# Fast sparsity adaptive matching pursuit algorithm for large-scale image reconstruction

**Part of the following topical collections:**

## Abstract

The accurate reconstruction of a signal within a reasonable period is the key process that enables the application of compressive sensing in large-scale image transmission. The sparsity adaptive matching pursuit (SAMP) algorithm does not need prior knowledge on signal sparsity and has high reconstruction accuracy but has low reconstruction efficiency. To overcome the low reconstruction efficiency, we propose the use of the fast sparsity adaptive matching pursuit (FSAMP) algorithm, where the number of atoms selected in each iteration increases in a nonlinear manner instead of undergoing linear growth. This form of increase reduces the number of iterations. Furthermore, we use an adaptive reselection strategy in the proposed algorithm to prevent the excessive selection of atom. Experimental results demonstrated that the FSAMP algorithm has more stable reconstruction performance and higher reconstruction accuracy than the SAMP algorithm.

## Abbreviations

- CoSaMP
Compressive sampling matching pursuit

- CS
Compressed sensing

- FSAMP
Fast sparsity adaptive matching pursuit

- MP
Matching pursuit

- OMP
Orthogonal matching pursuit

- PSNR
Peak signal-to-noise ratio

- ROMP
Regularized orthogonal matching pursuit

- RT
Reconstruction time

- SAMP
Sparsity adaptive matching pursuit

- SP
Subspace pursuit

- StOMP
Stagewise orthogonal matching pursuit

- SWOMP
Stagewise weak orthogonal matching pursuit

## 1 Introduction

The explosive growth of information has brought a great burden for signal processing and storage. In some application scenarios with resource strain on computing and bandwidth, the sampling frequency required in the tradition Nyquist sampling theorem makes signal acquisition, processing, storage, and transmission under the pressure of massive data. Particularly, the Nyquist sampling theorem increases the cost and lowers the effectiveness of data acquisition and processing equipment in the transmission and processing of large-scale image data [1, 2].

The emergence of the compressed sensing (CS) theory solves the problems caused by the limitation of sampling frequency and drives the signal processing into a new stage. Signal processing, data collection, and data compression are simultaneously performed through CS (synchronize) [3, 4]. That is, the methodology of signal processing in CS reduces the number of measurements during the sampling process but still retains sufficient information. Therefore, it has a great application prospect in large-scale image processing owing to its low measurement frequency and high reconstruction precision [5, 6, 7].

CS involves a three-part process, namely, signal sparse representation, signal compression under measurement matrix, and signal reconstruction. CS mainly addresses the issues regarding the improvement of reconstruction algorithm design. The performance of reconstruction algorithm is mainly reflected in the two aspects of reconstruction efficiency and reconstruction accuracy. Although the performance of signal acquisition process in CS is better, but the signal reconstruction accuracy or the signal reconstruction efficiency is low, CS is not practical. Therefore, the key in the application of CS is to design a good reconstruction algorithm that can balance the reconstruction efficiency and the reconstruction accuracy.

Presently, matching pursuit (MP) algorithms demonstrate excellent reconstruction performance in CS, although some of them require prior knowledge of signal sparsity and are thus less practical. Some MP algorithms, such as the sparsity adaptive matching pursuit (SAMP) algorithm, do not need this knowledge. The SAMP algorithm does not have constraints with atom selection threshold and has high reconstruction accuracy. However, its reconstruction time is extremely long because of repeated iteration when it approximates sparse signals.

