Blind single image restoration for ring-array photoacoustic tomography system via physics-aided unsupervised learning

Abstract

Ring-array photoacoustic tomography (PAT) has been successfully applied in noninvasive biomedical imaging. However, due to the influence of aperture shape and bandwidth of transducer, the reconstructed images usually suffer from spatially variant blur artifacts. Although deep learning methods can be employed for image restoration, they typically require large amounts of high-quality training data with ground truth to ensure good generalization, which is impractical for real-world ring-array PAT systems. In this study, we analyze and model the physics of blur formation in ring-array PAT system, integrate this with the unsupervised learning theory of deep prior (DP), and propose a blind single-image restoration framework. This framework consists of three generative networks designed based on the characteristics of the clear image and different types of point spread functions (PSFs). The parameters of the networks are optimized simultaneously to generate the optimal estimations of corresponding terms. Experimental results on both phantom and in vivo PAT images demonstrate that the proposed method is highly efficient and significantly improves the image resolution, outperforming state-of-the-art alternatives.

1. Introduction

Photoacoustic tomography (PAT) is a rapidly advancing imaging modality that has garnered significant attention in recent years [1,2]. In a PAT system, nanosecond laser pulses are used to illuminate biological tissues, inducing a thermoelastic effect that causes a rapid temperature rise and the emission of ultrasonic waves, these ultrasonic waves propagate through the medium, reach the surface of transducer array in the PAT system, and are recorded to reconstruct the energy absorption map using various methods [3], such as time reversal (TR) [4,5], filtered back projection (FBP) [6,7], and model based iterative reconstruction (IR) algorithms [8,9]. Compared to other optical or acoustic imaging techniques, PAT offers the dual advantages of high contrast and high resolution, making it highly promising for certain biomedical and clinical applications [10,11].

To accommodate various application scenarios, the transducer array in PAT systems can take on different geometries, such as linear, cylindrical, or spherical, among which the ring-array PAT system is one of the most widely used due to its isotropic imaging characteristics. However, there are two major challenges: First, the finite size of the transducer aperture often results in spatially rotational blur relative to the image center. In other words, the point spread function (PSF) varies for each pixel, and resolution decreases along the tangential direction at all locations [12,13]. Second, the finite detection bandwidth of transducer introduces spatially invariant blur, meaning that the induced PSF is identical for each pixel, leading to uniform resolution degradation across the image [12,13]. These two artifacts collectively impair image details and limit the performance of ring-array PAT system. Therefore, it is essential to develop effective methods to restore the degraded PAT images and achieve higher resolution.

Image deconvolution is the most commonly used method for restoring blurred images. Depending on whether the PSF is known, deconvolution methods can be classified as either non-blind or blind. Alternatively, based on the spatial consistency of the PSF across pixels, they can be categorized into spatially invariant and spatially variant deconvolution. All deconvolution methods are inherently ill-posed, meaning that the restored image tends to be contaminated by amplified noise and ringing artifacts. Therefore, regularization is essential, with the core idea of introducing additional constraints to suppress these negative artifacts. Numerous regularized deconvolution methods have been developed, such as those utilizing total variation (TV) [14,15], Tikhonov [16,17], hyper-Laplacian [18,19], and L₀-norm based [20] regularization. However, most of these methods are designed for spatially invariant deconvolution. For spatially variant deconvolution, some approaches employ complex auxiliary equipment to measure the spatially variant PSFs [21,22], while others are based on intricate mathematical models, which often suffer from low efficiency [[23], [24], [25], [26]]. These methods are generally unsuitable for addressing the combined spatially rotational and spatially invariant blur present in images obtained from the ring-array PAT systems. Recently, with the rapid advancement of deep learning, numerous image deconvolution methods based on supervised learning have emerged, including approaches leveraging generative adversarial networks (GANs) [27,28], attention mechanisms [29,30], and diffusion models [31,32]. These methods outperform traditional deconvolution methods in both restoration quality and efficiency. However, they typically require large amounts of paired degraded images and ground truths to ensure generalization, which is impractical for real-world ring-array PAT systems.

Deconvolution methods have indeed been applied to enhance the quality of PAT images. For instance, recorded photoacoustic signals have been deconvolved using either spatial impulse response (SIR) [33,34] or electrical impulse response (EIR) [35,36] to improve image resolution. In [37] and [38], a deconvolution based reconstruction method is proposed, where the formation of photoacoustic signals is modeled as the convolution of the system impulse response and the optical absorption map. In [39] and [40], a specialized deconvolution method is introduced, combining model based IR algorithm with the total impulse response to enhance resolution. Furthermore, deconvolution has been used in the image domain as a postprocessing method, e.g., the methods in [41] and [42] apply spatially invariant blind deconvolution for PAT image restoration. However, since the blur is spatially variant, these approaches do not effectively enhance image details. Recently, some spatially variant deconvolution methods have been proposed for PAT image deblurring with promising results. In [43] and [44], the authors use tiny acoustic-absorbing microspheres to measure spatially variant PSFs at prespecified positions in the PAT system. They then employ interpolation methods to estimate the PSFs for all pixels, using well-designed regularized non-blind deconvolution to restore the degraded PAT images. In [45], the authors analyze the mechanisms of spatially rotational blur and streak artifacts in the ring-array PAT system. They introduce the concept of rotational convolution to model the blur formation, and propose a rotational deconvolution problem model, which they solve using an alternating optimization algorithm. This method successfully removes rotational blur and streak artifacts, significantly improving image quality. Although many supervised deep learning based restoration methods have been developed for PAT image postprocessing [46], such as removing artifacts from sparse-view or limited-view reconstructions [[47], [48], [49]], suppressing reflection artifacts [50], aiding in radius calibration [51], and correcting speed-of-sound aberrations in heterogeneous media [52], few studies have focused on using supervised deep learning to address spatially variant blur in PAT images. This is primarily because supervised deep learning relies heavily on large datasets of high-quality training samples. In real-world PAT systems, it is impractical to obtain sufficient degraded images and their corresponding ground truths to construct such training datasets, which limits the generalization capability of these networks.

In recent years, an unsupervised deep learning framework known as deep prior (DP) has been introduced, which enables neural networks to learn directly from the input data, extracting the underlying information [53]. The DP framework has proven to be a highly effective tool for solving complex inverse problems [[54], [55], [56]]. Specifically, in the context of deblurring, e.g., the method in [56] uses neural networks to fit the potential distributions of both the clear image and PSF, optimizing them to achieve excellent deblurring results. However, this method is designed solely for handling spatially invariant blur. In [57], the authors propose a DP based method aiming at extracting spatially variant PSFs and the clear image from a degraded PAT image. However, similar to the approaches in [43] and [44], this method still relies on some premeasured PSFs to mitigate the ill-posedness and avoid suboptimal outcomes.

Inspired by the concept of DP framework, we modify and apply it for PAT image restoration. Our research makes three main contributions:

First, we analyze the physical imaging mechanism of ring-array PAT system and propose a comprehensive mathematical model to describe the spatially rotational blur caused by the SIR and the spatially invariant blur caused by the EIR. Based on this, we formulate a regularized single-image restoration problem to introduce learning constraints for the DP framework.

Second, following the learning principles of the DP framework, we design three generative neural networks tailored to the characteristics of the clear image, spatially rotational blur, and spatially invariant blur. We then propose a comprehensive optimization scheme to obtain the optimal solution for the restoration problem.

Third, we conduct experiments on six experimentally acquired PAT datasets, including both phantom and in vivo images, acquired with a real 256-element ring-array PAT system. The method is further evaluated under different noise levels and sparse-view acquisition conditions. The results demonstrate that our approach achieves performance comparable to or even better than state-of-the-art methods.

The principal novelty of our proposed method is the integration of ring-array PAT-specific physical degradation modeling with blind deep prior optimization under a unified re-degradation objective. The restored image and degradation-related components are jointly estimated from a single degraded PAT image. The spatially rotational PSF module and spatially invariant PSF module provide a description of the position-dependent rotational blur and locally shared blur, while the hybrid TVL₀ regularization imposes an explicit structural constraint to stabilize the ill-posed blind restoration problem. This formulation improves the physical consistency, stability, and interpretability of the restoration process without requiring paired training data or pre-measured dense PSFs.

The outline of the paper is organized as follows: Section 2 presents the physics of image blur in the ring-array PAT system, along with the mathematical models for blur degradation and restoration. Section 3 describes the principles of the DP framework and our modifications to it, detailing how we apply this framework to obtain optimal estimations of the clear image and other key variables. Section 4 provides comprehensive experimental results on phantom and in vivo biological tissue PAT images, including qualitative and quantitative comparisons with several state-of-the-art methods, and stability analysis of the estimated PSFs under different random initializations. Section 5 presents a comprehensive experimental evaluation, covering ablation study, quantitative evaluation with physical measurements, optimization stability evaluation, noise robustness evaluation, parameter sensitivity evaluation, sparse-view acquisition evaluation, and computational efficiency comparison. Section 6 presents a comprehensive discussion of the limitations of the proposed method. Finally, Section 7 concludes the paper.

2. Physics of image degradation in ring-array PAT system

Research has proven that the PAT images usually suffer from spatially variant blur, primarily due to the finite aperture and bandwidth of the transducer as well as the geometry of the transducer array [12,13]. Traditionally, spatially variant blur is modeled by convolving the pixels in the clear image with their corresponding PSFs. Specific to the ring-array PAT system, as illustrated in Fig. 1(a), a gel phantom wrapping an acoustic-absorbing microsphere is moved using a micro-displacement stage to measure photoacoustic signals at prespecified positions within a real ring-array PAT system. The PSFs at these positions are then reconstructed using FBP. The three rectangles in Fig. 1(b) show that these PSFs exhibit clear spatially rotational shapes and a certain degree of spatial expansion.

Fig. 1.

PSF measurement in a ring-array PAT system. (a) Schematic diagram for measuring the PSFs. (b) Map of the obtained PSFs.

However, using traditional models to restore the clear image and all the PSFs from a degraded PAT image faces a significant challenge due to the disparity between the number of unknown variables and the available data, making the restoration process severely ill-posed. Thus, it is crucial to develop a more concise model to accurately represent the spatially variant blur inherent in the ring-array PAT system.

In practical ring-array PAT systems, the effects induced by the finite detector aperture and limited detection bandwidth are physically coupled during acoustic detection. Nevertheless, these two mechanisms exhibit distinct dominant characteristics in the reconstructed image. The limited bandwidth mainly introduces an approximately spatially invariant blur, whereas the finite detector aperture and the rotationally symmetric structure of the ring-shaped transducer array mainly produce position-dependent rotational blur. Based on these dominant characteristics, the spatially variant blur is approximated by two successive degradation steps. Specifically, EIR-related spatially invariant blur is first applied to the clear image, followed by SIR-related spatially rotational blur applied to the intermediate result.

The bandwidth limitation of the transducer mainly determines the frequency response of the detection system and introduces a spatially invariant blur, which can be approximately described as:

$$\mathbf{g}=\mathbf{h}*\mathbf{x},$$ (1)

where x is the clear image, g is the blurred image resulting from the EIR, h is the spatially invariant PSF which is used to model the blur effect of EIR, and $*$ denotes the spatially invariant convolution operator.

In contrast, the finite aperture size and ring-array geometry mainly affect the spatial sensitivity distribution of the detectors. This effect varies with the spatial location of the absorber relative to the transducer array, leading to a position-dependent rotational blur. We introduce the concept of spatially rotational convolution, as proposed in our previous research [45], to model the spatially rotational blur in the ring-array PAT system, the blurred image is approximated by rotating the clear image through several equally spaced degrees and calculating a weighted sum of the rotated images.

