Preamble

Astronomical Techniques and Instruments, Vol. 2, September 2025, 280–287 Article Open Access Optimization of compressed sensing-based radio interferomet- ric imaging: hyperparameter selection Haoming Dai , Li Deng 1 State Key Laboratory of Solar Activity and Space Weather National Space Science Center Chinese Academy of Sciences Beijing 100190, China *Correspondence:

INTRODUCTION

Because of the good linearity of the laws governing salt-and-pepper noise, we believe that some of these laws can be exploited in interferometric imaging. Traditional sig- nal sampling must satisfy the Nyquist sampling theorem to effectively recover the original signal. However, sam- pling is usually sparse and noisy due to the limited num- ber of antennas. The compressed sensing theory pro- posed by Candès and Tao in 2006 aims to combine data acquisition with compression [ 1 – 3 ] . Its principle is to use the feature of signal sparsity to enable the efficient recon- struction of sparsely sampled noisy images.

Compressed sensing is also applied in the field of image denoising. Elad and Aharon proposed a dictionary- based denoising method [ 4 ] called the K-SVD algorithm.

The main drawback of such methods is that they require proximity between the reconstructed denoised image and the original noisy image, which may result in some resid- ual noise in the denoised image. Yang et al. proposed a dic- tionary learning method [ 5 ] , using a sparse representation model to construct and represent images. They reformu-

lated the problem as a dictionary learning issue by select- ing bases that can be sparsely combined to represent the processed image and achieve the minimum global recon- struction error (e.g., Mean Squared Error (MSE)). Peyr introduced an effective Compressed Sensing (CS) method based on tree-structured dictionaries , which uses itera- tive thresholding methods to recover signals and estimate the best basis. Shahdoosti proposed a block-matching- based algorithm that groups the noisy image using block-matching techniques and then employs the same sparse vector for all blocks within each group.

Bobin et al. applied compressed sensing algorithms to radio interferometric imaging reconstruction . Wenger et al. proposed a simulation of continuous observation of radio astronomical targets using compressed sensing algo- rithms . In their study, three methods based on com- pressed sensing, , and , were proposed, all of which showed good performance in reconstructing the images of observed dynamic targets. However, they did not consider noise in their analysis.

Iterative Shrinkage/Thresholding Algorithm (ISTA), Fast Iterative Shrinkage/Thresholding Algorithm (FISTA),

2 University of Chinese Academy of Sciences , Beijing 100190, China

© 2025 Editorial Office of Astronomical Techniques and Instruments, Yunnan Observatories, Chinese Academy of Sciences. This is an open access article under the CC BY 4.0 license ( Citation: Dai, H. M., Deng, L. 2025. Optimization of compressed sensing-based radio interferometric imaging: hyperparameter selection.

Astronomical Techniques and Instruments (5): 280−287. ati2025009

Abstract

Radio interferometric imaging samples visibility data in the spatial frequency domain and then reconstructs the image. Because of the limited number of antennas, the sampling is usually sparse and noisy. Compressed sensing- based on convex optimization is an effective reconstruction method for sparse sampling conditions. The hyperparameter for the regularization term is an important parameter that directly affects the quality of the reconstructed image. If its value is too high, the image structure will be missed. If its value is too low, the image will have a low signal-to-noise ratio. The selection of hyperparameters under different levels of image noise is studied in this paper, and solar radio images are used as examples to analyze the optimization results of compressed sensing algorithms under different noise conditions. The simulation results show that when the salt-and-pepper noise density is between 10% and 30%, the compressed sensing algorithm obtains good reconstruction results. Moreover, the optimal hyperparameter value has a linear relationship with the noise density, and the mean squared error of regression is approximately

Keywords

Astronomy image processing; Radio interferometers; Radio telescopes

and Two-Step Iterative Shrinkage/Thresholding (TwIST) are commonly used optimization algorithms in the field of compressed sensing. ISTA approximates the sparse solu- tion by iteratively applying soft thresholding to update the solution, with a convergence rate of . FISTA, pro- posed by Beck and Teboulle , accelerates convergence by introducing momentum in the iteration of ISTA, with a convergence rate of . TwIST, proposed by Biou- cas-Dias and Figueiredo , is a two-step iterative shrink- age/thresholding algorithm for image recovery. This method combines gradient descent and soft thresholding operations, thereby improving the convergence speed and stability of the algorithm. All of the above algorithms use regularization term for image reconstruction, but lack an analysis of the specific value that this hyperparame- ter should take. The selection process, which is based on noise density and the optimization of peak signal-to-noise ratio (PSNR) and structural similarity (SSIM), remains an area of ongoing research.

