#
Sparse Reconstruction Using Hyperbolic Tangent as Smooth l_{1}-Norm Approximation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Proposed Method

#### 2.2. Error Bounds for Proposed Smooth ${l}_{1}$-Norm

- $\left|z\right|={\left(z\right)}_{+}+{\left(-z\right)}_{+}$, where ${\left(z\right)}_{+}=\mathrm{max}\left\{z,0\right\}$ is the plus function;
- This plus function can be smoothly approximated as:$${\left(\mathit{z}\right)}_{+}\approx p\left(\mathit{z},\gamma \right)=\frac{1}{2}\left[\mathit{z}+\mathit{z}.\mathit{tanh}\left(\gamma \mathit{z}\right)\right]$$

**Lemma**

**1.**

**Lemma**

**2.**

**Proof.**

## 3. The Map Estimator and Proposed Thresholding Mechanism

Algorithm 1. Proposed Algorithms. |

Inputs: |

Sensing matrix ${\mathcal{F}}_{u}$, measurement vector $y\u03f5{\u2102}^{m}$, parameters $\gamma $, $\lambda $ $\mathrm{and}\text{}\beta $, |

Output: |

A k-sparse vector $\widehat{x}\in {R}^{n}$ |

Initialization: Initialize ${x}_{0}$, Index $i=0$ |

Step-1 (Sparse Representation): ${z}_{i}=\Psi {\mathit{x}}_{\mathit{i}}$ |

Step-1 (Gradient Computation): Find $\nabla f\left({z}_{i}\right)$ using Equation (5) |

Step-2 (Solution Update): Compute the update using Equations (6) and (7). |

Step-3 (Shrinkage): Estimate Solution using Equation (38), i.e., ${\widehat{\mathit{z}}}_{\mathit{i}+\mathbf{1}}={S}_{\beta}\left({z}_{i+1}\right)$ |

