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Abstract
Single photon counting compressive imaging, a combination of single-pixel-imaging and single-photon-counting technology, is provided with low cost and ultra-high sensitivity. However, it requires a long imaging time when applying traditional compressed sensing (CS) reconstruction algorithms. A deep-learning-based compressed reconstruction network refrains iterative computation while achieving efficient reconstruction. This paper proposes a compressed reconstruction network (OGTM) based on a generative model, adding sampling sub-network to achieve joint-optimization of sampling and generation for better reconstruction. To avoid the slow convergence caused by alternating training, initial weights of the sampling and generation sub-network are transferred from an autoencoder. The results indicate that the convergence speed and imaging quality are significantly improved. The OGTM validated on a single-photon compressive imaging system performs imaging experiments on specific and generalized targets. For specific targets, the results demonstrate that OGTM can quickly generate images from few measurements, and its reconstruction is better than the existing compressed sensing recovery algorithms, compensating defects of the generative models in compressed sensing.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With a single-photon detector, the classic single-pixel imaging can be extended to the single-photon level, and realize single-photon counting compressive imaging [1–5]. It has two main advantages: first, two-dimensional photon counting imaging can be realized by using single-pixel single photon detector, so the cost is low, especially in special wave bands such as infrared and terahertz. especially in special wave bands such as infrared [6] and terahertz [7]. Second, the point detector in the single-pixel imaging system could collect the light intensity of multiple pixels synchronously, thereby signal-to-noise ratio might be greatly improved. Ultrahigh-sensitivity imaging is achieved, namely the system’s imaging sensitivity surpassing the sensitivity of a single-photon point detector [8,9]. Therefore, single-photon counting compressive imaging is promising being widely used in medical diagnosis, astronomical observation, spectral measurement [10–12], etc.
Performing compressive sampling with few measurements could effectively shorten the sampling time, while the image can be restored only by combining the prior information of the scene during reconstruction. Reconstruction based on sparse-prior hypothesis mainly uses convex optimization [13], greedy matching pursuit [14], etc. These methods require a lot of iterative calculations, resulting in slow reconstruction. Solving reconstruction problems via deep learning (DL) has been widely studied, especially in computational ghost imaging [15,16]. Compared with traditional iterative algorithms, the trained reconstruction network can reconstruct high-dimensional images from low-dimensional measurements and effectively without huge computation, which provides a solution for the rapid imaging of single-photon counting compression imaging. In 2015, R.G.Baraniuk et al. used a stack denoising autoencoder (SDA) to regenerate the image from the down-sampling measurements [17]. In 2016, Kulkarni et al. proposed the ReconNet based on super-resolution image reconstruction [18]. In 2017, Yao et al. used the deep residual learning to build DR2-Net [19]. As one of the neural networks, the generative model has made a series of breakthroughs in recent years, which provides a new idea for compression reconstruction. Bora et al. combined CS with generative adversarial networks (GAN) and variational auto-encoder(VAE) respectively to propose CSGM [20], and achieved similar results to standard CS without usual sparsity constraint. Since the picture prior cannot adapt to measurement changes, it still requires a slow optimization process for reconstruction. Subsequently, the Task-aware GAN [21] and the DCS framework [22] succeeded in matching prior information with measurement changes. However, there are still two problems: one is that there are partial visual differences between reconstruction and ground truth in the absence of image quality assessment, which also exist in traditional GANs [23,24]. The other one is the complex architecture of the discriminator network in the GANs cannot be used as a measurement matrix, which limits its application in single-photon compressive imaging.
The rapid development of transfer learning, especially in unsupervised learning where pre-training can mine scene information from related domain, improves the result of fine-tuning [25–27]. Taking advantage of its effective acquisition of prior information, we propose a generative model optimized via sampling and transfer learning (OGTM) for single photon counting compressive imaging system, which can perform image reconstruction on the generator side. Experiments show that our OGTM outperforms most compressed sensing reconstruction algorithms. Our contributions could be summarized as follows:
- • We propose a compressed reconstruction network (OGTM) based on generative model. Different from the existing compressed sensing generative model, we add sampling sub-network to achieve joint-optimization of sampling and generation, thereby obtaining better imaging results.
