# Adaptive Algorithm on Block-Compressive Sensing and Noisy Data Estimation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}minimization have benefits on the reconstruction effect, but with large computational complexity and high time complexity. Compared with convex optimization algorithms, the non-convex algorithms, such as the greedy pursuit algorithm, operate quickly, with a slightly poor accuracy based on l

_{0}minimization, and can also meet the general requirements of practical applications. In addition, the iterative threshold method has also been widely used in both of them with excellent performance. However, the iterative threshold method is sensitive to the selection of the threshold and the initial value of the iteration that affects the efficiency and accuracy of the algorithm [16,17]. The selection of thresholds in this process often uses simple error values (including absolute or relative values) or quantitative iterations as stopping criterion of the algorithm, which does not guarantee algorithm optimization [18,19].

## 2. Preliminaries

#### 2.1. Compressive Sensing

#### 2.2. Block-Compressive Sensing (BCS)

#### 2.3. Problems of BCS

- Most existing research papers of BCS do not perform useful analysis on image partitioning and then segment according to the analysis result [21,22]. The common partitioning method (n = B × B) of BCS only considers reducing the computational complexity and storage space problem without considering the integrity of the algorithm and other potential effects, such as providing a better foundation for subsequent sampling and reconstructing by combining the structural features and the information entropy of the image.
- The basic sampling method used in BCS is to sample each sub-block uniformly according to the total sampling rate (TSR), while the adaptive sampling method selects different sampling rates according to the sampling feature of each sub-block [23]. Specifically, the detail block allocates a larger sampling rate, and the smooth block matches a smaller sampling rate, thereby improving the overall quality of the reconstructed image at the same TSR. But the crux is that the studies of criteria (feature) used to assign adaptive sampling rates are rarely seen in recent articles.
- Although there are many studies on the improvement of the BCS iterative construction algorithm [24], few articles focus on optimizing the performance of the algorithm from the aspect of iteration stop criterion in the image reconstruction process, especially in the noise background.

## 3. Problem Formulation and Important Factors

#### 3.1. Flexible Partitioning by Mean Information Entropy (MIE) and Texture Structure (TS)

#### 3.2. Adaptive Sampling with Variance and Local Salient Factor

#### 3.3. Error Estimation and Iterative Stop Criterion in Reconstruction Process

#### 3.3.1. Reconstruction Error Estimation in Noisy Background

#### 3.3.2. Optimal Iterative Recovery of Image in Noisy Background

#### 3.3.3. Application of Error Estimation on BCS

## 4. The Proposed Algorithm (FE-ABCS)

#### 4.1. The Workflow and Pseudocode of FE-ABCS

- Flexible partitioning: using the weighted MIE as the block basis to reduce the average complexity of the sub-images from the pixel domain and the spatial domain;
- Adaptive sampling: adopting synthetic feature and step-less sampling to ensure a reasonable sample rate for each subgraph;
- Optimal number of iterations: using the error estimate method to ensure the minimum error output of the reconstructed image in the noisy background.