Step-4 (Repeat): If stopping criterion is not met,$i=i+1$ & go to step 1 |

Output: $\widehat{\mathrm{x}}={\mathsf{\Psi}}^{\mathrm{H}}{\widehat{z}}_{i}$ |

## 4. Results and Discussions

#### 4.1. 1-D Sparse Signal Recovery

#### 4.2. 2-D Compressively Sampled MR Image Recovery

#### 4.3. Cardiac Cine Magnetic Resonance Imaging Recovery

^{3}, TR = 3 ms, TE = 1.5 ms. Five different acceleration rates R = (2, 4, 8, 12, 20) were used to assess the performance of the proposed method. For in vivo data, the following parameters were used: reconstruction matrix size: 256 × 256, 25 cardiac phases, with FOV of 375 mm. TE = 1 ms, TR = 3 ms and flip angle = 600. Five acceleration rates are $R=\left(2,4,8,12,20\right)$ are used to evaluate the performance of the proposed method. The reconstructed images are matched with the fully sampled original generated cardiac cine MRI as shown in Figure 11. All images are recovered in MATLAB by the proposed algorithm.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Donoho, D.L. Compressed sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Candes, E.J.; Wakin, M.B. An Introduction To Compressive Sampling. IEEE Signal Process. Mag.
**2008**, 25, 21–30. [Google Scholar] [CrossRef] - Candes, E.J.; Tao, T. Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? IEEE Trans. Inf. Theory
**2006**, 52, 5406–5425. [Google Scholar] [CrossRef][Green Version] - Lustig, M.; Donoho, D.; Pauly, J.M. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med.
**2007**, 58, 1182–1195. [Google Scholar] [CrossRef] - Lustig, M.; Donoho, D.L.; Santos, J.M.; Pauly, J.M. Compressed Sensing MRI. IEEE Signal Process. Mag.
**2008**, 25, 72–82. [Google Scholar] [CrossRef] - Auger, D.A.; Bilchick, K.C.; Gonzalez, J.A.; Cui, S.X.; Holmes, J.W.; Kramer, C.M.; Salerno, M.; Epstein, F.H. Imaging left-ventricular mechanical activation in heart failure patients using cine DENSE MRI: Validation and implications for cardiac resynchronization therapy. J. Magn. Reson. Imaging
**2017**, 46, 887–896. [Google Scholar] [CrossRef] - Liu, J.; Feng, L.; Shen, H.-W.; Zhu, C.; Wang, Y.; Mukai, K.; Brooks, G.C.; Ordovas, K.; Saloner, D. Highly-accelerated self-gated free-breathing 3D cardiac cine MRI: Validation in assessment of left ventricular function. Magn. Reson. Mater. Phys. Biol. Med.
**2017**, 30, 337–346. [Google Scholar] [CrossRef][Green Version] - van Amerom, J.F.P.; Lloyd, D.F.A.; Price, A.N.; Kuklisova Murgasova, M.; Aljabar, P.; Malik, S.J.; Lohezic, M.; Rutherford, M.A.; Pushparajah, K.; Razavi, R.; et al. Fetal cardiac cine imaging using highly accelerated dynamic MRI with retrospective motion correction and outlier rejection. Magn. Reson. Med.
**2018**, 79, 327–338. [Google Scholar] [CrossRef][Green Version] - Paulus, W.J.; Tschöpe, C.; Sanderson, J.E.; Rusconi, C.; Flachskampf, F.A.; Rademakers, F.E.; Marino, P.; Smiseth, O.A.; De Keulenaer, G.; Leite-Moreira, A.F.; et al. How to diagnose diastolic heart failure: A consensus statement on the diagnosis of heart failure with normal left ventricular ejection fraction by the Heart Failure and Echocardiography Associations of the European Society of Cardiology. Eur. Heart J.
**2007**, 28, 2539–2550. [Google Scholar] [CrossRef][Green Version] - Yerly, J.; Gubian, D.; Knebel, J.-F.; Schenk, A.; Chaptinel, J.; Ginami, G.; Stuber, M. A phantom study to determine the theoretical accuracy and precision of radial MRI to measure cross-sectional area differences for the application of coronary endothelial function assessment. Magn. Reson. Med.
**2018**, 79, 108–120. [Google Scholar] [CrossRef] - Ahmed, A.H.; Qureshi, I.M.; Shah, J.A.; Zaheer, M. Motion correction based reconstruction method for compressively sampled cardiac MR imaging. Magn. Reson. Imaging
**2017**, 36, 159–166. [Google Scholar] [CrossRef] - Gamper, U.; Boesiger, P.; Kozerke, S. Compressed sensing in dynamic MRI. Magn. Reson. Med.
**2008**, 59, 365–373. [Google Scholar] [CrossRef] [PubMed] - Donoho, D.L. De-noising by soft-thresholding. IEEE Trans. Inf. Theory
**1995**, 41, 613–627. [Google Scholar] [CrossRef][Green Version] - Blumensath, T.; Davies, M.E. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal.
**2009**, 27, 265–274. [Google Scholar] [CrossRef][Green Version] - Tropp, J.A.; Wakin, M.B.; Duarte, M.F.; Baron, D.; Baraniuk, R.G. Random Filters for Compressive Sampling and Reconstruction. In Proceedings of the 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings, Toulouse, France, 14–19 May 2006. [Google Scholar]
- Yin, W.; Osher, S.; Goldfarb, D.; Darbon, J. Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing. SIAM J. Imaging Sci.
**2008**, 1, 143–168. [Google Scholar] [CrossRef][Green Version] - Khajehnejad, M.A.; Xu, W.; Avestimehr, A.S.; Hassibi, B. Weighted ℓ<inf>1</inf>minimization for sparse recovery with prior information. In Proceedings of the 2009 IEEE International Symposium on Information Theory, Seoul, Korea, 28 June–3 July 2009; pp. 483–487. [Google Scholar]
- Shah, J.; Qureshi, I.; Omer, H.; Khaliq, A. A modified POCS-based reconstruction method for compressively sampled MR imaging. Int. J. Imaging Syst. Technol.
**2014**, 24, 203–207. [Google Scholar] [CrossRef] - Shah, J.A.; Qureshi, I.M.; Omer, H.; Khaliq, A.A.; Deng, Y. Compressively sampled magnetic resonance image reconstruction using separable surrogate functional method. Concepts Magn. Reson. Part A
**2014**, 43, 157–165. [Google Scholar] [CrossRef] - Bilal, M.; Shah, J.A.; Qureshi, I.M.; Kadir, K. Respiratory Motion Correction for Compressively Sampled Free Breathing Cardiac MRI Using Smooth l(1)-Norm Approximation. Int. J. Biomed. Imaging
**2018**, 2018, 7803067. [Google Scholar] [CrossRef][Green Version] - Shah, J.; Qureshi, I.M.; Deng, Y.; Kadir, K. Reconstruction of Sparse Signals and Compressively Sampled Images Based on Smooth l1-Norm Approximation. J. Signal Process. Syst.
**2017**, 88, 333–344. [Google Scholar] [CrossRef] - He, C.; Xing, J.; Li, J.; Yang, Q.; Wang, R. A New Wavelet Thresholding Function Based on Hyperbolic Tangent Function. Math. Probl. Eng.
**2015**, 2015, 528656. [Google Scholar] [CrossRef] - Lu, J.-y.; Lin, H.; Ye, D.; Zhang, Y.-s. A New Wavelet Threshold Function and Denoising Application. Math. Probl. Eng.
**2016**, 2016, 3195492. [Google Scholar] [CrossRef][Green Version] - Schmidt, M.; Fung, G.; Rosales, R. Fast Optimization Methods for L1 Regularization: A Comparative Study and Two New Approaches. In Proceedings of the Machine Learning: ECML 2007, Berlin/Heidelberg, Germany, 17–21 September 2007; pp. 286–297. [Google Scholar]
- Chang, S.G.; Bin, Y.; Vetterli, M. Spatially adaptive wavelet thresholding with context modeling for image denoising. IEEE Trans. Image Process.
**2000**, 9, 1522–1531. [Google Scholar] [CrossRef][Green Version] - Sendur, L.; Selesnick, I.W. Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency. IEEE Trans. Signal Process.
**2002**, 50, 2744–2756. [Google Scholar] [CrossRef][Green Version] - Nasri, M.; Nezamabadi-Pour, H. Image denoising in the wavelet domain using a new adaptive thresholding function. Neurocomputing
**2009**, 72, 1012–1025. [Google Scholar] [CrossRef] - Prinosil, J.; Smekal, Z.; Bartusek, K. Wavelet Thresholding Techniques in MRI Domain. In Proceedings of the 2010 International Conference on Biosciences, Online, 7–13 March 2010; pp. 58–63. [Google Scholar]
- Jin, J.; Yang, B.; Liang, K.; Wang, X. General image denoising framework based on compressive sensing theory. Comput. Graph.
**2014**, 38, 382–391. [Google Scholar] [CrossRef]

