Application of $L_{1}-L_{2}$ Regularization in Sparse-View Photoacoustic Imaging Reconstruction

Photoacoustic imaging (PAI) is an advanced technique used to reconstruct the distribution of energy absorption in tissues, even when ultrasound signals are incomplete and noisy. However, the reconstruction process is challenging due to the ill-posed nature of the problem. In order to address this challenge, regularization techniques are employed to obtain a meaningful solution. This article focuses on the significance of utilizing <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the difference of convex algorithm (DCA) in sparse photoacoustic image reconstruction. To assess the performance of <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the DCA, a comparative test was conducted using three evaluation indicators. The sampling amount and noise level were controlled to effectively evaluate its effectiveness. The results from tissue phantom experiments demonstrated that the <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the DCA method excelled in handling the reconstruction of highly noisy data with incomplete levels. Additionally, in a pig liver experiment, the <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the DCA method was compared to other methods and found to be superior in reducing errors and ensuring stability. Importantly, the <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the DCA method achieved similar image quality with a sampling number of 40, while other methods required a higher sampling number of 80. In scenarios with significant noise and the number of low sampling, the <inline-formula><tex-math notation="LaTeX">$L_{1}-L_{2}$</tex-math></inline-formula> norm based on the DCA method showcases its capability to deliver satisfactory reconstruction results. This discovery holds significant potential for enhancing sparse sampling photoacoustic tomography algorithms, and it offers valuable insights for future biomedical application development.

Application of L 1 − L 2 Regularization in Sparse-View Photoacoustic Imaging Reconstruction Mengyu Wang , Shuo Dai , Xin Wang , and Xueyan Liu , Member, IEEE Abstract-Photoacoustic imaging (PAI) is an advanced technique used to reconstruct the distribution of energy absorption in tissues, even when ultrasound signals are incomplete and noisy.However, the reconstruction process is challenging due to the illposed nature of the problem.In order to address this challenge, regularization techniques are employed to obtain a meaningful solution.This article focuses on the significance of utilizing L 1 − L 2 norm based on the difference of convex algorithm (DCA) in sparse photoacoustic image reconstruction.To assess the performance of L 1 − L 2 norm based on the DCA, a comparative test was conducted using three evaluation indicators.The sampling amount and noise level were controlled to effectively evaluate its effectiveness.The results from tissue phantom experiments demonstrated that the L 1 − L 2 norm based on the DCA method excelled in handling the reconstruction of highly noisy data with incomplete levels.Additionally, in a pig liver experiment, the L 1 − L 2 norm based on the DCA method was compared to other methods and found to be superior in reducing errors and ensuring stability.Importantly, the L 1 − L 2 norm based on the DCA method achieved similar image quality with a sampling number of 40, while other methods required a higher sampling number of 80.In scenarios with significant noise and the number of low sampling, the L 1 − L 2 norm based on the DCA method showcases its capability to deliver satisfactory reconstruction results.This discovery holds significant potential for enhancing sparse sampling photoacoustic tomography algorithms, and it offers valuable insights for future biomedical application development.

