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For signals reconstruction based on compressive sensing, to reconstruct signals of higher accuracy with lower compression rates, it is required that there is a smaller mutual coherence between the measurement matrix and the sparsifying matrix. Mutual coherence between the measurement matrix and sparsifying matrix can be expressed indirectly by the property of the Gram matrix. On the basis of the Gram matrix, a new optimization algorithm of acquiring a measurement matrix has been proposed in this paper. Firstly, a new mathematical model is designed and a new method of initializing measurement matrix is adopted to optimize the measurement matrix. Then, the loss function of the new algorithm model is solved by the gradient projection-based method of Gram matrix approximating an identity matrix. Finally, the optimized measurement matrix is generated by minimizing mutual coherence between measurement matrix and sparsifying matrix. Compared with the conventional measurement matrices and the traditional optimization methods, the proposed new algorithm effectively improves the performance of optimized measurement matrices in reconstructing one-dimensional sparse signals and two-dimensional image signals that are not sparse. The superior performance of the proposed method in this paper has been fully tested and verified by a large number of experiments.

The theory of compressive sensing (compressed sensing, CS) was proposed by Donoho et al. in 2006. The main idea of compressed sensing is to combine signal sampling and signal compression with the premise that the original signal is sparse or can be sparsely represented [

In the sampling process of CS, the original discrete signal is denoted as

The original signal

According to equation (

We use the orthogonal sparsifying basis

So in order to obtain

One can know that the sparsity level of the original signal, the design of the measurement matrix and the signal reconstruction algorithm are the three main parts of CS theory [

It has been proved that the lower mutual coherence between the measurement matrix and the sparsifying matrix is usually helpful to ensure the successful reconstruction of the signal [

Although these works have been contributed to optimizing measurement matrices, there are some limitations that usually require some specific experimental conditions and experimental objects in the previous works. Thus, based on the original works, it is necessary to develop a better optimization algorithm of generating the measurement matrix adaptive to a corresponding fixed sparsifying basis and improving the performance of measurement matrix with higher signals reconstruction accuracy under different measurement numbers and different signal sparsity levels, and the works of generating new optimized measurement matrices are contributed in this paper. Different from the conventional measurement matrices and the works in the literature [

The remainder of this paper is organized as follows: Section

From the above theoretical introduction of CS, one can know that the performance of the measurement matrix directly influences the results of signals reconstruction, and the measurement matrix is the important intermediate link between signal sparsifying and signal reconstruction. Candès have contributed pioneering works to the theory of compressed sensing. They have published a series of important papers about the conditions that measurement matrices should meet and the relationship between the number of measuring signals and the signal sparseness [

Generally, the sparsifying basis

According to the related mathematical theory [

The incoherence denotes the orthogonal property between the measurement matrix and the sparsifying matrix. According to the definition of the Gram matrix, minimizing the mutual coherence between

In this section, we give the mathematical definition of mutual coherence between the measurement matrix and the sparsifying matrix, and a new gradient projection based optimization algorithm is proposed to generate the new optimized measurement matrix.

On the basis of the Gram matrix, in [

Furthermore, from the perspective of between measurement matrix and sparsifying matrix, for the measurement matrix

Consequently, equation (

One can know that, to solve and minimize equation (

We can also update equation (

Our proposed new algorithm model (

For the two special terms in our new model (

In the process of solving

Then the second scalar parameter

Since loss function (

To acquire the value of

The use of parameter

Since local minimum solutions are likely to haunt us in the iterations of the algorithm, it is usually time-consuming to solve this new cost function (

First, we initialize the equivalent dictionary

Then, we suppose that the pseudo-inverse is equal to the true-inverse operation

According to

Finally, we can get the initialized measurement matrix

The above method of initializing

Consequently,

The specific and detailed steps of our proposed algorithm by gradient projection for solving problem (

The algorithm process.

