Construction of a Large Class of Deterministic Sensing Matrices That Satisfy a Statistical Isometry Property

Abstract

Abstract-Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard Compressed Sensing paradigm, the N × C measurement matrix Φ is required to act as a near isometry on the set of all k-sparse signals (Restricted Isometry Property or RIP). If Φ satisfies the RIP, then Basis Pursuit or Matching Pursuit recovery algorithms can be used to recover any k-sparse vector α from the N measurements Φα. Although it is known that certain probabilistic processes generate N × C matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix Φ has this property, crucial for the feasibility of the standard recovery algorithms. In contrast this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of k-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. An essential element in our construction is that we require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sublinear in C, and only quadratic in N , as compared to the superlinear complexity in C of the Basis Pursuit or Matching Pursuit algorithms; the focus on expected performance is more typical of mainstream signal processing than the worst-case analysis that prevails in standard Compressed Sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.

Stability of Efficient Deterministic Compressed Sensing for Images with Chirps and Reed-Muller Sequences

“…The advantage of using deterministic matrices for compressed sensing is that reconstruction can be very efficient [15,16]. The fast reconstruction algorithm [15][16][17] is called the Quadratic Reconstruction Algorithm. This algorithm takes advantage of the multivariable quadratic functions that appear as exponents of the entries in the sensing matrix, and therefore, only requires vector-vector multiplication instead of matrix-vector multiplication required in Basis and Matching Pursuit algorithms that are used for reconstructions with random sensing matrices.…”

“…This algorithm takes advantage of the multivariable quadratic functions that appear as exponents of the entries in the sensing matrix, and therefore, only requires vector-vector multiplication instead of matrix-vector multiplication required in Basis and Matching Pursuit algorithms that are used for reconstructions with random sensing matrices. In [17], Calderbank, Howard and Jafarpour set forth criteria on Φ that ensure a high probability that the mapping taking the k-sparse signal vector k x to the measurement vector y is injective, assuming a uniform probability distribution on the unit-magnitude k-sparse vectors in . They say that Φ has the Statistical Restricted Isometry Property (StRIP) with respect to parameters and δ if…”

“…They show that such deterministic sensing matrices sat-isfying StRIP can be constructed by chirps, Reed-Muller (RM) sequences, and BCH codes, as done in [15][16][17]. In the presence of noise, if Φ satisfies the StRIP property with parameters and δ, the reconstruction error due to the Quadratic Reconstruction Algorithm is given by…”

“…Our proposed algorithm [3] takes this into account and we have shown that it outperforms a standard compressed sensing method, which takes random noiselet measurements and uses minimization for reconstruction, for example, see [19]. 1  Authors in [17] also derived that the deterministic sensing matrices are resilient to noise. Suppose the noise comes from the measurements, say …”

“…where 0 x is the true signal, x is the recovered signal, k x is the best k-term approximation of 0 x , and μ is the measurement noise from a Gaussian distribution, see [17]. As mentioned there, this bound is tighter than bounds obtained by using random ensembles [12] and expanderbased methods [18].…”