I. INTRODUCTION
P HOTOACOUSTIC imaging (PAI) [1] is an advanced and promising non-invasive technique that utilizes the photoacoustic effect to provide robust and comprehensive imaging capabilities.By combining the high contrast of optical imaging with the deep penetration of ultrasound imaging [2], [3], this method is increasingly important in various fields, including tumor detection [4], vascular imaging [2], neurological research [5], and early-stage cancer screening [6].The procedure involves exposing biological tissue to a short pulse laser, generating a photoacoustic signal that is captured by an ultrasonic transducer positioned around the subject [7].Image reconstruction methods are then used to calculate the distribution of light absorption within the tissue, leading to valuable insights for clinical diagnosis and scientific investigations [8], [9].
The selection of a reconstruction method is a critical factor in determining the quality of image reconstruction.One widely used method is R.A. Kruger's filtering back-projection algorithm [10], which efficiently filters and processes the signal to produce superior imaging quality.The Fourier transform reconstruction method is another option, known for its speedy processing time, although it necessitates the collection of the complete range of signals, which can be a challenge.To overcome this challenge, Xu Y et al.Extensively investigated finite perspective PAI in 2004 [11].More recently, in 2008, J. Provost et al. introduced the concept of compressed sensing (CS) to PAI, which utilizes the sparse prior knowledge of tissues to significantly enhance finite perspective PAI.By offering an innovative perspective on image reconstruction, CS represents a new direction for improving imaging quality [12].
Improving image quality in the process of image reconstruction is a complex task, often hindered by noise and other undesirable effects.To address these issues, regularization techniques are commonly employed.However, traditional methods like the L 2 -norm constraint often result in over-smoothed images [13].In 2006, Emmanuel Candes and other researchers introduced CS to overcome this limitation [14].CS theory demonstrates that collecting sparse signals in a specific transform domain is possible using an observation matrix that is non-correlated with the transformation basis.This enables the use of non-uniform sampling frequencies.While CS surpasses the limitations of the Nyquist theorem [15], proper signal compression is crucial to meet the requirements of the restricted isometry property (RIP), uniform uncertainty principle, and exact reconstruction principle [16].Moreover, the use of an appropriate reconstruction algorithm is essential to retrieve the complete signal or image information from the collected measurements [17].
CS is a principle used to reconstruct images by obtaining sparser solutions.This is achieved by imposing L 0 -norm constraints on the model.However, solving the L 0 -norm is an NP-hard problem, which has led to the development of approximation algorithms such as orthogonal matching pursuit [18] and the ReSL0 algorithm [19].In order to overcome the challenges associated with solving the L 0 -norm, researchers have shifted their focus to solving the L 1 -norm instead [20].The L 1 -norm regularization, proposed by Candes and Donoho et al., is based on CS and adheres to the isometric principle, effectively restoring sparse solutions [21], [22].However, it may not always yield the most sparse solution.To address this limitation, non-convex L p -norm(0 ≤ p < 1) regularization has been investigated [23].These studies have shown that L p -norm regularization is widely used in PAI [23], [24], demonstrating superior reconstruction outcomes compared to L 1 -norm regularization.More sparse solutions are obtained when 1  2 ≤ p < 1 as p decreases, but cannot be obtained when 0 ≤ p < 1  2 .The L 1 − L 2 norm is found to be more effective in promoting sparsity compared to the L p -norm and can also tackle the issue of inadaptability that arises while solving the L p -norm [25].The difference of convex algorithm (DCA) is a descent technique which is introduced by Phan Thanh An in their research and it eliminates the need for line search [26], [27].In the first step, the L 1 -norm problem is solved to simplify it, followed by minimizing the L 1 − L 2 norm based on the DCA in addition to the L 1 -norm to avoid excessively sparse solutions obtained by the L 1 -norm.Multiple studies have shown that L 1 − L 2 norm regularization outperforms the L 1 -norm [28].Alternatively, by exploiting the variability of the L 1 -norm and L 2 -norm, the issue caused by a highly correlated observational matrix can also be mitigated [29].
In order to tackle the reconstruction problem in PAI, this article employs the L 1 − L 2 norm based on the DCA optimization effectively.The article is structured as follows.The second section introduces the photoacoustic theory and provides a comprehensive discussion of reconstruction methods, with a particular emphasis on the application of L 1 − L 2 norm based on the DCA regularization techniques.Moving on to the third section, offers an overview of the numerical simulations and the pork liver experiments conducted on tissue-mimicking phantoms, offering detailed insights into the experimental process and outcomes.Lastly, the fourth section presents a concise summary of the key findings and insightful observations derived from this study.

A. Photoacoustic Imaging
PAI is an advanced medical imaging technique that harnesses the photoacoustic effect to produce high-quality images of biological tissues.By directing a laser at the tissue, the chromophores in the tissue absorb the light energy at varying rates, leading to thermal expansion and the emission of mechanical ultrasound waves.These ultrasound waves are then captured by an external ultrasound transducer.Through a process called signal inversion, the initial distribution of optical absorption within the tissue is reconstructed, resulting in a detailed and precise image.
The following equation [12] can describe the imaging equation for PAI, provided that the thermal confinement condition and stress confinement condition in [1] are satisfied: where p(r, t) is the amplitude of the pressure in an ultrasonic wave at a specific time t and position r; c is the propagation speed of the ultrasonic pressure wave; β is the coefficient of volume expansion at constant pressure; H(r, t) is the heat source function; and C p is the specific heat capacity at constant pressure.
According to the actual situation, it's advantageous to contemplate the subsequent separable structure for the heating function [30]: where A(r)(J/m 3 ) represents the absorbed energy density, while δ(t) signifies the temporal profile of the illuminating pulse.δ(t) is a Dirichlet function.Based on the Green function, the solution of ( 1) can be expressed as the following equation [30]: where r 0 is the location of the ultrasonic detector.By letting k = ω/c and performing the Fourier transform with respect to time t, we can easily derive the frequency spectrum of the photoacoustic pressure [31]: Building upon the aforementioned theoretical analysis, we are now able to delve into the inherent forward problem present in PAI: y = (y 1 , y 2 , . . .y M ) is a column vector which represents the acoustic pressure p(r 0 , k). x = (x 1 , x 2 , . . .x N ) is a column vector representing the absorption coefficient A(r), Based on the distribution of the absorption coefficients, we can obtain the unknown reconstructed image.And K M ×N is a forward operator, and we discretize it into discrete components.