FE-ABCS Algorithm based on OMP (Orthogonal Matching Pursuit) | |

1: Input: Original image I, total sampling rate TSR, sub-image dimension n ($n={2}^{b},b=2,4,6,\cdots $), sparse matrix $\Psi \in {R}^{n\times n}$, initialized measurement matrix $\Phi \in {R}^{n\times n}$ 2: Initialization: $\left\{x|x\leftarrow I\text{\hspace{0.17em}}and\text{\hspace{0.17em}}x\in {R}^{N}\right\}$; ${T}_{1}=\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$n$}\right.\text{\hspace{0.17em}}$; //$quantity\text{\hspace{0.17em}}of\text{\hspace{0.17em}}subimages$ ${T}_{2}=1+{\mathrm{log}}_{2}n$; //$type\text{\hspace{0.17em}}of\text{\hspace{0.17em}}flexible\text{\hspace{0.17em}}partitioning$ step1: flexible partitioning (FP) 3: for j = 1,…,T _{2} do4: ${l}_{j}\times {h}_{j}={2}^{j-1}\times {2}^{{T}_{2}-j};{\left\{{I}_{i},i=1,\cdots ,{T}_{1}\right\}}_{j}\leftarrow I$; ${\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\}}_{j}\leftarrow x$; 5: ${g}_{MIE}^{j}=MIE\left({\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\}}_{j}\right)$; ${c}_{TS}^{j}={(f({g}_{TS}))}_{j}={(f([{g}_{TS}^{H},{g}_{TS}^{V}]))}_{j}$; 6: ${g}_{FB}^{j}={c}_{TS}^{j}\xb7{g}_{MIE}^{j}$; // Weighted MIE--Base of FP 7: end for 8:${j}_{opt}=\mathrm{arg}\underset{j}{\mathrm{min}}\left(\left\{{g}_{FB}^{j},j=1,\cdots ,{T}_{2}\right\}\right)$; 9:$l\times h={2}^{{j}_{opt}-1}\times {2}^{{T}_{2}-{j}_{opt}}$; $\left\{{I}_{i},i=1,\cdots ,{T}_{1}\right\}\leftarrow I$; $\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\}\leftarrow x$; step2: adaptive sampling (AS) 10: for i = 1,…,T _{1} do11: $D({x}_{i})\leftarrow {x}_{i}$; $L({x}_{i})\leftarrow {I}_{i}$; $J({x}_{i})=L{({x}_{i})}^{{\lambda}_{1}}\xb7D{({x}_{i})}^{{\lambda}_{2}}$; // synthetic feature (J)--Base of AS 12: end for 13: $\eta (\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\})=\frac{{\mathrm{log}}_{2}J(\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\})}{\frac{1}{{T}_{1}}{\displaystyle \sum _{i=1}^{{T}_{1}}{\mathrm{log}}_{2}J({x}_{i})}}$; 14: ${c}_{SR}(\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\})=\eta (\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\})\xb7TSR\text{\hspace{0.17em}}$; // ${c}_{SR}$--AS ratio of sub-images $\left\{{m}_{i},i=1,\cdots ,{T}_{1}\right\}={c}_{SR}(\left\{{x}_{i},i=1,\cdots ,{T}_{1}\right\})\xb7n$; 15: $\Phi ={\left({\varphi}_{1},\cdots ,{\varphi}_{n}\right)}^{T},{\chi}_{i}=randperm\left(n\right),{\Phi}_{{\chi}_{i}}=\Phi \left({\chi}_{i},\text{\hspace{0.17em}}:\right)$; 16: $\left\{{\Phi}_{i},i=1,\cdots ,{T}_{1}\right\}=\left\{{\Phi}_{{\chi}_{i}}(\left[1,\cdots ,{m}_{i}\right],:),i=1,\cdots ,{T}_{1}\right\}$; 17: $\left\{{y}_{i},i=1,\cdots ,{T}_{1}\right\}=\left\{{\Phi}_{i}\xb7{x}_{i},i=1,\cdots ,{T}_{1}\right\}$; | step3: restoring based on error estimation 18:$\left\{{\tilde{y}}_{i},i=1,\cdots ,{T}_{1}\right\}=\left\{{y}_{i}+{w}_{i},i=1,\cdots ,{T}_{1}\right\}$; // ${w}_{i}:{m}_{i}-dimension\text{}AWGN,\text{\hspace{0.17em}}{w}_{i}=0:noiseless$ 19: $\left\{{\Omega}_{i},i=1,\cdots ,{T}_{1}\right\}=\left\{{\Phi}_{i}\xb7\Psi ,i=1,\cdots ,{T}_{1}\right\}$; 20: for i = 1,…,T _{1} do21: ${\Omega}_{i}=\left\{{\omega}_{i1},\cdots ,{\omega}_{in}\right\},\text{\hspace{0.17em}}r={\tilde{y}}_{i},\text{\hspace{0.17em}}A=\varnothing ,\text{\hspace{0.17em}}{s}^{*}={0}^{n}$; //$\left\{{\omega}_{ij},j=1,\dots ,n\right\}$-- column vector of ${\Omega}_{i}$ 22: ${v}_{opt}^{i}=\left\{{v}^{i}|\text{\hspace{0.17em}}\mathrm{arg}\underset{{v}^{i}}{\mathrm{min}}\overline{{e}_{{y}_{i}}}\right\}$; // calculate optimal iterative of sub-images 23: for $j=1,\cdots ,{v}_{opt}^{i}$ do 24: $\wedge =\mathrm{arg}\underset{j}{\mathrm{min}}\left|\langle r,{w}_{ij}\rangle \right|$; 25: $A=A\cup \{\wedge \}$; 26: $r={\tilde{y}}_{i}-{\Omega}_{i}(:,A)\xb7{\left[{\Omega}_{i}(:,A)\right]}^{+}\xb7{\tilde{y}}_{i}$; 27: end for 28: ${s}_{i}^{\ast}={\left[{\Omega}_{i}(:,A)\right]}^{+}\xb7{\tilde{y}}_{i}$; // ${s}_{i}^{\ast}$--reconstructed sparse representation 29: ${x}_{i}^{*}=\Psi \xb7{s}_{i}^{\ast}$; // ${x}_{i}^{\ast}$--reconstructed original signal 30: end for 31: ${x}^{*}=\left\{{x}_{i}^{*},i=1,\cdots ,{T}_{1}\right\},\text{\hspace{0.17em}}{I}_{r}^{*}=\left\{{x}^{*},l,h\right\}$; // ${I}_{r}^{\ast}$--reconstructed image without filter step4: multimode filtering (MF) 32: $if\text{\hspace{0.17em}}\left(BEI\ge BE{I}^{*}\right)$; //$BE{I}^{*}-Threshod\text{\hspace{0.17em}}of\text{\hspace{0.17em}}block\text{\hspace{0.17em}}effect$ 33: ${I}_{r}^{*}=deblock({I}_{r}^{*})$; 34: $end\text{\hspace{0.17em}}if$ 35: $if\text{\hspace{0.17em}}\left(TSR\ge TS{R}^{*}\right)$; //$TS{R}^{*}-Threshod\text{\hspace{0.17em}}of\text{\hspace{0.17em}}TSR$ 36: ${I}_{F}^{*}=wienerfilter({I}_{r}^{*})$; 37: $else\text{\hspace{1em}}{I}_{F}^{*}=medfilter({I}_{r}^{*})$; 38: $end\text{\hspace{0.17em}}if$ 39: ${I}^{*}={I}_{F}^{*}$ // ${I}^{\ast}$--reconstructed image with M |

