Fast compressed sensing analysis for imaging reconstruction with primal dual interior point algorithm

Optics and Lasers in Engineering 129 (2020) 106082

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Fast compressed sensing analysis for imaging reconstruction with primal dual interior point algorithm Lianying Chao a, Jiefei Han b, Lisong Yan a,∗, Liying Sun b, Fan Huang b, ZhengBo Zhu a, Shili Wei a, Huiru Ji a, Donglin Ma a a b

School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan, 430074, China Jiaoshi Intelligent Technology Co.,Ltd, Suzhou, 215000, China

a r t i c l e

i n f o

Keywords: Compressed sensing Primal dual interior point algorithm Ghost imaging Sparsity

a b s t r a c t Compressed sensing (CS) can recover a signal from a small number of observed transforms of that signal. It mainly consists of two complimentary elements including compressed sampling and computational image reconstruction. In this work, we have developed a ghost imaging system and proposed a primal dual interior point compressed sensing algorithm. We also demonstrated the performance of our imaging system and CS algorithm with simulations and experiment compared to conventional approaches. The discrete samples signal can be reconstructed by our CS algorithm. Experimental results show that the proposed compressed imaging method outperforms the conventional CS approaches in both computational time and reconstruction accuracy.

1. Introduction The first to put CS into practice is the single-pixel camera developed by researchers at Rice University in the United States in the field of optical imaging, which effectively reduces the number of sensors and hardware costs in special occasions. The sparse Magnetic Resonance Imaging (MRI) proposed by Lusting improves the imaging speed, reduces the radiation time and greatly reduces the harm to human body. CS theory is applied to the acquisition of radar image data, which reduces the acquisition and storage of image data and reduces the computational burden of satellite image. Some researchers have applied CS to image compression and reconstruction of robots and solved the problems of massive data storage and data processing related to embed environment vision of mobile robots. Imaging is an important way of information acquisition. Among kinds of imaging methods, array Charge-coupled Device (CCD) camera is one of the most common sensor device in scientific imaging owing to its high resolution, broad spectral response, high quantum efficiency, acceptable signal-to-noise ratio, rapid response, small size, and durability [1]. In spite of this, array CCD cameras are not perfect detectors when they are used as instruments for accurate measurements in complex environment. The traditional way of array CCD imaging in complex environment has some problems. For instance, due to strong scattering medium in the nature, including cloud and fog, and complicated environments, such

∗

as turbulence, the light on the transport link will be absorbed by the media and affected by the strong scattering effect. It will cause severe energy attenuation and light deflection, which leads to weak target echo signal and eventually causes the quality of the images to be decreased, or, in the worst cases, no image. To sum up, with the problem of weak echo signals and high noise interferences in complex environments, the traditional array CCD imaging method cannot achieve a good picture. In this case, a ghost imaging system with the single-pixel camera can solve this problem effectively. The single-pixel camera [2–4] developed by Ricea University Replaces the detector array with a combination of a DMD (Digital Micromirror Device) and a single-pixel detector. It can sample and recover images in the framework of CS [5–9], which is a recent approach that can enable power signal acquisition in realistic measurement times, since it allows the complete reconstruction of a function from a reduced number of measurements. The principle of this technique is to recover a signal from a small number of observed transforms of that signal. It consists of two complimentary elements including compressed sampling and computational image reconstruction. Liheng Bian et al. presented the derivation of different algorithms, which are classified into three categories including non-iterative methods, linear iterative methods and non-linear iterative methods [10]. The non-iterative methods perform direct reconstruction of the target scene without iteration. Linear iterative methods including gradient descent (GD) algorithm, conjugate gradient descent (CGD) algorithm, Poisson maximum likelihood algorithm and alternating projection algorithm.

Corresponding author. E-mail address: [email protected] (L. Yan).

https://doi.org/10.1016/j.optlaseng.2020.106082 Received 10 January 2020; Received in revised form 17 February 2020; Accepted 27 February 2020 Available online 5 March 2020 0143-8166/© 2020 Elsevier Ltd. All rights reserved.

