]\displaystyle{ O(n^3) }[/math].

Conclusion

The compressive sensing algorithm can be applied to analog signals as well. The sensing technique finds many practical applications in image processing and similar fields. In this summary, we learned about compressive sensing which is a more efficient method compared to traditional techniques.

References

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