A PET image reconstruction method and system based on manifold prior constraint

Abstract

This invention discloses a PET image reconstruction method and system based on manifold prior constraints, belonging to the field of medical image reconstruction and intelligent information processing technology. It acquires historical PET image datasets and PET measurement data from positron emission tomography (PET), constructs a data consistency term based on a Poisson statistical model, and introduces the image manifold prior into the PET image reconstruction optimization model. The optimization model is solved using an alternating iterative approach based on variable splitting, including data consistency updates, manifold projection updates, and dual variable updates. Finally, the reconstructed PET image is output. This invention organically combines the PET imaging physical model with data-driven prior generation, which is beneficial for improving reconstructed image quality, suppressing noise, and preserving structural details in low-dose PET imaging scenarios.

Description

Technical Field

[0001] This invention belongs to the field of medical image reconstruction and intelligent information processing technology, specifically relating to a PET image reconstruction method and system based on manifold prior constraints. Background Technology

[0002] Positron emission tomography (PET) is a powerful medical imaging technique that provides metabolic information about physiological processes of interest by depicting the spatial distribution of radioactive tracers. It has wide applications in neuroimaging, oncology imaging, and cardiovascular imaging. However, PET image reconstruction is inherently an ill-posed inverse Poisson problem, making it particularly difficult to obtain high-quality PET images, especially under low-dose conditions designed to reduce radiation exposure. Over the years, researchers have proposed numerous PET reconstruction methods to address this ill-posed inverse problem. Classical iterative reconstruction methods (such as maximum likelihood (ML) reconstruction) can model the underlying imaging physics and noise statistics, but they often converge slowly and can suffer from severe noise amplification if early stopping is not employed. Penalized maximum likelihood (PML) or Bayesian reconstruction methods using manual priors can improve signal-to-noise ratio (SNR), but often result in over-smoothing and loss of fine anatomical details.

[0003] In recent years, deep learning-based methods have become a powerful paradigm for PET image reconstruction. Supervised learning methods (such as end-to-end reconstruction and depth unfolding reconstruction) can achieve good results by training on paired measurement data and high-quality reference datasets. However, their dependence on large-scale labeled datasets often limits their generalization ability. In contrast, unsupervised learning reconstruction methods circumvent this limitation by utilizing the statistical structure of the image to be reconstructed. In particular, probabilistic generative models (such as score-based diffusion models) have shown significant effectiveness in learning prior distributions from training data, demonstrating potential for high-quality PET reconstruction. Nevertheless, these methods typically require multiple steps of random sampling, rely on complex mechanisms to ensure data fidelity, and may introduce artifacts during reconstruction, resulting in poor accuracy of the reconstruction results. Summary of the Invention

[0004] To overcome the problems of high noise, insufficient detail preservation, and limited prior modeling ability in existing PET image reconstruction under low count conditions, this invention provides a PET image reconstruction method and system based on manifold prior constraints.

[0005] This invention specifically provides the following technical solution: a PET image reconstruction method based on manifold prior constraints, comprising the following steps: Acquire historical PET image datasets, PET images to be reconstructed, and PET measurement data; The manifold prior for radioactive tracer activity images is learned from the PET historical image dataset, and a PET image reconstruction problem satisfying the Poisson statistical model is obtained based on the relationship between PET measurement data and the PET image to be reconstructed. The manifold prior is introduced as a constraint in the PET image reconstruction problem to obtain an optimization model with manifold prior constraints. A scaled Lagrange multiplier is introduced into the optimization model with manifold prior constraints to construct an augmented Lagrange function. The optimization model is solved using an alternating iterative approach based on variable splitting. The reconstructed PET image is obtained when the stopping condition is met. The iterative process is as follows: under the condition of fixed manifold variables and Lagrange multipliers, a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term is obtained. The image variables are updated one or more times to obtain updated image variables. The intermediate variable formed by combining the updated image variables and the Lagrange multipliers is projected onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints. The Lagrange multipliers are updated according to the constraint residuals between the updated image variables and the manifold variables, and the reconstructed PET image is finally output.