In compressed sensing image reconstruction, the hyper- parameter of the regularization term is an important parameter that directly affects the quality of the recon- structed image. If the parameter value is too high, it may over-regularize to values near zero; if the parameter value is too low, this will cause a low PSNR in the image. There- fore, this study conducted a detailed investigation of the selection of hyperparameter values for the regulariza- tion term in radio interferometric image reconstruction, ana- lyzing which strategies are suitable under different noise conditions.

IMAGE RECONSTRUCTION COMPRESSED SENSING Interferometric Imaging Principle According to the principle of interferometric imaging in synthetic aperture radiometers, the original image is related to the visibility function , sam- pled in the spatial frequency domain using the following Fourier transform,

I ∈ R n × n V ∈ C n × n

V ( u , v ) = ∫

be a mapping from , which maps an pixel image to an pixel visibility through Equa- tion (1). Under ideal conditions, this mapping is repre- sented by , and the discrete Fourier transform can be writ- ten in the following form,

V = F ( I ) . (2)

Different antenna array distributions have different sam- pling distributions in the spatial frequency domain. Hence there exists a mask vector , representing the sampling capability of the antenna array in the spatial fre-

M ∈{ 0 , 1 } n × n

quency domain, defined as follows, is sampled is not sampled Because of the various factors that cause noise in actual observations, this study focuses on random salt- and-pepper noise, denoted as .The final obtained visibility function can be calculated as follows,

N ∈ R n × n

V = M ⊙F ( I + N ) , (4)

where represents element-wise multiplication, indicating that mask acts on the visibility across the entire space.

Compressed Sensing Image Reconstruction Algo rithm Design When the observations of an object have sparse charac- teristics, the compressed sensing method can effectively per- form image reconstruction. The compressed sensing algo- rithm is a nonlinear regularization optimization algorithm expressed as follows,

min x ∥ x ∥ 0 s.t. y = h ( x ) , (5)

y = h ( x )

where is the underdetermined constraint condi- tion for the optimization problem, and is the optimiza- tion constraint, which requires first-order differentiability (Lipschitz continuity) and must be convex. In fact, because of the difficulty of performing calculations related to the -norm while ensuring sparsity, this prob- lem is usually transformed into an -norm optimization problem, which converts it into a convex optimization prob- lem. When the observation matrix satisfies RIP, these two problems are equivalent,

min x ∥ x ∥ 1 s.t. y = h ( x ) . (6)

To simplify the problem, the problem in Equation (6) can be further transformed into an unconstrained optimiza- tion problem. In this step, a hyperparameter is intro- duced to represent the degree of image compression. The new unconstrained optimization problem is formulated as follows, The specific case considered in this study, i.e., that the visibility corresponding to the reconstructed image must be equal to the observed visibility, with . Thus, the image reconstruction problem is transformed into the following unconstrained optimiza- tion problem,

y = V h ( x ) = M ⊙F ( x )

For convenience, the relationship between functions

is given below, For the unconstrained optimization problem that includes the -regularized term , because ) is non- differentiable, the FISTA method can be used to solve . This method requires to be differentiable and to be Lipschitz continuous, but only needs to have a proximity operator. The optimization problem shown in Equation (8) can be solved using the three-step iterative algorithm given by the three following equations:

x k = p L ( y k ) , (10)

t k + 1 = 1 + √

  ������ )������

p L ( y ) = argmin x

p L L ∇ f ( y ) L = n 2 Here is the proximity operator. is the Lipschitz con- stant of . In this problem, . The proximity oper- ator shown in Equation (13) is the shrinkage operator in LASSO, which is the following closed-form solution,



ˆ x − γ L ˆ x > γ

p L ( y ) =

ˆ x + γ L ˆ x < − γ L

here, indicates the result of gradient descent, calculated as follows,

ˆ x = ( y − 1 L ∇ f ( y ) ) . (15)

x 1 = y 0 = 0 N × N , t 0 = 1 ∇ f ( y )

Here, we can set as the initial conditions, and end the iteration when tends to 0 or the maximum number of iterations has been reached.

