A signal X ∈ ℝ^N is said to be sparse, if the transform coefficient vector Θ = { θ_i, i = 1,2, …, N}, of X in an orthogonal basis Ψ_N×N are mostly zero. The sparse support of Θ (or X) is defined as S = {i: θ_i ≠ 0, i = 1,2,…, N}, and the length of S is the sparsity K = |S|, K ≪ N, so signal X is called as K-sparse signal. We consider a linear sparse case that K = αN, 0 < α ≪ 1 is a constant. If the most coefficients are not exactly zero but with small magnitude, signal X is called compressible signal. The elements with larger absolute value are regarded as significant. CS theory [1,2] states that M measurements of compressible signal X is enough for exact recovery where M < N. The sensing matrix Ф_M×N should be incoherent with the representation basis Ψ_N×N to guarantee the successful recovery. Fortunately, random matrices are largely incoherent with any fixed basis Ψ_N×N. Thus, the measurements usually are called random projections when a random matrix is used in CS encoding. The CS encoding is quite simple that: (1) Y = Φ M × N Xwhere Y ∈ ℝ^N is the measurement vector. The sensing rate is R = M/N. The reconstruction of CS is performed by solving the following constrained optimization problem: (2) ( ℓ 0 ) min ‖ Θ ‖ ℓ 0 s . t . Y = Φ X = Φ Ψ Θ

Although (l₀) can give the unique solution with only M > 2K measurements, it is a NP hard problem. A relaxation of (l₀) is the linear programming problem, also known as basis pursuit [7]: (3) ( ℓ 1 ) min ‖ Θ ‖ ℓ 1 s . t . Y = Φ X = Φ Ψ Θ

The restricted isometriy property (RIP) [8] is a sufficient condition for guarantee the equivalence of the (l₀) and (l₁) solutions with overwhelming probability. Let δ_k be a smallest positive number such that: (4) ( 1 − δ k ) ‖ X ‖ ℓ 2 2 ≤ ‖ Φ X ‖ ℓ 2 2 ≤ ( 1 + δ k ) ‖ X ‖ ℓ 2 2holds for all k-sparse signals, we say sensing matrix Ф satisfies the RIP with δ_k. Random generated sensing matrixes satisfy RIP with high probability [9]. When the measurements are polluted by noise, i.e., Y = ФX + e, where e ∈ ℝ^M, the (l₁) can be further improved by solving the Lasso [10], or basis pursuit denoising [7]: (5) ( Lasso ) min ‖ Θ ‖ ℓ 1 s . t . ‖ Y − Φ X ‖ ℓ 2 ≤ ɛwhere ε bounds the amount of noise e. The (Lasso) is again a convex programming problem that can be solved with many classical algorithms, e.g., gradient projection for sparse reconstruction (GPSR) [11] etc. Besides the convex optimization methods, CS recovery can be implemented by some iterative greedy algorithms, e.g., orthogonal matching pursuit (OMP) [12].

The investigation of utilizing compressible signals' statistic characteristics has already attracted a lot of interest. The most influential one is the Bayesian compressive sensing (BCS) [13] which recovers the signal by estimating its posterior density function. With different signal models, there are several BCS schemes. The sparse Bayesian learning algorithm in [13] invokes a hierarchical prior to simplify the Laplace prior of compressible signal. The authors in [14] also use Laplace priors. The BCS via belief propagation is proposed in [15] by assuming that the compressible signal obeys the mixture Gaussian model. In [16] the authors combine the belief propagation and Bayesian learning algorithms together to provide a more universal and low-complexity BCS scheme.

However, these compressible signal probability density functions are still based on the assumption that the significant elements are uniformly distributed. Considering the image signals with non-uniform distribution, the wavelet tree model is introduced in BCS in [17]. Besides BCS, the prior information of compressible signal is also used in other ways. In [18], a new weighted rule for the iterative reweighted l₁-norm minimization algorithm is integrated by explore the hidden Markov tree model of the wavelet coefficients. In [19], a weighted sampling scheme is proposed according to the statistics of image signals, where the weight matrix is determined by parameters of the direct-current and alternating-current components' distributions. The authors of [20] and [21] design unequal protection of the wavelet sub-bands in different scales by sampling with different random Gaussian matrices, and their difference is the measurement allocation strategy.