**Figure 1.**${\mathit{l}}_{\mathbf{1}}$ norm approximation using the hyperbolic tangent function for different values of $\mathsf{\gamma}$ = (1, 4, 6, and 10). As the value of gamma continues to increase and the approximation is closer to the actual ${\mathit{l}}_{\mathbf{1}}$ norm, however, it is less smooth. The proposed technique gives us the flexibility to choose between the level of smoothness and accuracy.

**Figure 2.**${\mathit{l}}_{\mathbf{1}}$ norm approximation error bounds for $\mathit{z}>\mathbf{0}$, the green line shows the upper bound proved mathematically in Equation (16), whereas the dotted red line shows the actual error between the proposed ${\mathit{l}}_{\mathbf{1}}$ norm smooth approximation and actual non-differentiable ${\mathit{l}}_{\mathbf{1}}$ norm. The error is maximum at approximately zero and approaches zero as $\mathit{\gamma}\to \infty $.

**Figure 3.**${\mathit{l}}_{\mathbf{1}}$ norm approximation error bounds for $\mathit{z}<\mathbf{0}$, the green line shows the upper bound proved mathematically in Equation (19), whereas the dotted red line shows the actual error between proposed ${\mathit{l}}_{\mathbf{1}}$ norm smooth approximation and actual non-differentiable ${\mathit{l}}_{\mathbf{1}}$ norm. The error is maximum at approximately zero and approaches zero as $\mathit{\gamma}\to \infty $.