Note that is an algorithm parameter used in FISTA and is not related to any practical physical meaning.

After performing the above optimization calculations, we use the continuity of the observed object to perform a Gaussian filtering on the reconstructed image to obtain the final image. Gaussian filtering effectively mitigates the significant differences in structural similarity caused by the sidelobe effect, making the structure of the recon- structed image close to that of the true structure.

As shown in Equation (14), hyperparameter repre- sents the compression capability of compressed sensing, and its selection strongly effects the performance of the model. Under different noise conditions, the value of must be adjusted so that the compressed sensing algo- rithm can suppress noise without losing the structure of the original image while retaining a higher PSNR.

EXPERIMENTS Simulation Scene One background image with a slowly moving point is given as an example to simulate a quiet sun with a sin- gle burst event. In this image, a bright target moves sev- eral pixels from the center of the image to the right and down. The image is normalized to , as shown in . In the remainder of the evaluation, we focus on the center 256 × 256 pixels of the view because the rest of the view is empty.

It is assumed that the image noise is discrete and ran- domly distributed, and the intensity of the noise will not (A) Image of the entire telescope’s field of view (2 048 × 2 048 pixels). (B) The center (256 × 256 pixels).

exceed the intensity of the signal. Therefore, salt-and-pep- per noise was selected for the simulation. Salt-and- pepper noise is predominantly associated with hardware malfunctions. For instance, certain pixel elements within a sensor may become defective, thereby consistently out- putting the maximum or minimum grayscale values. Dur- ing the transmission of image data, the communication channel is susceptible to external electromagnetic interfer- ence or signal attenuation, which may introduce errors that set some pixel values to extreme levels. Here, denotes the intensity of the noise. In this simulation, we set s = 0.5. The noisy image is calculated as follows,

s & p noise

I noise = (1 − s ) I + sN s & p . (16)

s & p s & p We denote the density of . The probabil- ity distribution of is given in the following,

p ( N s & p = I ) = 1 − d , no noise at this pixel



p ( N s & p = 1 ) = d 2 , salt noise at this pixel

p ( N s & p = 0 ) = d 2 , pepper noise at this pixel

p ( N s & p = otherwise ) = 0

This study focuses on the case with known noise densi- ties and standard deviations.

0 100 d =10% 200 250 200 150 100 50 0 A

0 100 d =15% 200 250 200 150 100 50 0 B

Image Reconstruction with Different Hyper ameters We reconstructed the visibility using the FISTA algo- rithm described above. In the experiments, we set the noise density to 10%, 15%, 20%, 25% and 30%, and we set hyperparameter to conduct multi- group experiments. The Fourier transform method was used as a comparison method. For convenient representa- tion, 0 denotes the Fourier transform. To facilitate the comparison of the signal-to-noise ratio and SSIM of the reconstructed images, the values of the reconstructed the images were scaled to .The reconstruction results for different noise densities are shown in . The images of ground truth with different noise density is shown in the last column.

γ ∈ [0 . 001 , 0 . 1]

, and displays the outcomes of the Fourier transformation under noise Noised image is shown in . Salt-and-pep- per noise often occurs because of hardware failures. For example, some pixel units of a sensor may be damaged and always output the highest or lowest gray level values.

During the transmission of the image data, the communica- tion channel may be affected by external electromagnetic interference or signal attenuation, and these errors may cause some pixel values to be incorrectly set to extreme val- ues. This study focuses on investigating this type of noise, which is less frequently considered in radio interfer- ometry.

The simulated baseline sampling is presented in . It is dense in the center and sparse around the edges. The minimum baseline is at least 10 times the wave- length, and the maximum baseline is about 1 000 times wavelength. The baseline samples are used to generate a mask vector . The distribution of these baselines is notably sparse, with only of the points sampled. In this paper, each pixel represents a wave- length, and hence

0 100 d =25% 200 250 200 150 100 50 0 D

0 100 d =30% 200 250 200 150 100 50 0 E

varying noise density. As the noise density increases, the clarity of the bright target features in the reconstructed images diminishes, and the signal-to-noise ratio drops. Addi-

Base line /Wavelenght −1 000 −750 −500 −250 /Wavelenght

0 100 d =20% 200 250 200 150 100 50 0 C

=10%, 15%, 20%, 25%, 30% (in pixels).

tionally, the background noise becomes more noticeable.