Compared with the unequal protection methods like the hidden Markov tree model in [18]. which uses an extra training process to estimate the exact statistics feature of the signal, the proposed EW-CS scheme firstly only need a coarse sparse support distribution trend as the prior knowledge, e.g., the well-known fact that the significant DCT or DWT coefficients appear at the low frequency. This makes the EW-CS scheme be widely used and easily implemented. From this point, the EW-CS is similar to the schemes presented in [20] and [21]. Nevertheless, the unequal protection mechanism of our scheme is quite different from theirs. They treat each sub-band (or parts with different importance) separately, but we provide nested windows to exploit the combined effect of different parts. With this design, the protection of the significant elements is further enhanced such that the overall performance of the EW-CS outperforms that of [21].

In the proposed EW-CS scheme, for clearer explanation, we assume that the signal X = {x_i | i = 1,2,…N}^T is compressible in the canonical basis, i.e., the orthogonal basis Ψ_N×N = I. The scheme can be directly applied to other basis. A more tractable definition of the compressible signal X is using the power law that the k-th largest entry of the signal obeys |x_(k)| ≤ C_p·k^−1/p, where C_p > 0 and 0 <p <1. C_p represents the energy of the signal which is a constant depending on p, and p is the speed of decay. The sparsity is defined as K when there are K significant elements containing the most energy of the compressible signal, i.e., the first K largest elements |x_(k)|, k = 1,…K.

The non-uniform compressible signal means that the distribution of the significant elements in the sparse support is non-uniform, that is, the support distribution has some structured characteristics. We use importance classes to describe this distribution. From the sparsity perspective, the importance of a class is corresponding to how many significant elements it contains. The compressible signal X is divided into W non-overlapped importance classes. The size of the w-th class is defined by s_w so that N = ∑ w = 1 W S w , w = 1, 2, …, W, 0 ≤ s_w ≤ N. We further assume that the importance of a class decreases with the class's index, i.e., the w-th class is more important than the (w + 1)-th class. The division can be described using a generating polynomial (GP) Π = ∑ w − 1 W Π w x w, where the parameter Π_k = (s_k/N) ϵ [0,1] is the ratio of the elements in the importance class w to all the N elements. The sparsity distribution is given as Λ ( x ) = ∑ w − 1 W Λ w x w , where 0 ≤ Λ_w ≤ 1 is the proportional factor representing the number of significant elements placed in the w-th class.

In the previous subsection, we described the proposed EW-CS scheme in general format. Nevertheless, the most common situation is that the significant elements of a natural compressible signal are concentrated at a certain part of the signal. Therefore, two importance classes are enough for the usual non-uniform compressible signal. The first importance class consists of more large elements than the second importance class. The first importance class is also the first window, and the entire signal is the second window. The two-window EW-CS has wider application prospects; thus, this special case of the EW-CS is discussed in detail below.

In the two-window EW-CS, the GP is Π(x) = Π₁x + (1 − Π₁)x² which is determined by only one coefficient Π₁ ∈ [0,1]. Similarly, the sparsity distribution can be written as Λ(x) = Λ₁x + (1 − Λ₁) x². If the first class really has more significant elements, 0.5 < Λ₁ ≤ 1; otherwise 0 < Λ₁ ≤ 0.5. For clarification, we adopt the sequential generation as shown in Figure 2, so the MP is Ω(x) = Ω₁x + (1−Ω₁)x². By increasing Ω₁, we progressively increase the protection of the first importance class. The extreme cases of Π₁ = M₁ = 0 and Π₁ = M₁ = 0 are the ordinary CS problems.

Based on above definitions, the joint sensing matrix is given as: (6) Φ M × N = ( Φ 1 m 1 × n 1 0 Φ 2 , 1 m 2 × n 2 Φ 2 , 2 m 2 × ( n 2 − n 1 ) ) , M = m 1 + m 2 and N = n 2where Φ 1 m 1 × n 1 is the sensing matrix for the first window (using Ф₁ for short form), and Φ 2 m 2 × n 2 = ( Φ 2 , 1 m 2 × n 1 Φ 2 , 2 m 2 × ( n 2 − n 1 ) ) is the sensing matrix for the second window (using Ф₂ = (Ф_2,1 Ф_2,2) for short form). Then the encoding procedure can be given as: (7) Y 1 = Φ 1 X ( 1 ) , Y 2 = Φ 2 X ( 2 ) = Φ 2 X , Y = [ Y 1 T Y 2 T ] Twhere Y₁ and Y₂ denote the measurement vectors for the two windows respectively. The encoding can also be written in joint coding format that: (8) Y = ( Y 1 Y 2 ) = ( Φ 1 0 Φ 2 , 1 Φ 2 , 2 ) ( X 1 X 2 ) = Φ X

It must be noted that X₁ and X₂ denote the signal elements of the two importance classes not the two windows, which are defined as different from X₍₁₎ and X₍₂₎. The joint coding format is necessary for facilitating the performance analysis of the EW-CS and contrast with the ordinary CS in the next section.