- • To avoid the slow convergence caused by alternating training, we propose to transfer the weights from an Autoencoder that have processed the same dataset, as the initial value of the sampling sub-network and generative reconstruction network. The results show that the convergence speed and imaging quality are significantly improved.
- • We test OGTM on the single-photon compression imaging system, and perform imaging experiments on specific targets and generalized targets. We find the characteristics of its reconstruction through actual experiments.
2. Background
2.1 Single photon counting compressive imaging system
Figure 1 briefly shows the single photon counting compressive imaging system that we proposed previously [4,28]. The light source consists of LED (CREE Q5), collimator (customized developed), attenuator (LOPF-25C-405), and diaphragm (APID25-1). The initial scattered light is collimated into parallel light by the collimator. With the coaction of attenuator and diaphragm, the intensity of parallel light is limited to the single photon level. The Object is illuminated by the parallel light and then imaged onto the DMD (0.7XGA 12° DDR) via the imaging lens (OLBQ25.4-050). The DMD consists of micro-mirror array sized 1024×768. Each mirror can be individually controlled to deflect ±12 degrees. We use FPGA (Altera DE2-115) to download each measurement vector of the binary matrix A from the PC to the DMD control module, which can control the flip frequency of DMD. The micro-mirror corresponding to the element “1” in the vector is deflected by +12°, and that of the element “0” in the vector is deflected by -12°. We set a focusing lens (OLBQ25.4-050) along the +12°direction of DMD to collect reflected light into a photon counting PMT (Hamamatsu H10682). And then PMT outputs the corresponding number of discrete pulses to FPGA for counting, and the count value is yi, which is the inner product of the i row of measurement matrix $A \in {R^{M\ast N}}$ and the image ${x} \in {{R}^{{N\ast 1}}}$. After M modulation, the final measurement value ${y} \in {{R}^{{M\ast 1}}}$ can be obtained, and finally the counting result is sent to the computer for reconstruction. The above modulation process possibly with added noise can be expressed as:
where A has been binarized, and ${e} \in {{R}^{{M\ast 1}}}$ is noise. Especially, the measurement dimension M ≤ N and M/N is called the measurement ratio. The reconstruction of high dimensional x after observing linear measurements of its entries, is the application of compressed sensing in the field of single photon imaging.
Fig. 1. single photon compressive imaging system, DMD: digital micro-mirror device, PMT: photomultiplier. FPGA: field programmable gate array.
2.2 Compressed sensing and sparse representation
Compressed sensing, also known as sparse sampling, is a way to find sparse solutions of underdetermined linear systems. It aims to reconstruct an unknown vector x* from a linear measurement value y:
Even without noise, this is an underdetermined system of linear equations, so recovery is not unique unless we make some assumptions on the structure of the unknown vector x*. Structural assumptions may be different according to different conditions, the most common one is that x is sparse. Traditional methods restrict random matrices to satisfy Restricted Isometry Property (RIP) properties such as Gaussian matrix, Bernoulli matrix, or Toeplitz matrix, so convex optimization can provably recover the sparse vector x. However, random matrices are inappropriate in practical. Sparse representation provides new ideas for compressed sensing, such as algorithms OMP [29], ROMP [30], IHT [31], TVAL3 [32]. They are mostly based on the assumption that image coefficients are sparse in a transformation domain or a set of basis:
$\psi$ is a transformation domain or a set of basis, z is the sparse representation of the original signal in $\psi$. The sparse recovery algorithm can be rephrased as follows:Therefore, the product of sparse transformation matrix $\psi$ and measurement matrix A, is the sparse dictionary that affects the recovery performance. z should be solved by an iterative strategy, and then multiplied by $\psi$ to recover x.