**Figure 4.**Hyperbolic tangent function based thresholding for alpha $\alpha $ = (2, 4, 8, 16), the value of $\alpha $ determines the slope of soft thresholding. The proposed method gives us the flexibility to shape the curves using $\alpha $ depending upon its application.

**Figure 5.**Fitness achieved by the soft thresholding and proposed algorithm. The proposed algorithm converges rapidly as compared to the soft thresholding technique.

**Figure 6.**Sparsity effect on successful recovery achieved by the soft thresholding and proposed algorithm. The proposed algorithm performs much better even with higher sparsity level as compared to the soft thresholding technique.

**Figure 7.**(

**a**) The recovered sparse signal from the proposed algorithm; (

**b**) The recovered sparse signal from soft thresholding.

**Figure 8.**Structural Similarity of proposed and soft thresholding algorithm for recovery of compressively sampled MR image against each iteration.

**Figure 9.**Correlation of proposed and soft thresholding algorithm of recovered compressively sample MR image.

**Figure 10.**(

**a**) Original 2D Brain MR Image, (

**b**) Conventional Soft Thresholding based recovered 2D MRI, (

**c**) 2D Brain MR Image recovered from undersampled image, (

**d**) Difference of original and soft thresholding image, (

**e**) Difference of proposed recovery method image with original image. The Difference is scaled up by the factor of 1000 in order to enhance its visibility.

**Figure 11.**(

**a**) Short axis cardiac cine MRI with completely sampled diastolic frame. (

**b**) Sparsifying transform of cine cardiac MRI diastolic frame with temporal Fourier transform ($\mathsf{\Psi}$), which results in sparse representation, (

**c**) Another sparse representation of cardiac cine MR image (diastolic frame) using total variation transform ($\mathsf{\Psi}$).

**Figure 12.**Simulated data (

**a**) Compares proposed method at bottom row with IST at top row with acceleration rate of 2. The arrow in (

**a**) depicts the very minute presence of artifacts. Here (

**b**) depicts the performance of proposed method at acceleration rate equal to 4. The arrow in (

**b**) depicts the presence of artefacts (

**c**) shows the results of both algorithms with acceleration rate set at 8. (

**d**) depicts the results when acceleration rate is set at 12. These artefacts are visible in the IST as mentioned by the white arrow in the figure (

**e**) shows very much degraded image quality of IST, while comparing it with the proposed method.

**Figure 13.**This figure depicts the efficiency of proposed algorithm by means of SSIM index. As acceleration rate increases, the SSIM of proposed algorithm degrades slowly while comparing it with IST algorithm.

**Figure 14.**Comparison of cardiac cine MRI recovery at number of iterations; it can be seen from results that proposed method converges to an optimal solution in lesser iterations as compared to traditional thresholding.

**Figure 15.**The performance using the peak signal-to-noise ratio (PSNR). PSNR of our method is better at all acceleration rates as compared to the soft thresholding method.

**Figure 16.**In real vivo data (

**a**) Compares proposed method at bottom row with IST at top row with acceleration rate of 2. The arrow in (

**a**) shows very minute artifacts. (

**b**) depicts the performance of proposed method at acceleration rate equal to 4. The arrow in (

**b**) depicts the presence of artefacts (

**c**) shows the results of both algorithms with acceleration rate set at 8. The artefacts due to subsampling become gradually more visible in IST results as highlighted by arrow mark (

**d**) depicts the results when acceleration rate is set at 12. Both techniques depicts the artefacts, however these artefacts are visible in the IST as mentioned by the white arrow in the figure (

**e**) show very much degraded image quality of IST, while comparing it with the proposed method. The subsampling artefacts dominate the traditional IST result when acceleration rate R is set at 20.

**Table 1.**Performance comparison of different sparsity transforms using mean squared error in the transform domain. Temporal FFT performs better in cardiac cine MRI.