To mathematically describe the rotational convolution process, we define a constant angular interval $\Delta \theta$and a finite number of rotational samples M. The rotation angle corresponding to the j-th rotational component is defined as:

$${\theta }_j=\left(j-\frac{M+1}{2}\right)\Delta \theta ,j=\mathrm{1},\mathrm{\dots },M,$$ (2)

where M denotes the number of discrete rotational samples. Accordingly, the rotational angles range from$-\frac{M-1}{2}\Delta \theta$ to $\frac{M-1}{2}\Delta \theta$. The corresponding spatially rotational PSF is represented by the weight vector $\mathbf{w}=[w_1,\cdots ,w_j,\cdots ,w_M]$, where $w_j$ represents the contribution of the rotated image at angle ${\theta }_j$. Similar to the spatially invariant PSF, all elements in w are constrained to be nonnegative and their summation equals one to satisfy energy conservation during the rotational blur process.

We further define the rotation operator ${\mathbf{R}}_{{\theta }_j}$, where ${\mathbf{R}}_{{\theta }_j}$represents the image x rotated by the angle ${\theta }_j$. Therefore, the spatially rotational convolution can be formulated as:

$$\mathbf{f}=\sum _{j=1}^M{w}_j{\mathbf{R}}_{{\theta }_j}\mathbf{x},$$ (3)

where f is the spatially rotational blurred image resulting from the SIR of the ring-array PAT system.

Based on the above analysis and combining (1) and (3), we can obtain the comprehensive model for image blur degradation in the ring-array PAT system due to EIR and SIR, which is shown as the following equation:

$$\mathbf{y}=\sum _{j=1}^M{w}_j{\mathbf{R}}_{{\theta }_j}\mathbf{x}\left(\mathbf{h}*\mathbf{x}\right)+\mathbf{n},$$ (4)

where y represents the degraded PAT image, n denotes the measurement error. This factorized representation reduces the degrees of freedom in blind degradation estimation and provides a more constrained and physically interpretable solution space, making the degradation estimation process more explicit, efficient, and easier to interpret.

This formulation does not imply that the EIR and SIR effects are completely independent. Instead, it assumes that their dominant characteristics can be approximately separated to construct a clear and tractable degradation model for the spatially variant blur in ring-array PAT systems. The coupling effects between aperture-induced and bandwidth-induced degradations may introduce modeling errors. However, in the proposed blind restoration framework, these errors can be partially compensated by the adaptive estimation of the unknown PSFs. Specifically, unlike conventional methods that rely on manually measured or predefined PSFs, our framework simultaneously estimates the spatially invariant PSF h and the spatially rotational PSF w from the degraded PAT image through the physics-aided deep prior optimization. Therefore, the estimated PSFs can implicitly absorb part of the residual coupling effects between the two degradation mechanisms.

3. Physics-aided DP framework for PAT image restoration

In this section, we first provide a brief overview of the basic DP framework. We then demonstrate how it has been modified using the aforementioned imaging physics to restore the degraded PAT image.

3.1. Principle of DP framework

Recently, the DP framework, an unsupervised learning method, has been proposed for addressing classic inverse problems such as image reconstruction [55], deblurring [56], and denoising [58]. Unlike supervised learning, which relies on training datasets, the DP framework uses neural networks to iteratively generate optimal estimations of the unknown variables through self-fitting. To introduce the DP framework accurately, we first present the general approach for solving inverse problems.

Consider the degradation model of a linear system, given by:

$$\mathbf{p}=\mathbf{A}\mathbf{s}+\mathbf{n},$$ (5)

where s is the original signal, A represents the linear response of the system, p is the output of the system, i.e., the obtained degraded signal, and n is also the measurement error.

The goal of an inverse problem is to derive s from p. This problem is often ill-posed and can be challenging to solve satisfactorily, especially when the condition number of A is large. To achieve a stable solution, the common approach is to introduce auxiliary constraints, i.e., regularization terms, to reformulate the inverse problem as an optimization problem as follow:

$$\mathbf{s}=\text{arg}{\min }_{\mathbf{s}}\frac{\lambda }{2}{\Vert \mathbf{A}\mathbf{s}-\mathbf{p}\Vert }_2^2+C(\mathbf{s}),$$ (6)

where $\lambda$ is the regularization coefficient and $C(\mathbf{s})$ is the regularization term.

In image restoration problems, such as image denoising and deblurring, which follow the same form as (6), the regularization term$C(\mathbf{s})$is crucial for determining image quality. According to the Hammersley-Clifford theorem [59], regularization terms can be derived from the prior probabilistic distribution of the clear image, leading to the regularization term often being referred to as image prior in image processing tasks. Since the problem in (6) is typically nonlinear and challenging to solve, some sophisticated iterative optimization methods have been proposed, such as the quadratic penalty method [60] and the alternating direction method of multipliers (ADMM) [61].

In contrast to the regularization methods that use explicit constraints, the DP framework provides an alternative approach by implicitly regularizing the ill-posed inverse problem. It achieves this by employing a generative neural network to approximate the distribution of the clear image. For the problem in (6), the DP framework omits the explicit regularization term$C(\mathbf{s})$and instead imposes the reparametrization $\mathbf{s}=G_{\theta }(\mathbf{z})$ as a constraint in the preliminary problem, i.e.,

$$\begin{array}{l}\theta =\text{arg}{\min }_{\theta }{\Vert \mathbf{p}-\mathbf{A}{G}_{\theta }(\mathbf{z})\Vert }_2^2,\\ s.t.\mathbf{s}=G_{\theta }(\mathbf{z}),\end{array}$$ (7)

where $G_{\theta }$ is a generative neural network parametrized by$\theta$, z is a random noise variable.

The parameters $\theta$ are iteratively updated based on the loss function in (7) until convergence is achieved, resulting in the final solution $\mathbf{s}=G_{\theta }(\mathbf{z})$. The basic DP framework is outlined in Algorithm 1, and recent research has shown that it outperforms traditional optimization methods in challenging tasks [[54], [55], [56]]. Therefore, we aim to modify the DP framework to achieve high-quality restoration for degraded images reconstructed from ring-array PAT systems.

Algorithm 1

The basic DP framework.

3.2. Modified DP framework

In this paper, we aim to develop a high-quality blind restoration algorithm that can accurately estimate the clear image and PSFs from the degraded PAT image, i.e., simultaneously restoring x, w, and h from (4) given y. To our knowledge, the most common approach for solving this problem is the alternating optimization scheme, which involves alternately fixing two of the three variables and finding the optimal estimation of the remaining one until convergence. However, despite the use of regularization methods, this approach often converges to suboptimal local solutions, making it challenging to achieve a stable result. Therefore, we propose to modify the basic DP framework using the degradation physics of the ring-array PAT system (i.e., (4)) to achieve the result. To derive the proposed method, we first construct the preliminary inverse optimization problem model based on (4):

$$(\mathbf{x},\mathbf{w},\mathbf{h})=\text{arg}{\min }_{(\mathbf{x},\mathbf{w},\mathbf{h})}{\Vert \mathbf{y}-\sum _{j=1}^M{w}_j{\mathbf{R}}_{{\theta }_j}\left(\mathbf{h}*\mathbf{x}\right)\Vert }_2^2.$$ (8)

There are three unknown variables, i.e., x, w, and h that need to be estimated. Inspired by the principle of the basic DP framework described in Subsection 3.1, we modify it by employing three neural networks $G_{\theta \mathbf{x}}$, $G_{\theta \mathbf{w}}$, and $G_{\theta \mathbf{h}}$to generate x, w, and h, respectively. Consequently, the problem model for the proposed physics-aided DP framework can be expressed as

$$({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})=\text{arg}{\min }_{({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})}{\Vert \mathbf{y}-\sum _{j=1}^M{[G_{\theta \mathbf{w}}({\mathbf{z}}_{\mathbf{w}})]}_j{\mathbf{R}}_{{\theta }_j}[G_{\theta \mathbf{h}}({\mathbf{z}}_{\mathbf{h}})*G_{\theta \mathbf{x}}(\mathbf{y}+{\mathbf{z}}_{\mathbf{x}})]\Vert }_2^2,$$ (9)

where ${\theta }_{\mathbf{x}}$, ${\theta }_{\mathbf{w}}$, and ${\theta }_{\mathbf{h}}$are the parameters of $G_{\theta \mathbf{x}}$, $G_{\theta \mathbf{w}}$, and $G_{\theta \mathbf{h}}$, respectively. The generator $G_{\theta \mathbf{x}}$uses y as a prior condition and takes $\mathbf{y}+{\mathbf{z}}_{\mathbf{x}}$ as input, where ${\mathbf{z}}_{\mathbf{x}}=\lambda \mathbf{\epsilon }$ is a Gaussian perturbation, $\mathbf{\epsilon }~N(0,1)$ is independently resampled at each iteration and λ is the perturbation coefficient. The perturbation ${\mathbf{z}}_{\mathbf{x}}$ is applied throughout the optimization process as a stochastic regularization strategy to improve optimization stability. In contrast, ${\mathbf{z}}_{\mathbf{w}}$and ${\mathbf{z}}_{\mathbf{h}}$ are fixed random vectors used as the inputs of $G_{\theta \mathbf{w}}$and $G_{\theta \mathbf{h}}$, respectively.

The problem model in (8) and (9) neglect the measurement error n. However, in PAT image restoration, this error significantly impacts the results due to the ill-posed nature of the problem, potentially leading to poor performance of the DP method. Therefore, it is essential to incorporate a robust explicit regularization term into the DP problem model in (9) to mitigate the negative artifacts caused by ill-posedness and stabilize the restoration process. To address this, we employ the robust hybrid TVL₀ regularization term proposed in [62], which is defined as follow:

$$\begin{array}{l}TV{L}_0\left(\mathbf{x}\right)=\alpha TV\left(\mathbf{x}\right)+\beta L_0\left(\mathbf{x}\right)\\ =\alpha \underset{\mathrm{TV}}{\underbrace{\sum _{i=1}^N\sqrt{{({\mathbf{d}}_h*\mathbf{x})}_i^2+{({\mathbf{d}}_v*\mathbf{x})}_i^2}}}+\beta \underset{L_0}{\underbrace{\sum _{i=1}^{N}sign\left(\left|{({\mathbf{d}}_h*\mathbf{x})}_i\right|+\left|{({\mathbf{d}}_v*\mathbf{x})}_i\right|\right)}}\end{array},$$ (10)

where d_h and d_v represent the horizontal and vertical derivative filters, sign denotes the sign function, $\alpha$and $\beta$ are the weights for the inner TV and L₀-norm based regularization components.

Based on (9) and (10), the modified DP problem model is given by

$$\begin{array}{l}({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})=\text{arg}{\min }_{({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})}L({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})\\ =\text{arg}{\min }_{({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})}{\Vert \mathbf{y}-\sum _{j=1}^M{[G_{\theta \mathbf{w}}({\mathbf{z}}_{\mathbf{w}})]}_j{\mathbf{R}}_{{\theta }_j}[G_{\theta \mathbf{h}}({\mathbf{z}}_{\mathbf{h}})*G_{\theta \mathbf{x}}(\mathbf{y}+{\mathbf{z}}_{\mathbf{x}})]\Vert }_2^2+TV{L}_0[G_{\theta \mathbf{x}}(\mathbf{y}+{\mathbf{z}}_{\mathbf{x}})]\end{array},$$ (11)

where $L({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})$ denotes the loss function.

Actually, there should be regularizations for $G_{\theta \mathbf{w}}({\mathbf{z}}_{\mathbf{w}})$ and$G_{\theta \mathbf{h}}({\mathbf{z}}_{\mathbf{h}})$ as well. However, we find that the hybrid TVL₀ is sufficiently robust to ensure the quality of the restored image. It effectively guarantees the accuracy of the outputs from $G_{\theta \mathbf{w}}({\mathbf{z}}_{\mathbf{w}})$and$G_{\theta \mathbf{h}}({\mathbf{z}}_{\mathbf{h}})$, so we do not apply additional regularizations for these variables.

