Low storage space for compressive sensing: semi-tensor product approach
- Jinming Wang^{1}Email author,
- Shiping Ye^{1},
- Yue Ruan^{1} and
- Chaoxiang Chen^{1}
https://doi.org/10.1186/s13640-017-0199-9
© The Author(s). 2017
Received: 22 February 2017
Accepted: 17 July 2017
Published: 28 July 2017
Abstract
Random measurement matrices play a critical role in successful recovery with the compressive sensing (CS) framework. However, due to its randomly generated elements, these matrices require massive amounts of storage space to implement a random matrix in CS applications. To effectively reduce the storage space of the random measurement matrix for CS, we propose a random sampling approach for the CS framework based on the semi-tensor product (STP). The proposed approach generates a random measurement matrix, where the dimensions of the random measurement matrix are reduced to a quarter (or 1/16, 1/64, and even 1/256) of the number of dimensions, which are used for conventional CS. We then estimate the values of the sparse vector with a modified iteratively re-weighted least-squares (IRLS) algorithm. The results of numerical simulations showed that the proposed approach can reduce the storage space of a random matrix to at least a quarter while maintaining quality of reconstruction. All results confirmed that the proposed approach significantly influences the physical implementation of the CS in images, especially on embedded system and field programmable gate array (FPGA), where storage is limited.
Keywords
Compressed sensing Compressive sensing Random measurement matrix Storage space Semi-tensor product1 Introduction
Compressive sensing (CS) [1] theory provides a new way to sample and compress data. The basic idea of CS is that a higher-dimensional signal is projected onto a measurement matrix, by which a low-dimensional sensed sequence is obtained. Meanwhile, [2–4] prove that if the sensed sequence consists of a small number of non-zero elements, then it can recover the original signal from the sensed sequence. CS applications confirm that random measurement matrices are suitable for compressed sensing. However, these applications require considerable storage space to realize random matrices [3]. As a result, much work has been done to reduce the storage space and improve performance.
In [5, 6], an intermediate structure for the measurement matrix is proposed based on random sampling, called block compressed sensing (block CS). Using the block CS, the data sampling is conducted in a block-by-block manner through the same measurement matrix, which overcomes the difficulties encountered in traditional CS technology, for which the random measurement ensembles are numerically unwieldy.
In [7, 8], Thong, et al. introduced a fast and efficient way to construct a measurement matrix, called the structurally random matrix (SRM), which attempted to improve the structure of an initial random measurement matrix using optimization techniques. SRM is related to large-scale, real-time CS applications for low requirements in storage space.
To reduce the storage space for CS, many deterministic measurement matrices have been designed [9–14]. They satisfy the restricted isometry property (RIP) and recovered the sparse signal successfully.
In [15, 16], low-dimensional orthogonal basis vectors or matrices were used to construct high-dimensional matrices according to the Kronecker product. The proposed algorithm effectively reduces the storage space of a measurement matrix.
Low-rank matrices and rank-one matrices are attractive because they need less storage space than general measurement matrices [17–20]. Indeed, if the measurement matrix is sparse, it takes less storage space and incurs less computational cost. Low-rank or rank-one matrices have been designed to sample and reconstruct the original signal, obtaining high-quality reconstructions.
All of the above findings are directed at how to reduce the storage space of a measurement matrix for CS. However, using the block CS, several block artifacts occur in the reconstructed images, owing to the block-by-block manner and the neglect of global sparsity. However, the SRM method is more complicated and difficult to achieve. Deterministic measurement matrices require little storage space and incur less computational cost, but the accuracy of the reconstruction is not as high as a random measurement matrix. To reconstruct the original signals, the Kronecker algorithm must generate an M × N dimensional measurement matrix, and this requires large-scale memory space.
For the same purpose, we propose a random sampling scheme for CS. The aim is to propose an algorithm that can maintain the same reconstruction performance as conventional compressive sensing, but requires less required storage for the measurement matrix and less memory for reconstructing.
The proposed algorithm is based on the semi-tensor product (STP) [21, 22], a novel matrix product that works by extending the conventional matrix product in cases of unequal dimensions. Our algorithm generates a random matrix, with dimensions that are smaller than M and N, where M is the length of the sampling vector and N is the length of signal that we want to reconstruct. Then, we use the iteratively re-weighted least squares (IRLS) algorithm to estimate the value of the sparse coefficients. Experiments were carried out using the sparse column signals and images, demonstrating that it outperforms other algorithms in terms of storage space and a suitable peak signal to noise ratio (PSNR) performance. The experimental results show that if we reduce the dimensions of the measurement matrix appropriately, there is almost no decline in the PSNR of the reconstruction, yet the storage space required by the measurement matrix can be reduced to a quarter (or even 16th) of the size.