B. Compressed Sensing
In 2006, Emmanuel Candes, David Donoho, and their colleagues introduced CS as a novel signal sampling technique.By utilizing just a few sampled points, CS can achieve the same level of accuracy as complete sampling.However, when it comes to PAI, collecting photoacoustic signals from every direction poses a significant challenge.To overcome this obstacle, a combination of CS and advanced image processing techniques can dramatically reduce the costs associated with compression and data acquisition.
The value of M represents the quantity of data collected for the measurement, while N denotes the number of pixels.It is noteworthy that the magnitude of M is considerably smaller than that of N .However, the CS theory holds that if the unknown x n×1 is sparse, x n×1 can be recovered by solving the L 0 -norm problem concerning x n×1 .
Most signals are typically not sparse.We can represent the signal x n×1 using a set of sparse bases.The four common sparse groups are: curvelets [12], wavelets [32], numerical derivatives (ND) [12], and Fourier [33].In this paper, we opt for wavelet bases, which exhibit superior properties [12]: s n×1 is a sparse representation of x n×1 .Φ is a collection of sparse basis.Based on the ( 5) and ( 7), we can get: Notably, the observation matrix K must satisfy the RIP condition.Meanwhile, the observation matrix K is uncorrelated with the sparse basis Φ.

C. L 1 − L 2 Regularization Based on the DCA Method
The recovery process of an image x n×1 involves an inherently ill-posed inverse problem.When the number of elements in M is smaller compared to N , it becomes challenging to obtain high-quality images.In order to achieve an acceptable image, regularization techniques are employed to mitigate the impact of noise and artifacts, effectively suppressing undesired elements.Tikhonov regularization is widely adopted as the go-to method for addressing undefined problems by improving problem conditions and reducing the influence of noise on the solution.To obtain the regularized solution of (5) within the Tikhonov framework, the objective is to minimize the resulting functions: where λ is the regularization parameter.I is an identity matrix, that can suppress the amplification of noise and have a certain smoothness.When the image is not continuous, the solution effect is not good, but the L 1 -norm of the reconstructed image under the sparse transformation [12] can well preserve these discontinuities.
CS is an effective method for reconstructing signals when faced with limited sampling conditions.By taking advantage of the sparse nature of the initial signal x n×1 , CS enables the successful reconstruction of the signal s n×1 using a reduced number of measurements M .The primary difficulty lies in identifying the signal with the highest degree of sparsity that also satisfies the M measurement vectors in y.To address this challenge, L 1 -norm regularization is commonly utilized.This regularization technique encourages a sparse solution and aids in finding an optimal reconstruction outcome.L 1 -norm regularization can be written as: Fig. 1.DCA-(L 1 − L 2 ) for solving (11).
CS presents the main difficulty of accurately reconstructing the original signal from a compressed version.The solution entails leveraging the fact that the compressed and original signals are independent, and a higher sparsity level facilitates accurate signal reconstruction with less data.It is also essential that the L 1 − L 2 norm complies with the RIP condition, with the L 1 − L 2 norm recovery being more demanding than that of the L 1 -norm.
where λ is the regularization parameter.L 1 − L 2 regularization utilizes the distinction between the L 1 norm and L 2 norm.When dealing with a highly correlated observation matrix, this discrepancy can help alleviate the challenge of obtaining sparser solutions that arise in such scenarios.By leveraging the L 1 − L 2 norm, it is possible to achieve more sparsity in the context of finite perspective sparse reconstruction.The flow chart of the L 1 − L 2 norm based on the DCA algorithm is shown in Fig. 1.