Based on (11), the workflow of the proposed DP framework is illustrated in Fig. 2. First, we add a Gaussian perturbation${\mathbf{z}}_x$to the degraded PAT image and input it into $G_{\theta \mathbf{x}}$ to generate the intermediate restored image. Meanwhile, we initialize the variables ${\mathbf{z}}_{\mathbf{w}}$ and ${\mathbf{z}}_{\mathbf{h}}$ with uniform noise and input them into $G_{\theta \mathbf{w}}$ and $G_{\theta \mathbf{h}}$, respectively, to generate the intermediate estimations of w and h. All outputs are used to calculate the loss as defined in (11). Next, we compute the gradients of the loss with respect to ${\theta }_{\mathbf{x}}$, ${\theta }_{\mathbf{w}}$, and ${\theta }_{\mathbf{h}}$, respectively, and update them simultaneously using the Adam algorithm [63].

Fig. 2.

Flowchart of the proposed physics-aided DP framework.

Similar to the basic DP framework, the proposed physics-aided DP framework also employs an iterative optimization scheme. Specifically, the steps outlined in Fig. 2 are iteratively implemented until convergence is achieved. After that, the final restored image can be obtained using the trained $G_{\theta \mathbf{x}}$, i.e., $\mathbf{x}=G_{\theta \mathbf{x}}(\mathbf{y}+{\mathbf{z}}_{\mathbf{x}})$. The entire algorithm framework is summarized in Algorithm 2:

Algorithm 2

The physics-aided DP framework for blind single PAT image restoration.

3.3. Network structures

In the proposed DP framework, three neural networks are utilized to generate the clear image, spatially invariant PSF, and spatially rotational PSF, respectively. Each network is designed with specific characteristics to suit its function. The details of these network structures are described as follows.

3.3.1. Structure of G_θx

Since the restored image is a two-dimensional matrix, we utilize a classic encoder-decoder architecture, i.e., a U-shaped network (U-net) with skip connections, to construct the generator $G_{\theta \mathbf{x}}$. The overall structure of $G_{\theta \mathbf{x}}$is shown in Fig. 3(a), the encoder part takes the variable $\mathbf{y}+{\mathbf{z}}_{\mathbf{x}}$as input, then it is processed by two sequential pointwise convolution and residual connection blocks (PRBs), followed by downsampling with a convolutional layer with a stride of two. This process is repeated four times until reaching the bottleneck of the U-Net. Within the data flow of the encoder, after every two PRBs, the height and width of the feature map are halved, while the number of channels is doubled. The decoder part also comprises PRBs, where the input of each PRB is the concatenation of the output from the previous PRB, upsampled using transposed convolution, and the output from the corresponding PRB in the encoder, via a skip connection. In the decoder, the height and width of the output feature map are doubled after every two PRBs, while the number of channels is halved. The final PRB is connected to a pointwise convolution layer to produce the final output.

Fig. 3.

The whole network structure of G_θx. (a) Structure and data flow of G_θx. (b) Structure of G_θh. (c) Structure of G_θw.

As shown in Fig. 3(a), the structure of PRB consists of three modules. The first and third modules share the same structure, centered on a pointwise convolution layer with 1 × 1 filters. The second module features a similar structure but utilizes 3 × 3 filters in its convolution layer. The output of the second module is residually connected to the original input of the PRB. This design choice is motivated by two main reasons: First, since the DP framework employs iterative optimization, limiting the filter size helps ensure computational efficiency. Thus, all convolution layers in the PRB use filters of small sizes. Second, relying solely on pointwise convolution layers can disrupt inter-pixel communication in feature maps, potentially degrading network performance. To address this, we include an auxiliary module with 3 × 3 filters, which experimentally proves sufficient for achieving satisfactory restoration of image details. We can also see from Algorithm 2 that the network parameters are updated through gradient backpropagation. Analogous to traditional supervised training, the residual connection is implemented to mitigate the risk of gradient disappearance during optimization process.

3.3.2. Structures of G_θh and G_θw

Since the degradation variables h and w have relatively small dimensions, two fully connected networks (FCNs) are adopted to estimate the spatially invariant PSF h and the spatially rotational PSF w, respectively. Both networks take fixed random vectors as inputs and consist of three fully connected layers.

The rotational PSF network $G_{\theta \mathbf{w}}$ consists of three fully connected layers with 200, 1000, and 100 neurons, respectively. Its 100-dimensional output represents the rotational weights w_j, which form the spatially rotational PSF w. The spatially invariant PSF network $G_{\theta \mathbf{h}}$consists of three fully connected layers with 100, 1000, and 9 neurons, respectively, and its nine-dimensional output is reshaped into a 3 × 3 spatially invariant PSF h.

Both networks employ ReLU activation in the hidden layers and a Softmax output layer to ensure non-negativity and unit-sum normalization of the estimated PSFs. The detailed structures of $G_{\theta \mathbf{h}}$ and $G_{\theta \mathbf{w}}$are illustrated in Fig. 3(b) and Fig. 3(c), respectively.

3.4. Implementation details

The proposed method is implemented in PyTorch, and all experiments are performed on a workstation equipped with two 10-core/20-thread CPUs operating at 2.4 GHz and an NVIDIA GeForce RTX 4090 GPU with 24 GB of memory.

The image restoration network, spatially rotational PSF network, and spatially invariant PSF network are randomly initialized without pretrained weights. For individual experiments, the random seed is set to 42 by default unless otherwise specified. For the 20 repeated experiments, random seeds from 42 to 61 are used to provide different initializations. Adam optimizer is used with a learning rate of 2 × 10⁻³ and default settings for other hyperparameters.

For the image restoration network, the input at each iteration is formed by adding a small Gaussian perturbation to the FBP-reconstructed PAT image, with the perturbation coefficient λ= 1 × 10⁻⁴. For the spatially rotational PSF network, the angular interval Δθ is set to 0.03° and the number of rotational components M is set to 100, corresponding to an overall angular range of approximately 3° and a 100 × 1 rotational PSF vector w. The size of spatially invariant PSF h is set to 3 × 3. These values provide a suitable representation of the spatially variant blur for the ring-array PAT system used in this study. Bilinear interpolation is used for the image rotation operation, and zero padding is applied to regions outside the image boundary. The convolution with the 3 × 3 spatially invariant PSF h also adopts zero padding with a padding size of 1. Besides, for the hybrid TVL₀ regularized loss function$L({\theta }_{\mathbf{x}},{\theta }_{\mathbf{w}},{\theta }_{\mathbf{h}})$in (11), we set $\alpha =30$and$\beta =20$in the experiments.

An unsupervised early-stopping criterion based on the windowed variance of consecutive restored images [64] is adopted. Optimization terminates when no new minimum is observed within a patience interval, and the image corresponding to the latest minimum is selected. The window length, starting iteration, and patience interval are set to 200, 1000, and 200, respectively. In practice, early stopping typically occurs around 1300–1800 iterations. For computational cost evaluation, the running time is measured over 2000 iterations, and the peak GPU memory usage is recorded. For input sizes of 512 × 512 and 1024 × 1024, the average running times are 279.71 s and 378.20 s per image, respectively. The corresponding peak GPU memory usages are 3.412 GB and 13.541 GB, respectively.

4. Experimental results

4.1. Experimental setup

We conduct experiments on PAT images of both phantoms and in vivo biological tissues to assess the performance of the proposed method. All degraded images are reconstructed from a real ring-array PAT system using the FBP algorithm. The system consists of a ring-shaped transducer array with a 25 mm radius, comprising 256 unfocused rectangular transducers. Each transducer is 0.51 mm in size, with a 0.1 mm interval between adjacent transducers. The photoacoustic signal is excited using a 750 nm wavelength laser. The transducer has a center detection frequency of 7 MHz and a bandwidth of 73%. In the FBP algorithm, we fix the speed of sound at 1500 m/s. Since 256 transducers do not provide sufficient spatial sampling of the photoacoustic signals, the reconstructed images suffer from streak artifacts, which can negatively affect the performance of restoration methods. Therefore, we adopt the preprocessing method from [45] to eliminate these streak artifacts prior to applying the restoration algorithms.

We also compare our proposed physics-aided unsupervised learning method with two state-of-the-art spatially variant deconvolution methods designed for PAT image restoration [43,45]. The first method, described in [43], involves measuring and reconstructing the PSFs at prespecified pixels in the PAT image using microspheres, with PSFs for other pixels obtained through bilinear interpolation. The blur effect is finally removed with a non-blind hyper-Laplacian regularized deconvolution algorithm [18]. Although this method is versatile across PAT systems with different geometrics, its requirement for precise PSF measurement makes it susceptible to experimental variability and time-consuming. The second method, detailed in [45], is a blind spatially variant deconvolution approach based on traditional optimization theory. It addresses spatially rotational blur in ring-array PAT systems without the need for pre-measured PSFs. While it avoids the use of additional equipment for PSF measurement, it does not account for the spatially invariant blur caused by EIR of the transducer. For simplicity, we refer to the two methods in [43] and [45] by measurement based non-blind spatially variant deconvolution (MNBSVD) and blind spatially rotational deconvolution (BSRD), respectively, in the following subsections.

In addition, we include two recent single-image blind restoration methods as additional baselines, i.e., the FFT-ReLU sparsity prior based blind deblurring method [65] and the cross-scale self-supervised blind deblurring method based on implicit neural representation [66], denoted as FRSP and CSINR, respectively. FRSP is an unsupervised optimization-based method that jointly estimates the latent image and blur kernel using an FFT- ReLU sparsity prior. CSINR is a self-supervised method based on implicit neural representation, which enforces cross-scale consistency to constrain the restoration process.

4.2. Evaluation metrics

To quantitatively evaluate the restoration performance, we employ complementary metrics including Full Width at Half Maximum (FWHM), Full Width at Tenth Maximum (FWTM), Centroid Shift (∆c), Line Energy Ratio (LER_2r), Contrast-to-Noise Ratio (CNR), and Signal-to-Noise Ratio (SNR).

Representative line profiles are extracted across target structures to characterize the spatial distribution of the restored responses. The corresponding FWHM and FWTM values are used to quantify the widths of the central response and its extended low-intensity region, respectively, while Δc is used to evaluate positional preservation after restoration. LER_2r is further introduced to characterize the concentration of signal energy within a prescribed linear interval. In addition, CNR and SNR are calculated to quantify target-background separability and signal quality, respectively.

All quantitative metrics are calculated directly from the original reconstructed images and the corresponding restored images, without additional normalization, intensity clipping, or grayscale enhancement. Together, these metrics provide a comprehensive assessment of spatial resolution, structural fidelity, localization accuracy, energy concentration, and image quality.

4.2.1. FWHM and FWTM

For each selected line profile, Full Width at Half Maximum (FWHM) is calculated as:

$$FWHM=x_R-x_L,$$ (12)

where x_R and x_L denote the two positions at which the intensity profile reaches the half-maximum level. The half-maximum intensity is defined as:

$$I_{half}=I_{baseline}+\frac{1}{2}(I_{peak}-I_{baseline}),$$ (13)

where I_peak is the maximum intensity of the selected target structure in the line profile, while I_baseline is calculated as median intensity of the local background regions on both sides of the target.

To obtain x_L and x_R with sub-pixel accuracy, linear interpolation is performed between adjacent sampling points around the half-maximum intensity level. Specifically, when I_half is located between two neighboring points (x_i, I_i) and (x_i+1, I_i+1), the corresponding half-maximum position is calculated as:

$$x_h=x_i+\frac{I_{half}-I_i}{I_{i+1}-I_i}(x_{i+1}-x_i),$$ (14)

where x_h represents the interpolated sub-pixel half-maximum position. This interpolation procedure is independently applied to the left and right sides of the intensity profile to obtain x_L and x_R, respectively.

