- Open Access
Reconstruction for block-based compressive sensing of image with reweighted double sparse constraint
© The Author(s). 2019
- Received: 18 September 2018
- Accepted: 30 April 2019
- Published: 24 May 2019
Block compressive sensing reduces the computational complexity by dividing the image into multiple patches for processing, but the performance of the reconstruction algorithm is decreased. Generally, the reconstruction algorithm improves the quality of reconstructed image by adding various constraints and regularization terms, namely prior information. In this paper, a reweighted double sparse constraint reconstruction model which combines the residual sparsity and ℓ1 regularization term is proposed. The residual sparsity aims to exploit the nonlocal similarity of image patches, and the ℓ1 regularization term is used to utilize the local sparsity of image patches. The resulting model is solved under the frame of split Bregman iteration (SBI). A large number of experiments show that the algorithm in this paper can reconstruct the original image efficiently and is comparable to the current representative compressive sensing reconstruction algorithm.
- Image reconstruction
- Compressive sensing
- Reweighted double sparse constraint
Compressive sensing (CS) theory proposed by Candès et al.  breaks the limitation of Nyquist sampling theorem, that is, it can recover the original signal at a sampling rate lower than twice the bandwidth if the signal is sparse in some domain. CS theory has been widely used in various fields since its birth, such as nuclear magnetic resonance, image processing, analog to information conversion, compressive radar, etc. 
Since natural images are almost always compressible, the application of compression sensing is natural. However, if compressive sensing is applied to the large-sized image directly, the measurement matrix would be very “large,” which leads to huge amounts of memory and further resulting in high computational complexity. Therefore, the block-based compressive sensing (BCS) method  is proposed. During the image processing of BCS, the image is divided into multiple image patches and each image patch is operated separately, so that the computational complexity is greatly reduced. Nevertheless, the quality of the reconstructed image is degraded compared to the quality of the image reconstructed by the entire image.
For the prior knowledge has a crucial influence on the performance of the image reconstruction algorithm, designing an effective regularization term is beneficial to make full use of image prior information and further improve the quality of the reconstructed image. The sparsity and nonlocal similarity, which are the most important properties of the images, are utilized to improve the quality of reconstructed images. The sparsity aims to represent the original image with a little nonzero value or approximate zero, more specifically, the sparsity is to organize the original image more sparsely in some domain [4, 5]. Currently, different predetermined transform basis, including discrete cosine transform (DCT), discrete wavelet transform (DWT), and so on, has been used to exploit the sparsity to further derive some reconstruction algorithm, such as smooth prediction Landweber of BCS based on DCT (BCS-SPL-DCT)  and BCS based on DWT (BCS-SPL-DWT) . Furthermore, to enrich the texture and structure in recovered images [7, 8], the multi-hypothesis (MH) prediction  method which explores the nonlocal similarities was proposed. By sharing the similar idea [10–12] where nonlocal similarities are exploited to design the local sparsifying transform, these methods can achieve better recovery performance than the algorithms that were previously designed for BCS without using nonlocal similarities. However, the recovered images still contain some visual artifacts.
For better CS recovery, some researchers used the idea of weighting the signal. Candès  proposed a weighted scheme based on the magnitude of signals to get closer to ℓ0 norm, while still using ℓ1 norm in the optimization problems. In a similar manner, Asif et al.  adaptively assigned weight values according to the homotopy of signals.
To better reconstruct the original image both in smooth region and edge region, this paper proposes a method that combines the sparsity and nonlocal similarity by the means of reweighting. Firstly, the residual which represents the difference between the target image patch and the similar image patch is calculated, and it should be more sparse. Therefore, residual sparsity was applied to exploit the nonlocal similarity. Secondly, reweighting applied in the sparse constraint is to consider the sparsity difference of the different image patches to enhance the sparsity. Thirdly, we use the frame of SBI to transform the proposed model into several sub problems to solve it effectively.
We propose a reconstruction model that combines the residual sparsity and l1 regularization term by the way of reweighting. It utilizes local sparsity and non-local self-similarity simultaneously so that the model can further improve the performance of the reconstruction algorithm.
To solve the model of reweighted double sparse constraint, we design an effective scheme based on the split Bregman iteration (SBI) algorithm. Extensive experiments of our method are conducted and compared with other typical algorithm using PSNR and SSIM.
The rest of this paper is organized as follows. Section 2 introduces the reconstruction model. There are four parts in Section 2, including residual model, reweighted sparse representation, weight estimation, and the solution of the proposed method. In Section 3, we present the plenty experiments and the results compared to other representative methods. Section 4 concludes the paper in the end.
The residual model ‖W1(x − u)‖1 exploits the nonlocal similarity and the ℓ1 regularization term of image patch ‖W2x‖1 exploits the local self-similarity. W1 is used to discriminatively weight different residual coefficients, W2 is used to reweight ℓ1 minimization. Sparsity can be further enhanced by combining W1 and W2. In the proposed model, the nonlocal similarity and the local self-similarity are combined effectively to further improve the reconstruction quality.
