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

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

doi:doi:10.11648/j.ijdst.20251103.11

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| Peer-Reviewed

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

Sarobidy Nomenjanahary Razafitsalama Fin Luc

, Marie Emile Randrianandrasana

, Hariony Bienvenu Rakotonirina^*

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

1. Introduction

Image reconstruction from a reduced number of measurements is a major challenge in numerous fields such as medical imaging, remote sensing, and embedded systems. Compressive sensing (CS) provides a solution to this problem by exploiting the sparsity of signals in appropriate bases, such as wavelets

[1]

. This work proposes an image reconstruction method based on the level-3 Symlet 4 (sym4) discrete wavelet transform, combined with three iterative reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The aim is to evaluate their performance in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost, across sampling rates ranging from 10% to 60%. By comparing these algorithms under identical experimental conditions using the standard Lena image, this study seeks to identify the most efficient approach for practical CS applications where both fidelity and speed are critical

[2]

2. Principle of Compressive Sensing Applied to an IMAGE

2.1. Discrete Wavelet Transform Phase

During this phase, the original image

I

of dimension

P \times Q

is processed by the following operations

[3, 4]

for j \leftarrow 1 to 3

I^{(j)} \leftarrow (↓_{2, col} ((↓_{2, row} (I^{(j - 1)} *_{row} h)) *_{col} h))

{cH}_{j} \leftarrow (↓_{2, col} ((↓_{2, row} (I^{(j - 1)} *_{row} h)) *_{col} g))

{cV}_{j} \leftarrow (↓_{2, col} ((↓_{2, row} (I^{(j - 1)} *_{row} g)) *_{col} h))

{cD}_{j} \leftarrow (↓_{2, col} ((↓_{2, row} (I^{(j - 1)} *_{row} g)) *_{col} g))

endFor

C \leftarrow {\begin{matrix} [vec (I^{(3)}), vec ({cH}_{3}), vec ({cV}_{3}), vec ({cD}_{3}), vec ({cH}_{2}), \\ vec ({cV}_{2}), vec ({cD}_{2}), \dots, vec ({cD}_{1})] \end{matrix}}^{T}

S \leftarrow {[size (I^{(3)}), size ({cH}_{3}), size ({cH}_{2}), size ({cH}_{1}), size (I)]}^{T}

Where:

I^{(j)}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing the image approximation at level

j

I^{(0)} = I

h

is an

8 \times

1 vector representing the coefficients of the Symlet 4 wavelet and is defined by the following formula:

h = [\begin{matrix} \begin{matrix} - 0.0758 \\ - 0.0296 \end{matrix} \\ \begin{matrix} 0.49760 \\ 0.80370 \end{matrix} \\ \begin{matrix} \begin{matrix} 0.29790 \\ - 0.0992 \end{matrix} \\ \begin{matrix} - 0.0126 \\ 0.03220 \end{matrix} \end{matrix} \end{matrix}]

(1)

g

is an

8 \times

1 vector defined by the following formula:

g = [\begin{matrix} g_{0} \\ ⋮ \\ g_{7} \end{matrix}]

(2)

Where:

g_{k} = {(- 1)}^{k + 1} h_{7 - k}

h_{7} = 0.03220

k \in \{0, 1, . . ., 7\}

{cH}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing horizontal detail coefficients at level

j

{cV}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing vertical detail coefficients at level

j

{cD}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing the diagonal detail coefficients at level

j

C

is a column vector concatenating all the coefficients

S

is a

5 \times 2

matrix representing the details of sub-bands

*_{row}

denotes the convolution product along the rows

10)

*_{col}

denotes the convolution product along the columns

11)

↓_{2, row}

denotes the horizontal sub-sampling operation

12)

↓_{2, col}

denotes the vertical sub-sampling operation

13)

vec ()

denotes the column-wise vectorization operator

14)

size ()

returns the dimensions of the matrix

2.2. Acquisition or Measurement Phase

During this phase, the signal represented by a column vector

C

of size

N \times 1

is recorded in a compressed form in a measurement vector represented by a column vector

y

of size

M \times 1

, using the following formula

[5]

y = AC

(3)

Where:

A

is a rectangular matrix of size

M \times N

, known as the measurement matrix

(M < N)

2.3. Reconstruction Phase

During this phase, the goal is to find the vector

C

that satisfies Equation (3). Since

A

is a rectangular matrix of size

M \times N

with

M < N

, there exists an infinite number of vectors

C

that satisfy the equation (3). However, assuming that

C

is sparse, the problem becomes finding the solution to the following minimization

[6, 7]

\hat{C} = \arg \min {‖C‖}_{1}

subjectto

y = AC

(4)

2.4. Inverse Discrete Wavelet Transform Phase

During this phase, the original image

I

of dimension

P \times Q

is reconstructed from the vector

\hat{C}

and the matrix

S

using the following operations

[8, 9]

for j \leftarrow 3 to 1

I^{(j - 1)} \leftarrow (↑_{2, col} ((↑_{2, row} (I^{(j)} *_{col} h)) *_{col} h))

+ (↑_{2, col} ((↑_{2, row} ({cH}_{j} *_{col} g)) *_{col} h))

+ (↑_{2, col} ((↑_{2, row} ({cV}_{j} *_{col} h)) *_{col} g))

+ (↑_{2, col} ((↑_{2, row} ({cD}_{j} *_{col} g)) *_{col} g))

endFor

I_{rec} \leftarrow I^{(0)}

Where:

I^{(j)}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing the image approximation at level

j

h

is an

8 \times

1 vector representing the coefficients of the Symlet 4 wavelet and is defined by the formula (1)

g

is an

8 \times

1 vector defined by the formula (2)

{cH}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing horizontal detail coefficients at level

j

{cV}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing vertical detail coefficients at level

j

{cD}_{j}

is a matrix of approximately

\frac{P}{2^{j}} \times \frac{Q}{2^{j}}

dimensions representing the diagonal detail coefficients at level

j

\hat{C}

is a column vector concatenating all the coefficients

S

is a

5 \times 2

matrix representing the details of sub-bands

*_{row}

denotes the convolution product along the rows

10)

*_{col}

denotes the convolution product along the columns

11)

↑_{2, row}

denotes the horizontal over-sampling operation

12)

↑_{2, col}

denotes the vertical over-sampling operation

13)

I_{rec}

denotes the reconstructed image.

3. Metric for Evaluating Reconstruction Quality

3.1. Mean Squared Error (MSE)

The

MSE = \frac{1}{P \times Q} \sum_{i = 0}^{P - 1} \sum_{j = 0}^{Q - 1} {(I_{ij} - {\hat{I}}_{ij})}^{2}

(5)

Where:

MSE

denotes the Mean Squared Error

P

denotes the number of rows of the image

Q

denotes the number of colomuns of the image

I_{ij}

denotes the pixel value at position

(i, j)

in the original image

{\hat{I}}_{ij}

denotes the pixel value at position

(i, j)

in the reconstructed image

3.2. Peak Signal-to-Noise Ratio (PSNR)

The following provides its definition

[12, 13]

PSNR = 10 \log_{10} (\frac{65025}{MSE})

(6)

Where:

PSNR

denotes the Peak Signal-to-Noise Ratio

MSE

denotes the Mean Squared Error

3.3. Structural Similarity Index (SSIM)

The

SSIM (x, y) = \frac{(2 μ_{x} μ_{y} + C_{1}) (2 σ_{xy} + C_{2})}{({μ_{x}}^{2} + {μ_{y}}^{2} + C_{1}) ({σ_{x}}^{2} + {σ_{y}}^{2} + C_{2})}

(7)

Where:

SSIM

denotes the Structural Similarity Index

μ_{x}

denotes the mean intensity value of image

x

μ_{y}

denotes the mean intensity value of image

y

σ_{x}

denotes the variance of image

x

σ_{y}

denotes the variance of image

y

σ_{xy}

denotes the covariance between

x

and

y

C_{1} = {(K_{1} L)}^{2}

C_{2} = {(K_{2} L)}^{2}

L

denotes the dynamic range of pixel values

10)

K_{1} = 0.01

11)

K_{2} = 0.03

4. Comparison of the Original and Reconstructed Images

In this section, the following points are specified:

1) The measurement matrix

A

of size

M \times 40000

is constructed by randomly selecting

M

rows from the identity matrix of size

40000 \times 40000

M

is expressed as a percentage (0% corresponds to 0 rows, while 100% corresponds to 40000 rows).