The remainder of this paper is organized as follows: In Section 2, the preliminaries of the STP and the conventional CS algorithm are introduced. In Section 3, we describe the proposed STP approach to the CS algorithm (STP-CS). In Section 4, we present the experimental results and a discussion. Finally, Section 5 concludes the paper and contains a discussion of our plans for future research.
2 Related works
In this section, the concepts of the conventional CS algorithm and some necessary preliminaries to the STP are briefly introduced. The STP of matrices was introduced by Cheng [21, 22].
2.1 Semi-tensor product
In [21, 22], the STP is presented as an extension of the conventional matrix product. For a conventional matrix product, if Col (A) ≠ Row (B), then matrices A and B are multiplicative. The STP of matrices, on the other hand, extends the conventional matrix product in cases of unequal dimensions. In [22], the STP is defined as follows:
In comparing the product of the conventional matrix with the STP of the matrix, it is easy to see that there are significant differences between them.
In recent years, the STP has been exploited by a wide range of applications: in nonlinear system control for structural analysis and control of Boolean networks [23, 24], in biological systems as a solution to Morgan’s Problem [25], in a linear system for nonlinear feedback-shift registers [26, 27], etc. However, we have not yet seen publicly reported applications for the STP in the field of CS, to the best of our knowledge.
2.2 Conventional CS algorithm
Here, the vector θ is called exact-sparse because it has k non-zeros and the rest of the elements are equal to zero. However, there might be cases where the coefficient vector θ includes only a few large components with many small coefficients. In this case, x is called a compressible signal, and sparse approximation methods can be applied.
As shown in (5), the higher dimensional signal x is projected onto the matrix Φ _{ M×N }, and a low-dimensional sensed sequence y _{ M×1} is obtained. Assume a signal is sampled using the above scheme and then the measurements y are transmitted. The crucial task (for the receiver) is to reconstruct the original samples x with knowledge about the measurements y _{ M×1} and the measurement matrix Φ _{ M×N }. The recovery problem is ill conditioned since M < N. However, several methods have been proposed to tackle this problem, such as the iteratively re-weighted least squares (IRLS) algorithm.
When the original signal is sampled and reconstructed, the role played by the measurement matrix is vital in order to faithfully reconstruct with precision and complexity [4]. However, this requires a lot of storage space in order to realize the measurement matrices in CS applications. This is especially true of random measurement matrices, because they are computationally expensive and require considerable memory [3]. Therefore, reducing the storage space of the measurement matrix is essential to practical CS applications, especially in terms of the feasibility of embedded hardware implementations.
3 Proposed algorithm
To effectively reduce the storage space of a random measurement matrix, we propose an STP approach for the CS algorithm (STP-CS), which can reduce storage space to at least a quarter of the size while maintaining the quality of reconstructed signals or images.
According to the example, we can see that the sparse signal x _{10×1} can project onto Φ _{3×5}, by which a low-dimensional sensed sequence y _{6×1} is obtained. The measurements are simply linear combinations of the elements of x _{10×1}.
If we assume t = 1, measurement y ^{’} _{6×1} is obtained by Φ _{6×10}, which is the same as in the conventional CS. Thus, Eq. (7) can be expressed as y _{ Mx1} = Φ _{ M × N } ⋅ x _{ N × 1}.
It should be pointed out that, for the same sparse signal, the measurements obtained from different measurement matrices are different. That is y _{6×1} ≠ y ^{’} _{6×1}.
If \( \mu \left({\varTheta}^t\right)\in \left[1,\kern0.5em \sqrt{N}\right],\kern0.5em \varPhi (t) \) is incoherent with basis Ψ with a high probability, such that \( {\varTheta}_{N\times N}^t \) satisfies the RIP with high probability [1–3]. This guarantees that a k-sparse or compressible signal can be fully represented by M measurements with the dimension-reduced measurement matrix Φ(t). The approach does not change the linear nature of the CS acquisition process, except for involving a smaller measurement matrix to obtain measurements. It is clear that the STP approach fits well with the conventional CS for t > 1.
When we assume t = 2, then the dimensions of Φ(t) are (M/2) × (N/2), and when we assume t = 4, the dimensions are (M/4) × (N/4), etc.. Thus, the storage space of Φ(t) is reduced quadratically. For example, to process a 1024 × 1024 image, when the sampling rate is 50% and the data is with double precision floating-point, there are 512 K measurements. A Gaussian random matrix requires 4096 K bytes when t = 1. While t = 2, the needed storage is 1024 K bytes, and when t = 4, this is reduced to 256 K bytes.