To extend the application of the CS theory to large-scale image signal processes, we focus on improving the reconstruction efficiency of the SAMP algorithm. In the SAMP algorithm, a high reconstruction accuracy is obtained by using a small initial step-size. Unfortunately, this setting causes the SAMP algorithm to run massive iteration in order to adjust the number of selected atoms for the adaptive approximation of signal sparsity. Therefore we propose a fast sparsity adaptive matching pursuit (FSAMP) algorithm where the number of selected atoms is changed from an original linear to a nonlinear growth form. In the FSAMP algorithm, the initial step-size is set at a large value, and the step-size gradually shrinks in each iteration until the number of selected atoms is precisely approximate to the signal sparsity. This method decreases the number of iterations and shortens reconstruction time. Meanwhile, to lower the impact on reconstruction accuracy because of the changes in atom selection method of FSAMP algorithm, we introduce a reselection strategy which prunes the selected atoms to ensure reconstruction accuracy.

The rest of the paper is organized as follows. In Section 2, we review the CS and provide the related works on image reconstruction algorithms. The detailed descriptions of the proposed FSAMP algorithm are provided in Section 3. In Section 4, we discuss some experimental results. Finally, the conclusion and future work are shown in Section 5.

## 2 Related work

The CS theory indicates that the high-dimensional sparse signal after sparse representation can be projected to a low-dimensional space by using a measurement matrix uncorrelated with the transform base when a signal is compressible or can be sparsely represented by a transform base. Thereafter, the original signal can be exactly constructed from the very small amount of projection signals by solving an optimization problem. The mathematical model of CS can be expressed as

The expression Ψ ∈ *R*^{S × M} is the measurement matrix, Φ ∈ *R*^{M × N} is the transform or dictionary base, and Θ = ΨΦ is the sensing matrix. The expression *Y* = Φ*X*shows that the original signal *Y* can be sparsely represented in Φ.

*Y*can be reconstructed by solving the L0-minimization problem.

*N*-dimensional reconstructed sparse signal,

*F*is

*S*-dimensional measurement signal, and Θ ∈

*R*

^{S × N}is sensing matrix. When the

*Y*∈

*R*

^{M × 1}is

*K*-sparse signal, the

*Y*do not need to be sparsely represented, so the dictionary base is an identity matrix. The

*Y*can be compressed to a smaller signal

*F*∈

*R*

^{S × 1}(

*S*< <

*M*) by measurement matrix Ψ [8] and can be expressed as follow:

*Y*∈

*R*

^{M × 1}is

*K*-sparse signal, Function (2) can be solved by the

*l*

_{0}-norm minimization based on CS [9, 10]. The

*Y*can be exactly reconstructed by measurement signal

*F*. Thus,

However, Donoho [11] indicated that the problem of the *l*_{0}-norm minimization is NP-hard; an exhaustive search on the \( {C}_M^K \) combinations of *Y* is necessary to the acquisition of a global optimal solution. Therefore, the algorithms for obtaining suboptimal solution are provided in succession, and these algorithms are divided into three kinds, namely, convex optimization algorithms, combination algorithms, and greedy algorithms [12]. Convex optimization algorithms have fairly high reconstruction accuracy and less measurement, but the complicated reconstruction process affects its practicability. Combination algorithms have shorter reconstruction time than convex optimization algorithms but need more measurements, which are hardly satisfied in practice. Greedy algorithms have low complexity and high reconstruction efficiency, although their reconstruction accuracy is inferior to the convex optimization algorithm. Nevertheless, greedy algorithms have better application prospect, and MP algorithms mostly represent greedy algorithms. Therefore, we investigated MP algorithms to increase reconstruction efficiency and reconstruction accuracy.

MP algorithm was first proposed by Mallat and Zhang [13]. In each iteration, MP algorithm selects a column vector (atom) from the measurement matrix that is maximally correlated with the current residual, where initial residual is the measurement signal, then the differences between the original sparse signal and reconstructed sparse signal diminish after the atom selection. When the residual reaches the preset threshold, the sparse signal can be concluded to be accurately reconstructed by the measurement matrix. The disadvantage of MP algorithm is that the residuals are only orthogonal to the current selected atom, and the selected atom has the possibility of being repeatedly selected during iterations. Such iterations render the MP algorithm difficult to converge. Pati et al. [14] proposed orthogonal matching pursuit (OMP) algorithm. OMP inherits the atom selection strategy of MP algorithm but makes the selected atoms to be orthogonal to each other. This improvement solves the problem of MP being hard to converge.