The l₂-norm recovery error defined as ‖X − X̂‖_l2 is adopted to judge the recovery performance. The performance analysis is presented by contrasting between the CS and the EW-CS. According to the joint coding format of the EW-CS in Subsection 3.3, the CS problem is also formulated by correspondingly divided the signal vector X, the measurement vector Y and the sensing matrix Ф into different parts. It is given as: (9) Y = ( Y 1 CS Y 2 CS ) = ( Φ 1 , 1 Φ 1 , 2 Φ 2 , 1 Φ 2 , 2 ) ( X 1 X 2 ) = Φ CS X

The most direct relationship between the CS and the EW-CS can be obtained by replacing the sub-matrix Ф_1,2 with all zero matrix of the same size, and then the CS problem becomes the EW-CS as shown in Equation (8). In CS, the measurement vectors of two windows are calculated as: (10) Y 1 CS = Φ 1 , 1 X 1 + Φ 1 , 2 X 2 Y 2 CS = Φ 2 , 1 X 1 + Φ 2 , 2 X 2while in the EW-CS they are: (11) Y 1 EW = Φ 1 , 1 X 1 Y 2 EW = Φ 2 , 1 X 1 + Φ 2 , 2 X 2

It can be found that the only difference between the CS and the EW-CS is the different generation of the first class of measurements, i.e., Y₁^CS and Y₁^EW. However, the joint recovery algorithms for the CS and the EW-CS use both of the two classes of measurements, so we need a joint coding representation to reveal the difference shown in Equations (10) and (11). Firstly, the compressible signal X can be written as X = X 1 ′ = X 2 ′. The N dimensional vector X 1 ′ is identical to the vector X for the first n₁ elements, and the remaining (n₂ − n₁) elements of X 1 ′ are set to zero. Similarly, the vector X 1 ′ is identical to X for the last (n₂ − n₁) elements, and the remaining elements of X 2 ′ are set to zero. Then, the encoding of CS is given as: (12) Y CS = Φ CS X = Φ CS ( ( X 1 0 ) + ( 0 X 2 ) ) = Φ CS ( X 1 0 ) + Φ CS ( 0 X 2 ) = Φ CS X 1 ′ + Φ CS X 2 ′while the encoding of the EW-CS is rewritten as: (13) Y EW = Φ EW X = Φ EW ( ( X 1 0 ) + ( 0 X 2 ) ) = Φ EW ( X 1 0 ) + Φ EW ( 0 X 2 ) = Φ EW X 1 ′ + Φ EW X 2 ′

For analyzing the discriminational protection of different importance classes, we should consider X₁ and X₂ separately. This is equal to considering X 1 ′ and X 2 ′. If the joint recovery is regarded for recovering X 1 ′ only, the Equations (12) and (13) are further rewritten as: (14) Y CS = Φ CS X 1 ′ + N e , 1 CS , where N e , 1 CS = Φ CS X 2 ′ = ( Φ 1 , 2 X 2 Φ 2 , 2 X 2 ) (15) Y EW = Φ EW X 1 ′ + N e , 1 EW , where N e , 1 EW = Φ EW X 2 ′ = ( 0 Φ 2 , 2 X 2 )

The first terms on the right hand of Equations (14) and (15) are the same according to Equations (8) and (9), so the difference between the CS and the EW-CS is the noise on the X 1 ′'s measurements. It is obvious that the energy of the noise in the CS scheme is larger than that in the EW-CS scheme. Intuitively, the recovery quality of X 1 ′ in the EW-CS is better than in the CS.