When noise is low, as in , increas- compresses weaker signals, preserving only the strong ones, which increases this reduction. shows that at , only the central 128 × 128 pixel area retains its signal, while the rest of the image is over- regularized to values near zero.

γ = 0 . 05

Conversely, when noise density is high, as in , noise suppression improves with increasing . However, even at , the image has a low signal-to-noise ratio. Comparing suggests that a high over-regularizes the image when noise density is low, but a higher is neces- sary for effective compression when the noise density is high. Comparing

γ γ = 0 . 05

GT with noise , better reconstruction is observed when the noise density and hyperparameter are well matched. Sec- tion 4 provides a quantitative analysis of the reconstruc- tion performance and optimal .

DISCUSSION

Reconstruction Performance Under Different Hyperparameter Values To evaluate the reconstruction performance of sparsely sampled images under noisy conditions, the PSNR and SSIM of the reconstructed images under five lev- els of noise density and different values of were com- puted. For convenience of notation, denote the best -values for PSNR and SSIM under the = 0.000 = 0.010 = 0.020 = 0.032 = 0.040 = 0.050 =0, 0.01, 0.02, 0.032, 0.04, 0.05 at noise density (column a to f). The last column shows the ground truth with noise. = 10%; (B) = 15%; (C) = 20%; (D) = 25%; (E) = 30%.

PSNR vs in different noise density

d = 10% d = 15% d = 20% d = 25% d = 30%

densities. given noise conditions. respectively show the variation in the PSNR and SSIM of the recon- structed image with respect to the value of hyperparame- ter . shows, the curves of the PSNR with respect to have a single peak. Except for the reconstruc- tion result at noise density which is close to the PSNR of the Fourier transform ( in the figure), as the noise density increases, the PSNR improvement at increases. As the noise density increases, simi- lar to the images in Section 3.2, increases, and the increases of the PSNR a become more signifi- cant (4–10 dB).

d = 10%

γ = 0

γ = γ ∗ PSNR

γ ∗ PSNR γ = γ ∗ PSNR

γ < γ shows, when , the curve changes slowly, the image structure is largely retained, only a small amount of edge noise is eliminated, and the similar- vs Noise density Regression MSE: 8.0e−08 Noise density

γ ∗ PSNR = 0 . 268 d − 0 . 024 6 , (18)

γ ∗ SSIM = 0 . 268 d − 0 . 026 2 . (19)

The equations for the regression (Equation (18) and Equation (19)), reveal that have linear regression coefficients that only differ by 0.02 in the con- stant term. The regression residuals are hence very small.

According to the characteristics of , when , there is less effect on the structural similarity, and

γ ∗ SSIM γ < γ ∗ SSIM

SSIM vs in different noise density

d = 10% d = 15% d = 20% d = 25% d = 30%

densities. ity is improved by a small amount. By contrast, when , the image structure is severely damaged and the structural similarity decreases dramatically, which is consistent with the reconstruction results in . The selection of should try not exceed

γ > γ ∗ SSIM

Based on the above study, the relationship between the noise density and is further analyzed.

Relationship Between Noise Density Hyperparameter A scatter plot of noise density and shown in . It can be clearly seen that are almost equal and have an approximate linear rela- tionship with the noise density . Moreover, are linearly regressed to , and the regression equa- tions are given as follows, vs Noise density Regression MSE: 7.2e−07 Noise density

γ ∗ = γ ∗ PSNR γ γ ∗ = 0 . 268 d − 0 . 024 6 8 . 10 × 10 − 8

hence we choose as the optimal hyperparame- ter . That is, we set , and the regres- sion MSE is ROBUSTNESS IN DIFFERENT ANT ENNA SCENARIOS To analyze the scalability of the method, we applied our approach to a variety of different antenna arrays and analyzed the results of the regression. These configura-