On the basis of OMP, Needell and Tropp proposed compressive sampling matching pursuit (CoSaMP) [15] algorithm. In contrast to the OMP algorithm, CoSaMP algorithm selects two K optimal-related atoms simultaneously then discards K atoms selected before the next iteration. The CoSaMP algorithm is robust noise because of its backtracking strategy for atom selection. A similar atom selection strategy is available between CoSaMP algorithm and subspace pursuit (SP) algorithm [16]. For a better reconstruction result, Needell and Vershynin introduced the regularization constraint to the atom selection strategy of regularized orthogonal matching pursuit (ROMP) algorithm, which selects K optimal-related atoms, then reselect atoms from the previous selected K atom based on the regularization constraint [17]. However, these reconstruction algorithms need to know signal sparsity in advance, which is extremely hard to obtain in practice. Therefore, the practical application of these reconstruction algorithms is not as successful as the theoretical research.

In order to break the constraint of signal sparsity on MP algorithms, Dohono et al. [18] proposed a stagewise orthogonal matching pursuit (StOMP) algorithm. StOMP algorithm uses a preset threshold to determine the process of atom selection, and do not need the prior knowledge of signal sparsity. Then, an improved algorithm of StOMP, a stagewise weak orthogonal matching pursuit (SWOMP) algorithm [19], was proposed by Blumensath and Davies. SWOMP algorithm changes the method of threshold setting during atom selection and lowers the requirements on the measurement matrix compared with StOMP algorithm. The SAMP algorithm is also an MP algorithm that does not depend on signal sparsity [20]. The atom selection process of SAMP algorithm is not constrained by the preset threshold in contrast to those of the StOMP and SWOMP algorithms, and the number of selected atoms in SAMP algorithm is determined by a fixed step-size. In case of a small step-size, the high reconstruction accuracy of SAMP algorithm corresponds with long reconstruction time. To shorten the reconstruction time of SAMP algorithm, some scholars proposed the improvements. Yu found that the fixed step-size of SAMP algorithm is the reason for the long reconstruction time, so they introduced a variable step-size and backtracking strategy to improve SAMP algorithm and decrease the number of iterations [21]. Huang et al. introduced a regularization constraint to atom selection and used different step-sizes in each subsection of iterations to shorten reconstruction time [22].

Based on the above algorithms, we conclude that obtaining good criteria for atom selection is the research priority for MPs. The SAMP algorithm has high reconstruction accuracy and low reconstruction efficiency. Therefore, further research must focus on the improvement of reconstruction efficiency. However, the two improved algorithms only consider one-dimensional signal but not verify their validity on large-scale image signal, while the improvements on reconstruction efficiency are limited.

In this paper, we aim to identify a highly efficient and accurate FSAMP algorithm large-scale image reconstruction. Specifically, the main contributions of this paper are as follows: (1) reduction of the number of iterations and reconstruction time of the SAMP algorithm, increase of the number of atoms selected in each iteration through nonlinear growth instead of linear growth, and gradual reduction of initial large step-size in the iterations until the number of selected atoms are precisely approximate to the signal sparsity; and (2) prevention of the excessive selection of atom, the introduction of an adaptive reselection strategy based on the varied residuals, and the deletion of the mismatching atoms for high reconstruction accuracy.

## 3 Fast sparsity adaptive matching pursuit algorithm

FSAMP preserves the atom selection method of SAMP algorithm. FSAMP still selects *L* atoms, which have the largest inner products with the current residual, then judges whether the number of atoms increase or not according to the changes between the current residual and the last residual. Compared with SAMP algorithm, in FSAMP, the number of atom *S*_{ t } increases at each iteration based on the current number of iterations, rather than the increase in the fixed step-size in SAMP algorithm, where *t* is the number of iterations. Detailed implementation of the process of FSAMP algorithm is as follows:

Input: the sensing matrix*A* = ΦΨ (*A* ∈ *R*^{M × N}), measurement signal*y*, and the parameter step-size sequence *s*.