D. The Evaluation Metrics
We assessed the precision of PAI reconstruction through multiple quantitative metrics, including normalized mean absolute error (NMAE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM).NMAE quantifies the reconstruction error.PSNR evaluates the quality of the resulting images.SSIM provides a comprehensive evaluation of luminance, contrast, and structural aspects, which accurately reflects human visual perception.To calculate NMAE, use the following formula: where x represents the reconstructed image, PSNR is defined by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
(14) where μ x, μ x respectively represent the mean of the reconstructed and real images; σ xx is the covariance of the reconstructed and real images; σ x, σ x respectively denote the variance of the reconstructed and real images; m 1 and m 1 are constants used to maintain stability, usually setting them as m 1 = (0.01 L) 2 , m 2 = (0.03 L) 2 , L is a dynamic range of the pixel values.

III. EXPERIMENTAL RESULT
In this section, we aim to verify the reliability of L 1 − L 2 based on the DCA by conducting numerical simulations and practical applications.We compare various methods, including the L 1 -norm, L 2 -norm, L 1 + L 2 norm, L 0 -norm, L 1/2norm, and L 1 − L 2 norm.Notably, the L 1 -norm based on the SPGL1 [34], the L 2 -norm and L 1 + L 2 norm based on the glmnet [35], the L 0 -norm based on the SL0 [36], L 1/2 -norm based on the AIT [37], L 1 − L 2 norm based on the DCA [3].All reconstruction methods were implemented in MATLAB.Five toolboxes [34], [35], [36], [37], [3] were employed for CS reconstruction.In each method, the parameter λ was set to the default value specified within the toolbox.To facilitate two-dimensional imaging, we perform both forward projection and backward reconstruction processes.Our focus lies in imaging objects confined within a slender plate.To achieve this, we employ the rice wavelet toolbox to configure the sparse transform operator.Furthermore, we incorporate a four-layer symmetric wavelet transformation, optimizing the imaging results.Since different algorithms use different model parameters, it is rather difficult to compare their relative performance completely and fairly.We simply terminated those six algorithms when the relative change of the two consecutive iterates became small.We define as follows: where δ is greater than 0, and iteration stops when δ = 0.005.All computations are carried out using MATLAB on a desktop computer equipped with a 3.6 GHz CPU and 32 GB of memory.

A. Experimental Simulation
We performed signal simulation using the phantom and the original phantom in Fig. 2. The phantom image with a size of 32 mm *32 mm and a resolution of 128 * 128.We used the wavelet transform to induce the sparsity of the variable x, with the variable x being a real image.Moreover, the maximum pixel value was one.In the experiment, the circular scan of our ultrasonic transducer was 45 nm in diameter.Additionally, the ultrasonic sound speed of 1500 m/s.In the course of ultrasound detection, the ultrasound detector samples 64 data points at each position.These data points were randomly selected from the frequency window of (0.15, 4) MHz.The objective is to formulate the projection matrix K.The phantom's grey scale  values are standardized within the range of (0, 1).This normalized information is utilized to create the simulation signal.The generation of measurement data is accomplished through the ( 8).This sophisticated approach involves the careful selection and processing of data points within a specified frequency range, contributing to the accurate representation of the ultrasound signal in the frequency domain.
Our research aimed to identify a suitable sparse reconstruction algorithm by comparing six methods, with a specific focus on phantom imaging, as illustrated in Fig. 3.When the channel number was 30, all six methods exhibited relatively poor imaging quality.However, when the channel numbers increased to 40 and 50, five of the methods showed good imaging quality, except for the Tikhonov method.To assess the imaging quality in a more objective manner, Fig. 5 presents the NMAE values of the phantom images under different conditions.As the number of channels increased, the Tikhonov method consistently yielded the largest NMAE values, while the SL0 method demonstrated the smallest NMAE values.Additionally, the second row of Fig. 5 displays the PSNR values of the phantom images under different conditions.As the channel number increased, the Tikhonov method consistently yielded the lowest PSNR value, whereas the SL0 method exhibited the fastest increase in PSNR.In the third row of Fig. 5, the SSIM values of the phantom images under different conditions are shown.When the channel number was below 40, all methods produced poor visual imaging results, with the Tikhonov method performing the worst.However, at a channel number of 40, the AIT method has the largest  SSIM value.As the channel number exceeded 40, the remaining five methods (excluding the Tikhonov method) produced visual imaging effects similar to the original images.
In order to evaluate the effectiveness and versatility of the L 1 − L 2 norm based on the DCA method, a study was conducted comparing it with five other methods using the original phantom, as shown in Fig. 4. When the channel number was set to 10, all six methods produced relatively poor imaging quality.However, with channel numbers of 20 and 30, the five alternative methods showed good imaging quality, except for the Tikhonov method.The first row of Fig. 5 illustrates the NMAE values of the original phantom images under different conditions, providing an objective evaluation of their imaging quality.It was consistently observed that the Tikhonov method produced the highest NMAE values, while the SL0 method consistently achieved the lowest NMAE values.The second row of Fig. 5 shows the PSNR values of the original phantom under various conditions.The PSNR values obtained with the Tikhonov method remained consistently below 20 as the number of channels increased.On the other hand, the L 1 − L 2 norm based on the DCA method and the SL0 method had slightly higher PSNR values with increasing channel numbers, and the SL0 method consistently outperformed the L 1 − L 2 norm based on the DCA method.The third row of Fig. 5 presents the SSIM values of the original phantom images under different conditions.When the channel number was below 40, the SL0 method and the L 1 + L 2 (glmnet) method had slightly higher SSIM values.However, when the channel number was not below 40, the SSIM values of the five methods (excluding Tikhonov) were close to 1, indicating that the images obtained with these methods closely resembled the original images.