This result can also be proved with the aid of mathematical formulae. From the Theorem 2 in [22], the l₂-norm recovery error from inaccurate measurements have the upper bound as: (16) ‖ f ^ − f ‖ ℓ 2 ≤ C 1 , S ⋅ ɛ + C 2 , S ⋅ ‖ f − f S ‖ ℓ 1 Swhere f_s denotes the truncated vector corresponding to the S largest values of compressible signal f (in absolute value). S is chosen such that δ_3S + 3δ_4S < 2, and f's sparsity K ≤ S. ε is the maximum l₂-norm of the noise e, i.e., ∥e∥_ℓ2 ≤ ε. The constants C₁,_S and C₂,_S depend on the reasonable value of δ_4S. The second term on the right hand side of Equation (16) is the approximation error in noiseless case which dominates the recovery error when the noise is small. It is obvious that this term becomes zero when the signal is exactly sparse. This upper bound is based on the assumption that the measurements are sufficient so that the signal is perfectly recovered.

For the X 1 ′, all the measurements contribute to the recovery, so we can assume that the measurements are enough. Based on the result in Equation (16), the error upper bounds for the CS and the EW-CS respectively are given as: (17) ‖ X ^ 1 ′ CS − X 1 ′ ‖ ℓ 2 ≤ C 1 , S 1 ⋅ ɛ 1 CS + C 2 , S 1 ⋅ ‖ X 1 ′ − X 1 , S 1 ′ ‖ ℓ 1 S 1 (18) ‖ X ^ 1 ′ EW − X 1 ′ ‖ ℓ 2 ≤ C 1 , S 1 ⋅ ɛ 1 EW + C 2 , S 1 ⋅ ‖ X 1 ′ − X 1 , S 1 ′ ‖ ℓ 1 S 1

Note that the second terms on the right side of Equations (17) and (18) are the same for the same X 1 ′. The only difference of the recovery performance of X 1 ′ between the CS and the EW-CS is on the noise error. According to Equations (14) and (15), the noise energy of the CS and the EW-CS has the relation as below: (19) ‖ N e , 1 CS ‖ ℓ 2 2 = ‖ Φ 1 , 2 X 2 ‖ ℓ 2 2 + ‖ Φ 2 , 2 X 2 ‖ ℓ 2 2 ≥ ‖ Φ 2 , 2 X 2 ‖ ℓ 2 2 = ‖ N e , 1 EW ‖ ℓ 2 2and the equality holds when X₂ is all zero vector. Based on Equation (19), it can be considered that ε₁^EW ≤ ε₁^CS. It means that the upper bound of the EW-CS for X 1 ′ is lower than CS, and it reflects the higher recovery quality in all probability. Therefore, the more important part of signal X, i.e., the subset of elements in X 1 ′, is indeed better protected using the EW-CS. Next, if the joint recovery is regarded for X 2 ′ only, the formulae (12) and (13) are rewritten as: (20) Y CS = Φ CS X 2 ′ + N e , 2 CS = ( Φ 1 , 2 X 2 Φ 2 , 2 X 2 ) + N e , 2 CS , where N e , 2 CS = Φ CS X 1 ′ = ( Φ 1 , 1 X 1 Φ 2 , 1 X 1 ) (21) Y EW = Φ EW X 2 ′ + N e , 2 EW = ( 0 Φ 2 , 2 X 2 ) + N e , 2 EW , where N e , 2 EW = Φ EW X 1 ′ = ( Φ 1 , 1 X 1 Φ 2 , 1 X 1 )

Comparing with Equations (14) and (15), the information term and the noise term are exchanged, so in this case it is found that the noise for the CS and the EW-CS is the same, while the measurements of X 2 ′ are different. It can be predicted from Equation (16) that the recovery error upper bounds for X 2 ′ of the CS and the EW-CS vary on the approximation error term. However, the approximation error term in Equation (16) is based on the assumption that the measurements are enough for best recovery. The measurements for X 2 ′ constitute only (1 − Ω₁) percentage of all the measurements, so they may not guarantee the best recovery. Oher looser bounds for approximation error are introduced here for analyzing X 2 ′. Of cause, the looser bounds can also be substituted into Equations (17) and (18) for analyzing X 1 ′, and results are also the same as we've already given.

