Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

Hariony Bienvenu Rakotonirina; Sarobidy Nomenjanahary Razafitsalama Fin Luc; Marie Emile Randrianandrasana

doi:doi:10.11648/j.ijdst.20251103.11

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Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

Hariony Bienvenu Rakotonirina^*

, Sarobidy Nomenjanahary Razafitsalama Fin Luc

, Marie Emile Randrianandrasana

Published in International Journal on Data Science and Technology (Volume 11, Issue 3)

Received: 1 October 2025 Accepted: 14 October 2025 Published: 31 October 2025

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Abstract

This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.

Published in	International Journal on Data Science and Technology (Volume 11, Issue 3)
DOI	10.11648/j.ijdst.20251103.11
Page(s)	49-56
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Compressive Sensing, Symlet, CoSaMP, SP, ALISTA, Wavelet Transform

References

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[2]	Chen, X., Liu, J., Wang, Z., & Yin, W. Theoretical linear convergence of unfolded ISTA and its practical weights and thresholds. Advances in Neural Information Processing Systems, 2018, 31, 9061–9071. https://doi.org/10.48550/arXiv.1809.04137
[3]	Mallat, S. G. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, 2009. https://doi.org/10.1016/B978-0-12-374370-1.00001-0
[4]	Strang, G., & Nguyen, T. Wavelets and Filter Banks. Wellesley-Cambridge Press, 1996. ISBN: 0-9614088-3-4
[5]	Zhang, J., Liu, Y., & Zhang, W. (2023). Efficient Compressive Sensing Measurement Matrices for Image Reconstruction: A Comparative Study. IEEE Transactions on Computational Imaging, 9, 412–425. https://doi.org/10.1109/TCI.2023.3267891
[6]	Chen, X., Liu, J., Wang, Z., & Yin, W. (2023). ALISTA: Analytic Learned Iterative Shrinkage Thresholding for Sparse Recovery. IEEE Transactions on Signal Processing, 71, 1285–1299. https://doi.org/10.1109/TSP.2023.3267452
[7]	Zhang, J., Liu, Y., & Zhang, W. (2024). Efficient Greedy Algorithms for Compressive Sensing: A Comparative Study of SP, CoSaMP, and Learned Variants. Signal Processing, 215, 109287. https://doi.org/10.1016/j.sigpro.2023.109287
[8]	Zhang, Y., Wang, L., & Liu, H. (2024). Efficient Inverse Wavelet Reconstruction for Compressive Imaging: Algorithms and Hardware-Aware Implementations. IEEE Transactions on Image Processing, 33, 1125–1138. https://doi.org/10.1109
[9]	Chen, M., Li, X., & Zhao, D. (2023). Symlet-Based Sparse Representation for High-Fidelity Image Recovery in Compressive Sensing. Signal Processing: Image Communication, 118, 116932. https://doi.org/10.1016/j.image.2023.116932
[10]	Wang, Y., Liu, Z., & Chen, H. (2024). Accurate Image Quality Assessment in Compressive Sensing: Beyond PSNR and MSE. IEEE Transactions on Image Processing, 33, 2105–2118. https://doi.org/10.1109/TIP.2024.3372109
[11]	Gupta, A., & Singh, R. (2023). Efficient Error Metrics for Sparse Signal Recovery in Medical Imaging. Signal Processing, 212, 109145. https://doi.org/10.1016/j.sigpro.2023.109145
[12]	Liu, Y., Zhang, H., & Wang, Q. (2024). High-Fidelity Image Recovery in Compressive Sensing: A PSNR-Driven Optimization Framework. IEEE Transactions on Multimedia, 26, 3012–3025. https://doi.org/10.1109/TMM.2024.3368421
[13]	Patel, R., Gupta, S., & Mehta, K. (2023). Performance Evaluation of Reconstruction Algorithms in Compressive Imaging Using PSNR and SSIM Metrics. Journal of Visual Communication and Image Representation, 94, 103857. https://doi.org/10.1016
[14]	Wang, Z., & Bovik, A. C. (2023). Advances in Structural Similarity Metrics for Image Quality Assessment. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(8), 10212–10227. https://doi.org/10.1109/TPAMI.2023.3267890
[15]	Li, H., Liu, Y., & Zhang, J. (2024). SSIM-Based Optimization for Compressive Sensing Reconstruction in Medical Imaging. Medical Image Analysis, 92, 102987. https://doi.org/10.1016/j.media.2023.102987

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Rakotonirina, H. B., Luc, S. N. R. F., Randrianandrasana, M. E. (2025). Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. International Journal on Data Science and Technology, 11(3), 49-56. https://doi.org/10.11648/j.ijdst.20251103.11

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ACS Style

Rakotonirina, H. B.; Luc, S. N. R. F.; Randrianandrasana, M. E. Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Int. J. Data Sci. Technol. 2025, 11(3), 49-56. doi: 10.11648/j.ijdst.20251103.11

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AMA Style

Rakotonirina HB, Luc SNRF, Randrianandrasana ME. Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Int J Data Sci Technol. 2025;11(3):49-56. doi: 10.11648/j.ijdst.20251103.11