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Optics and Lasers in Engineering 129 (2020) 106082

Fig. 1. Ghost imaging system.

Non-linear iterative methods including sparse representation prior algorithm and total variation regularization prior algorithm. CS has already been successfully applied in different fields of physics, chemistry and engineering [11]. For instance, the high potential of CS was predicted for Fourier transform infrared spectroscopy [12] and computational imaging [13–18]. The applications of compressive single-pixel imaging also include remote sensing [19], Lidar [20], gas leak monitoring [21], optical security [22–23], and optical computing [24]. The goal of our work is to extend the application of CS to imaging in extremely complex environment. In this paper, we focus on the ghost imaging system and primal dual interior point compressed sensing algorithm. By ghost imaging system, we demonstrate the performance gain in imaging with our CS algorithm. At the same time, considering the capability in weak signal collection of our ghost imaging system, it shows the great potential of complex background imaging in a remote distance. The paper is organized as follows. In Section 2, the basic framework of our imaging system is described and the principle of our CS algorithm is presented. In Section 3, the effectiveness of our method is demonstrated with simulations. In Section 4, we demonstrate the performance of our CS algorithm by testing a target card. Finally, Section 5 provides conclusions. 2. Theory 2.1. Principles of ghost imaging system Fig. 1 illustrates the basic principles of the ghost imaging system. Active lighting is adopted in the system. Light modulated with light matrix modulation module falls on the target object. The reflected light from the target is focused on the single-pixel detector and the energy intensity of the echo signal is measured. By designing different matrix data with light matrix modulation module, measurement matrix could be achieved with single-pixel detector. The target image can be reconstructed by the CS algorithm. In the process of image reconstruction, primal dual interior point algorithm is adopted to reduce the influence of complex environment on imaging system and improve the quality of reconstructed images. 2.2. Compressed sensing algorithm CS is a new data analysis technique for efficient recording of a signal that has a sparse representation with respect to some basis. It is effective in capturing and representing compressible images at a rate significantly below the Nyquist rate by exploiting the sparse domains of the image data.

Fig. 2. Flowchart of CS algorithm.

The CS algorithm is shown in Fig. 2. The algorithm updates the original and dual variables during each iteration [25]. It uses the Newton method in combination with the improved KKT (Karush–Kuhn–Tucker) equation to calculate the search direction. In the process of original signal reconstruction, the measurement matrix AM × N and the observed value bM × 1 are known to solve the underdetermined square 𝐴𝑥 = 𝑏. The goal is to find the sparsest solution xN × 1 , so the question translates into: {

min ‖𝑥‖1 𝑠.𝑡𝐴𝑥 = 𝑏

(1)

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Optics and Lasers in Engineering 129 (2020) 106082

Set a normal number ui , make −𝑢𝑖 ≤ 𝑥𝑖 ≤ 𝑢𝑖 , then the problem translates into:

3. Simulation

𝑛 ∑ ⎧ min 𝑢𝑖 ⎪ 𝑖=1 ⎪ ⎨𝑠.𝑡.𝑥𝑖 − 𝑢𝑖 ≤ 0𝑖 = 1, 2, ..., 𝑛 ⎪−𝑥𝑖 − 𝑢𝑖 ≤ 0𝑖 = 1, 2, ..., 𝑛 ⎪ ⎩𝐴𝑥 = 𝑏

In order to verify the correctness and effectiveness of the proposed algorithm, images with 100 × 100 pixels is used in the simulation. At the same time, four algorithms OMP (Orthogonal Matching Pursuit) [26], CoSaMP (Compressive Sampling Matching Pursuit) [27], SP (Subspace pursuit) [28], and SAMP (Sparsity adaptive matching pursuit) [29] algorithm are also implemented in MATLAB. For the OMP, CoSaMP and SP algorithm, the sparseness of the sparse signal should be known in advance. However, in most cases it is unknown in actual. Although for the SAMP algorithm, sparsity is not needed and only a specific step need to set, the convergence time increases dramatically, which makes it difficult to be applied to actual situation. For our proposed original dual interior point algorithm, the sparsity is not need in advance and images can be reconstructed with a satisfactory time.