B. Antinoise Ability Experiment
In the Antinoise Ability Experiment, simulated noise was intentionally introduced during the acquisition of photoacoustic signals.The main objective was to evaluate the accuracy and robustness of the L 1 − L 2 norm based on the DCA.Performance assessment involved introducing different levels of noise to the simulated data and examining its impact on various reconstruction methods.A dependable algorithm can be distinguished by higher PSNRs, lower NMAEs, and SSIMs.This study aimed to assess the algorithm's stability under diverse noise conditions, emphasizing the significance of achieving superior PSNRs, minimizing NMAEs, and maximizing SSIMs to ensure precise and stable image reconstruction.
Fig. 6 illustrates the behavior of different methods in terms of error rates, NMAE values, PSNR values, and SSIM values.The L 1 − L 2 norm based on the DCA method shows smaller NMAE values when the noise level is 40 and the number of channels is over 30.However, as the noise level increases to 30 and the number of channels exceeds 20, the L 1 − L 2 norm based on the DCA method outperforms other methods with significantly lower NAME values.Moreover, at a noise level of 20, the L 1 − L 2 norm based on the DCA method achieves the lowest NAME values.Compared to other methods, the L 1 − L 2 norm based on the DCA method demonstrates minimal variation in NMAE values when faced with different levels of noise, as long as the number of channels remains constant.Conversely, the other methods exhibit significant changes in NMAE values under varying noise conditions.Specifically, the SL0 method performs well at low noise levels but struggles with high noise levels, especially when the number of channels is involved, leading to instability.The L 1 − L 2 norm based on the DCA method does not exhibit notably higher PSNR and SSIM values at a noise level of 40.However, as the noise level decreases to 30, the L 1 − L 2 norm based on the DCA method shows significantly higher PSNR and SSIM values when the number of channels exceeds 30.Similarly, at a noise level of 20, the L 1 − L 2 norm based on the DCA method demonstrates significantly higher  PSNR and SSIM values when the number of channels is greater than 20.Although visually pleasing reconstructed images can be produced using the L 1 + L 2 (glmnet) method, it yields higher NMAE values and lower PSNR values at higher noise levels.Although the SL0 method performs well in reconstructing noisefree images, it faces challenges and instability when dealing with varying conditions.