Next, we will introduce the model in detail, including the residual model, reweighted sparse representation, and weight estimation.
2.1 Residual model
2.2 Reweighted sparse representation
2.3 Weight estimation
2.4 Solution to the proposed model
2.4.1 Z sub-problem
The gradient and optimal step are calculated by the above equation, and the output of the SD is an updated value of z(t, i _ max), where i _ max is 300 in our experimental setting.
2.4.2 X sub-problem
After solving the z sub-problem and the x sub-problem, update b through Eq. (16) and repeat the three steps until all iterations are completed. The detailed description of the algorithm was given in Algorithm 2.
In this section, extensive experiments are done to verify the performance of reconstructed image, and we compare the reconstruction quality between the proposed algorithm and other four algorithms. The model parameters are set as follows: the image patch size is 64, the size of the searching window is 20 × 20, and the rest of the parameters will be given in detail. All of the experiments in this paper are done on Intel (R) Core (TM) i3, 3.0 GHz CPU processor, and MATLAB 2012b on Windows 10 operating system.
3.1.1 Complexity of search window size
3.1.2 Effect of similar patches
3.1.3 Effect of regularization parameters
In this section, we test the effect of regularization parameters λ1 and λ2 to the performance of reconstructed image.
3.1.4 Effect of weight parameter
3.2 Reconstruction results
In this paper, we have the original image x ∈ ℝN and its corresponding measurement y ∈ ℝM. H is a measurement matrix whose size is M × N. Compressive sensing is intended to recover the original high-quality image from the measured value with high probability. The measurement rate is expressed by the ratio and is equal to M/N.
PSNR and SSIM comparisons with various image reconstruction algorithm
The figures show the PSNR and SSIM of the images reconstructed by different reconstruction algorithms at ratios of 20%, 30%, and 40%, respectively. According to experimental results, it can be seen that the images reconstructed by the BCS-SPL-DCT and BCS-SPL-DWT are generally fuzzy, the texture structure is not clear enough, and the visual effects are worse than the other three algorithms. Images reconstructed by MH algorithm and the CoS algorithm all have better image quality, and the visual intuition effects are not much different. Especially the CoS algorithm reconstructs the image with stronger texture and higher quality. In terms of the PSNR, the images reconstructed by proposed algorithm are slightly higher than images reconstructed by the MH algorithm and CoS algorithm, that is, the algorithm proposed in this paper can effectively improve the quality of image reconstruction and is generally better than the above four classic compressive sensing reconstruction algorithms.
In this paper, we propose a reweighted double sparse constraint reconstruction model. The model not only takes full advantage of the nonlocal similarity the and local self-similarity by using the residual model and ℓ1 regularization to enhance the sparsity but also establishes the model by the means of reweighting to further improve the quality of reconstructed image. The model is mathematically defined as a solution to the reweighted ℓ1 minimization problem; an effective solution based on the SBI framework was designed to reduce the computational complexity. Under the SBI framework, the model is transformed from an unconstrained problem to multiple simple sub-problems. In this paper, we use the steepest descent method and the soft threshold algorithm to solve these sub-problems, separately. A large number of experiments demonstrate that the proposed image reconstruction model has better reconstruction quality and visual effect than other four representative algorithms. Future work includes image block matching optimization and reducing computational complexity.
The authors thank the editor and reviewers.
Financial support for this work was provided by the National Natural Science Foundation of China [grant number 61501069] and the Fundamental Research Funds for the central Universities [Grant number 106112016CDJXZ168815].
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.
YZ conceived and designed the experiments. JZ and XC performed the experiments and analyzed the data. GH and ZZ wrote the paper. ZH put forward constructive comments. All authors read and approved the final manuscript.
Yuanhong Hong received his BS, MS, and PhD degrees in commutation engineering from Chongqing University, Chongqing, China in 2003, 2006, and 2011, respectively. He is currently an Associate Professor with the School of Microelectronics and Communication Engineering, Chongqing University. His research interests include image processing, machine learning and computer vision.
Jing Zhang received her BS degree in commutation engineering from Chongqing University, Chongqing, China in 2018. And now she is a postgraduate in School of Microelectronics and Communication Engineering at Chongqing University.
Xinyu Cheng received his BS degree in electronic information engineering from Chongqing University, Chongqing, China in 2018. And now he is a postgraduate in School of Microelectronics and Communication Engineering at Chongqing University.
Guan Huang received her BS degree in commutation engineering from Chongqing University, Chongqing, China in 2018. And now she is a postgraduate in School of Automotive Engineering at Chongqing university.
Zhaokun Zhou received his BS degree in commutation engineering from Chongqing University, Chongqing, China in 2018. And now he is a postgraduate in School of Automotive Engineering at Chongqing university.
Zhiyong Huang received the B.Sc. degree in electric engineering and the Ph.D. degree in electronic engineering from Chongqing University, Chongqing, China, in 2001 and 2009, respectively. He is currently an Associate Professor with the School of Microelectronics and Communication Engineering, Chongqing University. His research interests include image/video processing, computer vision, and machine learning.
The authors declare that they have no competing interests.
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