FWHM characterizes the width of the PSF core and is commonly used as a standard measure of spatial resolution. However, it does not capture the peripheral tails of the PSF. To complement this, Full Width at Tenth Maximum (FWTM) is additionally introduced to quantify a broader spatial extent of the PSF, including low-intensity tail regions. The tenth-maximum intensity level is defined as:

$$I_{10}=I_{baseline}+\frac{1}{10}(I_{peak}-I_{baseline}),$$ (15)

where I_peak and I_baseline are defined in the same manner as above. And the corresponding FWTM is calculated as:

$$FWTM=x_{10,R}-x_{10,L},$$ (16)

where $x_{10,L}$ and $x_{10,R}$ are the left and right coordinates at I₁₀, which are also determined via linear interpolation between adjacent sampling points.

4.2.2. Centroid Shift (∆c)

Centroid shift (∆c) is calculated to evaluate possible localization bias introduced during restoration. The intensity-weighted centroid is calculated as:

$$c=\frac{{\sum }_i{x}_i{I}_i}{{\sum }_i{I}_i},$$ (17)

where x_i is the spatial coordinate of pixel i, I_i is the corresponding intensity, and the summation is performed over all pixels in the selected line profile. ∆c is then defined as:

$$\Delta c=\left|c_{restored}-c_{input}\right|,$$ (18)

where c_input and c_restored represent the centroid positions before and after restoration, respectively. A small ∆c indicates that the restoration improves spatial concentration without introducing noticeable positional displacement.

4.2.3. LER_2r

Line Energy Ratio (LER_2r) is calculated to evaluate whether the restored energy remains concentrated around the target region. It is defined as:

$$LE{R}_{2r}=\frac{{\sum }_{\left|x_{\mathrm{i}}-c\right|\le r}{I}_i}{{\sum }_i{I}_i},$$ (19)

where c denotes the centroid position and r represents the evaluation radius. x_i is the spatial coordinate of pixel i, and I_i is the corresponding intensity. The numerator represents the signal energy within the interval |xᵢ − c| ≤ r, whose total width is 2r, while the denominator represents the total energy of the line profile. A higher LER_2r indicates that a larger proportion of the signal energy is concentrated around the centroid.

4.2.4. CNR and SNR

For CNR and SNR evaluation, signal ROIs are selected from representative target regions, while background ROIs are chosen from adjacent homogeneous regions without obvious structures or artifacts. The ROI locations and sizes are determined separately for different datasets. Within the same dataset, identical ROI positions and sizes are used for the input image and all restoration results to ensure a fair comparison.

For the i-th signal ROI and its corresponding background ROI, SNR and CNR are calculated as:

$$SNR=\frac{1}{n}\sum _{i=1}^{n}20{\log }_{10}\left(\frac{{\mu }_{s,i}}{{\sigma }_{b,i}}\right),$$ (20)

$$CNR=\frac{1}{n}\sum _{i=1}^{n}20{\log }_{10}\left(\frac{{\mu }_{s,i}-{\mu }_{\mathrm{b},i}}{{\sigma }_{b,i}}\right),$$ (21)

where μ_s,i denote the mean of the i-th signal ROI, while μ_b,i and σ_b,i denote the mean and standard deviation of the i-th background ROI. n represents the number of ROI pairs, which is set to 5 in our evaluation.

SNR reflects the relative signal intensity with respect to background fluctuations, whereas CNR further measures the intensity separability between the target and background regions. The signal and background ROIs are marked with green and pink rectangular boxes in the corresponding images, respectively.

4.3. Experiments on microsphere phantoms

An ideal ring-array PAT system should produce consistent point-like PSFs across all positions in the field of view. However, in a real ring-array PAT system, PSFs vary in shape at different positions, as illustrated in Fig. 1(b). If a restoration method can transform these varying PSFs into small, point-like PSFs, its effectiveness is demonstrated.

To validate this inference, we employ a similar approach to [43] to measure the PSFs in our experimental ring-array PAT system. As shown in Fig. 4, we use a gel phantom wrapping a single acoustic-absorbing microsphere with a diameter of 100 μm (measured to be approximately 125 μm), and move it using an electrically controlled micro-displacement stage to measure the PSFs at prespecified positions. According to the experimental setup, the ultimate resolution of the system, calculated using the method in [12], is approximately 250.4 μm, ensuring the size of the microsphere is sufficiently small for accurate PSF measurement. As shown in Fig. 1(b), the reconstructed PSF map reveals evidently comprehensive central rotational symmetric blur and spatially invariant blur.

Fig. 4.

Microscope-measured microsphere with an approximate diameter of 125 μm (1 div = 10 µm).

Fig. 5(a) displays the measured PSF map as Fig. 1(b), followed by the restoration results using MNBSVD, BSRD, FRSP, CSINR, and the proposed method in Fig. 5(b-f). To highlight the effectiveness of the proposed method, we focus on the contents in the bottom right corners of the corresponding figures, enlarging and exhibiting them in Fig. 5(g-l).

Fig. 5.

Comparison of the microsphere PSF map before and after restoration. (a) The original measured PSF map. (b-f) Restoration results of MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. (g-l) Enlarged views of the red rectangular region in (a-f). (m), (p), and (s) Locations of three representative PSFs in the original PSF map and the corresponding line-profile directions, with T and R denoting the local tangential and radial directions, respectively. (n), (q), and (t) Line profiles along T1, T2, and T3, respectively. (o), (r), and (u) Line profiles along R1, R2, and R3, respectively. The FWHM values calculated from line profiles of different methods are listed in the corresponding legends.

It should be noted that due to the inherently low pixel values of both the original and restored images, linear grayscale enhancement is applied to the relevant images solely for the purpose of visualization. All quantitative measurements and result analyses are performed using the original data and are not affected by the enhancement applied during the visualization process.

We can observe that MNBSVD and BSRD can reduce the spatially variant blur, but residual blur remains in the restored images, while FRSP mainly addresses the radial broadening but shows limited effectiveness in the tangential direction. CSINR reduces the PSF extent in both directions but produces inconsistent and distorted responses at some locations. In contrast, the proposed method restores the PSFs to compact, approximately circular responses with stable morphology across the field of view.

Three representative PSFs with progressively increasing tangential broadening are further selected for line-profile analysis in Fig. 5(m–u). The original PSFs exhibit increasingly broadened tangential profiles. In contrast, the proposed method achieves the lowest FWHM values while maintaining more consistent tangential and radial profiles. Moreover, the restored profiles retain smooth single-peak characteristics, with the background intensity remaining at zero and no noticeable secondary peaks, oscillations, sidelobes, or ringing artifacts. These results demonstrate the effectiveness of the proposed method in correcting spatially varying PSF deformation while maintaining morphological consistency.

For further quantitative evaluation, 17 representative PSFs distributed across different regions of the PSF map are selected, as shown in Fig. 6. At each spatial location, the FWHM values of different methods are calculated from the corresponding line profiles along the local tangential and radial directions.

Fig. 6.

Locations and measurement directions of representative PSFs used for FWHM evaluation. The local tangential and radial directions are denoted by T and R, respectively.

Since the physical size of each pixel in the PSF map image corresponds to 50 µm, Table 1 presents the FWHM values in both pixels and micrometers. Across the 17 representative positions, the original PSFs have tangential and radial FWHM values of 5.82 ± 2.15 pixels (291.0 ± 107.5 μm) and 3.13 ± 0.24 pixels (156.5 ± 12.0 μm). Both values are larger than the measured microsphere diameter of 125 μm, confirming the spatial broadening of the original point responses. Moreover, the substantially larger tangential FWHM and its higher standard deviation reveal pronounced position-dependent tangential broadening, whereas the smaller radial variation indicates a relatively consistent radial extent across different positions.

Table 1.

Tangential and radial FWHM values of the original measured and restored PSFs at 17 representative positions (Values are reported as mean ± standard deviation in pixels and micrometers).

FWHM	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
Tangential (px)	5.82 ± 2.15	4.42 ± 1.45	4.01 ± 1.29	5.21 ± 2.00	3.36 ± 0.59	2.67 ± 0.31
Radial (px)	3.13 ± 0.24	2.77 ± 0.45	3.23 ± 0.35	2.47 ± 0.39	2.67 ± 0.49	2.31 ± 0.25
Tangential (μm)	291.0 ± 107.5	221.0 ± 72.5	200.5 ± 64.5	260.5 ± 100.0	168.0 ± 29.5	133.5 ± 15.5
Radial (μm)	156.5 ± 12.0	138.5 ± 22.5	161.5 ± 17.5	123.5 ± 19.5	133.5 ± 24.5	115.5 ± 12.5

As shown in Table 1, the proposed method achieves the lowest FWHM values in both directions, with tangential and radial FWHM of 133.5 ± 15.5 μm and 115.5 ± 12.5 μm, respectively. Moreover, the proposed method yields the smallest difference between tangential and radial FWHM among all methods, demonstrating its effectiveness in restoring spatially variant blur and achieving the most isotropic point responses.

Although the radial FWHM is slightly smaller than the measured microsphere diameter of approximately 125 μm, this does not imply an artificial reduction in the physical target size, because FWHM characterizes only the width of the intensity response at half maximum rather than its full spatial extent.

To further assess whether the FWHM improvement is accompanied by excessive compression of the target response, Full Width at Tenth Maximum (FWTM) is evaluated in Table 2. As shown by the line profiles in Fig. 5, the background intensity of the PSF map is zero, and the nonzero pixels along the profiles therefore represent the effective PSF response. The proposed method achieves radial and tangential FWTM values of 4.07 ± 0.38 pixels (203.5 ± 19.0 μm) and 4.84 ± 0.75 pixels (242.0 ± 37.5 μm), respectively. Both values remain substantially larger than 2.5 pixels, corresponding to the measured microsphere diameter of approximately 125 μm. This indicates that the restored response retains a physically reasonable spatial extent while the system-induced broadening is effectively reduced.

Table 2.

Additional quantitative evaluation of the original measured and restored PSFs at 17 representative positions (Values are reported as mean ± standard deviation. FWTM and ∆c are given in pixels, and LER_2.5 is given as a percentage.).

Direction	Metric	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
Tangential	FWTM (px)	9.19 ± 3.49	7.44 ± 3.68	6.06 ± 1.85	7.68 ± 2.93	5.93 ± 0.70	4.84 ± 0.75
	∆c (px)	--	0.41 ± 0.28	0.27 ± 0.21	0.19 ± 0.14	0.75 ± 0.49	0.21 ± 0.17
	LER2.5 (%)	44.73 ± 20.24	63.72 ± 20.21	61.94 ± 12.53	51.39 ± 21.09	68.76 ± 9.33	74.59 ± 12.84
Radial	FWTM(px)	5.31 ± 0.49	4.46 ± 1.05	5.27 ± 0.45	4.20 ± 0.69	5.35 ± 0.66	4.07 ± 0.38
	∆c (px)	--	0.20 ± 0.19	0.10 ± 0.06	0.06 ± 0.08	0.60 ± 0.36	0.06 ± 0.05
	LER2.5 (%)	72.65 ± 8.66	83.22 ± 20.96	71.78 ± 10.39	82.91 ± 10.19	73.13 ± 11.44	84.66 ± 9.77

The Centroid Shift (∆c) in Table 2 further demonstrates stable localization after restoration. The proposed method achieves relatively small centroid shifts of 0.06 ± 0.05 pixels (3.0 ± 2.5 μm) in the radial direction and 0.21 ± 0.17 pixels (10.5 ± 8.5 μm) in the tangential direction, indicating good localization stability among the compared methods.