Output: sparse signal \( \overset{\wedge }{\theta } \).

- (1)
Initialization: the initial residual

*r*_{0}=*y*, the index set of selected atoms Λ_{0}=Ø, the set of selected atoms Ω_{0}=Ø,*S*_{ t }= arctan (*s*) ×*M*/4*π*,*En*=*s*,*Ds*=*M*/4,*ς*=*linspace*(*S*_{ t },*En*,*Ds*), the number of selected atoms*L*=*ς*_{ t }, and iteration count*t*= 1; - (2)
Compute the inner product of current residual and the sensing measurement matrix,

*u*_{ t }= |〈*A*,*r*_{t − 1}〉|, and find the index values*ϑ*_{t}corresponding to*L*maximum values from*u*_{ t }; - (3)
Add the index values

*ϑ*_{t}to index set ∧_{t}, ∧_{t}= ∧_{t ‐ 1}∪*ϑ*_{ t }, and let the set of selected atoms*ζ*_{t}correspond to the largest*L*elements of*A*, Ω_{ t }= Ω_{t − 1}∪*ξ*_{ t }; - (4)
Solve the least squares solution of \( y={\Omega}_t{\overset{\wedge }{\theta}}_t \), \( {\overset{\wedge }{\theta}}_t={\left({\Omega}_t^T{\Omega}_t\right)}^{-1}{\Omega}_t^Ty \);

- (5)
Update residual \( {r}_t=y-{\Omega}_t{\overset{\wedge }{\theta}}_t=y-{\Omega}_t{\left({\Omega}_t^T{\Omega}_t\right)}^{-1}{\Omega}_t^Ty \);

- (6)
If ‖

*r*_{ t }‖_{2}<*ε*, stop iteration and proceed to (8); otherwise, proceed to (7); - (7)
If ‖

*r*_{ t }‖_{2}≤ ‖*r*_{t ‐ 1}‖_{2}and*L*<*N*, update the number of atom selection*L*=*L*+*ς*_{ t },*t*=*t*+ 1, and proceed to (2); if ‖*r*_{ t }‖_{2}> ‖*r*_{t ‐ 1}‖_{2}and*L*<*N*, update*L*= ⌈*L*−*γL*⌉,*γ*= ‖*r*_{t − 1}‖_{2}/‖*r*_{t}‖_{2}; if none of the above is satisfied, stop iteration and proceed to (8); - (8)
Obtain the reconstructed original signal based on \( {\overset{\wedge }{\theta}}_t \) and the dictionary base Ψ.The

*ς*=*linspace*(*S*_{ t },*En*,*Ds*) is a linear descending sequence which is determined by three parameters,*S*_{ t },*En*, and*Ds. S*_{ t }= arctan (*s*) ×*M*/4*π*, which defines the initial value of the descending sequence*ς*.*En*defines the last value of the descending sequence*ς*, where the default value of*En*is*s*.*Ds*is used to determine the length of the descending sequence*ς*, where the default value of*Ds*is*M*/4, and*M*is the measurement frequency which is equal to the number of rows of the measurement matrix.

In the reconstruction process, FSAMP algorithm uses *S*_{ t } to determine the initial value of step-size. The arctan (*s*) in formula for *S*_{ t } is to ensure that the small parameter *s* corresponds to small initial step-size, and the large parameter *s* corresponds to convergence in initial step-size. The arctan (*s*) in formula for *S*_{ t } limits the increase of the step-size sequence whether the parameter *s* is large or small, and make the FSAMP algorithm robust for varying parameter *s*.