#### 4.2. Specific Parameter Setting of FE-ABCS

#### 4.2.1. Setting of the Weighting Coefficient ${c}_{TS}$

#### 4.2.2. Setting of the Adaptive Sampling Rate ${c}_{SR}$

- Initial value calculation of ${\eta}_{SR}({x}_{i})$: get the initial value of the sampling factor by Equation (10).
- Judgment of ${\eta}_{SR}({x}_{i})$ through MSRF (${\eta}_{\mathrm{min}}$): if the corresponding sampling rate factor of all image sub-blocks meets the minimum threshold requirement (${\eta}_{SR}({x}_{i})>{\eta}_{\mathrm{min}},i\in \left\{1,2,\cdots ,{T}_{1}\right\}$), there is no need for modification, however, if it is not satisfied, modify it.
- Modifying of ${\eta}_{SR}({x}_{i})$: if ${\eta}_{SR}({x}_{i})\le {\eta}_{\mathrm{min}}$, then ${\eta}_{SR}({x}_{i})={\eta}_{\mathrm{min}}$; if ${\eta}_{SR}({x}_{i})>{\eta}_{\mathrm{min}}$, then use the following equation to modify the value:$${\eta}_{SR}({x}_{i})==(1+(1-{\eta}_{\mathrm{min}})\frac{{T}_{1}-{T}_{1}{}^{\prime}}{{T}_{1}{}^{\prime}})\frac{{\mathrm{log}}_{2}J({x}_{i})}{\frac{1}{{T}_{1}{}^{\prime}}{\displaystyle \sum _{j=1}^{{{T}^{\prime}}_{1}}{\mathrm{log}}_{2}J({x}_{j})}}$$

#### 4.2.3. Setting of the Iteration Stop Condition ${v}_{opt}$

## 5. Experiments and Results Analysis

^{*}, ${\sigma}_{x}$ and ${\sigma}_{{x}^{\ast}}$ are the standard deviation of x and x

^{*}, ${\sigma}_{x{x}^{\ast}}$ represents the covariance of x and x

^{*}, constant ${c}_{1}={(0.01L)}^{2}$ and ${c}_{2}={(0.03L)}^{2}$, and L is the range of pixel values.

^{*}, ${m}_{x}(i)=\sqrt{{({h}_{H}\otimes x(i))}^{2}+{({h}_{V}\otimes x(i))}^{2}}$ and ${m}_{y}\left(\mathrm{i}\right)=\sqrt{{({h}_{H}\otimes {x}^{\ast}(i))}^{2}+{({h}_{V}\otimes {x}^{\ast}(i))}^{2}}$ are the gradient magnitude of x(i) and x

^{*}(i), ${h}_{H}$ and ${h}_{V}$ represent the Prewitt operator of horizontal and vertical direction, and ${c}_{3}$ is an adjustment constant.