[0006] Preferably, the step of obtaining a PET image reconstruction problem that satisfies the Poisson statistical model based on the relationship between PET measurement data and the PET image to be reconstructed, and introducing a manifold prior as a constraint into the PET image reconstruction problem, yields an optimization model with manifold prior constraints, specifically: Based on the Poisson statistical model, the relationship between PET measurement data, the PET image to be reconstructed, and background events is obtained, and the relationship is used as the PET image reconstruction problem that satisfies the Poisson statistical model. In the PET image reconstruction problem, a manifold prior is introduced as a constraint. Under the manifold prior constraint, the PET image to be reconstructed is taken as the optimization object, and the objective function is the minimization of the combination of linear terms and the logarithmic terms related to PET measurement data. This yields an optimization model with manifold prior constraints.

[0007] Preferably, the optimization model is solved using an alternating iterative approach based on variable splitting, specifically as follows: By introducing Lagrange multipliers and penalty parameters Construct the augmented Lagrange function, the specific expression of which is: ; in, This represents the augmented Lagrange function. Indicates the dimension of the projected data. Represents the system matrix. For deterministic differentiable mappings, Indicates the first A background event, Denotes the square of the 2-norm. Indicates the first One PET measurement data; Based on the augmented Lagrangian function, by adjusting image variables Latent space variables The optimization model is solved by alternating updates of the Lagrange multipliers.

[0008] Preferably, the step of obtaining the subproblem containing the Poisson negative log-likelihood term and the augmented Lagrange quadratic penalty term under the condition of fixed manifold variables and Lagrange multipliers specifically involves: The image variable subproblem can be written as: ; Among them, parameters We construct an expectation-maximizing EM substitution function for the negative Poisson log-likelihood term and update it using inner iterations. This represents the optimized estimate of the subproblem in the current iteration. It is the Lagrange multiplier from the previous iteration.

[0009] Preferably, the step of projecting the intermediate variable formed by combining the updated image variable and the Lagrange multiplier onto the image manifold prior to obtain a manifold variable that satisfies the manifold prior constraint specifically involves: The updated image variables are combined with the Lagrange multipliers to form intermediate variables, which are then projected onto the image manifold prior. In solving the nonlinear optimization problem, manifold variables satisfying the manifold prior constraints are obtained. Specifically, the nonlinear optimization problem involves determining the relationship between the deterministically differentiable output and the current estimated value. Add standardized Lagrange multipliers When the square of the 2-norm is minimized, we obtain the first... The estimated value after one iteration.

[0010] Preferably, the deterministic differentiable mapping The flow matching generation model defines a deterministic differentiable mapping from latent space variables to PET images by solving ordinary differential equations.

[0011] Preferably, it also includes training the flow matching generation model, specifically: The velocity field in the flow matching generation model is trained using conditional flow matching loss; Using a linear probability path ,in, These are samples from the PET image training set. For samples from the basic distribution, As the final sample, and The difference is taken as the target difference value; After training The conditional flow matching loss function is constructed by the square of the 2-norm between the difference between the lower velocity field and the target, and the trained flow matching generation model is obtained by minimizing the conditional flow matching loss function.

[0012] This invention provides a PET image reconstruction system based on manifold prior constraints, comprising: The data acquisition module is used to acquire historical positron emission tomography (PET) image datasets, PET images to be reconstructed, and PET measurement data. The problem construction module is used to learn the manifold prior about the activity images of radioactive tracers from the PET historical image dataset, and obtain the PET image reconstruction problem that satisfies the Poisson statistical model based on the relationship between PET measurement data and the PET image to be reconstructed. The manifold prior is introduced as a constraint in the PET image reconstruction problem to obtain an optimization model with manifold prior constraints. The iterative reconstruction module introduces scaled Lagrange multipliers into the optimization model with manifold prior constraints, constructs an augmented Lagrange function, and solves the optimization model using an alternating iterative approach based on variable splitting. The reconstructed PET image is obtained when the stopping condition is met. Specifically, the iterative process involves: under fixed manifold variables and Lagrange multipliers, obtaining a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term; updating the image variables one or more times to obtain updated image variables; projecting the intermediate variable formed by combining the updated image variables and the Lagrange multipliers onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints; updating the Lagrange multipliers based on the constraint residuals between the updated image variables and the manifold variables; and finally outputting the reconstructed PET image.