As listed in Table 2, LER₂.₅ is calculated from the tangential and radial line profiles. The total evaluation width 2r is set to 2.5 pixels, corresponding to the measured microsphere diameter of approximately 125 μm. Therefore, LER₂.₅ represents the percentage of the total profile energy contained within a 2.5-pixel-wide interval centered at the intensity-weighted centroid. The proposed method achieves LER₂.₅ values of 84.66 ± 9.77% in the radial direction and 74.59 ± 12.84% in the tangential direction, indicating a higher concentration of the restored response within the physical scale of the microsphere.

Overall, the combined FWHM, FWTM, ∆c, LER_2.5, and line profile analyses demonstrate that the proposed method reduces spatial broadening and improves the concentration of the reconstructed response while preserving a physically reasonable spatial extent, energy distribution, and localization accuracy.

4.4. Experiments on other phantoms

Two representative phantoms, including a randomly scattered microsphere phantom and a blood vessel phantom, are used to further evaluate the proposed method.

In Fig. 7, the microspheres are randomly distributed within the three-dimensional phantom, and most of them are not strictly aligned with the imaging plane of the ring transducer array. As a result, the degradation of these microspheres involves not only in-plane spatially rotational blur but also additional three-dimensional effects, such as defocus, propagation path differences, and medium scattering. These complex effects cannot be fully described by the two-dimensional degradation model proposed in this study. Accordingly, microspheres that substantially deviate from the imaging plane are not used for strict restoration performance comparison, whereas microspheres closer to the imaging plane provide a more reliable basis for evaluating the proposed method.

Fig. 7.

Experimental results of the microsphere phantom. (a) Photograph of the phantom; (b) The original input PAT image, where five pink and five green boxes are drawn to indicate the background and signal regions of interests (ROIs) used for CNR and SNR calculations, respectively. (c-g) Restoration results obtained using MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. (h-n) Enlarged views of the regions enclosed by the red rectangles in (a-g), respectively. (o) and (p) Line profile comparisons along L1 and L2, respectively, where L1 (horizontal yellow line) and L2 (vertical blue line) are the lines marked in the enlarged view (i). The corresponding FWHM values are listed in the legends of the line-profile plots.

It should be noted that, five pink and five green boxes drawn on the original input PAT image denote the background and signal regions of interest (ROIs) used for the CNR and SNR calculations, respectively. The horizontal yellow line L1 and vertical blue line L2 indicate the locations selected for line-profile extractions and FWHM calculations. The corresponding FWHM values are listed in the legends of the line profile plots. The same labeling and evaluation conventions are used throughout the subsequent experiments.

As shown in Fig. 7 and Table 3, in the original reconstructed PAT image, spatially variant blur causes broadened and partially overlapping microsphere responses, leading to degraded spatial resolution. Compared with all competing methods, the proposed method achieves the best performance in separating adjacent microspheres, restoring realistic morphological shapes, and removing background noise.

Table 3.

Quantitative comparison of different restoration methods on the randomly scattered microsphere phantom.

Metric	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
FWHM L1 (pixels)	3.81	3.18	3.24	3.13	3.04	2.62
FWHM L2 (pixels)	4.54	3.74	4.17	3.15	3.26	2.27
CNR (dB)	29.74	35.27	35.76	29.29	30.94	40.80
SNR (dB)	29.94	35.44	35.93	29.52	31.09	40.93

Quantitatively, the proposed method yields lower FWHM values together with higher CNR and SNR than the compared methods. The line profiles also exhibit narrower target responses and lower background fluctuations. The FWHM values along L1 and L2 decrease from 3.81 pixels and 4.54 pixels in the original PAT image to 2.62 pixels and 2.27 pixels after restoration, respectively. At a pixel size of 50 μm, these values correspond to reductions from 190.5 μm to 131 μm along L1 and from 227 μm to 113.5 μm along L2. Given the measured microsphere diameter of approximately 125 μm, the restored response widths are closer to the physical microsphere size. These results indicate that the proposed method effectively reduces response broadening without excessive sharpening or background noise amplification.

To further assess the physical reasonableness of the response narrowing, three representative microspheres near the imaging plane are selected for direct size measurement, as shown in Fig. 8. Their apparent diameters are measured on the image grid and converted to physical lengths using the pixel size of 50 μm. In the reconstructed PAT image, the measured diameters are approximately 320, 250, and 250 μm, corresponding to 6.4 pixels, 5.0 pixels, and 5.0 pixels, respectively. After restoration, they decrease to approximately 250 μm, 150 μm, and 212 μm, corresponding to 5.0 pixels, 3.0 pixels, and 4.24 pixels, respectively. Although the restored diameters remain larger than the measured physical diameter in most cases, they are consistently reduced and become closer to the actual microsphere size. This comparison further confirms that the reduced FWHM values reflect effective blur reduction rather than excessive geometric shrinkage.

Fig. 8.

Comparison of the microsphere phantom results before and after restoration. (a) Photograph of the phantom. (b) The original PAT image. (c) Restoration result of the proposed method. (d-f) Enlarged views of the contents in the red rectangles in (a-c), respectively.

In addition to the microsphere phantoms, we also create a blood vessel phantom to further evaluate the algorithms, as shown in Fig. 9(a). The restoration results of all methods are presented in Fig. 9(b-g), and the quantitative results are presented in Table 4. The original reconstructed image exhibits blurred vessel boundaries and low contrast between adjacent vascular structures. MNBSVD, BSRD, and FRSP improve vessel visibility to different degrees, although residual boundary blurring and local intensity fluctuations remain. CSINR partially sharpens the vascular structures but introduces local morphological distortion, resulting in inconsistent improvements in the quantitative metrics. In contrast, the proposed method provides clearer vessel boundaries, improved structural continuity, and more stable background intensity while preserving the overall vascular morphology.

Fig. 9.

Experimental results of the blood vessel phantom. (a) Photograph of the phantom; (b) The original input PAT image. (c-g) Restoration results obtained using MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. (h-n) Enlarged views of the regions enclosed by the red rectangles in (a-g), respectively. (o) and (p) Line profile comparisons along L1 and L2, respectively.

Table 4.

Quantitative comparison of different restoration methods on the blood vessel phantom.

Metric	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
FWHM L1 (pixels)	10.04	7.96	9.14	10.01	8.84	7.74
FWHM L2 (pixels)	7.79	5.52	6.40	6.90	8.14	5.18
CNR (dB)	16.50	20.37	18.38	17.46	17.31	21.24
SNR (dB)	17.61	21.32	19.36	18.51	18.36	22.27

Overall, the quantitative results are consistent with the visual observations. In phantom experiments, the proposed method achieves lower FWHM values while maintaining higher CNR and SNR than the compared methods. The line profile results further show that the proposed method produces more compact and distinguishable target responses, together with lower background fluctuations. These results indicate that the proposed method provides a more favorable balance among spatial response narrowing, contrast preservation, background noise reduction, and structural fidelity.

4.5. Experiments on in vivo biological tissues

In addition to the phantom experiments, the proposed method is further evaluated on in vivo biological tissue images acquired by the ring-array PAT system, including male mouse abdominal blood vessels and a human finger. Compared with phantom images, in vivo PAT images contain more complex anatomical structures and stronger background fluctuations, which makes restoration more challenging.

Fig. 10 (a) presents a PAT image of the abdominal blood vessels of a male mouse, which is rich in intricate details. Recovering such details from spatially variant blurred images is known to be a particularly challenging task in image restoration. Table 5 presents the corresponding quantitative results. After restoration, MNBSVD effectively reduces background fluctuations and achieves the highest SNR among the compared methods. However, its improvement in vessel contrast and spatial response width remains limited. BSRD enhances vascular structures to some extent, but residual edge blurring and local structural inconsistency are still observed. FRSP and CSINR partially improve vessel visibility, whereas background fluctuations and unstable local intensity distributions remain evident. In contrast, the proposed method provides more continuous vascular morphology, clearer vessel boundaries, and better target-background separability, resulting in the best FWHM and CNR performance while maintaining a competitive SNR.

Fig. 10.

Experimental results of abdominal blood vessels of a male mouse. (a) The original input PAT image. (b-f) Restoration results obtained using MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. (g-l) Enlarged views of the regions enclosed by the red rectangles in (a-f), respectively. (m) and (n) Line profile comparisons along L1 and L2, respectively.

Table 5.

Quantitative comparison of different restoration methods on the in vivo male mouse abdominal blood vessel image.

Metric	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
FWHM L1 (pixels)	6.50	6.05	6.13	6.86	6.21	6.03
FWHM L2 (pixels)	2.68	2.56	2.59	2.92	2.63	2.52
CNR (dB)	10.35	15.65	13.52	14.20	14.32	16.13
SNR (dB)	27.98	33.65	31.15	30.88	30.06	32.70

Fig. 11 presents the visual comparison for a human finger image, and Table 6 presents the corresponding quantitative results. Noticeable spatially variant blur is also observed in the original PAT image, resulting in broadened structural responses and reduced local contrast. MNBSVD and BSRD effectively reduce background fluctuations and achieve relatively high SNR values. However, their CNR and FWHM results remain inferior to those of the proposed method, indicating limited recovery of target-background contrast and local spatial response. FRSP reduces the FWHM values to some extent, but the accompanying background fluctuations limit its improvement in CNR and SNR. CSINR shows less stable intensity distributions and relatively weak quantitative performance. In comparison, the proposed method better preserves local tissue structures, improves target-background separability, and achieves the most favorable FWHM and CNR results.

Fig. 11.

Experimental results of a human finger. (a) The original input PAT image. (b-f) Restoration results obtained using MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. (g-l) Enlarged views of the regions enclosed by the red rectangles in (a-f), respectively. (m) and (n) Line profile comparisons along L1 and L2, respectively.

Table 6.

Quantitative comparison of different restoration methods on the human finger image.

Metric	Input	MNBSVD	BSRD	FRSP	CSINR	Proposed
FWHM L1 (pixels)	10.00	9.77	9.75	9.00	9.57	8.80
FWHM L2 (pixels)	5.21	4.86	5.25	4.75	5.18	4.73
CNR (dB)	23.86	26.18	25.89	25.95	24.18	26.78
SNR (dB)	33.21	36.34	36.26	34.82	32.93	35.54

Overall, the quantitative results are consistent with the visual observations. Across all in vivo experiments, the proposed method achieves the best FWHM and CNR while maintaining a competitive SNR. Although MNBSVD attains the highest SNR in some cases, its CNR and FWHM results remain less favorable. The line profile analysis further confirms that the proposed method enhances target responses while preserving smoother background regions, without introducing redundant fluctuations or noise-driven artifacts. These results demonstrate that the proposed method provides a more balanced restoration performance in terms of spatial resolution, contrast, background stability, and structural fidelity for complex in vivo PAT images.

The animal experiment in Fig. 10 was performed using one male mouse in accordance with the National Institutes of Health (NIH) Guide for the Care and Use of Laboratory Animals, after approval of the laboratory animal protocol by the Institutional Animal Care and Use Committee (IACUC) of the University of Science and Technology of China (Protocol Number USTCACUC1803065).

The experiment on a human subject in Fig. 11 was approved by the Institutional Review Board (IRB) of the First Affiliated Hospital of the University of Science and Technology of China (IRB: 2022-ky357) and adhered to the tenets of the Declaration of Helsinki. All subjects were provided with written informed consent prior to the study.

