Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) 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.cssp.20251201.12

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Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm

Sarobidy Nomenjanahary Razafitsalama Fin Luc

, Marie Emile Randrianandrasana

, Hariony Bienvenu Rakotonirina^*

Published in Science Journal of Circuits, Systems and Signal Processing (Volume 12, Issue 1)

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

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Abstract

This paper presents an efficient image reconstruction method based on Compressive Sensing (CS) theory, leveraging the level-3 Reverse Biorthogonal 4.4 (rbio4.4) discrete wavelet transform in combination with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach exploits the sparsity of images in a suitable wavelet basis, enabling compressed acquisition from a reduced number of random linear measurements. The process consists of four stages: (1) decomposition of the original image using the rbio4.4 wavelet transform to obtain sparse coefficients, (2) compressed sampling via a random measurement matrix, (3) reconstruction of the sparse signal using SP, CoSaMP, or ALISTA, and (4) final image reconstruction through the inverse wavelet transform. Experimental evaluation was conducted on the classic Lena image (200 × 200 pixels), comparing the three algorithms in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost (reconstruction time in minutes) across sampling rates ranging from 10% to 60%. Results show that all three algorithms achieve nearly identical reconstruction quality (virtually indistinguishable SSIM values at each sampling rate), confirming their effectiveness within the CS. However, ALISTA stands out significantly due to its exceptional speed, exhibiting substantially lower reconstruction times thanks to its learned nature, which replaces iterative procedures with fixed, optimized operations. In contrast, CoSaMP demonstrates higher and sometimes unpredictable computational times depending on the sampling rate. These findings highlight ALISTA’s strong potential for real-time or embedded applications.

Published in	Science Journal of Circuits, Systems and Signal Processing (Volume 12, Issue 1)
DOI	10.11648/j.cssp.20251201.12
Page(s)	8-15
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, Reverse Biorthogonal, CoSaMP, SP; ALISTA, Wavelet Transform

1. Introduction

Reconstructing images from a limited number of measurements poses a significant challenge in various domains, including medical imaging, remote sensing, and embedded systems. Compressive sensing (CS) addresses this issue by leveraging the sparsity of signals in suitable bases, such as wavelets

[1]

. In this work, we propose an image reconstruction approach that employs the level-3 reverse biorthogonal 4.4 (rbior4.4) wavelet transform in conjunction with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and ALISTA. The objective is to assess their performance in terms of reconstruction quality quantified using SSIM and computational cost, across sampling rates varying 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

10 \times 1

vector representing the coefficients of the Reverse Biorthogonal 4.4 wavelet and is defined by the following formula:

h = [\begin{matrix} \begin{matrix} 0 \\ 0 \end{matrix} \\ \begin{matrix} - 0.0645 \\ - 0.0407 \end{matrix} \\ \begin{matrix} \begin{matrix} 0.41810 \\ 0.78850 \end{matrix} \\ \begin{matrix} 0.41810 \\ \begin{matrix} - 0.0407 \\ \begin{matrix} - 0.0645 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]

(1)

g

is an

10 \times 1

vector defined by the following formula:

g = [\begin{matrix} \begin{matrix} - 0.0378 \\ - 0.0238 \end{matrix} \\ \begin{matrix} 0.11060 \\ 0.37740 \end{matrix} \\ \begin{matrix} \begin{matrix} - 0.8527 \\ 0.37740 \end{matrix} \\ \begin{matrix} 0.11060 \\ \begin{matrix} - 0.0238 \\ \begin{matrix} - 0.0378 \\ 0 \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix}]

(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

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

*_{col}

denotes the convolution product along the columns

↓_{2, row}

denotes the horizontal sub-sampling operation

↓_{2, col}

denotes the vertical sub-sampling operation

10)

vec ()

denotes the column-wise vectorization operator

11)

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

10 \times 1

vector representing the coefficients of the Reverse Biorthogonal 4.4 wavelet and is defined by the formula (1)

g

is an

10 \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 CoSaMP, SP et 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

Table 1. Images reconstructed.