3) The resolution of Equation (4) is carried out using the CoSaMP, SP, ALISTA algorithm

4) The Mean Squared Error (MSE) is computed using Equation (5)

5) The Peak Signal-to-Noise Ratio (PSNR) is computed using Equation (6)

6) The Structural Similarity Index (SSIM) is computed using Equation (7)

7) Processed image: Lena excerpted from the original publication: A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, 1989, p. 400. Original source: photograph from Playboy magazine, November 1972. Used here for non-commercial educational and research purposes.

Download: Download full-size image

Figure 1. Processed image.

1) Image size: 200×200 pixels

2) Programming environment: MATLAB 2023

3) Hardware specifications: CPU: Intel(R) Core i7-9750H, GPU: NVIDIA GetForce RTX 2060, RAM: 16Go

Table 1 presents the images reconstructed by the CoSaMP, SP, and ALISTA algorithms for sampling rates ranging from 10% to 60%.

Table 1. Images reconstructed.

M (%)

CosAmp

ALISTA

SSIM: 0.7849

Time (min): 1.5223

SSIM: 0.7849

Time (min): 0.69

SSIM: 0.7848

Time (min): 0.052

SSIM: 0.8833

Time (min): 5.4176

SSIM: 0.8833

Time (min): 2.3729

SSIM: 0.8831

Time (min): 0.0925

SSIM: 0.9421

Time (min): 20.94

SSIM: 0.9421

Time (min): 3.2572

SSIM: 0.9418

Time (min): 0.708

SSIM: 0.9490

Time (min): 34.28

SSIM: 0.9490

Time (min): 4.825

SSIM: 0.9487

Time (min): 0.29

SSIM: 0.9532

Time (min): 20.32

SSIM: 0.9532

Time (min): 6.2886

SSIM: 0.9530

Time (min): 4.97

SSIM: 0.9635

Time (min): 26.89

SSIM: 0.9635

Time (min): 7.6473

SSIM: 0.9633

Time (min): 7.4

The results obtained are summarized in Table 2.

Table 2. Summary of the results.

$M (%)$	SSIM			Reconstruction Time
$M (%)$	CoSaMP	SP	ALISTA	CoSaMP	SP	ALISTA
10%	0.78495	0.78495	0.7848	1.5223	0.69075	0.052
20%	0.88334	0.88334	0.88315	5.4176	2.3729	0.092525
30%	0.94218	0.94218	0.94189	20.94	3.2572	0.70858
40%	0.94901	0.94901	0.94877	34.28	4.8251	0.29
50%	0.95325	0.95325	0.95302	20.32	6.2886	4.97
60%	0.96351	0.96351	0.9633	26.89	7.6473	7.4

5. Conclusion

This paper presents a robust compressed sensing method for reconstructing images. It utilizes the Daubechies 4 (db4) discrete wavelet transform to generate a sparse image representation and reconstructs the original data from a limited number of measurements through the CoSaMP, SP, and ALISTA algorithms. The proposed approach was assessed in terms of reconstruction fidelity and computational efficiency across various sampling ratios. A summary of the experimental outcomes is provided in Figures 2 and 3.

Download: Download full-size image

Figure 2. SSIM vs% of measurements.

Download: Download full-size image

Figure 3. Reconstruction time vs% of measurements.

Figure 2 shows the evolution of SSIM as a function of the sampling rate for the CoSaMP, SP, and ALISTA algorithms. Overall, reconstruction quality improves as the percentage of measurements increases, which aligns with the principles of compressive sensing. The three methods exhibit nearly identical performance across the entire range, with curves that are visually indistinguishable suggesting structural fidelity is robustly maintained regardless of the algorithm chosen. At low sampling rates (10–20%), ALISTA and SP slightly outperform CoSaMP, though the differences are marginal (within 0.005 SSIM). All three algorithms converge to near-perfect reconstruction quality (SSIM > 0.96) at 60% sampling. These results indicate that while algorithmic design varies, their perceptual reconstruction outcomes are remarkably consistent, with no single method offering a clear structural advantage under these experimental conditions.