Initialization: the number of iterations

Calculate the value of

Find the parameter

The line backtracking search to minimize

Update

When the algorithm performs convergence and meets the termination condition:

Stop the iteration, and

In our proposed algorithm, after the measurement matrix is initialized, and then the measurement matrix is trained and optimized by making full use of the information of the known sparsifying matrix. After a large number of experiments (the experiments in Section

It is worth noting that, because there may be both positive and negative elements in the measurement matrix

Finally, after acquiring the optimized solution of

According to the above analysis, the measurement matrix obtained by our optimization algorithm may have lower coherence with the specific sparsifying basis theoretically. Next, we will test the performance of the optimized measurement matrix by specific experiments compared with the conventional random measurement matrices and the classic optimization methods in literature [

In the section, to test the performance of the optimized measurement matrices after adding the new regularization penalty term to the traditional cost function (

According to equation (

Distribution of the off-diagonal elements of the Gram matrix obtained using different measurement matrices and different methods at

In the experiments, in order to facilitate the statistics of the number of off-diagonal elements, the absolute value of off-diagonal elements was rounded to two significant digits with the accuracy of 0.01, which means that there are 100 evenly distributed points from 0 to 1 and the number of off-diagonal elements is counted at every point. The known square DWT (discrete wavelet transform) matrix was given as the sparsifying dictionary

According to the results of Figure

The following experiments were conducted to test the performance of our proposed algorithm in practical signals reconstruction, OMP algorithm was used as the CS-based signals reconstruction algorithm. The measurement value

For the reconstruction of one-dimensional sparse signals, the mean square error (MSE) is used to evaluate the accuracy of reconstructed one-dimensional signals. The smaller MSE is, the higher the reconstruction accuracy of one-dimensional signals is MSE is defined as follows:

In the experiments, since the one-dimensional sparse signals have already been sparse, there is no need to sparsify the original signals again, so the

Results of reconstructing one-dimensional sparse signals by different measurement matrices and different methods at

As shown in Figure

The relationships between MSE and sparsity level

As shown in Figure

The relationships between MSE and measurement number (

As shown in Figure

In this subsection, we conduct the following experiments to test the performance of our method in reconstructing two-dimensional images. We choose eight representative images that are not sparse with the size of

Eight representative images of (a) Peppers, (b) Boats, (c) Building (d) Cameraman, (e) Barbara, (f) Mandrill, (g) Goldhill, and (h) Fingerprint.

For two-dimensional nonsparse images, the iteration number in the OMP algorithm is set to the measurement number

It is worth noting that, in the reconstruction of one-dimensional sparse signals, the sparsifying basis

Table

PSNRs/SSIMs of reconstructed images by different measurement matrices at

Images | Sparse random | Gaussian | Bernoulli | Part Hadamard | NoReg | Elad’s | Wang’s | Optimized matrix |
---|---|---|---|---|---|---|---|---|

Peppers | 21.2336/0.33377 | 21.2234/0.3352 | 20.9420/0.32786 | 21.4817/0.34782 | 21.4106/0.33903 | 21.4965/0.34668 | 22.3604/0.4139 | |

Boats | 20.1211/0.32113 | 20.0598/0.31796 | 20.1037/0.32355 | 20.4870/0.34079 | 21.1489/0.32162 | 20.3158/0.33196 | 21.1873/0.3976 | |

Building | 24.0236/0.43797 | 24.4098/0.44937 | 24.0987/0.44321 | 24.7301/0.47496 | 24.2669/0.44778 | 24.8264/0.47467 | 26.1096/0.57190 | |

Cameraman | 21.8334/0.43157 | 21.5152/0.41978 | 21.6693/0.41437 | 22.0669/0.46524 | 21.6392/0.42525 | 22.1898/0.45114 | 22.7035/0.52447 | |

Barbara | 20.1602/0.42315 | 20.4094/0.43275 | 20.0468/0.41262 | 20.5244/0.44034 | 20.3292/0.43005 | 20.4563/0.4436 | 21.8251/0.5315 | |

Mandrill | 16.4434/0.20460 | 16.4298/0.20306 | 16.4234/0.20594 | 16.6225/0.21432 | 16.6581/0.21474 | 16.4741/0.20784 | 16.9468/0.2518 | |

Goldhill | 20.2144/0.29227 | 20.1329/0.28863 | 20.1679/0.29005 | 20.5636/0.31045 | 20.3922/0.30052 | 20.4659/0.30194 | 21.2499/0.37623 | |

Fingerprint | 15.6476/0.25184 | 15.5595/0.24256 | 15.5394/0.24149 | 15.7223/0.25411 | 15.6447/0.24709 | 15.6734/0.25524 | 17.0946/0.38497 |

The statistical PSNRs/SSIMs of five experiment runs’ results in reconstructing image

Times | Sparse random | Gaussian | Bernoulli | Part Hadamard | NoReg | Elad’s | Wang’s | Optimized matrix |
---|---|---|---|---|---|---|---|---|