M (%)

CosAmp

ALISTA

SSIM: 0.764

Time (min): 1.617

SSIM: 0.764

Time (min): 0.6236

SSIM: 0.764

Time (min): 0.053

SSIM: 0.8684

Time (min): 5.69

SSIM: 0.8684

Time (min): 2.55

SSIM: 0.8683

Time (min): 0.098

SSIM: 0.9357

Time (min): 22.34

SSIM: 0.9357

Time (min): 3.32

SSIM: 0.9356

Time (min): 0.38

SSIM: 0.9421

Time (min): 36.03

SSIM: 0.9421

Time (min): 4.93

SSIM: 0.9420

Time (min): 0.44

SSIM: 0.9469

Time (min): 21.32

SSIM: 0.9469

Time (min): 6.4646

SSIM: 0.9468

Time (min): 6.69

SSIM: 0.9569

Time (min): 27.81

SSIM: 0.9569

Time (min): 8.7055

SSIM: 0.9569

Time (min): 8.14

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%	0.764	0.764	0.764	1.617	0.6236	0.053
20%	0.86846	0.86846	0.8683	5.6993	2.5555	0.098
30%	0.93574	0.93574	0.9356	22.34	3.3232	0.38
40%	0.94216	0.94216	0.94209	36.03	4.9337	0.448
50%	0.9469	0.9469	0.9468	21.32	6.4646	6.69
60%	0.9569	0.9569	0.9569	27.81	8.7055	8.14

5. Conclusion

This study proposes an efficient compressed sensing approach for image reconstruction. It exploits the Reverse Biorthogonal 4.4 (rbior4.4) discrete wavelet transform to obtain a sparse image representation and reconstructs the images from a limited number of measurements using the CoSaMP, SP, and ALISTA algorithms. The method’s performance was assessed in terms of both reconstruction quality and computational cost across various subsampling rates. A summary of the results is presented in Figures 2 and 3 below.

Download: Download full-size image

Figure 2. SSIM vs% of measurements.

Download: Download full-size image

Figure 3. Reconstruction time vs% of measurements.

From Figure 2, we deduce that the three algorithms CoSaMP (red), SP (green), and ALISTA (blue) produce reconstructions whose SSIM quality increases with the measurement rate, which is fully consistent with compressive sensing (CS) theory: the more measurements are acquired, the better the reconstruction. Across all values of

M

, the curves for the three algorithms are either overlapping or very close, indicating that they achieve comparable performance in terms of structural fidelity. Notably, ALISTA despite being a learned algorithm does not exhibit any significant degradation compared to classical iterative methods (CoSaMP, SP). The slope of the curves is particularly steep between 10% and 30% of measurements and then flattens, reflecting the well-known reconstruction threshold phenomenon in CS.

Figure 3 compares the computational efficiency of the three algorithms by plotting reconstruction time (in minutes) as a function of the sampling rate. ALISTA (blue) is consistently the fastest, with reconstruction times below 9 minutes even at 60% measurements, whereas CoSaMP (red) exhibits substantially higher runtimes, peaking at 36 minutes for

M = 40 %

. SP (green) lies between the two, showing moderate computational cost. ALISTA’s exceptional speed is expected, as it is a learned algorithm in which iterative operations are replaced by fixed, data-driven layers optimized during training, thereby drastically reducing the number of required iterations. In contrast, CoSaMP appears to suffer from a suboptimal implementation or practical behavior particularly around

M = 40 %

, where runtime sharply increases. This may be due to a high number of iterations required for convergence combined with the computational overhead of support-set updates. SP, meanwhile, maintains a moderately linear increase in runtime, reflecting its algorithmic simplicity. These results highlight that ALISTA provides a massive speedup without sacrificing reconstruction quality, making it an ideal candidate for real-time or embedded applications.

A promising avenue to enhance both computational efficiency and sparsity in compressive sensing image representation would be to substitute the conventional Discrete Wavelet Transform (DWT) employed here with the Lifting Wavelet Transform (LWT). Unlike DWT, LWT supports in-place computation without requiring explicit convolution operations, thereby substantially lowering both memory footprint and computational complexity.