For compressible signals obeying the power law, there is an earlier result in [1] and [2] which states that the estimation from accurate measurements with minimal l₁-norm obeys the following result with overwhelming probability: (22) ‖ f ^ − f ‖ ℓ 2 ≤ C p , α ( M log N ) − ( 1 p − 1 2 )where M ≥ logN is the number of measurements, and N is the length of signal f. C p , α = C P , α ′ ⋅ C p is a constant depending only on the signal energy C_p, the fixed decay speed p and α. The parameter α is a sufficiently small positive number to ensure Equation (22) is behaved with high probability. The bound in Equation (22) also holds for the compressible signal with ∥f∥_ℓ1R and p = 1 when C p , α = C P , α ′ ⋅ R. By substituting Equation (22) into Equation (16), we get the upper bound as: (23) ‖ f ^ − f ‖ ℓ 2 ≤ C 1 , S ⋅ ɛ + C p , α ( M log N ) − ( 1 p − 1 2 )

This is a looser one than Equation (16), for there is a probability of failure although it is quite trivial. If we assume that the component signal X 2 ′ is still a compressible signal obeying the power law, the recovery error upper bounds of X 2 ′ for the CS and the EW-CS then can be given as: (24) ‖ X ^ 2 ′ C S − X 2 ′ ‖ ℓ 2 ≤ C 1 , S 2 ⋅ ɛ 2 CS + C p 2 , α ( M 2 CS log N ) − ( 1 p 2 − 1 2 ) (25) ‖ X ^ 2 ′ EW − X 2 ′ ‖ ℓ 2 ≤ C 1 , S 2 ⋅ ɛ 2 EW + C p 2 , α ( M 2 EW log N ) − ( 1 p 2 − 1 2 )where M₂ is the number of effectual measurements for X 2 ′. From Equations (20) and (21) we know that the noise error terms of Equations (24) and (25) are the same, i.e., ɛ 2 EW = ɛ 2 CS = ɛ 2; nevertheless, M 2 CS = M for ordinary CS, and M 2 EW = ( 1 − Ω 1 ) ⋅ M, where Ω₁ is defined in previous section as the ratio of number of measurements assigned to the first window (i.e., the more important part X₁). Obviously, M 2 EW < M 2 CS, so weighing the recovery quality of X 2 ′ in the CS and the EW-CS cases against each other depends on the value of p₂. It is easy to find that when 0 < p₂ < 2, the CS's bound is lower than the EW-CS's; when p₂ = 2, they are equal; when p₂ > 2, the EW-CS's bound is lower than CS's in return. However, the error bound is behaved for compressible signal usually with 0 < p₂ < 1 or with 0 < p₂ < 2 [2] in the broadest sense. So the aforementioned comparison leads to the result that the recovery quality of X 2 ′ in the EW-CS is worse than that in CS with high probability.

Now, the comparison between the CS and the EW-CS for the entire signal X is discussed. The square error of signal X is given as: (26) ‖ X ^ − X ‖ ℓ 2 2 = ‖ X ^ 1 − X 1 ‖ ℓ 2 2 + ‖ X ^ 2 − X 2 ‖ ℓ 2 2 = ‖ X ^ 1 ′ − X 1 ′ ‖ ℓ 2 2 + ‖ X ^ 2 ′ − X 2 ′ ‖ ℓ 2 2 − ‖ X ^ ‖ ℓ 2 2

Thus, the overall recovery error upper bounds for X using the CS and the EW-CS can be derived by substituting Equations (17), (18), (24) and (25) into Equation (26) such that: (27) ‖ X ^ CS − X ‖ ℓ 2 2 ≤ [ C 1 , S 1 2 ⋅ ( ɛ 1 CS ) 2 + A ⋅ ɛ 1 CS ] + [ B ( M 2 CS ) − ( 2 p 2 − 1 ) + D ( M 2 CS ) − ( 1 p 2 − 1 2 ) ] + E − ‖ X ^ CS ‖ ℓ 2 2 (28) ‖ X ^ EW − X ‖ ℓ 2 2 ≤ [ C 1 , S 1 2 ⋅ ( ɛ 1 EW ) 2 + A ⋅ ɛ 1 EW ] + [ B ( M 2 EW ) − ( 2 p 2 − 1 ) + D ( M 2 EW ) − ( 1 p 2 − 1 2 ) ] + E − ‖ X ^ EW ‖ ℓ 2 2where A = 2 C 1 , S 1 C 2 , S 1 ‖ X 1 ′ − X 1 , S 1 ′ ‖ ℓ 2 S 1, B = C p , α 2 ( log N ) ( 2 p 2 − 1 ), D = 2 C 1 , S 2 C p , α ( log N ) ( 1 p 2 − 1 2 ) ɛ 2, E = C 2 , S 1 2 ⋅ ‖ X 1 ′ − X 1 , S 1 ′ ‖ ℓ 2 S 1 + C 1 , S 2 2 ⋅ ( ɛ 2 ) 2.