2.3 Compressed sensing using generative models
Even though the sparsity assumption on x is the most common choice, it is not the only possible one. Indeed, other approaches, such as combining sparsity with additional model-based constraints [33] or graph structures [34], have been developed. In 2017, CSGM achieves compressive sensing image reconstruction without the usual sparsity constrain. In 2019, Yan Wu further proposed DCS framework via meta-learning. As shown in Fig. 2, using the compressed measurements error to update $\widehat {z}$ and Gθ, G implicitly constrain output $\widehat {x}$ in a low-dimensional manifold via its architecture and the weights adapted from data:
The estimated $\widehat x = {G_\theta }(\widehat z)$ is the reconstructed signal. Measurement process A that produces measurement value can be a complex discriminator network or a simple measurement matrix. The discriminator network output, one-dimensional 0 or 1, is a limiting case of compressed sensing, and its multilayer structure cannot be loaded on our single-photon imaging system. But the constant one-layer measurement matrix does not possess efficient sampling property. DCS still has problems such as simple dataset and lack of image quality assessment, but there is still broad prospect in compressed sensing. Therefore, we improve and propose a generative model optimized via sampling and transfer learning.
Fig. 2. Illustration of Deep Compressed Sensing. $\widehat z$ latent representation. G: a generator that reconstructs the signal from a latent representation $\widehat z$. A: measurement process. x: original image. $\widehat x$:reconstructed image. y and $\widehat {y}$:corresponding measurement value.
3. Proposed network
3.1 Network architecture
Figure 3 shows our optimized generative model named OGTM. Compared with the traditional sparse recovery algorithm in eq.4, we replace the sparse transformation matrix $\psi$ with a generator and the measurement matrix A with a sampling sub-network. The generator network G is a continuous mapping from low dimension Rk to high dimension space Rn. Unlike DCS framework [22], our model uses latent z at 50% generation ratio (k/n=0.5). When the generation ratio ≥ the measurement ratio, the model has better reconstructions. Considering the single-photon compression imaging system, the sampling network can only be designed as a single fully connected layer, and the back-propagating strategy needs to be modified during the training process because of the binarized measurement matrix Aφ. Unlike complex structures such as convolution and residuals that are widely used in deep learning, we have found in many experiments that a generator composed of two fully connected layers can not only ensure sufficient weights to fit the data distribution, but also enable z to update quickly according to measurements. The network structure is simplified, greatly shortening the training time. Remarkably, inspired by the strategy of GAN, we use joint-optimization to alternately update the sampling network Aφ and generator network Gθ, which improves the sampling efficiency and reconstruction quality.
Fig. 3. Our proposed model. z:latent representation. Aφ:binarized measurement matrix that can jointly optimize with generator network Gθ. FC: Fully-connected layer of the generator. The parameter update corresponds to the same color loss function.
The model input are batches of 64*64*1 images. They can be multiplied by measurement matrix Aφ to simulate the compression sampling process on the DMD as shown in Fig. 1, and the measurement value can be obtained. The reconstruction from generator can also get the corresponding measurement value. Here we take the measurement error as the loss function, and update z continuously with Tz=3 gradient descent, to make z approach latent representation of the current input x. Then two different loss functions are used to jointly optimize Gθ and Aφ. After training, Aφ can be loaded on the DMD for compression sampling of real images, and the measurement values obtained are input into OGTM for generative reconstruction. It is worth noting that whether training or testing, Gθ and Aφ should be constant when $\; $z is updating. This is outlined in Algorithm 1.
3.2 Transfer learning
Transfer learning is mostly applied to transfer annotated data or weights of same network structure in related fields to perform on the target field. We design an Autoencoder to assign pre-training values to the initial weights of OGTM separately as shown in Fig. 4, with the purpose of accelerating the training convergence speed and improving the final reconstruction quality.
Fig. 4. Autoencoder pre-trained network at 0.5 coding ratio. WE: Weight of Encoder. WD: Weight of Decoder.
Differ in public knowledge, our transfer mode is inverted according to the encoding and decoding function. The encoding weight WE is used as the initial values of the sampling network Aφ. The decoding weights WD1 and WD2 are used as the initial values of the OGTM generator Gθ. This end-to-end training can effectively obtain image information and trained weights can be transferred to OGTM as initial values for further fine-tuning. In the pre-training process, we will train several autoencoders at different coding rates. This is outlined in Algorithm 2.