#### 5.1. Experiment and Analysis without Noise

#### 5.1.1. Performance Comparison of Various Algorithms

- Analysis of the performance indicators of the first four algorithms shows that for the BCS algorithm, BCS with a fixed column block is inferior to BCS with a fixed square block because square partitioning makes good use of the correlation of intra-block regions. MIE-based partitioning minimizes the average amount of information entropy of the sub-images. However, when the overall image has obvious texture characteristics, simply using MIE as the partitioning basis may not necessarily achieve a good effect, and the BCS algorithm based on weighted MIE combined with the overall texture feature can achieve better performance indicators.
- Comparing the adaptive BCS algorithms under different features in Table 1, variance has obvious superiority to IE among the single features, because the variance not only contains the dispersion of gray distribution but also the relative difference of the individual gray distribution of sub-images. In addition, the synthetic feature (combined local saliency) has a better effect than a single feature. The main reason for this is that the synthetic feature not only considers the overall difference of the subgraphs, but also the inner local-difference of the subgraphs.
- Combining experimental results of the eight BCS algorithms in Table 1 reveals that using adaptive sampling or flexible partitioning alone does not provide the best results, but the proposed algorithm combining the two steps will have a significant effect on both PSNR and SSIM.

#### 5.1.2. Parametric Analysis of the Proposed Algorithm

#### 5.2. Experiment and Analysis Under Noisy Conditions

#### 5.2.1. Effect Analysis of Different Iteration Stop Conditions on Performance

#### 5.2.2. Impact of Noise-Std and TSR on ${v}_{opt}$

#### 5.3. Application and Comparison Experiment of FE-ABCS Algorithm in Image Compression

#### 5.3.1. Application of FE-ABCS Algorithm in Image Compression

_{1}× n = N), that is different from the modules of FDWT and IDWT in which dimensions of the input and output signals are the same (both T

_{1}× n = N). These differences make the proposed algorithm have a larger compression ratio (CR) and smaller bits per pixel (bpp) than JPEG2000 under the same quantization and encoding conditions.

#### 5.3.2. Comparison Experiment between the Proposed Algorithm and the JPEG2000 Algorithm

**b**) and (

**c**) of Figure 13, the image generated by the FE-ABCS-QC algorithm is slightly better than the one of the JPEG2000 algorithm, either from the perception of objective data or subjective sense.

- Small Rate (bpp): the reason why the performance of the FE-ABCS-QC algorithm is worse than the JPEG2000 algorithm at this condition is that the small value of M which changes with Rate causes the observing process to fail to cover the overall information of the image.
- Medium or slightly larger Rate (bpp): the explanation for the phenomenon that the performance of the FE-ABCS-QC algorithm is better than the JPEG2000 algorithm in this situation is that the appropriate M can ensure the complete acquisition of image information and can also provide a certain image compression ratio to generate a better basis for quantization and encoding.
- Large Rate (bpp): this case of the FE-ABCS-QC algorithm is not considered because the algorithm belongs to the CS algorithm and requires M << N itself.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Effect of different partitioning methods on mean information entropy (MIE) of images (the abscissa represents flexible partitioning with shape $n=l\times h={2}^{i-1}\times {2}^{9-i}=256$).

**Figure 2.**The workflow of two block-compressive sensing (BCS) algorithms. (

**a**) Typical BCS algorithm, (

**b**) FE-ABCS algorithm.

**Figure 3.**Reconstructed images of Cameraman and performance indicators with different BCS algorithms (TSR = 0.5).

**Figure 4.**The comparison of the proposed algorithm with 4 reconstruction algorithms (OMP, IRLS, BP, SP): (

**a**) TSR = 0.4, (

**b**) TSR = 0.6.

**Figure 7.**The correlation between the PSB, Noise-std, and TSR under the six different iteration stop conditions of Lena: (

**a**) PSB changes with Noise-std, (

**b**) PSB changes with TSR.

**Figure 8.**Iterative reconstruction images based on $\gamma $ and ${v}_{opt}$ at the condition of Noise-std = 20 and TSR = 0.5.