Thus far, the STP approach could be an effective way to reduce the storage space of the measurement matrix. There remains a key question, however, regarding how the original signal can be reconstructed based on the STP approach acquisition process.
To reconstruct the original signal, we adopted the IRLS to reconstruct the original signal [28–35]. In [29], it was shown empirically that using ℓ ^{ q }-minimization with 0 < q < 1 can do with fewer measurements than ℓ ^{1}-minimization. In case of a noisy k-sparse vector, using ℓ ^{ q }-minimization with 0 < q < 1 is more stable than ℓ ^{1}-minimization [31, 32]. In [33–35], an approximate ℓ ^{0}-norm minimization algorithm was proposed. The approximate ℓ ^{0}-norm minimization shows attractive convergence properties, which is capable of very fast signal recovery, thereby reducing retrieval latency when handling high-dimensional signals.
When we derive a vector of measurements y _{ M×1} by a random measurement matrix Φ(t), we initialize the algorithm by taking w ^{0} = (1, ⋯ , 1)_{1 × N } , x ^{0} = (1, ⋯1)_{1 × N }, and ε _{0} = 1. The k-sparse signal x is then reconstructed by iterations.
Therefore, in Section 4, we experimentally reconstruct the original sparse signal with ℓ ^{ q }-norm (0 < q < 1) minimization and approximate ℓ ^{0}-norm minimization, respectively.
4 Experiments results and discussion
In this section, we verify the performance of the proposed STP-CS. Our intent is to determine tradeoffs between recovery performance and the reduction ratio of the measurement matrix. We also compared the performance of STP-CS with that of CS with ℓ ^{ q }-minimization (0 < q < 1) and approximate ℓ ^{0}-minimization. We begin the numerical experiments with some N × 1 column-sparse vectors and some N × N gray-scale images. In our experiments the dimensions of the measurement matrix Φ(t) are (M/t) × (N/t), where t could be 1, 2, 4, or even larger, and the matrix Φ(t) is Gaussian N(0, 1/(M/t)) i.i.d. entries, which approximately satisfy the RIP with high probability [2, 32]. In addition, as we have shown in Section 3, when t = 1, there is no reduction to the dimensions of the matrix Φ(t). As such, it can be treated as conventional CS, whereas when t = 2, 4 or higher, the dimensions of the matrix Φ(t) are reduced. Therefore, we performed the comparison with different t. Our experiments and comparisons were implemented in Matlab R2010b on an Intel i7–4600 laptop with 8 GB of memory, running Windows 8.
4.1 Comparison with one-dimensional sparse signal vectors
To compare the performance with the matrices Φ(t), we measured the rate of convergence and the probability of an exact reconstruction for different sparsity values k and for different numbers of measurements.
First, we considered one-dimensional sparse vectors x _{ N×1}, where N is the length of the sparse vector.
When N = 256, M = 128, and k = 40, a 40-sparse vector is generated with a random positioning of the non-zeros. Here, according to [1], if a sparse vector is used to ensure the uniqueness of a sparse solution, the number of non-zero elements can reach a limit of M/2. Therefore, we give a maximum number of k (k = 1, 2, ⋯, M/2).
Different dimensional measurement matrices Φ(t) for STP-CS (N = 256)
M | t = 1 | t = 2 | t = 4 |
---|---|---|---|
32 | Φ _{32×256} | Φ _{16×128} | Φ _{8×64} |
64 | Φ _{64×256} | Φ _{32×128} | Φ _{16×64} |
128 | Φ _{128×256} | Φ _{64×128} | Φ _{32×64} |
As shown in Table 1, when t = 1, the matrix Φ(1) is M × N, whereas when, t = 2, 4, the dimensions of the matrices Φ(2) and Φ(4) are (M/2) × (N/2) and (M/4) × (N/4), respectively. Hence, the storage space needed for the matrix Φ is reduced effectively. Meanwhile, there is also a significant reduction in the memory requirements for reconstruction.
If the relative error is less than \( {10}^{-5},\kern0.5em \widehat{x} \) can be considered the correct solution, and the recovery is successful. Otherwise, the recovery is considered to have failed.
It needs to be pointed out that matrices Φ(t) were generated only once during the trails.
As shown in Fig. 1, when the sparsity value k is relatively small, namely k ≤ 20, the probability of an exact reconstruction remains almost 100%, regardless of whether the dimensions of the measurement matrix are reduced. When we increase the value k, the probability of an exact reconstruction declines. Compared to the probability curve with t = 1, the probability curves with t = 2 or 4 decline quickly. However, they nevertheless maintain a high probability of an exact reconstruction. It is clear that we can recover the original sparse vector in the matrices with reduced dimensions. Furthermore, by contrasting frames (a) and (b), we can see that when sparsity value k approaches the limit value of M/2, (a) still has a higher probability of reconstruction than (b).