Performance Metrics | Soft Thresholding | Proposed Algorithm |
---|---|---|

MSE | 1.00 × 10^{−2} | 1.61 × 10^{−4} |

Fitness | 0.8664 | 0.0224 |

SNR | 12.6712 | 30.6259 |

Correlation | 0.9787 | 0.9995 |

**Table 2.**Performance comparison of conventional soft thresholding and proposed method with different compression levels, i.e., 5% to 50% of subsampling of the original 2-D MR image. These results show that the proposed method achieves better results in terms of SSIM and PSNR at varying compression ratios.

Compression Ratio | Soft Thresholding | Proposed Algorithm | ||
---|---|---|---|---|

SSIM | PSNR | SSIM | PSNR | |

5 % | 0.6843 | 75.9056 | 0.7048 | 76.1609 |

10% | 0.7786 | 78.9320 | 0.8175 | 79.6580 |

20% | 0.8994 | 82.0316 | 0.8472 | 83.7628 |

30% | 0.9407 | 87.3535 | 0.9790 | 91.1620 |

40% | 0.9724 | 91.2540 | 0.9920 | 96.1281 |

50% | 0.9884 | 95.4245 | 0.9955 | 99.5496 |

**Table 3.**Performance comparison of different sparsity transforms using mean squared error in the transform domain. Temporal FFT performs better in cardiac Cine MRI.

Performance Metrics | Soft Thresholding | Proposed Algorithm |
---|---|---|

MSE | 1.38 × 10^{−4} | 0.73 × 10^{−4} |

PSNR | 86.7195 | 89.4497 |

ISNR | 28.3832 | 31.1135 |

SSIM | 0.9346 | 0.9711 |

SNR | 26.0298 | 28.7491 |

Correlation | 0.9980 | 0.9989 |

**Table 4.**Performance comparison of different sparsity transforms using mean squared error in the transform domain. Temporal FFT performs better in cardiac cine MRI.

Acceleration Rates | Spatial Domain | Total Variation | Temporal FFT |
---|---|---|---|

2 | 0.1096 | 0.1123 | 0.0728 |

4 | 0.2321 | 0.1849 | 0.0848 |

8 | 0.2810 | 0.2438 | 0.0948 |

12 | 0.3533 | 0.2684 | 0.1043 |

20 | 0.4756 | 0.2982 | 0.1150 |

**Table 5.**Comparison of proposed method with conventional IST algorithm with RMSE. Proposed method performance is much better as acceleration rates are increased.

Acceleration Rates | Undersampled Image | Iterative Soft Thresholding | Proposed Method | |
---|---|---|---|---|

Simulated Data | 2 | 0.081 | 0.0365 | 0.0353 |

4 | 0.1218 | 0.0472 | 0.0372 | |

8 | 0.1498 | 0.0702 | 0.0419 | |

12 | 0.1583 | 0.0775 | 0.0485 | |

20 | 0.1782 | 0.0941 | 0.0606 | |

In vivo Data | 2 | 0.085 | 0.0099 | 0.0056 |

4 | 0.106 | 0.0241 | 0.0172 | |

8 | 0.1170 | 0.0495 | 0.0206 | |

12 | 0.120 | 0.0567 | 0.0338 | |

20 | 0.1398 | 0.0585 | 0.0551 |

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**MDPI and ACS Style**

Haider, H.; Shah, J.A.; Kadir, K.; Khan, N.
Sparse Reconstruction Using Hyperbolic Tangent as Smooth *l*_{1}-Norm Approximation. *Computation* **2023**, *11*, 7.
https://doi.org/10.3390/computation11010007

**AMA Style**

Haider H, Shah JA, Kadir K, Khan N.
Sparse Reconstruction Using Hyperbolic Tangent as Smooth *l*_{1}-Norm Approximation. *Computation*. 2023; 11(1):7.
https://doi.org/10.3390/computation11010007

**Chicago/Turabian Style**

Haider, Hassaan, Jawad Ali Shah, Kushsairy Kadir, and Najeeb Khan.
2023. "Sparse Reconstruction Using Hyperbolic Tangent as Smooth *l*_{1}-Norm Approximation" *Computation* 11, no. 1: 7.
https://doi.org/10.3390/computation11010007