**Figure 9.**Correlation between ${v}_{opt}$, TSR, and Noise-std of sub-images: (

**a**) Noise-std = 20, (

**b**) TSR = 0.4.

**Figure 13.**The two algorithms’ comparison of test image Bikes at the condition of bpp = 0.25: (

**a**) original image, (

**b**) JPEG2000 image (PSNR = 29.80, SSIM = 0.9069, GMSD = 0.1964), (

**c**) image by the FE-ABCS-QC algorithm (PSNR = 30.50, SSIM = 0.9366, GMSD = 0.1574).

**Table 1.**The Peak Signal to Noise Ratio (PSNR) and Structural Similarity (SSIM) of reconstructed images with eight BCS algorithms based on OMP. (TSR = total sampling rate).

Images | Algorithms | TSR = 0.2 | TSR = 0.3 | TSR = 0.4 | TSR = 0.5 | TSR = 0.6 |
---|---|---|---|---|---|---|

PSNR/SSIM | PSNR/SSIM | PSNR/SSIM | PSNR/SSIM | PSNR/SSIM | ||

Lena | M-B_C | 29.0945/0.7684 | 29.8095/0.8720 | 30.6667/0.9281 | 31.8944/0.9572 | 32.9854/0.9719 |

M-B_S | 31.1390/0.8866 | 31.7478/0.9227 | 32.4328/0.9480 | 33.2460/0.9651 | 34.2614/0.9763 | |

M-FB_MIE | 30.7091/0.8613 | 31.2850/0.9115 | 32.0093/0.9413 | 32.9032/0.9600 | 33.8147/0.9737 | |

M-FB_WM | 31.1636/0.8880 | 31.7524/0.9236 | 32.4623/0.9479 | 33.2906/0.9645 | 34.2691/0.9763 | |

M-B_C-A_I | 29.1187/0.7838 | 29.8433/0.8803 | 30.8898/0.9305 | 32.0023/0.9577 | 33.2193/0.9732 | |

M-FB_WM-A_I | 31.1763/0.8967 | 31.8872/0.9344 | 32.7542/0.9584 | 33.7353/0.9732 | 34.8647/0.9827 | |

M-FB_WM-A_V | 31.2286/0.9087 | 32.0579/0.9433 | 33.0168/0.9643 | 34.1341/0.9775 | 35.4341/0.9856 | |

M-FB_WM-A_S | 31.3609/0.9138 | 32.0943/0.9487 | 33.1958/0.9681 | 34.3334/0.9807 | 35.8423/0.9878 | |

Goldhill | M-B_C | 28.4533/0.7747 | 28.9144/0.8718 | 29.3894/0.9080 | 29.7706/0.9315 | 30.2421/0.9495 |

M-B_S | 29.5494/0.8785 | 29.9517/0.9089 | 30.3330/0.9341 | 30.8857/0.9514 | 31.4439/0.9640 | |

M-FB_MIE | 29.7012/0.8882 | 29.9811/0.9154 | 30.4465/0.9364 | 30.9347/0.9516 | 31.5153/0.9642 | |

M-FB_WM | 29.7029/0.8867 | 30.0277/0.9151 | 30.4827/0.9361 | 30.9555/0.9516 | 31.5333/0.9649 | |

M-B_C-A_I | 28.4436/0.7809 | 28.8691/0.8693 | 29.3048/0.9089 | 29.7046/0.9321 | 30.2355/0.9499 | |

M-FB_WM-A_I | 29.6708/0.8918 | 30.0833/0.9215 | 30.5120/0.9424 | 31.0667/0.9574 | 31.6899/0.9697 | |

M-FB_WM-A_V | 29.5370/0.8957 | 30.0891/0.9253 | 30.5379/0.9456 | 31.0922/0.9607 | 31.8011/0.9724 | |

M-FB_WM-A_S | 29.7786/0.8975 | 30.1482/0.9272 | 30.5689/0.9472 | 31.1310/0.9622 | 31.8379/0.9736 | |

Cameraman | M-B_C | 28.5347/0.7787 | 29.0078/0.8559 | 29.3971/0.9051 | 29.9417/0.9379 | 30.6612/0.9592 |