Fit line: γ = 0.268 0 d − 0.024 6 Fit line: γ = 0.268 0 d − 0.026 2

Base line of X120_8Y720_30 /Wavelengh /Wavelengh −1 000 −1 000 /Wavelengh −1 000 −500 Base line of X140_10Y720_30 /Wavelengh /Wavelengh −1 000 −1 000 /Wavelengh −1 000 −500 Base line of O100_10Y720_30 /Wavelengh /Wavelengh −1 000 /Wavelengh −1 000 −500 O100_10Y600_30 X120_8Y720_30 O100_10X510_30 Hex100_10T750_30 X140_10Y720_30 O80_10Y600_30 O120_10Y600_30 O100_10Y720_30 O100_10Y480_30 −0.01 −0.02 tions included various combinations of circular-shaped, cross-shaped, Y-shaped, T-shaped, and hexagonal-shaped arrays, as well as different array sizes. We selected nine arrays for experimentation, and the array sampling num- bers and sampling rates are listed in . The array baseline distribution is shown in . and the experimen- Base line of Hex100_10T750_30 /Wavelengh −1 000 /Wavelengh −1 000 −500 Base line of O120_10Y600_30 /Wavelengh −1 000 /Wavelengh −1 000 −500 Base line of O100_10X510_30 /Wavelengh −1 000 /Wavelengh −1 000 −500 O100_10Y600_30 X120_8Y720_30 O100_10X510_30 Hex100_10T750_30 X140_10Y720_30 O80_10Y600_30 O120_10Y600_30 O100_10Y720_30 O100_10Y480_30 −0.005 −0.010 −0.015 −0.020 −0.025 tal results are reported in . Note that the regression result is denoted as for both PSNR and SSIM.

The results of supplementary experiments indicate that for different types of telescope arrays, as long as the minimum antenna spacing is the same and the array is

Base line of O100_10X510_30 /Wavelengh −1 000 −500 Base line of O80_10Y600_30 /Wavelengh −1 000 −500 Base line of O100_10Y480_30 /Wavelengh Regression result for noise + Regression result for noise +

0.015 A

Data name Data name Array mentioned above Array mentioned above

Array name Samples Sample rate/(%) PSNR_k PSNR_b SSIM_k SSIM_b Array above −0.024 −0.026 X120_8Y720_30 −0.003 O100_10X510_30 −0.016 −0.018 Hex100_10T750_30 X140_10Y720_30 −0.003 −0.008 O80_10Y600_30 −0.017 −0.021 O120_10Y600_30 −0.018 −0.022 O100_10Y720_30 −0.022 −0.024 O100_10Y480_30 −0.019 −0.026 O100_10X510_30 −0.012 −0.02

quasi-symmetric (e.g., Y-shaped, cross-shaped, or circu- lar-shaped arrays with approximately equal distributions in the U and V directions), the slope of the regression results exhibits good robustness, yielding a slope for the lin- ear fitting results of 0.26 ± 5%. However, when the mini- mum antenna spacing changes or there is a significant bias in the baseline distribution (e.g., Hex100_10T750_ 30), the robustness of the results changes substantially.

In this paper, we analyzed the effect of noise density and hyperparameter on the reconstruction perfor- mance of compressed sensing algorithm during interferome- tric imaging image reconstruction. We chose images with salt-and-pepper noise as our simulation inputs. The results show that when the noise density is constant, if too small, the image cannot be effectively denoised, and if it is too large, the image structure may be seriously affected. Hence, it is very important to choose a suitable value for . Under the conditions of noise density , when the optimal value of is selected, the compressed sensing method can improve the PSNR by 6–10 dB, and substantially improve the retention of image structure, increasing the SSIM to about 0.9. The opti- mal hyperparameter has a linear relationship with noise density , this is . More- over, the regression MSE is . The noise type, noise intensity, and characteristics are important factors affecting the selection of , and further research on these factors should be carried out.

γ d = 10% − 30% γ

γ ∗ PSNR = 0 . 268 d − 0 . 024 6 8 . 10 × 10 − 8

AI DISCLOSURE STATEMENT Deepl and Kimi was employed for language and gram- mar checks within the article. The authors carefully reviewed, edited, and revised the Deepl and Kimi-gener- ated texts to their own preferences, assuming ultimate responsibility for the content of the publication.

AUTHOR CONTRIBUTIONS Haoming Dai conceived the ideas, implemented the study, and wrote the paper. Li Deng supervised this paper. All authors read and approved the final manuscript.

DECLARATION OF INTERESTS The authors declare no competing interests.

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