We assume that ‖ X ^ CS ‖ ℓ 2 2 ≈ ‖ X ^ EW ‖ ℓ 2 2. From the analysis above, we've known that the first term in right hand side of Equation (28) is smaller than Equation (27) but the second term is larger, so the comparison between Equation (27) and Equation (28) is undetermined. With the same settings, the difference between the upper bounds of the CS and the EW-CS is a function depending only on ε₁ and M₂ which are determined by sparsity distribution and MP, i.e., the parameters Λ₁ and Ω₁. However, if there are really more significant elements placed in X 1 ′, i.e., 0.5 < Λ₁ ≤ 1, the first term will dominate the overall error upper bounds; otherwise, the second term will do so. Thus, the EW-CS is superior to the CS for a non-uniform compressible signal with proper importance classes assignment.

Based on the ana1ysis of the two-window EW-CS, the results are easily extended to W ≥ 3 windows case. The encoding of the W-window EW-CS is written as: (29) Y EW = ( Φ 1 , 1 ⋯ 0 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋮ Φ 1 , w ⋯ Φ w , w ⋯ 0 ⋮ ⋰ ⋮ ⋱ ⋮ Φ W , 1 ⋯ Φ W , w ⋯ Φ W , w ) ( X 1 ⋮ X w ⋮ X W ) = Φ EW ( ( X 1 ⋮ 0 ⋮ 0 ) + ⋯ + ( 0 ⋮ X w ⋮ 0 ) + ⋯ + ( 0 ⋮ 0 ⋮ X w ) ) = Φ EW ( X 1 ′ + ⋯ + X w ′ + ⋯ + X W ′ )

Note that the EW-CS's sensing matrix Φ^EW is a lower triangular matrix. We only consider one importance class X w ′ ∈ ℝ N at a time, and assume that it is compressible obeying the power law; thus, the looser bound in Equation (23) is used. So the recovery error upper bound of the elements in this class is: (30) ‖ X ^ w ′ C S − X w ′ ‖ ℓ 2 ≤ C 1 , S w ⋅ ɛ w CS + C p w , α ( M w CS log N ) − ( 1 p w − 1 2 ) (31) ‖ X ^ w ′ EW − X w ′ ‖ ℓ 2 ≤ C 1 , S w ⋅ ɛ w EW + C p w , α ( M w EW log N ) − ( 1 p w − 1 2 )

Comparing with the conventional CS, only for the first importance class vector X 1 ′, the second terms on the right hand sides of Equations (30) and (31) are the same, so the first class is always better protected in the EW-CS than in the CS. For the last importance class vector X W ′, the first terms are the same, and the second term of the EW-CS has high probability to be larger than the CS; thus, the last importance class is usually sacrificed. For other importance classes, the two terms on the right hand side are both different from the CS, so it is hard to judge the protection level comparing with the CS. However, it can be instinctively known that the smaller w is, the w-th importance class has better recovery quality. Finally, the overall error upper bound for the EW-CS is given as: (32) ‖ X ^ EW − X ‖ ℓ 2 2 = ∑ w = 1 W ‖ X ^ w EW − X w ‖ ℓ 2 2 = ∑ w = 1 W ‖ X ^ w ′ EW − X w ′ ‖ ℓ 2 2 − ( W − 1 ) + ‖ X ^ EW ‖ ℓ 2 2

In our simulation, the l₂-norm of the recovery error, i.e., ∥ X − X̂ ∥_ℓ2, is used to evaluate the performance of the proposed EW-CS algorithm. A compressible signal X ∈ ℝ^N is generated obeying the power law with the parameters C_p = 10 and p = 1 The length of the signal is N = 512, and the sparsity is set as K = 50. The two-window EW-CS is used to compare with the ordinary CS. For constructing the signal in a non-uniform format, the randomly selected Λ₁K elements from the first K K largest elements are placed in the first importance class, and the rest (1−Λ₁)K elements are placed in the second importance class. Other (N−K) smaller elements are distributed randomly. The parameters defining the size of the first importance class and its corresponding number of measurements are fixed as Π₁ = 0.3 and Ω₁ = 0.5.

The overall sensing rate is R, and the total number of measurements is M. The sensing matrix Ф^EW is constructed by first creating a M × N matrix, i.e., the sensing matrix Ф^CS, with i.i.d. draws of a Gaussian distribution N(0,1), then the right upper Ω₁M × (1−Π₁)N sub-matrix is replaced by all zero matrix with the same size. The recovery algorithm we used is the l₁ based method which is implemented by using a package for specifying and solving convex programs [23,24].