M-B_S | 31.1796/0.8763 | 31.4929/0.9121 | 31.9203/0.9391 | 32.3009/0.9581 | 32.7879/0.9704 | |

M-FB_MIE | 31.1487/0.8782 | 31.5067/0.9123 | 31.8644/0.9403 | 32.3170/0.9577 | 32.7946/0.9703 | |

M-FB_WM | 31.2118/0.8675 | 31.4645/0.9072 | 31.8130/0.9365 | 32.2050/0.9559 | 32.6811/0.9686 | |

M-B_C-A_I | 28.5669/0.7852 | 28.8807/0.8612 | 29.3928/0.9164 | 29.9924/0.9461 | 30.6130/0.9639 | |

M-FB_WM-A_I | 31.2554/0.8901 | 31.5975/0.9296 | 32.0955/0.9533 | 32.6859/0.9701 | 33.4007/0.9802 | |

M-FB_WM-A_V | 31.2869/0.9085 | 31.8762/0.9550 | 32.5052/0.9746 | 33.3531/0.9848 | 34.4449/0.9904 | |

M-FB_WM-A_S | 31.3916/0.9287 | 31.9731/0.9621 | 32.6508/0.9790 | 33.6779/0.9877 | 34.8958/0.9918 | |

Couple | M-B_C | 28.6592/0.7582 | 29.0162/0.8557 | 29.5471/0.9109 | 30.2260/0.9440 | 30.9136/0.9640 |

M-B_S | 30.1529/0.8912 | 30.6910/0.9289 | 31.2853/0.9541 | 31.9693/0.9695 | 32.7464/0.9796 | |

M-FB_MIE | 30.1920/0.8895 | 30.7257/0.9282 | 31.2948/0.9531 | 31.9509/0.9692 | 32.7424/0.9794 | |

M-FB_WM | 30.1357/0.8917 | 30.6890/0.9259 | 31.3185/0.9539 | 31.9520/0.9691 | 32.7622/0.9793 | |

M-B_C-A_I | 28.5694/0.7428 | 29.0442/0.8589 | 29.5828/0.9088 | 30.2127/0.9444 | 30.9839/0.9642 | |

M-FB_WM-A_I | 30.2105/0.9027 | 30.7783/0.9413 | 31.4680/0.9630 | 32.3143/0.9759 | 33.2604/0.9840 | |

M-FB_WM-A_V | 30.1896/0.9099 | 30.8541/0.9454 | 31.4990/0.9670 | 32.3769/0.9792 | 33.3260/0.9864 | |

M-FB_WM-A_S | 30.3340/0.9117 | 30.9047/0.9475 | 31.5496/0.9686 | 32.3788/0.9798 | 33.3561/0.9869 |

**Table 2.**The PSNR and SSIM of reconstructed images with eight BCS algorithms based on iteratively reweighted least square (IRLS) and basis pursuit (BP).

Restoring Method | Algorithms | TSR = 0.4 | TSR = 0.6 | ||||||
---|---|---|---|---|---|---|---|---|---|