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
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[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 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Science Journal of Circuits, Systems and Signal Processing, 12(1), 8-15. https://doi.org/10.11648/j.cssp.20251201.12

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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 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Sci. J. Circuits Syst. Signal Process. 2025, 12(1), 8-15. doi: 10.11648/j.cssp.20251201.12

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AMA Style

Luc SNRF, Randrianandrasana ME, Rakotonirina HB. Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Sci J Circuits Syst Signal Process. 2025;12(1):8-15. doi: 10.11648/j.cssp.20251201.12

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@article{10.11648/j.cssp.20251201.12,
  author = {Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana and Hariony Bienvenu Rakotonirina},
  title = {Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm
},
  journal = {Science Journal of Circuits, Systems and Signal Processing},
  volume = {12},
  number = {1},
  pages = {8-15},
  doi = {10.11648/j.cssp.20251201.12},
  url = {https://doi.org/10.11648/j.cssp.20251201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.cssp.20251201.12},
  abstract = {This paper presents an efficient image reconstruction method based on Compressive Sensing (CS) theory, leveraging the level-3 Reverse Biorthogonal 4.4 (rbio4.4) discrete wavelet transform in combination with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach exploits the sparsity of images in a suitable wavelet basis, enabling compressed acquisition from a reduced number of random linear measurements. The process consists of four stages: (1) decomposition of the original image using the rbio4.4 wavelet transform to obtain sparse coefficients, (2) compressed sampling via a random measurement matrix, (3) reconstruction of the sparse signal using SP, CoSaMP, or ALISTA, and (4) final image reconstruction through the inverse wavelet transform. Experimental evaluation was conducted on the classic Lena image (200 × 200 pixels), comparing the three algorithms in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost (reconstruction time in minutes) across sampling rates ranging from 10% to 60%. Results show that all three algorithms achieve nearly identical reconstruction quality (virtually indistinguishable SSIM values at each sampling rate), confirming their effectiveness within the CS. However, ALISTA stands out significantly due to its exceptional speed, exhibiting substantially lower reconstruction times thanks to its learned nature, which replaces iterative procedures with fixed, optimized operations. In contrast, CoSaMP demonstrates higher and sometimes unpredictable computational times depending on the sampling rate. These findings highlight ALISTA’s strong potential for real-time or embedded applications.
},
 year = {2025}
}

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TY  - JOUR
T1  - Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) 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.cssp.20251201.12
DO  - 10.11648/j.cssp.20251201.12
T2  - Science Journal of Circuits, Systems and Signal Processing
JF  - Science Journal of Circuits, Systems and Signal Processing
JO  - Science Journal of Circuits, Systems and Signal Processing
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EP  - 15
PB  - Science Publishing Group
SN  - 2326-9073
UR  - https://doi.org/10.11648/j.cssp.20251201.12
AB  - This paper presents an efficient image reconstruction method based on Compressive Sensing (CS) theory, leveraging the level-3 Reverse Biorthogonal 4.4 (rbio4.4) discrete wavelet transform in combination with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach exploits the sparsity of images in a suitable wavelet basis, enabling compressed acquisition from a reduced number of random linear measurements. The process consists of four stages: (1) decomposition of the original image using the rbio4.4 wavelet transform to obtain sparse coefficients, (2) compressed sampling via a random measurement matrix, (3) reconstruction of the sparse signal using SP, CoSaMP, or ALISTA, and (4) final image reconstruction through the inverse wavelet transform. Experimental evaluation was conducted on the classic Lena image (200 × 200 pixels), comparing the three algorithms in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost (reconstruction time in minutes) across sampling rates ranging from 10% to 60%. Results show that all three algorithms achieve nearly identical reconstruction quality (virtually indistinguishable SSIM values at each sampling rate), confirming their effectiveness within the CS. However, ALISTA stands out significantly due to its exceptional speed, exhibiting substantially lower reconstruction times thanks to its learned nature, which replaces iterative procedures with fixed, optimized operations. In contrast, CoSaMP demonstrates higher and sometimes unpredictable computational times depending on the sampling rate. These findings highlight ALISTA’s strong potential for real-time or embedded applications.