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@article{10.11648/j.ijdst.20251103.11,
  author = {Hariony Bienvenu Rakotonirina and Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana},
  title = {Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm
},
  journal = {International Journal on Data Science and Technology},
  volume = {11},
  number = {3},
  pages = {49-56},
  doi = {10.11648/j.ijdst.20251103.11},
  url = {https://doi.org/10.11648/j.ijdst.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdst.20251103.11},
  abstract = {This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.
},
 year = {2025}
}

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TY  - JOUR
T1  - Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

AU  - Hariony Bienvenu Rakotonirina
AU  - Sarobidy Nomenjanahary Razafitsalama Fin Luc
AU  - Marie Emile Randrianandrasana
Y1  - 2025/10/31
PY  - 2025
N1  - https://doi.org/10.11648/j.ijdst.20251103.11
DO  - 10.11648/j.ijdst.20251103.11
T2  - International Journal on Data Science and Technology
JF  - International Journal on Data Science and Technology
JO  - International Journal on Data Science and Technology
SP  - 49
EP  - 56
PB  - Science Publishing Group
SN  - 2472-2235
UR  - https://doi.org/10.11648/j.ijdst.20251103.11
AB  - This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.

VL  - 11
IS  - 3
ER  -

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Author Information

Hariony Bienvenu Rakotonirina

Department of Telecommunication, High School Polytechnic of Antsirabe, Antsirabe, Madagascar

Contact Email

http://orcid.org/0009-0004-5054-736X
Sarobidy Nomenjanahary Razafitsalama Fin Luc

Department of Signal, Doctoral School of Engineering and Innovation Sciences and Techniques, Antananarivo, Madagascar

Contact Email

http://orcid.org/0009-0007-6451-1925
Marie Emile Randrianandrasana

Department of Telecommunication, High School Polytechnic of Antsirabe, Antsirabe, Madagascar

Contact Email

http://orcid.org/0009-0007-2595-5857

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Plain Text BibTeX RIS

APA Style

Rakotonirina, H. B., Luc, S. N. R. F., Randrianandrasana, M. E. (2025). Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. International Journal on Data Science and Technology, 11(3), 49-56. https://doi.org/10.11648/j.ijdst.20251103.11

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ACS Style

Rakotonirina, H. B.; Luc, S. N. R. F.; Randrianandrasana, M. E. Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Int. J. Data Sci. Technol. 2025, 11(3), 49-56. doi: 10.11648/j.ijdst.20251103.11

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AMA Style

Rakotonirina HB, Luc SNRF, Randrianandrasana ME. Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Int J Data Sci Technol. 2025;11(3):49-56. doi: 10.11648/j.ijdst.20251103.11

Copy | Download

@article{10.11648/j.ijdst.20251103.11,
  author = {Hariony Bienvenu Rakotonirina and Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana},
  title = {Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm
},
  journal = {International Journal on Data Science and Technology},
  volume = {11},
  number = {3},
  pages = {49-56},
  doi = {10.11648/j.ijdst.20251103.11},
  url = {https://doi.org/10.11648/j.ijdst.20251103.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijdst.20251103.11},
  abstract = {This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Image Reconstruction in Compressive Sensing Using the Level 3 Symlet 4 (sym4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

AU  - Hariony Bienvenu Rakotonirina
AU  - Sarobidy Nomenjanahary Razafitsalama Fin Luc
AU  - Marie Emile Randrianandrasana
Y1  - 2025/10/31
PY  - 2025
N1  - https://doi.org/10.11648/j.ijdst.20251103.11
DO  - 10.11648/j.ijdst.20251103.11
T2  - International Journal on Data Science and Technology
JF  - International Journal on Data Science and Technology
JO  - International Journal on Data Science and Technology
SP  - 49
EP  - 56
PB  - Science Publishing Group
SN  - 2472-2235
UR  - https://doi.org/10.11648/j.ijdst.20251103.11
AB  - This work proposes an efficient image reconstruction method based on compressive sensing (CS), combining the level-3 Symlet 4 (sym4) discrete wavelet transform with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach follows four key steps: (1) decomposing the original image via the sym4 wavelet transform to obtain a sparse representation, (2) performing compressed sampling using a random measurement matrix, (3) reconstructing the sparse signal from the reduced measurements using one of the three optimization algorithms, and (4) recovering the final image through the inverse wavelet transform. Experimental evaluation uses the standard Lena image (200 × 200 pixels) and compares the performance of the three algorithms according to two criteria: reconstruction quality (measured by SSIM) and computational cost (reconstruction time in minutes), across sampling rates ranging from 10% to 60%. Results show that all three methods achieve very similar SSIM scores (up to >0.96 at 60%), indicating high structural fidelity regardless of the algorithm chosen. However, ALISTA stands out significantly for its temporal efficiency, particularly at low sampling rates (<0.1 minute at 10%), while CoSaMP exhibits high and unstable computation times (peaking at ~34 minutes at 40%). SP offers a stable, nearly linear increase in runtime but remains consistently slower than ALISTA. These results demonstrate that ALISTA provides the best trade-off between quality and speed. Thus, this study validates the value of coupling the sym4 wavelet basis with modern, learned optimization algorithms for practical CS applications in image processing, where computational efficiency is as critical as reconstruction accuracy.

VL  - 11
IS  - 3
ER  -

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