Figure 3 compares the reconstruction time (in minutes) of the three algorithms as a function of the sampling rate. ALISTA demonstrates exceptional computational efficiency: its runtime remains consistently low (under 8 minutes), increasing smoothly from less than 1 minute at 10% to approximately 8 minutes at 60%. SP follows a stable, nearly linear trend, rising gradually from 1 to 8 minutes over the same range. In contrast, CoSaMP exhibits erratic and significantly higher computational cost, peaking sharply at 40% (≈35 minutes) before dropping unexpectedly at 50% (≈20 minutes) and rising again at 60% (≈28 minutes). This non-monotonic behavior suggests CoSaMP’s convergence is highly sensitive to measurement density, potentially due to increased iteration counts. These observations underscore that ALISTA not only matches the reconstruction quality of its competitors but also provides superior speed and predictability making it particularly suitable for real-time or resource-constrained applications.

A promising direction for simultaneously improving both the efficiency and sparsity of image representation in compressive sensing would be to replace the conventional DWT used in this work with the Lifting Wavelet Transform (LWT). Unlike the DWT, the LWT enables an in-place implementation without explicit convolution, significantly reducing both memory complexity and computational cost which could further amplify the runtime advantages already demonstrated by ALISTA.

Abbreviations

ALISTA	Analytic Learned Iterative Shrinkage Thresholding Algorithm
CoSaMP	Compressive Sampling Matched Pursuit
CS	Compressive Sensing
LWT	Lifting Wavelet Transform
MSE	Mean Squared Error
PSNR	Peak Signal-to-Noise Ratio
SSIM	Structural Similarity Index
SP	Subspace Pursuit

Author Contributions

Hariniony Bienvenu Rakotonirina: Conceptualization, Formal Analysis, Investigation, Methodology, Project administration, Resources, Software, Validation, Writing – original draft, Writing – review & editing

Sarobidy Nomenjanahary Razafitsalama Fin Luc: Investigation, Methodology, Software

Marie Emile Randrianandrasana: Supervision, Validation

Data Availability Statement

The datasets and code used for reconstruction are available upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

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
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[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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APA Style

Luc, S. N. R. F., Randrianandrasana, M. E., Rakotonirina, H. B. (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

Luc, S. N. R. F.; Randrianandrasana, M. E.; Rakotonirina, H. B. 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

Luc SNRF, Randrianandrasana ME, Rakotonirina HB. 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 = {Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana and Hariony Bienvenu Rakotonirina},
  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  - Sarobidy Nomenjanahary Razafitsalama Fin Luc
AU  - Marie Emile Randrianandrasana
AU  - Hariony Bienvenu Rakotonirina
Y1  - 2025/10/31
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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
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EP  - 56
PB  - Science Publishing Group
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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
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Author Information

Sarobidy Nomenjanahary Razafitsalama Fin Luc

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

Research Fields: Mathematics, Algorithmics, Compressive Sensing, Artificial intelligence, Data compression

Contact Email

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

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

Research Fields: Telecommunication, Signal processing, Compressive Sensing, Radar, Electromagnetic wave

Contact Email

http://orcid.org/0009-0007-2595-5857
Hariony Bienvenu Rakotonirina

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

Research Fields: Telecommunication, Signal processing, Compressive Sensing, Artificial intelligence, High Amplifier Power Nonlinearity

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http://orcid.org/0009-0004-5054-736X

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APA Style

Luc, S. N. R. F., Randrianandrasana, M. E., Rakotonirina, H. B. (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

Luc, S. N. R. F.; Randrianandrasana, M. E.; Rakotonirina, H. B. 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

Luc SNRF, Randrianandrasana ME, Rakotonirina HB. 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 = {Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana and Hariony Bienvenu Rakotonirina},
  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  - Sarobidy Nomenjanahary Razafitsalama Fin Luc
AU  - Marie Emile Randrianandrasana
AU  - Hariony Bienvenu Rakotonirina
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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