In iterations, when the difference between the signal sparsity and the number of atom selection adjust to large step-size sequence, the corresponding residual will suddenly increase. To avoid this situation, FSAMP algorithm adjusts the number of atom selection based on the changes of residuals in step (7). When L_{2}-norm of the current residual is larger than L_{2}-norm of the last residual of the iteration, FSAMP algorithm reduces the number of atoms. The number of deleted atoms depends on the ratio between L_{2}-norm of the current residual and L_{2}-norm of the last residual of the iteration. ⌈ ⌉ is a top integral function and ⌈*I* − *γI*⌉ is the smallest integer greater than or equal to (*I* − *γI*).

## 4 Simulation results and disscussion

*M*is measurement frequency. The Φ generated by Eq. (5) also has a very strong randomness. When the measurement frequency

*M*≥

*cK*log(

*N*/

*K*), the Bernoulli random matrix is able to satisfy the RIP criterion with great probability [23], where

*c*is a small constant,

*K*is the sparsity of signal, and

*N*is the signal dimension, as well as the number of columns in measurement matrix. Compared with Gaussian random measurement, the elements of Bernoulli random measurement matrix are relatively simple which make Bernoulli random measurement matrix easier to store in practical applications.

### 4.1 The comparison of reconstruction performance under Gaussian random measurement matrix

The PSNR (dB) averages of all test images of all reconstruction algorithm under different compression ratios using Gaussian random measurement matrix

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 6.08 | 11.71 | 16.07 | 21.39 |

StOMP | 9.27 | 9.27 | 9.27 | 9.27 |

SWOMP | 2.46 | 5.99 | 17.51 | 26.08 |

SAMP | 5.52 | 11.47 | 18.54 | 28.40 |

FSAMP | 5.45 | 12.13 | 21.10 | 29.58 |

The RT (reconstruction time) averages of all test images of all reconstruction algorithm under different compression ratios using Gaussian random measurement matrix

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 0.01076 | 0.05755 | 0.21668 | 0.64998 |

StOMP | 0.00015 | 0.00017 | 0.00023 | 0.00040 |

SWOMP | 0.00291 | 0.03464 | 0.26225 | 0.55218 |

SAMP | 0.23592 | 2.20282 | 10.07340 | 29.93917 |

FSAMP | 0.06924 | 0.21072 | 0.51435 | 1.05438 |

The average number of iterations of all test images under different compression ratios using Gaussian random measurement matrix

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 51 | 102 | 154 | 205 |

StOMP | 1 | 1 | 1 | 1 |

SWOMP | 3.5 | 6.7 | 9.6 | 10.1 |

SAMP | 205.0 | 410.0 | 613.9 | 818.5 |

FSAMP | 39.4 | 29.0 | 25.6 | 24.1 |

### 4.2 Comparison of reconstruction performance under Bernoulli random measurement matrix

The PSNR (dB) averages of all test images of all reconstruction algorithm under different compression ratios using Bernoulli random measurement matrix

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 9.05 | 14.20 | 17.92 | 21.06 |

StOMP | 8.89 | 14.46 | 18.42 | 21.97 |

SWOMP | 3.05 | 15.44 | 16.38 | 24.53 |

SAMP | 8.46 | 13.64 | 18.50 | 26.13 |

FSAMP | 8.31 | 12.77 | 21.14 | 27.49 |

However, the reconstruction result of Table 4 shows that the reconstruction accuracy of FSAMP is the highest, and its PSNR averages are 2 dB higher than other reconstruction algorithms.