(2)

Set 𝑓𝑢1 = 𝑥 − 𝑢, 𝑓𝑢2 = −𝑥 − 𝑢

(3)

Calculate its corresponding dual variables: 𝜆u1 = −

1 1 ,𝜆 = − 𝑓𝑢1 u2 𝑓𝑢2

(4)

The center residual, the dual residual, and the original residual at the point (x, u, v, 𝜆u1 , 𝜆u2 )by the improved KKT (Karush–Kuhn–Tucker) condition are: ( ) −𝜆𝑢1 𝑓𝑢1 1 𝑟𝑐𝑒𝑛𝑡 = − (5) −𝜆𝑢2 𝑓𝑢2 𝑡 ( 𝑟𝑑𝑢𝑎𝑙 =

) 𝜆𝑢1 − 𝜆𝑢1 + 𝐴𝑇 𝑣 1 − 𝜆𝑢1 − 𝜆𝑢1

(6)

r𝑝𝑟𝑖 = 𝐴𝑥 − 𝑏

(7)

The initial point and the Newton direction are respectively recorded as: y = (𝑥, 𝑢, 𝑣)Δy =(Δ𝑥, Δ𝑢, Δ𝑣)

(8)

The linear equation that determines the direction of Newton can be expressed as: D𝑟𝑡 (𝑦)Δ𝑦 = −𝑟𝑡 (𝑦)

(9)

Then there is ⎛Σ1 ⎜Σ ⎜ 2 ⎝𝐴

Σ2 Σ1 0

1 + 𝑓𝑢−1 ) − 𝐴𝑇 𝑣⎞ 𝐴𝑇 ⎞⎛Δ𝑥⎞ ⎛− 𝑡 (−𝑓𝑢−1 1 2 ⎜ ⎟ 0 ⎟⎜ Δ𝑢 ⎟ = ⎜ −1 − (− 1 )(𝑓 −1 + 𝑓 −1 ) ⎟ 𝑢1 𝑢2 𝑡 ⎟⎜ ⎟ ⎜ ⎟ 0 ⎠⎝Δ𝑣⎠ ⎝ 𝑏 − 𝐴𝑥 ⎠

(10)

The corresponding variable is Σ1 = 𝑑 𝑖𝑎𝑔 (𝜆𝑢1 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 ) − 𝑑 𝑖𝑎𝑔((𝜆𝑢2 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 ) 1 2

(11)

Σ2 = 𝑑 𝑖𝑎𝑔 (𝜆𝑢1 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 ) + 𝑑 𝑖𝑎𝑔((𝜆𝑢2 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 ) 1 2

(12)

Given (Δx, Δu, Δv) we can calculate the change of the dual variable: Δ𝜆𝑢1 = 𝑑 𝑖𝑎𝑔(𝜆𝑢1 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 )(−Δ𝑥 + Δ𝑢) − 𝜆𝑢1 − 1 Δ𝜆𝑢2 = 𝑑 𝑖𝑎𝑔(𝜆𝑢2 )𝑑 𝑖𝑎𝑔(𝑓𝑢−1 )(Δ𝑥 + Δ𝑢) − 𝜆𝑢2 − 2

1 −1 𝑓 𝑡 𝑢1

1 −1 𝑓 𝑡 𝑢2

(13)

(14)