1 | 24.2901/0.44622 | 24.3995/0.44873 | 24.0885/0.43630 | 24.6580/0.47171 | 24.5528/0.46479 | 24.7666/0.47067 | 26.3642/0.58124 | |

2 | 23.5791/0.41832 | 24.4172/0.45150 | 24.1424/0.44034 | 24.3653/0.45435 | 24.5539/0.45525 | 24.8598/0.47935 | 25.9501/0.56853 | |

3 | 24.0906/0.44499 | 24.5570/0.45856 | 24.0547/0.44028 | 24.8548/0.47995 | 24.1901/0.44146 | 24.9355/0.47619 | 25.9409/0.56286 | |

4 | 24.0453/0.43432 | 24.0803/0.43536 | 24.3196/0.44327 | 24.6074/0.46906 | 23.8855/0.43135 | 24.9349/0.48395 | 26.2562/0.56896 | |

5 | 24.1129/0.44600 | 24.5950/0.45270 | 23.8883/0.45586 | 25.1650/0.49973 | 24.1520/0.44606 | 24.6354/0.46319 | 26.0366/0.57790 | |

Means | 24.0236/0.43797 | 24.4098/0.44937 | 24.0987/0.44321 | 24.7301/0.47496 | 24.2669/0.44778 | 24.8264/0.47467 | 26.1096/0.57190 | |

Variances | 0.0704/1.4532e-4 | 0.0412/7.4193e-5 | 0.0243/5.6149e-5 | 0.0895/2.7715e-4 | 0.0822/1.6450e-4 | 0.0162/6.4530e-5 | 0.0364/5.6239e-5 |

From Table

In order to visually show the improvement of the reconstructed images by our method, Figure

The reconstructed images Building by measurement matrices of (a) sparse random (PSNR: 24.0906 dB, SSIM: 0.44499), (b) Gaussian (PSNR: 24.5570 dB, SSIM: 0.45856), (c) Bernoulli (PSNR: 24.0547 dB, SSIM: 0.44028), (d) part Hadamard (PSNR: 24.8548 dB, SSIM: 0.47995), (e) No Regularization (PSNR: 24.1901 dB, SSIM: 0.44146), (f) Elad’s method (PSNR: 24.9355 dB, SSIM: 0.47619), (g) Wang’s method (PSNR: 25.9409 dB, SSIM: 0.56286), and (h) our proposed method (PSNR: 27.6707 dB, SSIM: 0.61185) at

From the experimental results of Tables

In order to further demonstrate the effectiveness of our method in reconstructing nonsparse images at a higher sampling rate, a person image Cameraman is reconstructed at the compression rate of 0.5 (

The reconstructed images Cameraman by measurement matrices of (a) sparse random (PSNR: 29.9159 dB, SSIM: 0.78685), (b) Gaussian (PSNR: 29.4204 dB, SSIM: 0.77486), (c) Bernoulli (PSNR: 29.3394 dB, SSIM: 0.77426), (d) part Hadamard (PSNR: 31.0353 dB, SSIM: 0.82327), (e) No Regularization (PSNR: 29.8907 dB, SSIM: 0.78744), (f) Elad’s method (PSNR: 31.1338 dB, SSIM: 0.8232), (g) Wang’s method (PSNR: 30.6554 dB, SSIM: 0.82174), and (h) our proposed method (PSNR: 32.0550 dB, SSIM: 0.83511) at

As shown in Figure

In addition, compressed sensing is widely applied in biomedical image processing, such as the image of magnetic resonance imaging (MRI). To verify that our method has universal performance for different types and different sizes of images and also has a good performance in reconstructing MRI images, we conducted the experiments on smaller data and reconstruct the image Brain acquired by MRI with the size of

(a) Original MRI image Brain, reconstructed images by measurement matrices of (b) sparse random (PSNR: 20.15 dB, SSIM: 0.523), (c) Gaussian (PSNR: 19.24 dB, SSIM: 0.448), (d) Bernoulli (PSNR: 19.36 dB, SSIM: 0.481), (e) part Hadamard (PSNR: 18.38 dB, SSIM: 0.443), (f) No Regularization (PSNR: 18.62 dB, SSIM: 0.438), (g) Elad’s method (PSNR: 19.29 dB, SSIM: 0.480), (h) Wang’s method (PSNR: 19.15 dB, SSIM: 0.407), and (i) our proposed method (PSNR: 23.83 dB, SSIM: 0.574) at