During the comparisons, an issue emerged that caught our attention, regarding why the probability of an exact reconstruction declines so quickly, when we reduced the dimensions of matrix Φ(t) (t > 1).
In (18), we see that all the coefficients for different measurements in the ith group are the same—namely, (φ _{ i,1}, …, φ _{ i,N/t }). As shown in (10), (y _{1}, y _{2})^{T}, (y _{3}, y _{4})^{T}, and (y _{5}, y _{6})^{T} are the groups.
The sparse vector we used in this experiment had a value of k—namely, k = 40 or 60. In these comparisons, 500 attempts were executed for generating the three different measurement matrices. We generated the sparse vector x only once with a given k. The numerical results on the curves represent the mean of these 500 attempts.
As shown in Fig. 2, for k = 40, the rate of convergence was roughly the same for different matrices. For k = 60, more iterations were needed to achieve a sparse solution when increasing the value of t. This showed that if the original signal is sufficiently sparse, the rate of convergence was still fast, despite a reduction in the number of dimensions of the measurement matrix.
As shown in Fig. 3, for the same number of measurements, the probabilities of exact reconstruction differed little with different measurement matrices. Hence, there was no need to increase the number of measurements to derive the solution when we reduced the number of dimensions of the measurement matrix.
According to the comparisons of one-dimensional sparse signals, we can see that the STP approach can reconstruct a sparse signal with a randomly measurement matrix Φ(t) (t > 1). Moreover, performance with dimensionality reduction for a Gaussian random measurement matrix Φ(t) is generally comparable to that of the random matrix without any reduced dimensions. Furthermore, the performance with the dimensionality reduction to matrix Φ(t) depends on the sparsity of the signal x. In particular, if the original signal is sufficiently sparse, its performance with dimensionality reduction was relative good compared to that of the matrix without reduced dimensions. On the other hand, if the original signal is not sufficiently sparse, the performance declines. Therefore, there is a tradeoff between the performance of the reconstruction and the dimensionality of the measurement matrix.
4.2 Comparisons with two-dimensional signals
Here, to compare the performance with the matrices Φ(t) for two-dimensional signals, we measured the PSNR values of the reconstructed images.
In these comparisons, the signals were two-dimensional natural images. We know that signals and natural images must be sparse in a certain transform domain or dictionary, in order for them to be reconstructed exactly within the CS framework. In our experiments, we employed coefficients from the wavelet transform as two-dimensional compressible signals, and projected the coefficients onto a Gaussian random measurement matrix. When we derived the measurements y with a matrix Φ(t), the coefficients were reconstructed by IRLS with approximate ℓ ^{0}-minimization. Three natural images of different sizes were used in our experiments. Lena (size: 256 × 256), Peppers (size: 256 × 256), and OT-Colon (size: 512 × 512), OT-Colon is a DICOM gray-scale medical image, and it can be retrieved from [36].
Different dimensional measurement matrices Φ(t) for images (M/N = 0.5)
Size of image (N × N) | t = 1 | t = 2 | t = 4 | t = 8 | t = 16 |
---|---|---|---|---|---|
256 × 256 | Φ _{128×256} | Φ _{64×128} | Φ _{32×64} | Φ _{16×32} | Φ _{8×16} |
512 × 512 | Φ _{256×512} | Φ _{128×256} | Φ _{64×128} | Φ _{32×64} | Φ _{16×32} |
As shown in Table 2, with the increase in the value of t, the storage space was reduced quadratically, such that the storage space of Φ(16) was 1/256th that of Φ(1).