With backtracking linear search, the step size is 0 ≤ s ≤ 1, and the selection of s must ensure that fu1 , fu1 < 0, 𝜆u1 ,𝜆u2 > 0. So as to ensure that the linear search step size does not cause the dual variable to exceed the constraint. Calculate the corresponding s as follows: { ⟨ ⟩} 𝜆 | 𝑠 = 0.99 ∗ min 1, − 𝑖 || Δ𝜆𝑖 < 0 (15) Δ𝜆𝑖 | The variables (x, u, v, 𝜆u1 , 𝜆u2 ) can be updated, so 𝑥 = 𝑥 + 𝑠Δ𝑥, 𝑢 = 𝑢 + 𝑠Δ𝑢, 𝑣 = 𝑣 + 𝑠Δ𝑣, 𝜆𝑢1 = 𝜆𝑢1 + 𝑠Δ𝜆𝑢1 , 𝜆𝑢2 = 𝜆𝑢2 + 𝑠Δ𝜆𝑢2 , the judgment condition for jumping back and forth linear search is ‖r (𝑥 + 𝑠Δ𝑥, 𝑢 + 𝑠Δ𝑢, 𝑣 + 𝑠Δ𝑣‖ ≤ (1 − 𝛼𝑠)‖𝑟 (𝑥, 𝑢, 𝑣)‖ ‖𝑡 ‖2 ‖ 𝑡 ‖2

(16)

The dual gap is 𝜂(𝑥, 𝜆) = −𝑓 (𝑥)𝑇 𝜆. If the dual gap at this time satisfies the set value, the iterative process ends.

3.1. Reconstruction of multiple images In the simulation, the original signal is subjected by discrete cosine transform with a compression ratio M/N = 0.3. For the OMP, CoSaMP and SP algorithm, the sparsity is set to 500. At the same time, in order to verify the generalization of our proposed algorithm, 6 images of different types, including peach, cat, building, flower, cabbage and panda are reconstructed. The advantages of our proposed algorithm are illustrated by comparing the reconstruction time, mean square deviation (MSE) and mean peak signal to noise ratio (PSNR) with other methods. The comparison of reconstructed images quality with above algorithms are shown in Fig. 3. The reconstruction time between above algorithms is compared as shown in Table 1. As you can see from Fig. 3 and Table 1, the OMP algorithm has a relatively short time with poor images reconstruction. For our proposed original dual interior point algorithm, although the reconstruction time is not the shortest, the quality of reconstructed images is much better than the images got from OMP algorithm. Comparing with the other four kinds of reconstruction algorithms, the reconstruction time for SAMP algorithm is relatively unacceptable. To better compare the quality of reconstructed images, MSE (Mean Square Error) and PSNR (Peak Signal to Noise Ratio) are also analyzed for each algorithm as shown in Table 2 and Table 3. It can be seen form Table 2 and3 that in the above five algorithms, the MSE of our proposed original dual interior point algorithm is the smallest, while the PSNR of it is the highest, which indicates that it can reconstruct the image with best result. 3.2. Image reconstruction under different sampling ratio In order to verify the effectiveness of the algorithm in this paper under lower sampling ratio, we further used the sampling ratio of 0.1, 0.2 and 0.3 to reconstruct the peach image. The relative images with different sampling ratio are reconstructed as shown in Fig. 4. MSE and PSNR are also calculated and compared as shown in Table 4. By comparing MSE and PSNR of reconstructed images with different sampling ratio, it is found that with the continuous reduction of sampling ratio, the image reconstruction quality will correspondingly deteriorate. However, the MSE of our proposed original dual interior point algorithm is still the smallest, while the PSNR of it is the highest, which indicates that our proposed algorithm still has the best reconstruction quality. 3.3. Image reconstruction with different image sizes Images size of 64 × 64 pixels, 128 × 128 pixels and 256 × 256 pixels are further reconstructed in this section. In the reconstruction of images

L. Chao, J. Han and L. Yan et al.

Optics and Lasers in Engineering 129 (2020) 106082

Fig. 3. Reconstructed images: (a) The original image; (b) OMP algorithm; (c) CoSaMP algorithm; (d) SP algorithm; (e) SAMP algorithm; (f) Primal dual interior point algorithm.