VL  - 12
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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: Telecommunication, Signal processing, Compressive Sensing, Artificial intelligence, Mathematics

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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

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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 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Science Journal of Circuits, Systems and Signal Processing, 12(1), 8-15. https://doi.org/10.11648/j.cssp.20251201.12

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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 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Sci. J. Circuits Syst. Signal Process. 2025, 12(1), 8-15. doi: 10.11648/j.cssp.20251201.12

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AMA Style

Luc SNRF, Randrianandrasana ME, Rakotonirina HB. Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm. Sci J Circuits Syst Signal Process. 2025;12(1):8-15. doi: 10.11648/j.cssp.20251201.12

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@article{10.11648/j.cssp.20251201.12,
  author = {Sarobidy Nomenjanahary Razafitsalama Fin Luc and Marie Emile Randrianandrasana and Hariony Bienvenu Rakotonirina},
  title = {Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) Discrete Wavelet Transform and SP, CoSaMP and ALISTA Algorithm
},
  journal = {Science Journal of Circuits, Systems and Signal Processing},
  volume = {12},
  number = {1},
  pages = {8-15},
  doi = {10.11648/j.cssp.20251201.12},
  url = {https://doi.org/10.11648/j.cssp.20251201.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.cssp.20251201.12},
  abstract = {This paper presents an efficient image reconstruction method based on Compressive Sensing (CS) theory, leveraging the level-3 Reverse Biorthogonal 4.4 (rbio4.4) discrete wavelet transform in combination with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach exploits the sparsity of images in a suitable wavelet basis, enabling compressed acquisition from a reduced number of random linear measurements. The process consists of four stages: (1) decomposition of the original image using the rbio4.4 wavelet transform to obtain sparse coefficients, (2) compressed sampling via a random measurement matrix, (3) reconstruction of the sparse signal using SP, CoSaMP, or ALISTA, and (4) final image reconstruction through the inverse wavelet transform. Experimental evaluation was conducted on the classic Lena image (200 × 200 pixels), comparing the three algorithms in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost (reconstruction time in minutes) across sampling rates ranging from 10% to 60%. Results show that all three algorithms achieve nearly identical reconstruction quality (virtually indistinguishable SSIM values at each sampling rate), confirming their effectiveness within the CS. However, ALISTA stands out significantly due to its exceptional speed, exhibiting substantially lower reconstruction times thanks to its learned nature, which replaces iterative procedures with fixed, optimized operations. In contrast, CoSaMP demonstrates higher and sometimes unpredictable computational times depending on the sampling rate. These findings highlight ALISTA’s strong potential for real-time or embedded applications.
},
 year = {2025}
}

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TY  - JOUR
T1  - Image Reconstruction in Compressive Sensing Using the Level 3 Reverse Biorthogonal 4.4 (rbio4.4) 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.cssp.20251201.12
DO  - 10.11648/j.cssp.20251201.12
T2  - Science Journal of Circuits, Systems and Signal Processing
JF  - Science Journal of Circuits, Systems and Signal Processing
JO  - Science Journal of Circuits, Systems and Signal Processing
SP  - 8
EP  - 15
PB  - Science Publishing Group
SN  - 2326-9073
UR  - https://doi.org/10.11648/j.cssp.20251201.12
AB  - This paper presents an efficient image reconstruction method based on Compressive Sensing (CS) theory, leveraging the level-3 Reverse Biorthogonal 4.4 (rbio4.4) discrete wavelet transform in combination with three reconstruction algorithms: Subspace Pursuit (SP), Compressive Sampling Matched Pursuit (CoSaMP), and the Analytic Learned Iterative Shrinkage Thresholding Algorithm (ALISTA). The approach exploits the sparsity of images in a suitable wavelet basis, enabling compressed acquisition from a reduced number of random linear measurements. The process consists of four stages: (1) decomposition of the original image using the rbio4.4 wavelet transform to obtain sparse coefficients, (2) compressed sampling via a random measurement matrix, (3) reconstruction of the sparse signal using SP, CoSaMP, or ALISTA, and (4) final image reconstruction through the inverse wavelet transform. Experimental evaluation was conducted on the classic Lena image (200 × 200 pixels), comparing the three algorithms in terms of reconstruction quality measured by the Structural Similarity Index (SSIM) and computational cost (reconstruction time in minutes) across sampling rates ranging from 10% to 60%. Results show that all three algorithms achieve nearly identical reconstruction quality (virtually indistinguishable SSIM values at each sampling rate), confirming their effectiveness within the CS. However, ALISTA stands out significantly due to its exceptional speed, exhibiting substantially lower reconstruction times thanks to its learned nature, which replaces iterative procedures with fixed, optimized operations. In contrast, CoSaMP demonstrates higher and sometimes unpredictable computational times depending on the sampling rate. These findings highlight ALISTA’s strong potential for real-time or embedded applications.

VL  - 12
IS  - 1
ER  -

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