3.3 Binarization strategy and training method
In this section, we introduce several hyperparameters and the training method. Both OGTM and autoencoder select the same dataset. As for aAutoencoder hyperparameters, we set the batch size to 64, the learning rate to $0.00001$, and iterate with one Adam optimizer. As for that of OGTM, we set the batch size to 16, αz to 0.01,TZ to 3, the learning rate αG and αA to 0.00001 and iterate with two Adam optimizers to realize joint-optimization.
To apply the network to a single-photon imaging system, the binary measurement matrix is needed. Inspired by the binarized neural network, we binarize the floating-point weights of the encoder with sign function before training:
4. Results and discussion
4.1 Performance verification test
In this section, simulations will verify the effect of transfer learning and sampling-generation joint optimization. According to the initialization method of the generating network G and the sampling sub-network A, we set up four cases for comparison. G(Ran)-A(Ran): The generator network G and sampling sub-network A are initialized by random numbers. G(WD)-A(Ran): G is initialized by the weight of the autoencoder, A is initialized by random numbers. G(Ran)-A(WE): G is initialized by random numbers and A is initialized by the weight of the autoencoder. G(WD)-A(WE): Both G and A are initialized by the weight of the autoencoder. The input image from CelebA dataset provided by Liu et al. [36] is reshaped to 1 ∗ 4096 size after the central cropping and gray-scale processing. The number of measurements m in OGTM varies with the Measurement Ratio(MR),1024,409,163, and 40 corresponding to MR = 0.25, 0.10, 0.04,and 0.01, respectively. Eight images are extracted from the CelebA dataset as the test set and do not participate in training. We use peak signal to noise ratio (PSNR) value to evaluate reconstruction quality.
Figure 5 shows imaging result of the four cases, when the joint optimization of sampling and generating network is not performed. We find weight transfer learning did not achieve better results, when the sampling rate is greater than 0.05. Reconstruction images present in Fig. 7(b) and Fig. 7 (c). The reason may be that when the measurement rate is low, the transferred weight can map the main features of the images. However, as the measurement ratio increases, the constant sampling sub-network cannot follow the update of the generator, and the transferred WD makes the loss function encounter local optimization.
Fig. 5. PSNR values for imaging results of four different initialization cases, when Only Gθ is optimized.
Figure 6 shows PSNR values for imaging results of four different initialization cases, when Gθ and Aφ are joint-optimized. In the case of G(Ran) – A(Ran) and G(WD)-A(Ran), the joint optimization seem not to work. But the PSNR of the two cases containing A(WE) surpasses the others containing A(Ran) at each measurement ratio. The joint optimization strategy further fine-tunes the transferred weights from sampling sub-network and generator. Therefore, their multiplication has good sparse recovery performance than the random Gaussian measurement matrix that satisfies the RIP property. Reconstruction images present in Fig. 7(d) and Fig. 7(e).
Fig. 6. PSNR values for imaging results of four different initialization cases, when Gθ and Aφ joint-optimized.
Fig. 7. Reconstructed images of models upon CelebA-Cropped dataset at MR=0.10. Top to bottom rows are the(a)original images, and reconstructions by(b) G(Ran)-A(Ran), (c) G(WD)-A(WE), (d) joint-optimized G(Ran)-A(Ran) and (e) joint-optimized G(WD)-A(WE). © [2015] IEEE. Reprinted, with permission, from [International Conference on Computer Vision (ICCV)].
The model optimized via sampling and transfer learning not only has a better reconstruction, but also greatly accelerates the convergence to save training time. Figure 8 shows the convergence process of the above four models. The reconstruction loss of the two models containing A(WE) converge rapidly in the early training stage, and finally stabilize at a lower value, corresponding to the result of MR=0.01 in Fig. 6. In summary, the combination of joint optimization and transfer learning makes the trained OGTM have good sparse reconstruction performance.
Fig. 8. Smoothed reconstruction error (per pixel) of four models in the training iteration (MR=0.01).