Comparison of the PSNR of reconstructions with different dimensions of Gaussian measurement matrices (ℓ ^{0}-minimization with ρ = 0.8)
Ratio of dimensionality reduction | M/N | PSNR | |||
---|---|---|---|---|---|
Lena | Peppers | OT-colon | |||
Max | t = 1 | 0.8125 | 41.4152 | 45.1162 | 41.4891 |
t = 2 | 41.2529 | 45.2535 | 41.2479 | ||
t = 4 | 41.2816 | 45.0404 | 41.2863 | ||
t = 8 | 41.6383 | 44.4560 | 41.3755 | ||
t = 16 | 41.6666 | 43.9702 | 41.0504 | ||
t = 1 | 0.7500 | 39.3990 | 42.5244 | 39.2847 | |
t = 2 | 39.3532 | 42.5405 | 39.1986 | ||
t = 4 | 39.4859 | 42.8989 | 39.1524 | ||
t = 8 | 39.3292 | 41.9847 | 39.1286 | ||
t = 16 | 39.4723 | 41.0188 | 39.0994 | ||
t = 1 | 0.5000 | 35.4444 | 36.4737 | 30.5470 | |
t = 2 | 35.3266 | 36.6717 | 30.3621 | ||
t = 4 | 35.2764 | 37.3340 | 30.6017 | ||
t = 8 | 35.0658 | 36.3877 | 30.5098 | ||
t = 16 | 35.5209 | 36.6022 | 30.3819 | ||
t = 1 | 0.4375 | 26.2675 | 27.5362 | 28.5509 | |
t = 2 | 26.4057 | 27.7582 | 28.4599 | ||
t = 4 | 27.3657 | 28.1248 | 28.2993 | ||
t = 8 | 26.8054 | 27.9102 | 27.9184 | ||
t = 16 | 27.4303 | 29.2851 | 27.5377 | ||
Min | t = 1 | 0.8125 | 40.6046 | 44.2813 | 41.0969 |
t = 2 | 40.6000 | 44.0375 | 40.5387 | ||
t = 4 | 39.7232 | 43.0134 | 40.6741 | ||
t = 8 | 38.3074 | 39.8453 | 38.9174 | ||
t = 16 | 36.2093 | 37.5271 | 36.5640 | ||
t = 1 | 0.7500 | 38.6814 | 42.0387 | 38.6598 | |
t = 2 | 38.4750 | 41.6335 | 38.1730 | ||
t = 4 | 38.4480 | 39.8260 | 38.7530 | ||
t = 8 | 37.1189 | 38.5990 | 38.1803 | ||
t = 16 | 34.8398 | 33.6085 | 33.4261 | ||
t = 1 | 0.5000 | 34.9586 | 36.1277 | 30.1636 | |
t = 2 | 34.7745 | 35.5442 | 29.9858 | ||
t = 4 | 34.4683 | 35.7925 | 29.7738 | ||
t = 8 | 34.0103 | 34.7392 | 29.3071 | ||
t = 16 | 17.1770 | 8.2242 | 28.3865 | ||
t = 1 | 0.4375 | 25.7846 | 27.1451 | 28.1552 | |
t = 2 | 25.5322 | 26.9129 | 27.8469 | ||
t = 4 | 25.5191 | 26.4387 | 28.0847 | ||
t = 8 | 24.9147 | 21.1803 | 27.5269 | ||
t = 16 | 5.9977 | 21.0325 | 24.9710 | ||
Mean | t = 1 | 0.8125 | 41.0056 | 44.8147 | 41.1039 |
t = 2 | 40.9707 | 44.5737 | 41.1145 | ||
t = 4 | 40.6192 | 44.2741 | 41.1011 | ||
t = 8 | 40.1403 | 43.8523 | 41.0180 | ||
t = 16 | 39.7578 | 42.6253 | 40.9543 | ||
t = 1 | 0.7500 | 39.0722 | 42.2699 | 39.1848 | |
t = 2 | 38.9532 | 42.0751 | 39.1056 | ||
t = 4 | 38.8577 | 41.6732 | 39.0522 | ||
t = 8 | 38.3079 | 41.3472 | 38.6734 | ||
t = 16 | 36.8636 | 40.0724 | 38.3980 | ||
t = 1 | 0.5000 | 35.1674 | 36.3082 | 30.3101 | |
t = 2 | 35.0681 | 36.1250 | 30.2036 | ||
t = 4 | 35.1892 | 36.2985 | 30.1288 | ||
t = 8 | 34.8385 | 35.6255 | 30.1171 | ||
t = 16 | 34.4991 | 35.2709 | 29.1196 | ||
t = 1 | 0.4375 | 25.9959 | 27.3607 | 28.3191 | |
t = 2 | 25.9664 | 27.3312 | 28.2886 | ||
t = 4 | 25.9391 | 27.0231 | 28.2801 | ||
t = 8 | 25.6793 | 26.1873 | 27.7021 | ||
t = 16 | 25.7151 | 26.8239 | 27.3460 |
In the maximum values listed, we see that there is almost no difference among the PSNR values, regardless of whether the number of dimensions of the measurement matrix was reduced by t ^{2} times, even 256 times. Moreover, some values from t > 1 were greater than those from t = 1. This indicates that the proposed algorithm is effective at sampling and reconstructing sparse signals with measurement matrices where the number of dimensions was reduced while maintaining a high level of quality. We can therefore confirm that the quality of the reconstructed image relies significantly on the random matrix generated, rather than the dimensions of the random matrix. This means that if we generate a suitable random matrix (that is, if it satisfies RIP and NSP), we can also obtain a precise reconstruction, even if the dimensions of the matrix are reduced.