PSNR | SSIM | GMSD | CT | PSNR | SSIM | GMSD | CT | ||

IRLS | R-B_C | 32.38 | 0.9361 | 0.1943 | 5.729 | 33.42 | 0.9790 | 0.1348 | 13.97 |

R-B_S | 32.67 | 0.9634 | 0.1468 | 5.928 | 35.08 | 0.9843 | 0.0987 | 14.94 | |

R-FB_MIE | 32.14 | 0.9593 | 0.1658 | 5.986 | 34.44 | 0.9825 | 0.1094 | 13.79 | |

R-FB_WM | 32.46 | 0.9631 | 0.1441 | 6.071 | 34.80 | 0.9841 | 0.0992 | 14.25 | |

R-B_C-A_I | 30.55 | 0.9460 | 0.1882 | 6.011 | 34.08 | 0.9825 | 0.1238 | 14.46 | |

R-FB_WM-A_I | 33.05 | 0.9714 | 0.1346 | 6.507 | 36.01 | 0.9894 | 0.0863 | 14.98 | |

R-FB_WM-A_V | 33.25 | 0.9751 | 0.1216 | 6.994 | 36.71 | 0.9914 | 0.0691 | 17.34 | |

R-FB_WM-A_S | 33.59 | 0.9787 | 0.1188 | 7.456 | 37.23 | 0.9927 | 0.0661 | 19.38 | |

BP | P-B_C | 30.56 | 0.9380 | 0.1984 | 33.47 | 33.35 | 0.9787 | 0.1378 | 69.67 |

P-B_S | 32.72 | 0.9638 | 0.1484 | 34.22 | 34.61 | 0.9823 | 0.1072 | 71.00 | |

P-FB_MIE | 32.00 | 0.9531 | 0.1627 | 34.04 | 34.14 | 0.9810 | 0.1149 | 68.78 | |

P-FB_WM | 32.82 | 0.9635 | 0.1512 | 35.08 | 34.57 | 0.9832 | 0.1070 | 69.48 | |

P-B_C-A_I | 30.72 | 0.9428 | 0.1973 | 33.84 | 33.57 | 0.9795 | 0.1335 | 70.27 | |

P-FB_WM-A_I | 33.01 | 0.9705 | 0.1442 | 35.63 | 35.70 | 0.9884 | 0.0888 | 70.73 | |

P-FB_WM-A_V | 33.32 | 0.9750 | 0.1277 | 36.98 | 36.37 | 0.9909 | 0.0742 | 72.59 | |

P-FB_WM-A_S | 33.49 | 0.9773 | 0.1210 | 37.85 | 37.45 | 0.9932 | 0.0662 | 74.41 |

**Table 3.**The experimental results of Lena at different stop conditions and noise background (TSR = 0.4).

Sparsity | ${\mathit{v}}_{\mathit{\varsigma}=0.1}$ | ${\mathit{v}}_{\mathit{\varsigma}=0.2}$ | ${\mathit{v}}_{\mathit{\varsigma}=0.3}$ | ${\mathit{v}}_{\mathit{\varsigma}=0.4}$ | ${\mathit{v}}_{\mathit{\varsigma}=0.5}$ | ${\mathit{v}}_{\mathit{o}\mathit{p}\mathit{t}}$ | |
---|---|---|---|---|---|---|---|

Noise-std | PSNR and SSIM and GMSD and BEI and CT | ||||||

5 | 32.95 | 32.71 | 32.39 | 32.12 | 31.96 | 32.48 | |

0.961 | 0.965 | 0.961 | 0.957 | 0.955 | 0.960 | ||

0.165 | 0.160 | 0.169 | 0.173 | 0.178 | 0.171 | ||

10.63 | 10.25 | 10.04 | 9.92 | 9.98 | 10.09 | ||

0.741 | 0.949 | 1.166 | 1.567 | 1.911 | 1.481 | ||

10 | 32.40 | 31.81 | 31.30 | 31.07 | 30.92 | 32.23 | |

0.957 | 0.955 | 0.949 | 0.941 | 0.937 | 0.956 | ||

0.169 | 0.179 | 0.190 | 0.194 | 0.200 | 0.177 | ||

10.29 | 10.04 | 10.14 | 10.15 | 10.20 | 10.21 | ||

0.741 | 0.922 | 1.233 | 1.637 | 1.961 | 0.926 | ||

15 | 31.63 | 30.87 | 30.46 | 30.27 | 30.12 | 31.81 | |

0.948 | 0.937 | 0.926 | 0.917 | 0.912 | 0.949 | ||

0.180 | 0.202 | 0.213 | 0.220 | 0.223 | 0.185 | ||

10.32 | 10.25 | 10.32 | 10.38 | 10.39 | 10.25 | ||

0.727 | 0.933 | 1.154 | 1.550 | 2.063 | 0.798 | ||

20 | 30.97 | 30.08 | 29.83 | 29.66 | 29.61 | 31.48 | |

0.936 | 0.916 | 0.898 | 0.887 | 0.879 | 0.941 | ||

0.197 | 0.219 | 0.236 | 0.244 | 0.247 | 0.191 | ||

10.45 | 10.34 | 10.39 | 10.43 | 10.43 | 10.50 | ||

0.720 | 0.914 | 1.203 | 1.476 | 2.348 | 0.739 | ||

30 | 30.03 | 29.34 | 29.04 | 28.94 | 28.89 | 30.75 | |

0.901 | 0.862 | 0.832 | 0.816 | 0.803 | 0.920 | ||

0.227 | 0.257 | 0.266 | 0.272 | 0.275 | 0.204 | ||

10.59 | 10.52 | 10.54 | 10.55 | 10.67 | 10.62 | ||

0.901 | 0.947 | 1.237 | 1.500 | 1.996 | 0.690 | ||

40 | 29.38 | 28.77 | 28.57 | 28.50 | 28.45 | 30.14 | |

0.856 | 0.795 | 0.756 | 0.734 | 0.717 | 0.899 | ||

0.252 | 0.277 | 0.286 | 0.290 | 0.291 | 0.221 | ||

10.79 | 10.57 | 10.71 | 10.68 | 10.67 | 10.72 | ||

0.736 | 0.791 | 1.169 | 1.534 | 2.047 | 0.678 |

Images | Lena | Baboon | Flowers | Oriental Gate | Index | |
---|---|---|---|---|---|---|