The reconstructed images Brain by measurement matrices of (a) sparse random (PSNR: 27.21 dB, SSIM: 0.708), (b) Gaussian (PSNR: 25.82 dB, SSIM: 0.679), (c) Bernoulli (PSNR: 26.54 dB, SSIM: 0.697), (d) part Hadamard (PSNR: 27.60 dB, SSIM: 0.722), (e) No Regularization (PSNR: 27.35 dB, SSIM: 0.720), (f) Elad’s method (PSNR: 28.23 dB, SSIM: 0.747), (g) Wang’s method (PSNR: 28.59 dB, SSIM: 0.729), and (h) our proposed method (PSNR: 30.03 dB, SSIM: 0.773) at

As shown in Figure

These above experiments have verified the superior performance of our new algorithm in optimizing measurement matrix by reconstructing one-dimensional sparse signals and two-dimensional nonsparse images under different sparsity levels and different measurement numbers. Our proposed method can make the elements distribution of the Gram matrix closer to the identity matrix than other methods. In addition, through comparing the experimental results of the new loss function (

Our optimized measurement matrices perform well in extracting information from original signals and restoring information from sampled signals so that our new method can reconstruct original signals of higher precision with higher probability. The main contributions of this paper are as follows:

The new algorithm model (

A new measurement matrix initialization method is designed based on the characteristics of our proposed new algorithm model (

Due to the coupling of measurement matrix and sparsifying basis, it is usually time-consuming to solve the model (

Based on the characteristics of our method, our proposed algorithm can find a solvable measurement matrix adaptive to the corresponding fixed sparsifying basis and have good flexibility and adaptability in reconstructing CS-based signals.

In addition, here are two important matters worth being discussed:

First, the amount of information the measurement matrix extracts or samples from original signals increases with the rise of measurements number, which makes it easier to recover the original signals for the reason that more valuable information can be obtained from original signals. This is why the improvement degree of PSNR and SSIM in reconstructing two-dimensional images by our method is smaller at a compression rate 0.5 than at a compression rate 0.25. Thus, the trade-off between compression rate and recovery accuracy has to be considered fully in the process of measuring and recovering signals.

Second, small mutual coherence between the measurement matrix and sparsifying basis would ensure that original signals may be recovered successfully with high probability. The incoherence property is only a necessary condition and one of many factors (such as signal reconstruction algorithm, signal sparsifying transform, the original signal itself, and so on) that affect the final accuracy of signal reconstruction and the main work of this paper is to only optimize the measurement matrix, so the incoherence property cannot ensure that original signals are reconstructed successfully and precisely with a probability of 100% but just a large probability. Thus, the higher frequency of the off-diagonal elements with low absolute value in Figure

In this paper, the gradient projection based strategy is used to solve our proposed new algorithm model with the new method of initializing measurement matrix for compressively sensed signals reconstruction, which is a new idea for acquiring optimized measurement matrices via minimizing the mutual coherence between measurement matrix and sparsifying basis. According to the theoretical analysis of our method and the experimental results, we can conclude that our proposed measurement matrix optimization method has good performance and outperforms the conventional random measurement matrices and some existing optimization methods. Not only one-dimensional sparse signals but also two-dimensional nonsparse images can be reconstructed by our optimized measurement matrices with less error, higher accuracy, and better quality. In addition, this new algorithm also has good performance in reconstructing the MRI image. Thus, our research by proposing the new algorithm in this paper may enlighten wide exploration in this direction of the field and has potential application value in the broader field of signal processing. It is of great worth to make further research in the incoherence property and the optimization algorithm of the measurement matrix in compressive sensing. This is also the focus of our future works to extend our research to broader applications.

The data used to support the findings of this study are included within the article.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Ziran Wei made significant contributions to this study regarding conception, methodology and analysis, and writing the manuscript. Jianlin Zhang contributed much to the research through his general guidance and advice and optimized the paper. Zhiyong Xu made research plans and managed the study project. Yong Liu gave lots of advice and helped in conducting experiments.

The authors are grateful to the Fraunhofer Institute for Computer Graphics Research IGD of Germany for full support. This research was funded by the National High Technology Research and Development Program of China (863 Program, grant no. G158207), the West Light Foundation for Innovative Talents (grant no. YA18K001), and the UCAS-Fraunhofer Joint Training Ph.D. Scholarship.