4.6. 4.6. PSF quantification

Joint estimation of the latent image and degradation-related variables from a single degraded PAT image is inherently ill-posed, and trivial or mutually compensated solutions may arise if the degradation model is insufficiently constrained. In the proposed framework, this ambiguity is alleviated by the physics-guided parameterization of the degradation process, the non-negativity and normalization constraints imposed on h and w, and the deep image prior and hybrid TVL₀ regularization imposed on x. These constraints restrict the feasible solution space and reduce the tendency toward degenerate solutions.

To visualize the spatial distribution of the PSF estimated by the proposed method, we present the PSF estimation results corresponding to Fig. 5, Fig. 7, Fig. 9, Fig. 10, and Fig. 11, respectively. Specifically, the spatially invariant PSF h is parameterized as a compact 3 × 3 kernel, while the spatially rotational PSF w is represented by 100 rotational components with an angular interval of Δθ = 0.03°, corresponding to an overall angular range of approximately 3°. The full-field PSF map is then generated by applying the estimated h and w to an ideal 1 × 1-pixel point source with unit intensity at different spatial positions.

In Fig. 12, linear contrast enhancement is applied to Columns (III)–(VI) due to their low original contrast, whereas the rotational PSF w in Column (II) is displayed using its original estimated values. The enhancement is applied solely for visualization and is not involved in any computation or image restoration.

Fig. 12.

Visualization of the PSFs corresponding to different PAT images (50 μm per pixel). The rows (a-e) correspond to the results from Fig. 5, Fig. 7, Fig. 9, Fig. 10, and Fig. 11, respectively. The Columns (I-VI) sequentially present the following: (I) The original PAT images. (II) Spatially rotational PSF w. (III) Spatially invariant PSF h. (IV) Full-field PSF maps. (V) Enlarged views of the yellow rectangular regions in (IV) (center position). (VI) Enlarged views of the blue rectangular regions in (IV) (bottom-right corner position). For visualization, linear intensity enhancement is applied to the images in Columns (III)–(VI) only and is not involved in image restoration or quantitative evaluation.

As shown in Fig. 12, the PSF near the imaging center forms a compact spot with concentrated energy and no obvious directional bias. As the spatial position moves away from the center, the PSF gradually becomes elliptical and is stretched tangentially along concentric circles, with its orientation changing continuously and the rotational blur becoming more pronounced. Meanwhile, each PSF retains a local diffusion response, reflecting the combined effects of spatially rotational and spatially invariant blur. Each pixel in the reconstructed PAT image corresponds to a physical size of 50 µm.

It should be noted that the synthesized full-field PSFs in Fig. 12 are responses to an ideal single-pixel point source, whereas the experimentally measured microsphere PSF map in Fig. 5 is obtained from finite-sized microspheres with a diameter of approximately 125 µm. In addition, practical ring-array PAT degradation is affected by detector bandwidth, finite aperture, acoustic propagation, reconstruction processes, and medium conditions. Accordingly, h and w are used to describe the dominant spatially invariant and spatially rotational degradation characteristics rather than an exhaustive decomposition of all physical effects.

In the proposed framework, the latent image x, h, and w are jointly optimized under the re-degradation constraint, with TVL₀ regularization imposed on x. Therefore, h and w should be regarded as image-dependent effective degradation representations rather than fixed calibrated system parameters, and their estimates may vary across different PAT images. For example, as shown in Fig. 12(b), the responses of randomly distributed microspheres are influenced by their spatial positions, acoustic propagation paths, surrounding medium, and overlapping signals from neighboring microspheres. Consequently, the measured response at a given position cannot be regarded as an isolated local PSF, which may also contribute to variations in the estimated degradation characteristics across different PAT images.

To evaluate the stability of the estimated PSFs, 20 independent experiments with different random seeds are conducted on the microsphere PSF map in Fig. 12(a). As shown in Fig. 13, the estimated h and w exhibit consistent mean values and small standard deviations across the repeated experiments, indicating stable estimation of the dominant degradation characteristics.

Fig. 13.

Stability analysis of the estimated PSFs over 20 independent experiments. (a) Estimated spatially rotational PSF weights w, where the solid line represents the mean value and the shaded region represents the mean ± standard deviation. (b) Estimated spatially invariant PSF weights h, where the blue bars represent the mean values, the numerical labels indicate the corresponding mean values, and the error bars denote the standard deviations. (c) Element index map of the 3 × 3 spatially invariant PSF h. The values shown in (a) and (b) are the original outputs of the PSF estimation, without any additional intensity enhancement or rescaling.

5. Comprehensive performance evaluation

5.1. Ablation study

The proposed DP framework employs three neural networks,$G_{\theta \mathbf{x}}$, $G_{\theta \mathbf{h}}$, and$G_{\theta \mathbf{w}}$, to generate the restored image, spatially invariant PSF, and spatially rotational PSF, respectively.$G_{\theta \mathbf{x}}$serves as the image generator and provides the restored image, whereas$G_{\theta \mathbf{h}}$and$G_{\theta \mathbf{w}}$jointly construct the physics-based degradation model by characterizing the spatially invariant blur and the position-dependent directional blur associated with the ring-array imaging geometry, respectively. In addition, the hybrid TVL₀ regularization provides an explicit structural constraint to preserve coherent image structures, limit background fluctuations, and stabilize the blind single-image restoration process.

To evaluate the contribution of each component in the proposed framework, we conducted ablation experiments on a human finger PAT image and a vessel phantom PAT image. The corresponding visual results are shown in Fig. 14 and Fig. 15, and the quantitative results are presented in Table 7 and Table 8. The complete framework is compared with three reduced variants obtained by removing the spatially rotational PSF module, the spatially invariant PSF module, and the hybrid TVL₀ regularization, denoted as without Rot. PSF, without Inv. PSF, and without TVL₀, respectively.

Fig. 14.

Ablation results of the human finger. (a) Input PAT image (b-e) restoration results obtained using without Rot. PSF, without Inv. PSF, without TVL₀, and the full model, respectively. (f-j) enlarged views of the regions marked by the red rectangles in (a-e). (k) and (l) line profiles along L1 and L2, respectively.

Fig. 15.

Ablation results of the vessel phantom. (a) Photograph of the phantom; (b) Input PAT image (c-f) restoration results obtained using without Rot. PSF, without Inv. PSF, without TVL₀, and the full model, respectively. (g-l) enlarged views of the regions marked by the red rectangles in (a-f). (m) and (n) line profiles along L1 and L2, respectively.

Table 7.

Quantitative ablation results of the human finger.

Metric	Input	without Rot. PSF	without Inv. PSF	without TVL0	Full model
FWHM L1 (pixels)	10.00	9.56	9.09	9.66	8.80
FWHM L2 (pixels)	5.21	5.11	5.06	4.83	4.73
CNR (dB)	23.86	22.91	25.33	24.25	26.78
SNR (dB)	33.21	32.89	35.12	34.06	35.54

Table 8.

Quantitative ablation results of the vessel phantom.

Metric	Input	without Rot. PSF	without Inv. PSF	without TVL0	Full model
FWHM L1 (pixels)	27.76	27.65	27.54	27.30	27.27
FWHM L2 (pixels)	18.40	17.23	17.48	17.30	17.15
CNR (dB)	11.36	12.20	18.27	17.17	18.69
SNR (dB)	28.04	28.06	34.82	33.71	35.09

The rotational PSF module generates position-dependent PSFs that characterize the directional blur inherent to the ring-array geometry, which cannot be captured by a single spatially invariant PSF. Removing this module leads to residual directional spreading and reduced structural definition, with CNR and SNR decreasing from 26.78/35.54 dB to 22.91/32.89 dB for the finger image and from 18.69/35.09 dB to 12.20/28.06 dB for the vessel phantom. These results demonstrate its critical role in compensating spatially varying blur.

The spatially invariant PSF module generates a locally shared PSF that complements the rotational PSF by modeling the blur component common to neighboring regions. Removing it results in wider line-profile responses and moderate reductions in CNR and SNR, indicating its contribution to local blur compensation.

The hybrid TVL₀ regularization provides an explicit structural prior for the ill-posed blind restoration. Without it, the restored images exhibit less distinct boundaries and stronger background fluctuations, confirming its role in structural preservation and background stability.

Overall, the full model achieves the best performance across all metrics. The rotational PSF generates position-dependent PSFs for directional blur, the invariant PSF generates a locally shared PSF for common blur, and the hybrid TVL₀ regularization stabilizes structural recovery. Together, these components provide more complete degradation modeling and improve restoration robustness.

5.2. Quantitative evaluation with physical measurements

To further evaluate structural fidelity, vessel-width measurements are additionally performed using a vessel phantom. The physical vessel widths are measured at nine representative cross-sections, and the corresponding measurement locations are marked on the optical photograph, which is presented as the ground truth (GT) image, as shown in Fig. 16(a). The physically measured widths are recorded as the corresponding GT values for quantitative comparison. At the same cross-sectional locations, profiles approximately perpendicular to the local vessel directions are selected in the input PAT image and the images restored by different methods. For each cross-section, the two boundary points on opposite sides of the vessel are marked, and the Euclidean distance between them is calculated as the vessel width. The measured pixel distances in the PAT images are then converted to physical dimensions using the reconstruction scale of 50 μm/pixel and compared with the corresponding GT values. The same measurement locations and procedure are applied to the input image and all restored images to ensure a consistent comparison.

Fig. 16.

Comparison of vessel-width measurements for the vessel phantom. (a) Optical photograph of the vessel phantom used as the ground truth (GT). (b) Original PAT image. (c–g) Restoration results obtained using MNBSVD, BSRD, FRSP, CSINR, and the proposed method, respectively. Nine representative vessel cross-sections used for quantitative width measurements are marked in each image. The lower three rows show enlarged views of the representative vessel cross-sections D1, D2, and D5, respectively.

Fig. 16 shows the vessel-width measurements at the nine evaluated cross-sections, while Table 9 presents the corresponding quantitative widths and relative errors. Compared with the GT, the input PAT image consistently overestimates the vessel widths, with relative errors ranging from 7.06% to 70.29%. The proposed method substantially reduces this overestimation, with relative errors ranging from 3.17% to 37.66%, and achieves the lowest relative error among all compared restoration methods at seven of the nine evaluated cross-sections. For cross-Sections 3–9, the relative errors of the proposed method are all below 10%.

Table 9.

Quantitative comparison of vessel widths and corresponding relative errors. (Values in parentheses indicate the percentage relative error with respect to the GT vessel width.).

Index	GT (μm)	Input (μm, %)	MNBSVD (μm, %)	BSRD (μm, %)	FRSP (μm, %)	CSINR (μm, %)	Proposed (μm, %)
1	478	814 (70.29%)	687 (43.72%)	741 (55.02%)	810 (69.46%)	712 (48.95%)	658 (37.66%)
2	644	881 (36.80%)	761 (18.17%)	723 (12.27%)	797 (23.76%)	821 (27.48%)	737 (14.44%)
3	775	900 (16.13%)	868 (12.00%)	890 (14.84%)	854 (10.19%)	938 (21.03%)	850 (9.68%)
4	649	695 (7.09%)	689 (6.16%)	697 (7.40%)	706 (8.78%)	702 (8.17%)	695 (7.09%)
5	697	851 (22.09%)	772 (10.76%)	759 (8.90%)	844 (21.09%)	837 (20.09%)	752 (7.89%)
6	652	750 (15.03%)	791 (21.32%)	757 (16.10%)	768 (17.79%)	726 (11.35%)	701 (7.52%)
7	683	762 (11.57%)	760 (11.27%)	746 (9.22%)	762 (11.57%)	750 (9.81%)	743 (8.78%)
8	585	671 (14.70%)	638 (9.06%)	636 (8.72%)	674 (15.21%)	630 (7.69%)	626 (7.01%)
9	694	743 (7.06%)	729 (5.04%)	735 (5.91%)	734 (5.76%)	770 (10.95%)	716 (3.17%)

Notably, although the proposed method reduces the overestimated vessel widths in the input PAT image, the restored widths remain larger than the corresponding GT values at all nine cross-sections, indicating no excessive narrowing of the vessels. As shown in Fig. 16, the restored vascular structures also preserve their overall morphology without noticeable distortion. These results demonstrate improved vessel-width accuracy while avoiding excessive sharpening.