In the first example, we consider a compressible signal with Λ₁ = 0.8 and R = 0.4. The original signal and its recovered version are plotted in Figure 5. The l₂-norm error for the component signals X 1 ′ and X 1 ′ are 4.9488 and 11.2609 respectively, and the l₂-norm error for the signal X is 1.1808. Note that the significant elements of the signal are almost well recovered and the elements in the first importance class have better recovery quality than that in the second class.

For various choices of Λ₁, the l₂-norm error of the signal X and the component signals X 1 ′ and X 2 ′ using the proposed EW-CS and ordinary CS is shown in Figure 6. We ran each value 100 times, and simulations are presented for overall sensing rates R = 0.2 and R = 0.4. The results indicate that the performance of the EW-CS varies with the ratio Λ₁, while the ordinary CS is not impacted by Λ₁. With the increase of Λ₁, the recovery error of the EW-CS decreases. Roughly speaking, when 0 ≤ Λ₁ ≤ 0.5, the CS outperforms the EW-CS. This because that the first importance class is not the really most important one. It means that using the EW-CS with wrong matched window settings will introduce more errors.

Nevertheless, when 0.5 ≤ Λ₁ ≤ 1, the EW-CS begins to outperform the CS. In this case, the window settings match the signal's features, so the superiority of the EW-CS is demonstrated. It can also be noticed that the performance for the component signal X 1 ′ in the EW-CS is always superior to that in the CS. It verifies the theoretical result proved in Subsection 4.1 that the elements in the first class are better protected. However, the performance for the component signal X 2 ′ in the EW-CS is worse than that in the CS. This also agrees with our analysis.

For testing the practical performance of the proposed EW-CS, instead of using artificially generated signals, we used real-world non-uniform compressible signals. The image signals' DCT or DWT coefficients constitute non-uniform compressible signals. With this prior knowledge, the random sensing with the EW-CS can be accomplished. The 512 × 512 test gray images with different features, i.e., “Lena”, “Boat”, “Peppers” and “Aerial”, are used for simulation. The images are divided into 32 × 32 blocks for computational facility.

The M-CS scheme in [21] is simulated here for comparison. M-CS needs to do 2-D DWT at the encoder for exploiting the frequency bands feature of wavelet coefficients. Then they used the l₁ based method to recover the wavelet coefficients, and did the inverse wavelet transform to recover the original image. In our simulation, the three level DWT is used in M-CS with the measurement allocation rule in [21]. The level-3 (the lowest band) is assigned 60 measurements, and the number of measurements of level-1 is approximately four times of level-2.

Although the importance class design in the EW-CS can also directly use the sub-bands of 2-D DWT coefficients, the additional transform process is not willing to do. Thus, we choose 1-D DCT as the sparse representation basis which will be used only at the decoder so that the low-complexity encoder is guaranteed. The ordinary CS is also simulated with the 1-D DCT.

Firstly, we use the two-window EW-CS with the parameters Π₁ = 0.3 and Ω₁ = 0.5. The visual quality comparison for “Lena” is given in Figure 7 and the PSNR performances with different sensing rates of the four images are further shown in Figure 8. It is obvious that the proposed EW-CS scheme outperforms the other methods expect for a small number of low sensing rates. The significant coefficients of image signal really appear in the first window of the EW-CS such that they are well recovered. Thus, the EW-CS achieves better overall performance than the ordinary CS.

In M-CS, the significant coefficients are only sampled by the measurements allocated to the sub-band they belong, not all the measurements, so its performance is worse than the EW-CS at most sensing rates. At low sensing rate, the number of measurements for the most significant coefficients in the M-CS is fixed to guarantee successful recovery, and these measurements are noiseless. However, the EW-CS cannot recover the elements in the first importance class from the noisy measurements even though they are more. When the sensing rate increases, this problem will not exist.

In the second simulation, we reconstruct the images from the noisy measurements which are transmitted through an AWGN channel. The sensing rate is fixed as R = 0.4. In order to compare the PSNR performance of the original CS method and the EW-CS method, we fix the signal power, and compensate the power of the all zero matrix Φ_1,2 to the Φ_1,1 to enlarge the signal power of the most significant class. The PSNR performances with different signal-to-noise ratio (SNR) for the four images are shown in Figure 9.