4.2 Comparison with existing methods
In this section, we design a series of comparative experiments to evaluate OGTM and existing algorithms such as Autoencoder, Lasso(Wavelet) as was done in [20], TVAL3 and DR2. DR2 is a multis-scale network which consists of two fully-connected layers and four residual blocks. As for TVAL3, its transformation domain $\psi$ is analogous to our generator. As for Lasso (Wavelet), its wavelet coefficients is analogous to our latent z and wavelet bases is analogous to our generator. Our model requires the optimal distribution of latent z and generator Gθ without sparsity constraints. In order to adapt to our experiment system and ensure the fairness of comparison, we binarize the measurement matrix A when testing these algorithms. The eight images mentioned above are still used as test sets for simulations to evaluate the performance of the network in the real imaging system. We evaluate reconstruction quality with peak signal to noise ratio (PSNR)and structural similarity (SSIM) values. The following data are the average of 8 images.
Table 1 and Fig. 9 fully show the simulation results of the test set on five different methods. Compared with Lasso(wavelet), OGTM has the obvious advantage at each measurement ratio. Compared with Autoencoder, OGTM has a slight advantage at low measurement rates, and the advantage becomes greater as the measurement rate increases. Compared with TVAL3and DR2, OGTM has the obvious advantage at low measurement rates. For instance, at MR=0.01, the OGTM outperforms TVAL3 by 7.8 dB on the PSNR and outperforms DR2 by 3.5 dB on the PSNR. But in the high measurement ratio, it is not as good as TVAL3.
Table 1. PSNR and SSIM of different algorithms upon the CelebA-cropped dataset at different measurement ratios.Note: The bold values indicate the best PSNR or SSIM of these models
Visually, Fig. 10 shows that the Autoencoder has a balanced reconstruction, but there are errors in reconstruction at low measurement rates. The reconstruction quality of Lasso (Wavelet) and TVAL3 are greatly influenced by the measurement ratio. The reconstruction of DR2 has a blocking artifact. Figure 10(I) also shows that OGTM has better image reconstruction than other algorithms at low measurement rate (MR=0.04). In summary, our OGTM optimized via sampling and transfer learning, has certain advantages over traditional sparse recovery algorithms in the field of single-photon counting compressive imaging.
Fig. 10. Reconstructed images of various algorithms upon CelebA-cropped dataset at different measurement ratios. Top to bottom rows are the(a)original images, and reconstructions by (b) Autoencoder, (c) Lasso (Wavelet), (d) TVAL3, (e)DR2 and (f) our OGTM. © [2015] IEEE. Reprinted, with permission, from [International Conference on Computer Vision (ICCV)].
4.3 Actual imaging results of OGTM
In this section, we give the imaging experiment results of OGTM. The time of each measurement is set to 1.0s. The photon count rate detected by PMT is about 20,000 counts/ s. We use Fashion-Mnist dataset which contains ten types of clothing and is expanded to 64*64. Different from simulations, OGTM uses the measurement values from the single-photon compressive imaging system for reconstruction, and the update of z has a significant impact on the final reconstruction. After many experiments, we modified some parameters to achieve the best effect: increase TZ to 10, and decrease αz to 0.01. We image object “skirt” in the single-photon compressive imaging system at four measurement rates and get the results shown in Fig. 11. OGTM obtains higher mean PSNR value than TVAL3. It is worth noting that OGTM rebuilds the contour of the object at MR=0.025. This generative property brings advantages to the reconstruction of clothing targets at low measurement rates.
Fig. 11. Single photo counting compressive imaging results using traditional method and proposed method,(a) Reconstructed algorithm is OGTM; (b) Reconstructed algorithm is TVAL3.
To verify the generalization of OGTM, we select other categories of pictures that are not belong to the dataset for imaging. We compare with other sparse restoration algorithms such as BPDN as was done in [37] and FISTA as was done in [38] . We find that the dataset used greatly affects the reconstruction properties of OGTM and imaging targets similar to the dataset category will have better reconstruction results. As shown in Fig. 12(a), when MR=0.025, the image “5” and the picture “NCU” show the contour of clothing on the reconstruction, which are not the information of the objects. When MR=0.25, generative noise around the imaging objects reduces the reconstruction quality. The imaging of BPDN (L1-magic) and FISTA algorithms also have Gaussian white noise, which affects PSNR at different measurement ratios. In contrast, the sparse reconstruction performance of OGTM at low measurement ratios is better. It has broad prospects in special fields such as medical diagnosis, astronomical observation, spectral measurement, etc.