5.3. Evaluation of optimization stability

To evaluate the optimization stability of the proposed DP-based restoration framework, we analyze the convergence behavior under different initial learning rates and examine the effectiveness of the early-stopping criterion.

During DP optimization, the network first recovers dominant image structures and progressively fits noise-like components in later iterations, making early stopping necessary to avoid overfitting. The windowed output-variance criterion [64] is adopted to monitor the temporal consistency of consecutive restoration outputs. Specifically, this criterion calculates the variance of intermediate restored images within a sliding window of length W. A lower variance indicates greater consistency among consecutive outputs, while a subsequent increase signals the onset of noise overfitting. The stopping iteration is determined when no lower variance is observed within a patience interval of P iterations, with early-stopping detection initiated from t₀ to avoid premature termination. The optimization terminates at t_stop, and the restoration result at t_choose=t_stop - P is taken as the output of the early-stopping criterion, where P is the patience interval. In this study, the sliding-window length, detection starting iteration, and patience interval are set to W= 200, t₀= 1000, and P = 200, respectively.

For each learning rate, 20 repeated experiments are conducted with different random seeds, providing diverse initialization configurations and enabling systematic assessment of the proposed method's robustness to initialization randomness. The optimization runs for 4000 iterations, and the early-stopping criterion is applied to mark the recommended stopping iteration for each repeated experiment. The early-stopping iteration t_stop is determined independently for each repeated experiment, and the corresponding t_choose = t_stop - P is the iteration used to calculate CNR and SNR.

As shown in Fig. 17, both the windowed output variance and total loss decrease rapidly and exhibit noticeable transient fluctuations during the early optimization stage, reflecting the rapid restoration of the dominant structural components. The curves subsequently decrease more gradually and approach relatively stable ranges. Larger learning rates produce more pronounced fluctuations, whereas smaller learning rates result in smoother but slower optimization trajectories. Despite these differences, all tested learning rates eventually reach relatively stable loss and output-variance levels.

Fig. 17.

Convergence behavior of the proposed method under different learning rates (a) Windowed output variance. (b) Total loss. The solid curves and shaded regions represent the mean values and corresponding standard deviations over repeated experiments, respectively. The color-matched vertical lines indicate the mean termination iteration t_stop of the 20 repeated experiments for each learning rate, respectively.

Table 10 reports the mean values and standard deviations (SD) of CNR, SNR, and t_stop across the 20 experiments for each learning rate. For all tested learning rates, the restoration results selected by the early-stopping criterion exhibit higher mean CNR and SNR than the original PAT input. Larger learning rates such as 0.02 produce more pronounced optimization fluctuations, whereas excessively small learning rates such as 0.0002 lead to slower convergence and lower restoration performance. However, the quantitative results under learning rates between 0.0006 and 0.01 are relatively close, indicating a certain degree of robustness to learning rate. Considering both restoration performance and optimization stability, an initial learning rate of 0.002 is adopted in the subsequent experiments.

Table 10.

Quantitative results and early-stopping statistics under different initial learning rates (Mean and SD denote the mean and standard deviation computed from 20 independent experiments).

Metric	Input	0.02	0.01	0.005	0.002	0.001	0.0006	0.0004	0.0002
Mean CNR (dB)	23.86	26.41	26.59	26.57	26.74	26.68	26.56	25.87	25.45
CNR SD (dB)	-	0.64	0.58	0.53	0.48	0.47	0.48	0.46	0.42
Mean SNR (dB)	33.21	35.15	35.36	35.23	35.44	35.48	35.36	35.17	34.64
SNR SD (dB)	-	0.62	0.55	0.52	0.46	0.48	0.46	0.45	0.43
Mean stop iteration	-	1439	1498	1456	1569	1593	1670	1621	1827
Stop iteration SD	-	117	136	142	148	175	262	212	354

To further evaluate the early-stopping behavior at the selected learning rate, an iteration-sensitivity analysis is performed at LR= 0.002 (LR denotes the learning rate) based on the 20 repeated experiments. As shown in Table 11, the quantitative results obtained at t_choose and t_stop, which are determined independently in each experiment, are compared with those obtained at fixed iterations of 200, 500, 1000, 1250, 1500, 1750, 2000, 3000, and 4000. For each setting, the reported CNR and SNR values represent the means over 20 repeated experiments. The quantitative results remain relatively stable and consistently high from approximately 1250–2000 iterations.

Table 11.

Quantitative results for early stopping and fixed iterations at LR= 0.002 (Mean and SD denote the mean and standard deviation computed from 20 independent experiments).

Metric	Input	200	500	1000	1250	1500	1750	2000	3000	4000	tchoose	tstop
Mean CNR (dB)	23.86	24.68	25.66	26.28	26.69	26.77	26.68	26.61	26.23	25.14	26.74	26.71
CNR SD (dB)	-	1.43	0.86	0.67	0.52	0.49	0.49	0.51	0.54	0.59	0.48	0.47
Mean SNR (dB)	33.21	33.42	33.68	34.85	35.39	35.48	35.40	35.31	34.82	34.08	35.45	35.43
SNR SD (dB)	-	1.22	0.78	0.61	0.49	0.48	0.49	0.50	0.53	0.57	0.46	0.47

Both the mean selected iteration ${\overline{t}}_{choose}=1369$ and the mean termination iteration ${\overline{t}}_{stop}=1569$fall within this interval, with CNR and SNR values close to the best observed results. With further optimization beyond this range, both metrics gradually decline, indicating progressive fitting of noise-like components and increased local intensity fluctuations. These results validate that the adopted early-stopping criterion, based on windowed output variance, effectively identifies a stable high-performance stage while mitigating the risk of overfitting.

5.4. Evaluation of parameter sensitivity

To evaluate the sensitivity of the proposed framework to the major parameters, the in vivo human finger PAT image shown in Fig. 11(a) is used for evaluation. The parameter settings used in the sensitivity evaluation are summarized in Table 12 and Table 13. Index 1 represents the default configuration, with M= 100, Δθ= 0.03°, h size = 3 × 3, λ= 1 × 10⁻⁴, α= 30, and β= 20. For all other settings, the “Parameter setting” column specifies the parameters varied relative to the default configuration, while the remaining parameters are kept unchanged. For each setting, 20 independent runs with different random initializations are performed, and the mean CNR and SNR values are reported. The same early-stopping strategy described in Section 5.3 is used for quantitative evaluation. For computational cost comparison, the computation time is evaluated separately with the number of optimization iterations fixed at 2000 for all settings.

Table 12.

Sensitivity analysis of rotational degradation modeling parameters.

Index	Parameter setting	CNR (dB)	SNR (dB)	Time (s)
1	Default: M= 100, Δθ= 0.03°	26.61	35.35	279.52
2	Δθ= 0.01°	24.14	34.03	279.49
3	Δθ= 0.02°	25.75	34.61	279.33
4	Δθ= 0.04°	25.97	35.07	279.68
5	Δθ= 0.05°	25.32	34.59	279.55
6	M= 50	24.49	33.36	218.96
7	M= 75	25.73	34.87	248.77
8	M= 150	26.04	35.18	344.66
9	M= 200	25.23	34.74	412.37
10	M= 50, Δθ= 0.06°	25.11	34.86	218.67
11	M= 75, Δθ= 0.04°	25.87	35.03	247.65
12	M= 150, Δθ= 0.02°	26.71	35.42	344.88
13	M= 200, Δθ= 0.015°	26.69	35.56	409.28

Table 13.

Sensitivity analysis of restoration and regularization parameters.

Index	Parameter setting	CNR (dB)	SNR (dB)	Time (s)
1	Default: h size = 3 × 3, λ= 10⁻⁴, α= 30, β= 20	26.61	35.35	279.52
2	h size = 1 × 1	25.31	35.14	277.81
3	h size = 5 × 5	24.95	34.26	278.53
4	h size = 7 × 7	21.93	26.32	278.21
5	h size = 9 × 9	19.88	23.60	280.40
6	λ= 0	25.17	34.63	277.87
7	λ= 10⁻⁵	26.51	35.26	278.45
8	λ= 10⁻³	26.23	35.05	278.67
9	λ= 10⁻²	25.34	34.27	280.51
10	α= 1	25.87	34.61	279.15
11	α= 10	26.45	35.12	279.97
12	α= 100	26.32	35.01	279.07
13	β= 1	26.09	34.75	278.53
14	β= 10	26.57	35.49	278.83
15	β= 100	26.38	35.16	278.69

Table 12 summarizes the sensitivity to the rotational degradation parameters. With M= 100, variations in Δθ result in moderate changes in CNR and SNR, with Δθ= 0.03° providing favorable restoration performance. Increasing M generally increases the computational cost without consistently improving the restoration metrics. When the overall angular range is maintained at approximately 3º, different combinations of M and Δθ produce relatively similar restoration performance, although larger M requires substantially more computation. Therefore, M= 100 and Δθ= 0.03° are adopted as a practical balance between degradation representation and computational efficiency.

Table 13 further evaluates the spatially invariant PSF size and the parameters λ, α, and β. The 3 × 3 PSF kernel provides better restoration performance than either the 1 × 1 kernel or larger kernels, while moderate Gaussian perturbation improves the restoration results compared with no perturbation or excessive perturbation. The results also show that the proposed framework remains relatively stable over different TV and L₀ regularization settings.

Overall, the sensitivity analyses indicate that the proposed method remains relatively robust to moderate variations in the major parameters. The default settings are adopted considering degradation representation capability, restoration performance, optimization stability, and computational efficiency.

5.5. Evaluation of noise robustness

To evaluate the robustness of the proposed framework to input noise, Gaussian noise with different intensities is added to the original in vivo human finger PAT image. The noise level is defined on a normalized [0,1] intensity scale and then rescaled to the original image intensity range. A fixed random seed of 0 is used for noise generation and network initialization. The noisy input image is generated as:

$${\mathbf{y}}_{\sigma }=\mathbf{y}+{\mathbf{n}}_{\sigma },$$ (22)

where y denotes the normalized blurred input image, ${\mathbf{n}}_{\sigma }~N(\mathbf{0},{\sigma }^2\mathbf{I})$denotes rescaled Gaussian noise, and σ is the noise standard deviation in the normalized intensity scale. The noise levels are set to σ = 0, 0.01, 0.02, 0.04, 0.06, 0.08 and 0.10.

As shown in Fig. 18, increasing the noise level progressively degrades the input image. Stronger noise introduces more background fluctuations and reduces target-background separability, leading to lower CNR and SNR values. After restoration, the main structural features remain more distinguishable than those in the corresponding noisy inputs, although the visual quality gradually decreases as the noise level increases.

Fig. 18.

Noise robustness evaluation under different Gaussian noise levels. (a-g) Noisy input PAT images and corresponding enlarged views at σ = 0, 0.01, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. (h-n) corresponding restoration results obtained using the proposed method and their enlarged views. The red rectangles indicate the enlarged regions, while the yellow and blue arrows indicate representative structural features. The pink and green boxes in (a) indicate the five background and signal ROIs used for CNR and SNR calculations, respectively.

The quantitative results in Table 14 are consistent with the visual observations. As σ increases, the CNR and SNR values of both the noisy input and the restored result decrease. Nevertheless, the restored images consistently achieve higher CNR and SNR than their noisy inputs at all tested noise levels. These results indicate that the proposed method maintains its ability to improve target-background separability and structural visibility under noisy input conditions. Under severe noise degradation, however, the recovery of weak structures and fine details remains limited, resulting in reduced overall restoration performance.