The average RT values of all test images of the reconstruction algorithm under different compression ratios

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 0.01177 | 0.05907 | 0.23321 | 0.71140 |

StOMP | 0.00593 | 0.01115 | 0.02812 | 0.06358 |

SWOMP | 0.00351 | 0.03551 | 0.26978 | 0.56118 |

SAMP | 0.23588 | 2.42514 | 10.58985 | 29.67760 |

FSAMP | 0.06889 | 0.22308 | 0.53588 | 1.05328 |

The average number of iterations of each test image under different compression ratios

PSNR | Compression ratio | |||
---|---|---|---|---|

0.2 | 0.4 | 0.6 | 0.8 | |

OMP | 51 | 102 | 154 | 205 |

StOMP | 9.9 | 9.7 | 9.8 | 10.1 |

SWOMP | 3.6 | 6.7 | 9.6 | 10.0 |

SAMP | 205.0 | 410.1 | 614.0 | 818.6 |

FSAMP | 39.4 | 29.1 | 25.7 | 24.1 |

Basing on the PSNR results, reconstruction time, and number of iterations, we conclude that the FSAMP algorithm has the best reconstruction performance among the five reconstruction algorithms whether under Gaussian or Bernoulli random measurement matrices. Furthermore, given the sufficient randomness of the Gaussian random measurement matrix elements, the reconstruction performance of the five reconstruction algorithms under the Gaussian random measurement matrix are better than those under the Bernoulli random measurement matrix. The Gaussian random measurement matrix is more consistent with respect to the design requirement of the measurement matrix for CS.

## 5 Conclusions

Obtaining timely and accurate reconstruction results is the key focus of CS application for large-scale image transmission. MP algorithms exhibit optimal reconstruction performance with respect to reconstruction accuracy and reconstruction time. However, some MP algorithms require prior knowledge of the signal sparsity, and other MP algorithms that do not require this knowledge have unstable reconstruction accuracy or long reconstruction time.

In this regard, we focus on the reconstruction time of the SAMP algorithm, which does not need signal sparsity in advance and demonstrates high reconstruction accuracy. Therefore, we propose an FSAMP algorithm where the number of selected atoms is changed from the original linear growth model to a nonlinear one. The FSAMP algorithm starts at a large step-size and gradually shrinks in iterations until the number of selected atoms is precisely approximate to the signal sparsity. To prevent the excessive selection of atom, the FSAMP introduces the adaptive reselection strategy on the basis of varied residuals and delete mismatching atoms to increase its reconstruction accuracy. Overall, the FSAMP algorithm exhibits optimal reconstruction performance among the above five reconstruction algorithms whether under Gaussian or Bernoulli random measurement matrices.

## Notes

### Acknowledgements

We would like to thank the anonymous reviewers for their insightful comments on the paper, as these comments led us to an improvement of the work.

### Funding

This work was supported by the Fundamental Research Funds for the Central Universities (2042017kf0044), China Postdoctoral Science Foundation (Grant No. 2016M592409, No. 2017M612511), the National Natural Science Foundation of China (Nos. 61701453, 61572372, 41671408), and Hubei Provincial Natural Science Foundation of China (No. 2017CFA041).

### Authors’ contributions

SHY and SW contributed to the main idea; SHY designed and implemented the algorithms and drafted manuscript; QFG contributed to the algorithm design, performance analysis, and simulations. XX helped revise the manuscript. All authors read and approved the final manuscript.

### Authors’ information

SHY is a post doctor with the Faculty of Information Engineering, China University of Geosciences, Wuhan, China. Her research interests include image processing, machine vision, and wireless communication.

QFG is a professor at the Faculty of Information Engineering at China University of Geosciences. His research topics include geographic information systems (GIS) and science (GIScience), land-use and land-cover change, and human-environment relationships and interactions.

WS is a graduate student at the Faculty of Information Engineering at China University of Geosciences. His research interests include image processing and machine vision.

XX is serving as an assistant professor in Urban and Environmental Computation at the Research Center for Industrial Ecology & Sustainability, Institute of Applied Ecology, Chinese Academy of Sciences and the vice director in Key Laboratory for Environment Computation & Sustainability of Liaoning Province. Her research interests mainly focus on the virtual geographic environments (VGE) and dynamic and multi-dimensional GIS (DMGIS).

### 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.

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