C. Experiments Simulating Biological Processes in an Artificial Environment
To assess the feasibility of the L 1 − L 2 norm based on the DCA approach.We conducted a comprehensive investigation employing a tissue-mimicking phantom experiment.Fig. 7(a) is the representation of our experimental setup.The illuminating source utilized was a Q-switched Nd: YAG(LS-2137 U/2, Lotis TII Ltd, Belarus) laser operating at a wavelength of 532 nm, with a frequency resolution of 10 Hz.The initial laser pulse underwent amplification through a concave lens, followed by homogenization with ground glass, before being directed onto a pork liver.To capture the ensuing photoacoustic signal, we utilized a 5 MHz single-element ultrasonic transducer featuring a 12.7 mm diameter (V309, Panametrics).Both the transducer and sample were immersed in the water tank.To facilitate the coupling of photoacoustic waves with the transducer.The stepper motor (PMC100-3) was deployed to orchestrate controlled rotation of the transducer around the sample during the sampling process.The rotation radius is 40 mm.
The ultrasound signal underwent initial amplification through a Panametrics pulse amplifier.Subsequently, it was meticulously captured and averaged 30 times using an oscilloscope (MSO4000B; Tektronix).The entire experimental setup was intricately managed by a personal computer, overseeing the stepper motors' operation.More importantly, the personal computer can handle the signal acquisition process.
In Fig. 7(b), three circular slices of pork liver were submerged approximately 3 mm deep within a slab of fresh pork fat.The pork liver slices exhibited a thickness of around 1 mm and a diameter of roughly 1 mm in the imaging plane.
For the pork liver experiment, we carefully collected the signals at 40 and 80 uniformly distributed locations, respectively.The subsequent reconstruction, depicted in Fig. 8 and Fig. 9, employed 64 frequency samples randomly selected from within a (0.25, 5) MHz window.It's noteworthy that the transducer response, denoted as was intentionally constrained to a specific value of one.
Fig. 8 illustrates the imaging profiles of six different methods applied to pork liver phantom images at 40 sampling points.The outlines of the Tikhonov and AIT methods are characterized by a lack of clarity and significant noise.Conversely, the remaining four methods yield images reconstructed using the L 1 − L 2 norm based on the DCA method, which exhibit comparatively clear profiles with reduced noise.
Shifting focus to Fig. 9, which showcases the imaging profiles of the same six methods for pork liver phantom images, but with 80 sampling points.The outline of the Tikhonov method

C. CONCLUSION
The main objective of this study is to assess the effectiveness of the L 1 − L 2 norm based on the DCA in sparse reconstruction techniques for photoacoustic images.By utilizing the concept of CS theory, the L 1 − L 2 norm based on the DCA showcases its ability to reconstruct images using fewer signals.To evaluate the performance of these reconstructed images, we conducted a comprehensive analysis using specific numerical metrics, comparing them to five alternative regularization methods.The results of our study suggest that the imaging quality of SL0 is excellent when there is no noise present.However, as the level of noise increases, SL0 does not maintain the same high quality.On the other hand, the L 1 − L 2 norm based on the DCA method consistently achieves the best imaging quality in environments with high levels of noise.Furthermore, the imaging quality evaluation index remains stable for the L 1 − L 2 norm based on the DCA method regardless of the level of noise, demonstrating its robustness in noise-intensive scenarios.These groundbreaking results offer invaluable inspiration for advancing sparse perspective photoacoustic imaging techniques.By capitalizing on the insights gained from the superior performance of L 1 − L 2 norm based on the DCA, we can further enhance the precision and effectiveness of sparse perspective photoacoustic imaging approaches, leading to significant improvements in this field.These attributes L 1 − L 2 norm based on the DCA as a promising solution for future biomedical applications, underscoring its potential impact and versatility in advancing imaging technologies.

Fig. 3 .
Fig. 3. Phantom rebuilding results indicators by different methods.The first to the third rows are reconstructed images under channel numbers of 30, 40 and 50.

Fig. 4 .
Fig. 4. Reconstruction of the original phantom under six different methods.The first to the third rows are reconstructed images under channel numbers of 10, 20 and 30.

Fig. 5 .
Fig. 5. First column is the three evaluation indexes of phantom under the six methods, the second column is the three evaluation indexes of original under the six methods, and the three evaluation indexes are NMAE, PSNR and SSIM.

Fig. 6 .
Fig. 6.Reconstruction of the original phantom under six different methods.From left to right are the experimental results at different noise levels.From top to bottom are the evaluation indexes of reconstructed images by different methods.

Fig. 7 .
Fig. 7. (a) Schematic representation of PAI.(b) Cross-sectional representation of a pork liver, showcasing the three circular slices of pork liver.

Fig. 8 .
Fig. 8. Results were obtained from the pork liver experiments using six different reconstruction methods.The reconstructed pork liver images with 40 views.

Fig. 9 .
Fig. 9. Results were obtained from the pork liver experiments using six different reconstruction methods.The reconstructed pork liver images with 80 views.

Fig. 10 .
Fig. 10.First figure shows the PSNR values of the reconstructed liver images at sampling points 40 and 80.The second graph shows the SSIM values of the reconstructed liver images at sampling points 40 and 80.