In the minimum values listed, some values from t > 1 were significantly lower than those from t = 1. For instance, the sampling rate of Lena was 0.4375 and t = 16, and the PSNR was only 8.2242 dB. By calculating the corresponding mutual coherence (μ(Θ ^{16})), we found that this μ(Θ ^{16}) was considerably greater than others. This confirmed that the random matrix Φ(t) (t > 1) we generated should satisfy RIP and NSP appropriately. Thus, we can improve the stability of reconstruction quality.
Tests on other random measurement matrices with different dimensions. (M/N = 0.5, ℓ ^{0}-minimization with ρ = 0.8)
Images | Ratio of dimensionality reduction | PSNR | |||
---|---|---|---|---|---|
Bernoulli | Hadamard | Toeplitz | |||
Lena (256 × 256) | Max | t = 1 | 35.3431 | 35.2931 | 35.6554 |
t = 2 | 35.3431 | 35.3935 | 35.7550 | ||
t = 4 | 35.6510 | 35.2870 | 35.4465 | ||
t = 8 | 35.2639 | 35.3398 | 35.8770 | ||
t = 16 | 36.2619 | 35.0118 | 35.9155 | ||
Min | t = 1 | 34.9759 | 35.0136 | 34.5715 | |
t = 2 | 34.8563 | 35.0056 | 34.3634 | ||
t = 4 | 34.8053 | 34.7791 | 34.1974 | ||
t = 8 | 34.1163 | 34.1614 | 33.8705 | ||
t = 16 | 15.8923 | 33.0596 | 13.8632 | ||
Mean | t = 1 | 35.0819 | 35.1956 | 35.1887 | |
t = 2 | 35.1102 | 35.2093 | 35.1094 | ||
t = 4 | 35.0779 | 35.1018 | 34.9980 | ||
t = 8 | 34.6943 | 34.7907 | 34.9373 | ||
t = 16 | 34.5464 | 34.4744 | 34.1420 | ||
Peppers (256 × 256) | Max | t = 1 | 36.4659 | 36.5146 | 36.7284 |
t = 2 | 36.8068 | 36.5124 | 37.1701 | ||
t = 4 | 36.6284 | 36.4531 | 37.0168 | ||
t = 8 | 37.1965 | 36.6096 | 37.0168 | ||
t = 16 | 37.6965 | 36.7270 | 37.8743 | ||
Min | t = 1 | 36.1021 | 35.9480 | 36.0183 | |
t = 2 | 35.9979 | 36.0554 | 35.2744 | ||
t = 4 | 35.4798 | 35.7992 | 35.6017 | ||
t = 8 | 34.2548 | 35.9329 | 35.6017 | ||
t = 16 | 17.7105 | 34.3432 | 15.4983 | ||
Mean | t = 1 | 36.3395 | 36.2944 | 36.3954 | |
t = 2 | 36.3561 | 36.3110 | 36.2393 | ||
t = 4 | 36.0967 | 36.1344 | 36.4202 | ||
t = 8 | 35.9423 | 36.1665 | 36.4202 | ||
t = 16 | 35.3987 | 35.7520 | 35.4352 |
As demonstrated by the results in Table 4, the proposed STP approach is suitable for other random measurement matrices. It produces high-quality images with a suitable number of dimensions in the random matrix.
As shown in Fig. 6, we can see that the PSNR of the reconstructed Lena at different sampling ratios was better than the other two low-memory techniques.
During these experiments, we focused on the performance of the reconstructed one- and two-dimensional signals, where the dimensions of the random measurement matrices we used were reduced by t ^{2} times (t ≥ 1). As mentioned above, when t = 1, the dimensions of the random matrix were not reduced. This can be treated as equivalent to conventional CS. When t > 1, such as t = 2, the dimensions of the random matrix were reduced by four times, and if t = 8, the dimensions can be reduced by 64 times. From the results, we can see that increasing the value of t is an effective way to reduce the storage space of the random measurement matrix and the memory required for reconstruction. The tradeoff between more precise reconstructions and constrained storage requirements depends on the specific application, and this can significantly influence the physical implementation of CS in images, especially with embedded systems and FPGAs, where storage is limited.
5 Conclusions
In this paper, a novel approach to STP-CS was proposed. Our work aimed at reducing the amount of storage space needed with conventional compressive sensing. We provided a theoretical analysis of the acquisition process with STP-CS and that of the recovery algorithm with IRLS. Furthermore, numerical experiments were conducted on one-dimensional sparse signals and two-dimensional compressible signals, where the two-dimensional signals were the coefficients from wavelet transforms. A comparison of the numerical experiments demonstrated the effectiveness of the STP approach. Moreover, they show that our proposed STP approach for compressive sensing did not improve the quality of the reconstructed signal, yet reduced the storage space of the measurement matrix and the memory requirements for sampling and reconstruction. With a suitable reduction to the dimensionality of the random measurement matrix (e.g., when t = 2, 4, 8, or 16), we achieved a recovery performance similar to that obtained when t = 1, while the storage requirements reduced by t ^{2} times.