Table 14.

Quantitative results under different Gaussian noise levels.

Noise level σ	PAT CNR (dB)	Restoration CNR (dB)	PAT SNR (dB)	Restoration SNR (dB)
0	23.86	26.29	33.21	35.08
0.01	21.79	25.61	31.11	34.26
0.02	18.62	24.38	27.95	33.21
0.04	13.74	21.04	23.10	29.87
0.06	10.48	17.99	19.85	26.92
0.08	8.04	14.85	17.44	23.88
0.10	6.12	9.45	15.55	18.30

Overall, the added convergence curves, repeated run statistics, early stopping sensitivity analysis, and noise-robustness evaluation demonstrate the stable optimization behavior and reproducible restoration performance of the proposed framework. The results further indicate that the proposed method improves target-background separability and preserves the main structural features under different noise intensities, although severe input noise limits the achievable recovery of weak structures and fine details.

5.6. Evaluation of sparse-view acquisition condition

The performance of the proposed method under sparse-view acquisition conditions is evaluated using a vessel phantom. PAT data is acquired using a 256-element ring-array transducer with 64, 128, and 256 active elements. Fig. 19 (a) and Fig. 19 (i) display the same photograph of the vessel phantom for reference. Fig. 19 (b-d) display the original input PAT images reconstructed with 64, 128, and 256 active elements, respectively, and Fig. 19 (j-l) display the corresponding restoration results obtained using the proposed method. Fig. 20 presents the corresponding line profile analyses. As shown in these figures and quantified in Table 15, reducing the number of active elements progressively degrades the reconstructed images, introducing stronger background fluctuations, more severe sparse-view artifacts, and increased structural discontinuities. After restoration, the images exhibit improved vessel visibility, better target-background separability, reduced background fluctuations, and higher CNR and SNR than the corresponding inputs. However, restoration remains challenging under highly sparse conditions. With only 64 active elements, the proposed method improves the visibility of main vascular structures, but fine-detail recovery remains limited due to severe artifacts and background disturbances. Future work will focus on improving degradation modeling and constraint strategies.

Fig. 19.

Comparison of the vessel phantom using different numbers of active transducer elements before and after restoration. (a) and (i) Photographs of the vessel phantom. (b-d) the original input PAT images reconstructed with 64, 128, and 256 active elements, respectively. (j-l) Restoration results obtained using the proposed method, corresponding to 64, 128, and 256 active elements, respectively. (e-h) and (m-p) enlarged views of the regions marked by red rectangles in (a-d) and (i-l), respectively.

Fig. 20.

Line profile comparisons of the input and restored vessel phantom images using different numbers of active transducer elements. (a) and (b) L1 and L2 line profiles for 64 active elements, respectively. (c) and (d) L1 and L2 line profiles for 128 active elements, respectively. (e) and (f) L1 and L2 line profiles for 256 active elements, respectively. The gray and red curves represent the input and restored images, respectively. The locations of L1 and L2 are indicated by the yellow and blue lines in Fig. 19 (f), respectively.

Table 15.

Quantitative results under different numbers of active transducer elements.

Element number	PAT CNR (dB)	Restoration CNR (dB)	PAT SNR (dB)	Restoration SNR (dB)
64	1.62	7.99	17.68	22.77
128	5.67	13.91	22.18	29.27
256	11.36	18.69	28.04	35.09

5.7. Computational efficiency comparison

Besides image quality, computational efficiency is also a crucial metric for evaluating the performance of restoration algorithms. Therefore, we conduct experiments to compare the running time of the five methods, using 512 × 512 and 1024 × 1024 PAT images for testing. All methods are run on a workstation equipped with two 10-core 20-thread CPUs with a base clock speed of 2.4 GHz, and a NVIDIA GeForce RTX 4090 GPU with 24 GB of memory.

The time required for image preprocessing to remove streak artifacts is not included, as it remains the same for all five methods: 51.80 s for 512 × 512 images and 108.79 s for 1024 × 1024 images. The running time is listed in Table 16, with the shortest time highlighted in bold.

Table 16.

Running time of different methods.

Image size	MNBSVD	BSRD	FRSP	CSINR	Proposed
512 × 512	326.15 s	748.59 s	74.33 s	321.66 s	279.71 s
1024 × 1024	1337.80 s	1124.29 s	442.01 s	889.18 s	378.20 s

The runtime differences mainly result from the computational structures of the compared methods. MNBSVD requires numerous pixel-wise convolutions using spatially varying PSFs, while BSRD repeatedly updates the degradation model and latent image through alternating minimization. FRSP achieves the shortest runtime among all methods for 512 × 512 images with 74.33 s, but its runtime increases substantially to 442.01 s for 1024 × 1024 images due to repeated FFT-based deconvolution and kernel estimation over the entire image domain. CSINR jointly optimizes coordinate-based representations of the latent image and blur kernel through cross-scale self-supervision, also resulting in increased computational cost for larger images.

The proposed method simultaneously updates the image restoration network, rotational PSF network, and spatially invariant PSF network through GPU-accelerated backpropagation. The outputs of the three networks are combined in the forward degradation model, enabling joint estimation without alternating subproblem optimization and allowing more efficient use of GPU parallelism. Although FRSP is faster for the 512 × 512 image, the proposed method achieves the shortest runtime for the 1024 × 1024 image, demonstrating its computational advantage for high-resolution PAT images.

6. Discussions

As demonstrated by the experimental results in this work, the proposed method can effectively restore images reconstructed by a ring-array PAT system. It successfully removes spatially rotational blur caused by the finite aperture of the transducers, as well as spatially invariant blur induced by the limited detection bandwidth, thereby significantly improving image resolution. However, several limitations still remain in practical use.

First, as observed from the results in Fig. 5 and Fig. 7, the proposed method achieves satisfactory restoration performance for images located within the imaging plane of the ring transducer array. In contrast, the restoration performance degrades for regions deviating from this plane. This limitation arises from the image degradation model proposed in this work. As indicated by its derivation, the model is two-dimensional and is only applicable to the degradation process within the imaging plane of the ring transducer array. For out-of-plane regions, the degradation is no longer strictly characterized by the degradation model, and thus cannot be effectively handled by the proposed method.

Second, as shown in the restored images in Fig. 9 and Fig. 11, the proposed method performs less effectively in high-intensity regions, where brightness saturation and spatial expansion artifacts may occur. The phenomenon is closely related to a well-known challenge in inverse problems such as deconvolution and tomographic reconstruction, where strong or high-intensity structures are prone to over-enhancement, ringing, or spatial spreading during inversion. This effect is typically caused by the ill-posed nature of the inverse problem and the amplification of high-frequency components and noise during reconstruction, where bright or high-contrast regions often exhibit overshoot or halo artifacts after inversion [[67], [68], [69], [70]]. We agree that handling saturation and over-enhancement in high-intensity regions is an important and challenging problem. However, this typically requires explicit modeling of nonlinearity, intensity constraints, or specialized regularization, which is beyond the primary scope of the present work.

Third, the noise-robustness experiment in Section 5.5 demonstrates that the proposed method maintains effective restoration performance under different input SNR levels. However, severe noise still degrades performance and hinders fine-detail recovery. Similarly, the sparse-view experiments in Section 5.6 demonstrate that the proposed method can improve visibility and reduce background fluctuations under sparse acquisition conditions. Nevertheless, restoration remains challenging under highly sparse settings, such as with only 64 active elements, where fine vascular details are difficult to recover due to severe artifacts and background disturbances. In addition, although the proposed method can partially mitigate reflection-related artifacts commonly present in PAT images, severe reflections are not explicitly modeled in the current formulation and require further investigation. Addressing these limitations through enhanced degradation modeling and adaptive restoration strategies will be an important direction for future work.

Finally, the proposed method is still an iterative reconstruction approach with relatively high computational cost. When applied to larger images, the computational burden may increase significantly, which could limit its practical applicability. Therefore, developing lightweight network architectures and improving computational efficiency will be important directions for future work.

To address the above limitations, we will conduct further research to improve the performance and broaden the applicability of the proposed method.

7. Conclusions

In this paper, we analyze the physics of image blur degradation caused by the finite bandwidth and aperture of the transducer in ring-array PAT system. We propose a mathematical model that incorporates both spatially invariant and spatially rotational convolutions to represent the degradation process. To restore the degraded PAT image, we extend the basic DP framework, an unsupervised learning method, in two significant ways: First, by formulating a blind regularized restoration problem based on the derived degradation model. Second, by designing three deep neural networks tailored to generate the clear image, spatially invariant PSF, and spatially rotational PSF, respectively. These parameters of the networks are updated simultaneously using the error backpropagation scheme. The experiments on phantoms and in vivo biological tissues demonstrate that the proposed method effectively improves restoration quality while maintaining favorable computational efficiency, particularly for high-resolution PAT images.

CRediT authorship contribution statement

Wende Dong: Writing – review & editing, Supervision, Methodology, Conceptualization. Gang Luo: Writing – original draft, Software. Xiaoyu Han: Writing – original draft, Software. Chenlong Zhu: Data curation. Songde Liu: Supervision. Chao Tian: Supervision.

Ethics declaration

Written informed consent to take part in the study and to publish the article has been obtained from all participants or their legal representatives. The privacy rights of participants have been observed.

This study was performed in compliance with relevant laws, regulatory frameworks and guidelines where the research took place. This study was approved by the Institutional Review Board (IRB) of the First Affiliated Hospital of the University of Science and Technology of China. (Approval No. IRB: 2022-ky357)

This study was conducted in accordance with the following guidelines for animal welfare and/or reporting: National Institutes of Health (NIH) Guide for the Care and Use of Laboratory Animals. This study was approved by the Institutional Animal Care and Use Committee (IACUC) of the University of Science and Technology of China. (Approval No. USTCACUC1803065)

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Biographies

Wende Dong is an associate researcher in the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. He obtained his B.S. degree and PhD degree in Optical Engineering from Zhejiang University in 2008 and 2013, respectively. His research interests focus on image restoration, deep learning.

Gang Luo is a master student in the College of Automation Engineering, Nanjing University of Aeronautics and Astronautics. He obtained his B.S. degree in Beijing University of Chemical Technology in 2024. His research interests focus on image restoration and deep learning.

Xiaoyu Han is a graduate from Nanjing University of Aeronautics and Astronautics, majoring in electronic information. He obtained his B.S. degree in Measurement and Control Technology and Instruments from Nanjing University of Aeronautics and Astronautics in 2025. His research interests are on photoacoustic tomography and deep learning.

Chenlong Zhu is a doctoral student at the School of Engineering Science, University of Science and Technology of China. He obtained his master’s degree in Instrument Science and Technology from Nanjing University of Aeronautics and Astronautics in 2024, and B.S. degree in Electrical Engineering and Automation from China Jiliang University in 2021. His research interests focus on image restoration and deep learning.

Songde Liu is an associate research fellow at the School of Engineering Science, University of Science and Technology of China (USTC), Hefei, China. He received the B.S. degree in Measurement and Control Technology and Instrument and the Ph.D. degree in Instrument Science and Technology from USTC. His research interests focus on photoacoustic computed tomography, ultrasound computed tomography, and their biomedical applications.

Chao Tian is a professor at the School of Engineering Science, University of Science and Technology of China. He received the B.S. degree in Electrical Engineering and the PhD degree in Optical Engineering from Zhejiang University, Hangzhou, China. From 2013–2017, he worked as a Post-Doctoral Research Fellow at the University of Michigan, Ann Arbor. He is a Senior Member of Optica and a member of IEEE and SPIE. His research interests focus on biomedical imaging and image processing.

Data availability

Data will be made available on request.