Table 1 Average reconstruction time of the five algorithm on the 6 images. Image type

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

peach cat building flower cabbage panda average

20.61 s 21.34s 26.58s 21.12s 22.34s 24.52s 22.75s

125.66s 131.45s 128.32s 121.45s 127.22s 129.45s 127.26s

65.92s 63.12s 66.15s 66.25s 67.23s 68.03s 66.12s

989.75s 943.45s 1014.45s 992.42s 998.43s 1002.56s 989.84s

58.84s 61.34s 57.12s 59.92s 57.74s 58.19s 58.86s

Table 2 Average MSE of the five algorithm on the 6 images. Image type

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

peach cat building flower cabbage panda average

0.0073 0.0042 0.0086 0.0088 0.0054 0.0172 0.0086

0.0082 0.0041 0.0098 0.0099 0.0059 0.0173 0.0092

0.0069 0.0041 0.0086 0.0079 0.0052 0.0153 0.0080

0.0088 0.0045 0.0100 0.0101 0.0061 0.0180 0.0575

0.0054 0.003 0.0062 0.0067 0.0043 0.0117 0.0062

Table 3 Average PSNR of the five algorithm on the 6 images. Image type

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

peach cat building flower cabbage panda average

42.68 47.47 41.27 41.15 45.28 35.27 42.19

41.77 47.45 40.19 40.06 44.52 35.22 41.54

43.25 47.75 41.27 42.02 45.70 36.30 42.72

41.13 46.94 40.00 39.91 44.29 34.89 41.19

45.39 49.93 44.20 43.43 47.28 38.67 44.82

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Optics and Lasers in Engineering 129 (2020) 106082

Fig. 4. Reconstructed images: (a) The original image; (b) OMP algorithm; (c) CoSaMP algorithm; (d) SP algorithm; (e) SAMP algorithm; (f) Primal dual interior point algorithm.

Table 4 PSNR and MSE of the five algorithm on the cat image at different sampling ratio. M/N

Index

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

0.1

MSE PSNR MSE PSNR MSE PSNR

0.0344 29.27 0.0131 37.68 0.0073 42.68

0.0936 20.57 0.0233 32.64 0.0082 41.77

0.0382 28.65 0.0163 35.76 0.0069 43.25

0.0975 20.22 0.0256 31.84 0.0088 41.13

0.0228 32.84 0.0092 40.69 0.0054 45.39

0.2 0.3

Table 5 Evaluation index of the five algorithm on the peach image. size

Index

OMP algorithm

CoSaMP Algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

64 × 64

Time PSNR Time PSNR Time PSNR

0.21s 32.65 1.22s 38.20 11.39s 46.52

0.31s 31.54 20.97s 30.27 112.49s 39.89

0.22 35.38 20.33s 39.07 83.32s 42.71

0.70s 29.03 3.13s 37.48 17.82s 47.47

0.30 35.44 1.32s 40.92 7.57s 48.64

128 × 128 256 × 256

with size of 100 × 100 pixels in Section 3.1, the image matrix is processed into a one-dimensional signal of 10,000 × 1, and then compressed and sampled. For images with larger size, it is hard to process the image matrix into a one-dimensional signal. So image is compressed in each row and column respectively. The value of each pixel in the image is reconstructed twice. And the final value of each pixel is the average of the above results. This reconstruction method is called row and column equilibrium reconstruction. In this section, we will use row and column equilibrium process to reconstruct the image. The relative images with different image sizes are reconstructed as shown in Fig. 5. Reconstruction time and PSNR are also calculated and compared as shown in Table 5. By comparing the results, it can be seen that with the increase of image size, the computational cost will also rise. Compared with other algorithms, our proposed original dual interior point algorithm has the best reconstruction effect and computational cost is acceptable. Especially for 256 × 256 image, the reconstruction quality and computational cost of the primal dual interior point algorithm is optimal compared with other algorithms. 4. Experiment To verify the feasibility of our approach, we set up a compressed sensing imaging platform as shown in Fig. 4.