Although the dimensions of the random matrix were reduced and the PSNR of the reconstructed signal declined somewhat, it was also possible to improve the accuracy, provided that the generated random matrix satisfies the RIP and NSP appropriately. Moreover, the proposed algorithm is easy to execute, and additional operations for sampling and reconstruction are unnecessary. This can significantly influence physical implementations of the CS.
However, more investigation is required to improve the recovery performance and optimize the sampling and reconstruction processes. Further work remains in constructing the so-called independent identically distributed random matrix with fewer dimensions. Moreover, we shall attempt to optimize the matrix based on QR decomposition, which could help to improve the incoherence between the measurement matrix and the sparse basis. This will help to improve the performance of reconstruction. This has also motivated us to employ other measurement matrices, such as the structurally random matrix, low-rank matrix, rank-one matrix, and etc., in order to reduce required storage while maintaining or improving performance quality. Parallel framework [37, 38] will also be considered to reduce the time consuming during the reconstruction.
Declarations
Acknowledgements
The authors wish to thank the anonymous reviewers for their valuable comments and for their help in finding errors. We appreciate their assistance in improving the quality and the clarity of the manuscript.
Funding
This work was supported by the Science and Technology Project of Zhejiang Province, China (Grant No. 2015C33074, 2015C33083).
Authors’ contributions
Wang and Ye drafted the manuscript and implemented the experiments for verifying the feasibility of the proposed scheme. Ruan and Chen participated in the design of the proposed scheme and drafted the manuscript. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- D.L. Donoho, Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)MathSciNetView ArticleMATHGoogle Scholar
- E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)MathSciNetView ArticleMATHGoogle Scholar
- E. Candes, J. Romberg, Sparsity and incoherence in compressive sampling. Inverse problems 23(3), 969–985 (2007)MathSciNetView ArticleMATHGoogle Scholar
- E.J. Candes, J.K. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)MathSciNetView ArticleMATHGoogle Scholar
- L. Gan, Block compressed sensing of natural images, digital signal processing, 2007 15th international conference on. IEEE, 403–406 (2007)Google Scholar
- V. Abolghasemi, S. Ferdowsi, S. Sanei, A block-wise random sampling approach: Compressed sensing problem. Journal of AI and Data Mining 3(1), 93–100 (2015)Google Scholar
- N. Cleju, Optimized projections for compressed sensing via rank-constrained nearest correlation matrix. Appl. Comput. Harmon. Anal. 36(3), 495–507 (2014)MathSciNetView ArticleMATHGoogle Scholar
- T.T. Do, L. Gan, N.H. Nguyen, et al., Fast and efficient compressive sensing using structurally random matrices. IEEE Trans. Signal Process. 60(1), 139–154 (2012)MathSciNetView ArticleGoogle Scholar
- A. Amini, F. Marvasti, Deterministic construction of binary, bipolar, and ternary compressed sensing matrices. IEEE Trans. Inf. Theory 57(4), 2360–2370 (2011)MathSciNetView ArticleMATHGoogle Scholar
- R. Calderbank, S. Howard, S. Jafarpour, Construction of a large class of deterministic sensing matrices that satisfy a statistical isometry property. IEEE Journal of Selected Topics in Signal Processing 4(2), 358–374 (2010)View ArticleGoogle Scholar
- L Gan, T T Do, T D Tran. Fast compressive imaging using scrambled block Hadamard ensemble, signal processing conference, 2008 16th European. IEEE, 2008, 1-5Google Scholar
- H. Yuan, H. Song, X. Sun, et al., Compressive sensing measurement matrix construction based on improved size compatible array LDPC code. IET Image Process. 9(11), 993–1001 (2015)View ArticleGoogle Scholar
- Xu, Yangyang, W. Yin, and S. Osher. Learning circulant sensing kernels, Inverse Problems & Imaging, 8.3(2014) 901-923Google Scholar
- V. Tiwari, P.P. Bansod, A. Kumar. Designing sparse sensing matrix for compressive sensing to reconstruct high resolution medical images. Cogent Eng. 2(1),1-13 (2015)Google Scholar