Stop Condition | ||||||

γ = 300 | 32.17 | 29.84 | 31.40 | 33.05 | PSNR | |

0.9562 | 0.8703 | 0.9744 | 0.9698 | SSIM | ||

0.1661 | 0.2095 | 0.1812 | 0.1636 | GMSD | ||

10.16 | 10.72 | 9.866 | 8.845 | BEI | ||

0.8310 | 0.855 0 | 0.844 0 | 1.050 | CT | ||

21.93 | 13.52 | 20.28 | 21.09 | PSGBC | ||

γ =1 | 31.99 | 29.75 | 31.50 | 32.66 | PSNR | |

0.9574 | 0.8731 | 0.9768 | 0.9693 | SSIM | ||

0.1638 | 0.1738 | 0.1571 | 0.1459 | GMSD | ||

10.01 | 10.58 | 9.583 | 8.745 | BEI | ||

2.905 | 2.704 | 2.823 | 3.578 | CT | ||

6.429 | 5.224 | 7.240 | 6.934 | PSGBC | ||

${v}_{opt}$ | 32.75 | 30.03 | 31.78 | 33.15 | PSNR | |

0.9603 | 0.8729 | 0.9771 | 0.9637 | SSIM | ||

0.1471 | 0.1956 | 0.1538 | 0.1471 | GMSD | ||

9.898 | 10.68 | 9.693 | 9.025 | BEI | ||

0.7630 | 0.8900 | 0.8500 | 0.8740 | CT | ||

28.31 | 14.10 | 24.50 | 27.53 | PSGBC |

**Table 5.**The comparison results of different test-images under the various conditions (bits per pixel (bpp)) based on the JPEG2000 algorithm and the FE-ABCS-QC algorithm.

Method | JPEG2000 (PSNR/SSIM/GMSD) | FE-ABCS-QC (PSNR/SSIM/GMSD) | ∆P/∆S/∆G | |
---|---|---|---|---|

Test Image | ||||

Lena | bpp = 0.0625 | 31.64/0.9387/0.1842 | 30.58/0.7341/0.2478 | −1.06/−0.2046/0.0636 |

bpp = 0.125 | 33.38/0.9697/0.1399 | 32.79/0.9413/0.1702 | −0.59/−0.0284/0.0303 | |

bpp = 0.2 | 34.59/0.9807/0.1161 | 34.20/0.9710/0.1339 | −0.39/−0.0097/0.0178 | |

bpp = 0.25 | 35.25/0.9850/0.0996 | 37.80/0.9932/0.0612 | 2.55/0.0082/−0.0384 | |

bpp = 0.3 | 35.73/0.9875/0.0917 | 38.28/0.9941/0.0554 | 2.55/0.0066/−0.0363 | |

Monarch | bpp = 0.0625 | 30.56/0.8184/0.2335 | 27.52/0.3615/0.2726 | −3.04/−0.4569/0.0391 |

bpp = 0.125 | 31.31/0.9074/0.1881 | 29.12/0.6388/0.2568 | −2.19/−0.2686/0.0687 | |

bpp = 0.2 | 32.49/0.9466/0.1554 | 32.91/0.9473/0.1507 | 0.42/0.0007/−0.0047 | |

bpp = 0.25 | 32.81/0.9572/0.1476 | 35.77/0.9886/0.0682 | 2.96/0.0314/−0.0794 | |

bpp = 0.3 | 33.40/0.9679/0.1305 | 36.17/0.9896/0.0664 | 2.77/0.0217/−0.0641 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zhu, Y.; Liu, W.; Shen, Q.
Adaptive Algorithm on Block-Compressive Sensing and Noisy Data Estimation. *Electronics* **2019**, *8*, 753.
https://doi.org/10.3390/electronics8070753

**AMA Style**

Zhu Y, Liu W, Shen Q.
Adaptive Algorithm on Block-Compressive Sensing and Noisy Data Estimation. *Electronics*. 2019; 8(7):753.
https://doi.org/10.3390/electronics8070753

**Chicago/Turabian Style**

Zhu, Yongjun, Wenbo Liu, and Qian Shen.
2019. "Adaptive Algorithm on Block-Compressive Sensing and Noisy Data Estimation" *Electronics* 8, no. 7: 753.
https://doi.org/10.3390/electronics8070753