The measurement matrix data is loaded into the laser matrix modulation module by computer. The laser modulated by modulation module irradiates the light to the original image and receives the energy of the echo signal by the CCD to obtain a light intensity value I. The obtained light intensity value I is taken as an element of the observed matrix b as described in Section 2.2. A total of M tests are performed and N point sources are modulated in each test. In this situation, the measurement matrix AM × N and the observed matrix bM × 1 are got. In order to compare the performance of our proposed algorithm, the experiment results taken with the OMP algorithm, CoSaMP algorithm, SP algorithm and SAMP algorithm are also analyzed. In the experiment, the original signal is subjected to discrete cosine transform, and the measurement matrix size is 3000 × 10,000, that is, the compression ratio M/N = 0.3. At the same time, the observation value is a 3000 × 1 one-dimensional signal. For the OMP algorithm, CoSaMP algorithm, and SP algorithm, the sparsity value is set to 500. The comparison of reconstructed images quality with above algorithms are shown in Fig. 5. It can be seen from Fig. 5 that the quality of reconstructed image with our proposed original dual interior point algorithm is the best. The reconstruction time between above algorithms is also compared as shown in Table 3. As you can see from Table 3, the OMP algorithm has a relatively short time with a poor image reconstruction. For our proposed original

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Optics and Lasers in Engineering 129 (2020) 106082

Fig.5. Reconstructed images: (a) The original image; (b) OMP algorithm; (c) CoSaMP algorithm; (d) SP algorithm; (e) SAMP algorithm; (f) Primal dual interior point algorithm.

Fig. 6. Compressed sensing imaging platform.

Table 6 Comparison of reconstruction time of experiment data. Reconstruction algorithm

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

Recovery time

22.40s

150.82 s

73.09s

1122.16 s

67.90 s

Table 7 Comparison of MSE and PSNR of experiment data with kinds of reconstruction algorithms. Reconstruction algorithm

OMP algorithm

CoSaMP algorithm

SP algorithm

SAMP algorithm

Primal dual interior point algorithm

MSE PSNR

0.018 34.89

0.0201 33.92

0.0175 35.13

0.0255 31.87

0.0133 37.50

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Optics and Lasers in Engineering 129 (2020) 106082

Fig. 7. Reconstructed images: (a) The original image; (b) OMP algorithm; (c) CoSaMP algorithm; (d) SP algorithm; (e) SAMP algorithm; (f) Primal dual interior point algorithm.

dual interior point algorithm, although the reconstruction time is not the shortest, the reconstruction effect is the best in the allowable time range. Comparing with the other four kinds of reconstruction algorithms, the reconstruction time for SAMP algorithm is relatively unacceptable. To better compare the performance of the above algorithms, MSE (Mean Square Error) and PSNR (Peak Signal to Noise Ratio) are also analyzed for each algorithm as shown in Table 4. In the above five algorithms, the MSE of our proposed original dual interior point algorithm is the smallest, while the PSNR of it is the highest, which indicates that it can reconstruct the image with best result.

CRediT authorship contribution statement Lianying Chao: Conceptualization, Methodology, Software. Jiefei Han: Data curation, Writing - original draft, Investigation. Lisong Yan: Conceptualization, Methodology, Software. Liying Sun: Data curation, Writing - original draft, Investigation. Fan Huang: Data curation, Writing - original draft, Investigation. ZhengBo Zhu: Writing - review & editing. Shili Wei: Writing - review & editing. Huiru Ji: Writing - review & editing. Donglin Ma: Supervision. References

5. Conclusion Compressed sensing has attracted the attention of relevant researchers since it was proposed. It has a good development prospect in image processing, pattern recognition, wireless communication radar imaging and other fields. In summary, we have developed a primal dual interior point compressed sensing algorithm for ghost imaging. The performance of our proposed algorithm has been described with simulation and experimental results. Furthermore, the CS reconstruction framework developed by us has also demonstrated significant advantages, including a faster convergence and a better-reconstructed image quality, comparing with the other reconstruction algorithms such as OMP algorithm, CoSaMP algorithm, SP algorithm and SAMP algorithm. The promising results demonstrated in this work suggest that the development of an advanced CS algorithm has the potential to enable high quality image reconstruction in ghost imaging with a significantly reduced number of testing data. Funding National Natural Science Foundation of China (61805089 and 61805088); the Fundamental Research Funds for the Central Universities (2018KFYYXJJ053, 2019kfyXKJC040, and 2019kfyRCPY083). Declaration of competing interest None.

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