- B. Zhang, X. Tong, W. Wang, et al., The research of Kronecker product-based measurement matrix of compressive sensing. EURASIP J. Wirel. Commun. Netw. 1(2013), 1–5 (2013)Google Scholar
- M.F. Duarte, R.G. Baraniuk, Kronecker compressive sensing. IEEE Trans. Image Process. 21(2), 494–504 (2012)MathSciNetView ArticleGoogle Scholar
- R. Otazo, E. Candès, D.K. Sodickson, Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components. Magn. Reson. Med. 73(3), 1125–1136 (2015)View ArticleGoogle Scholar
- T.T. Cai, A. Zhang, Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inf. Theory 60(1), 122–132 (2014)MathSciNetView ArticleMATHGoogle Scholar
- E Riegler, D Stotz, H Bolcskei. Information-theoretic limits of matrix completion, information theory (ISIT), 2015 IEEE international symposium on. IEEE. 1836–1840 (2015)Google Scholar
- K. Lee, Y. Wu, Y. Bresler, Near optimal compressed sensing of sparse rank-one matrices via sparse power factorization. Computer Science 92(4), 621–624 (2013)Google Scholar
- D. Cheng, H. Qi, Z. Li, Analysis and Control of Boolean Networks: A Semi-Tensor Product Approach (Springer Science & Business Media, London, 2011), pp. 19–53Google Scholar
- D.Z. Cheng, H. Qi, Y. Zhao, An Introduction to Semi-Tensor Product of Matrices and Its Applications (World Scientific, Singapore, 2012)View ArticleMATHGoogle Scholar
- D.Z. Cheng, H. Qi, A linear representation of dynamics of Boolean networks. IEEE Trans. Autom. Control 55(10), 2251–2258 (2010)MathSciNetView ArticleGoogle Scholar
- J.E. Feng, J. Yao, P. Cui, Singular Boolean networks: Semi-tensor product approach. SCIENCE CHINA Inf. Sci. 56(11), 1–14 (2013)MathSciNetView ArticleGoogle Scholar
- E. Jurrus, S. Watanabe, R.J. Giuly, et al., Semi-automated neuron boundary detection and nonbranching process segmentation in electron microscopy images. Neuroinformatics 11(1), 5–29 (2013)View ArticleGoogle Scholar
- J. Zhong, D. Lin, On maximum length nonlinear feedback shift registers using a Boolean network approach, control conference (CCC), 2014 33rd Chinese. IEEE. 2502–2507 (2014)Google Scholar
- H Wang, D Lin. Stability and linearization of multi-valued nonlinear feedback shift registers, IACR Cryptol. ePrint Arch. 253 (2015)Google Scholar
- R Chartrand, W Yin. Iteratively reweighted algorithms for compressive sensing, Acoustics, speech and signal processing, 2008. ICASSP 2008. IEEE international conference on. IEEE. 3869-3872 (2008)Google Scholar
- I. Daubechies, R. DeVore, M. Fornasier, et al., Iteratively reweighted least squares minimization for sparse recovery. Commun. Pure Appl. Math. 63(1), 1–38 (2010)MathSciNetView ArticleMATHGoogle Scholar
- E.J. Candes, M.B. Wakin, S.P. Boyd, Enhancing sparsity by reweighted ℓ_{1}-minimization. J. Fourier Anal. Appl. 14(5), 877–905 (2008)MathSciNetView ArticleMATHGoogle Scholar
- R Saab, Özgür Yılmaz. Sparse recovery by non-convex optimization--instance optimality, Applied & Computational Harmonic Analysis, 29.1(2010) 30-48Google Scholar
- E.J. Candes, T. Tao, Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52(12), 5406–5425 (2006)MathSciNetView ArticleMATHGoogle Scholar
- X.L. Cheng, X. Zheng, W.M. Han, Algorithms on the sparse solution of under-determined linear systems. Applied Mathematics A Journal of Chinese Universities 28(2), 235–248 (2013)MathSciNetMATHGoogle Scholar
- H Bu, R Tao, X Bai, et al. Regularized smoothed ℓ0 norm algorithm and its application to CS-based radar imaging, Signal Process., l.122(2016) 115-122Google Scholar
- C. Zhang, S. Song, X. Wen, et al., Improved sparse decomposition based on a smoothed L0 norm using a Laplacian kernel to select features from fMRI data. J. Neurosci. Methods 245, 15–24 (2015)View ArticleGoogle Scholar
- S. Barr. (2013) Medical image samples. [online]. Available: http://www.barre.nom.fr/medical/samples/
- C. Yan, Y. Zhang, J. Xu, et al., A highly parallel framework for HEVC coding unit partitioning tree decision on many-core processors. IEEE Signal Processing Letters 21(5), 573–576 (2014)View ArticleGoogle Scholar
- C. Yan, Y. Zhang, J. Xu, et al., Efficient parallel framework for HEVC motion estimation on many-Core processors. IEEE Transactions on Circuits & Systems for Video Technology 24(12), 2077–2089 (2014)View ArticleGoogle Scholar