Fig. 12. Single-photon compression imaging results of other objects. (a) Reconstructed algorithm is OGTM; (b) Reconstructed algorithm is BPDN(L1-magic). (c) Reconstructed algorithm is FISTA.
5. Conclusion
This paper proposes a generative model optimized via sampling and transfer learning for single-photon counting compressive imaging system, which is mainly oriented to small-scale reconstruction imaging at low measurement ratio. In the simulation, a series of verification tests show that OGTM which combines sparsity with model structure has better sparse recovery performance. The OGTM using transfer learning also has faster convergence and better reconstructions. The experimental results show that the reconstruction quality of OGTM is much better than LASSO (wavelet) algorithm for sparse recovery. In contrast to the traditional compressive sensing reconstruction algorithm TVAL3, the proposed network has a certain advantage. The lower the measurement ratio, the more obvious the advantage. In addition, the jointly optimized binary measurement matrix can be loaded on the DMD for actual imaging so that OGTM can be directly applied to the single-photon compressive imaging system. The actual imaging results show that compared with the traditional single-photon compressive reconstruction method, OGTM has obvious advantages at low measurement rates for specific objects.
Funding
the Funding Scheme to Outstanding Young Talents of Jiangxi Province (20171BCB23007); National Natural Science Foundation of China (61865010); National Natural Science Foundation of China (61565012).
Disclosures
The authors declare no conflicts of interest.
References
1. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). [CrossRef]
2. M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019). [CrossRef]
3. R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3(12), 696–705 (2009). [CrossRef]
4. H. Wang, Q. Yan, B. Li, C. Yuan, and Y. Wang, “Sampling Time Adaptive Single-Photon Compressive Imaging,” IEEE Photonics J. 11(3), 1–10 (2019). [CrossRef]
5. W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, S.-Q. Gao, G.-J. Zhai, Q. Zhao, and M.-L. Ge, “Single photon counting imaging system via compressive sensing,” arXiv preprint arXiv:1202.5866 (2012).
6. S. Komiyama, O. Astafiev, V. Antonov, T. Kutsuwa, and H. Hirai, “A single-photon detector in the far-infrared range,” Nature 403(6768), 405–407 (2000). [CrossRef]
7. S. Komiyama, “Single-photon detectors in the terahertz range,” IEEE J. Sel. Top. Quantum Electron. 17(1), 54–66 (2011). [CrossRef]
8. R. Gabet, G. Stéphan, M. Bondiou, P. Besnard, and D. Kilper, “Ultrahigh sensitivity detector for coherent light: the laser,” Opt. Commun. 185(1-3), 109–114 (2000). [CrossRef]
9. R. Sobolewski, A. Verevkin, G. Gol’Tsman, A. Lipatov, and K. Wilsher, “Ultrafast superconducting single-photon optical detectors and their applications,” IEEE Trans. Appl. Supercond. 13(2), 1151–1157 (2003). [CrossRef]
10. C. Barbieri, G. Naletto, T. Occhipinti, C. Facchinetti, E. Verroi, E. Giro, A. Di Paola, S. Billotta, P. Zoccarato, and P. Bolli, “AquEYE, a single photon counting photometer for astronomy,” J. Mod. Opt. 56(2-3), 261–272 (2009). [CrossRef]
11. J. S. Dam, P. Tidemand-Lichtenberg, and C. Pedersen, “Room-temperature mid-infrared single-photon spectral imaging,” Nat. Photonics 6(11), 788–793 (2012). [CrossRef]
12. A. Pourmorteza, R. Symons, V. Sandfort, M. Mallek, M. K. Fuld, G. Henderson, E. C. Jones, A. A. Malayeri, L. R. Folio, and D. A. Bluemke, “Abdominal imaging with contrast-enhanced photon-counting CT: first human experience,” Radiology 279(1), 239–245 (2016). [CrossRef]
13. R. Gribonval, G. Chardon, and L. Daudet, “Blind calibration for compressed sensing by convex optimization,” in 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (IEEE, 2012), 2713–2716. [CrossRef]
14. T. T. Do, L. Gan, N. Nguyen, and T. D. Tran, “Sparsity adaptive matching pursuit algorithm for practical compressed sensing,” in 2008 42nd Asilomar Conference on Signals, Systems and Computers, (IEEE, 2008), 581–587.
15. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2008). [CrossRef]
16. F. Wang, H. Wang, H. Wang, G. Li, and G. Situ, “Learning from simulation: An end-to-end deep-learning approach for computational ghost imaging,” Opt. Express 27(18), 25560–25572 (2019). [CrossRef]
17. P. Vincent, H. Larochelle, I. Lajoie, Y. Bengio, P.-A. Manzagol, and L. Bottou, “Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion,” J. Mach. Learn. Res. 11(3), 3371–3408 (2010). [CrossRef]
18. K. Kulkarni, S. Lohit, P. Turaga, R. Kerviche, and A. Ashok, “Reconnet: Non-iterative reconstruction of images from compressively sensed measurements,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 449–458.
19. H. Yao, F. Dai, S. Zhang, Y. Zhang, Q. Tian, and C. Xu, “Dr2-net: Deep residual reconstruction network for image compressive sensing,” Neurocomputing 359, 483–493 (2019). [CrossRef]
20. A. Bora, A. Jalal, E. Price, and A. G. Dimakis, “Compressed sensing using generative models,” arXiv preprint arXiv:1703.03208 (2017).
21. M. Kabkab, P. Samangouei, and R. Chellappa, “Task-aware compressed sensing with generative adversarial networks,” arXiv preprint arXiv:1802.01284 (2018).
22. Y. Wu, M. Rosca, and T. Lillicrap, “Deep compressed sensing,” arXiv preprint arXiv:1905.06723 (2019).
23. I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Networks,” Commun. ACM 63(11), 139–144 (2020). [CrossRef]
24. A. Radford, L. Metz, and S. Chintala, “Unsupervised representation learning with deep convolutional generative adversarial networks,” arXiv preprint arXiv:1511.06434 (2015).
25. M. Long, Y. Cao, J. Wang, and M. Jordan, “Learning transferable features with deep adaptation networks,” in International conference on machine learning, (PMLR, 2015), 97–105.
26. M. Oquab, L. Bottou, I. Laptev, and J. Sivic, “Learning and transferring mid-level image representations using convolutional neural networks,” Proceedings of the IEEE conference on computer vision and pattern recognition1717–1724 (2014).
27. E. Tzeng, J. Hoffman, T. Darrell, and K. Saenko, “Simultaneous deep transfer across domains and tasks,” Proceedings of the IEEE International Conference on Computer Vision4068–4076 (2015).
28. Q. Wang, Q. Yan, S. Deng, H. Wang, C. Yuan, and Y. Wang, “Iterative Adaptive Photon-Counting Compressive Imaging Based on Wavelet Entropy Automatic Threshold Acquisition,” IEEE Photonics J. 11(5), 1–13 (2019). [CrossRef]
29. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). [CrossRef]
30. D. Needell and R. Vershynin, “Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit,” IEEE J. Sel. Top. Signal Process. 4(2), 310–316 (2010). [CrossRef]
31. T. Blumensath and M. E. Davies, “Iterative hard thresholding for compressed sensing,” Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009). [CrossRef]
32. C. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” (2010).
33. R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010). [CrossRef]
34. C. Hegde, P. Indyk, and L. Schmidt, “A nearly-linear time framework for graph-structured sparsity,” in International Conference on Machine Learning, (2015), 928–937.
35. M. Courbariaux, I. Hubara, D. Soudry, R. El-Yaniv, and Y. Bengio, “Binarized neural networks: Training deep neural networks with weights and activations constrained to+ 1 or-1,” arXiv preprint arXiv:1602.02830 (2016).
36. Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep Learning Face Attributes in the Wild,” Computer Science (2014).
37. E. Candes and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” URL: www.acm.caltech.edu/l1magic/downloads/l1magic.pdf4, 14 (2005).
38. J. Huang, S. Zhang, and D. Metaxas, “Efficient MR image reconstruction for compressed MR imaging,” Med. Image Anal. 15(5), 670–679 (2011). [CrossRef]