We can observe that the EW-CS is worse than the CS at lower SNR, but outperforms the CS when the SNR increases. The M-CS's PSNR is lowest at the same SNR, but its decay speed is similar to the CS and is slower than the EW-CS. When the channel noise energy gets greater, the advantage of recovery the first importance class of the EW-CS is weaken, and the recovery quality of the second importance class degrades further. Thus, the EW-CS algorithm is more applicable to the mild transmission environments.

Then, we investigate the impact of changing window size on the recovery quality of the two-window EW-CS. The PSNR performance of “Lena” with different Π₁ is given in Figure 10. The window design must follow a basic rule that the rank of the joint sensing matrix of the EW-CS must be larger than the total number of measurements, or the convex programming problem is an unsolved one. So the ranges of Π₁ for different measurement allocation parameters Ω₁ will be different. That's why the x axis for each curve in Figure 10 is different. For a fixed Ω₁, there is a optimal first window size, but the general trend is the first window should not be too large.

We also evaluate the PSNR performance of the EW-CS with different Ω₁ when Π₁ = 0.2, Π₁ = 0.3 and Π₁ = 0.4. The available ranges of Ω₁ for different Π₁ are different according to the design rule of the joint sensing matrix. The simulation results are plotted in Figure 11. It can be found that for the fixed first window size with Π₁ = 0.2 and Π₁ = 0.3, there is an optimal number of measurements that should be assigned to it. For Π₁ = 0.4, however, the more measurements, the better.

Finally, we design the EW-CS with more than two windows. The rank law should also be obeyed. So the three-window EW-CS is designed with the GP Π(x) = 0.3x + 0.2x² + 0.5x³ and MP Ω(x) = 0.5x + 0.2x² + 0.3x. The four-window EW-CS is designed withΠ(x) = 0.1x + 0.2x² + 0.3x³ + 0.4x⁴ and Ω(x) = 0.1x + 0.2x² + 0.3x³ + 0.4x⁴. The five-window EW-CS is designed with Π(x) = 0.1x + 0.1x² + 0.1x³ + 0.2x⁴ + 0.5x⁵ and Ω(x) = 0.1x + 0.1x² + 0.3x³ + 0.2x⁴ + 0.3x⁵. The PSNR performance for different window designs for image “Lena” is shown in Figure 12. The differences among the four different window designs are trivial. It means that the simple EW-CS with two windows is competent in most image signal compression applications.

The variation of the EW-CS applied in wireless sensor networks is implemented here. We define a 500 m × 500 m sensing area. N = 256 sensors are randomly distributed within this area to monitor the events of interest. When E events happen, the sensor's data is affected by all the events, and the relationship between them is assumed to be a distance decaying model, which is determined by the distance d_i,j from the sensor i to the event j, where i = 1, 2, …, N and j = 1, 2, …, E, and the attenuation factor. If the j-th event's magnitude is C_j, the i-th sensor's data is calculated as S i = ∑ j − 1 E C j d i j − α, where α is the propagation loss factor. Here we assume that E = 3, and all C_j are equal to 100, α = 3.8.

For the two-window EW-CS, Figure 13 shows a scenario under our settings with the first window ratio Π₁ = 0.3. The first window size is determined by the threshold given to each sensor from the collection centre (not shown in Figure 13). All the sensors transmit their data to the centre using one hop. As the EW-CS using less number of transmissions than the ordinary CS with the same sensing rate, we define the transmission rate (the ratio between the total number of transmissions and the number of sensors) as the criterion. The networked data in Figure 13 and its recovered version with the transmission rate 0.4 are shown in Figure 14, where Ω₁ = 0.5. The l₂-norm recovery error of this example is 0.2401. More simulation results with different transmission rates are plotted in Figure 15. The performance of the EW-CS is quite superior to the ordinary CS. Considering the transmission noise through the AWGN channel, the l₂-norm recovery error with different SNR is given in Figure 16, where the transmission rate is fixed as 0.4. In the noisy case, the EW-CS performs similar as in image compression that it holds its advantage only at high SNR.

Finally, the performance of the EW-CS with changing first window size is shown in Figure 17, where Ω₁ = 0.5 and the transmission channel is noiseless. The EW-CS in CWS case has the feature that the Π₁N elements in the first window are expected to be the first Π₁N largest ones. With the same ratio Ω₁, the smaller the first window is, the more measurements are assigned to the most largest or significant elements, so the overall performance is inversely proportional to the size of the first window as shown in Figure 17. When the first window size approaches the entire network data, the EW-CS's performance gets closer to the ordinary CS.