[0013] The present invention provides a computer device, including a memory and a processor. The memory stores a program, and when the program is executed by the processor, the processor performs the steps of the above-described PET image reconstruction method based on manifold prior constraints.

[0014] The present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the steps of the above-described PET image reconstruction method based on manifold prior constraints.

[0015] Compared with the prior art, the present invention has the following significant advantages: This invention learns the manifold prior of radioactive tracer activity images from historical PET image datasets and introduces it as a constraint into the PET image reconstruction problem that satisfies the Poisson statistical model. By combining augmented Lagrangian function and variable splitting alternating iterative optimization, it can effectively suppress noise and artifacts in reconstructed images, especially significantly improving the image signal-to-noise ratio under low count statistics or high noise levels. At the same time, the introduction of manifold prior preserves the essential nonlinear structural features of the image, avoiding the problem of excessive smoothing leading to loss of edge details in traditional analytical or iterative methods. Thus, it obtains high-quality PET reconstructed images with clear edges and realistic textures, improving the sensitivity of lesion detection and the accuracy of quantitative analysis. Attached Figure Description

[0016] Figure 1 This is a comparison diagram of different reconstruction methods under low-dose (15%) conditions provided by the present invention; wherein, Figure 1 (a) is the real image. Figure 1 (b) shows the reconstructed images and corresponding error maps obtained using ML, TV, FBSEM, DPS and the proposed method; Figure 2 This is a comparison chart of reconstruction indicators provided by the present invention; wherein Figure 2 (a) represents the normalized root mean square error (NRMSE). Figure 2 (b) represents the peak signal-to-noise ratio (PSNR). Figure 2 (c) represents the structural similarity index SSIM; Figure 3 This is a comparison chart of different reconstruction methods provided by this invention on a real dataset at a 20% dose level; wherein, Figure 3 (a) is the real image. Figure 3 (b) shows the reconstructed images and corresponding error maps obtained using ML, TV, FBSEM, DPS and the proposed method; Figure 4 This is a flowchart of a PET image reconstruction method based on manifold prior constraints provided by the present invention. Detailed Implementation

[0017] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention. The present invention proposes a new PET reconstruction method that utilizes a modern generative model—flow matching. Flow matching can achieve deterministic sampling through an ordinary differential equation (ODE) solver, thereby avoiding the multi-step stochastic reverse process typically required by diffusion models. The proposed method is an optimization-based inference method: first, a flow matching generation prior is learned from a set of high-quality PET images, and then this prior is incorporated into the reconstruction model as a manifold constraint. To solve this optimization problem, a computationally efficient algorithm is developed based on the alternating direction multiplier method (ADMM), which alternates between two steps: strengthening data consistency through an expectation-maximization (EM) process and projecting the solution onto the learned image manifold prior. The effectiveness of the proposed method is verified by numerical simulation and real data experiments. This method utilizes a pre-trained flow matching generative model to establish a deterministic differentiable mapping from the latent space to the PET image space, and introduces the image manifold prior corresponding to this mapping as a prior constraint into the PET reconstruction problem. Furthermore, through data consistency update, manifold projection update, and dual variable update under an alternating iterative framework based on variable splitting, the PET reconstruction model with manifold constraints is iteratively solved to obtain the reconstructed image.

[0018] Problem Description: In PET, the goal is to obtain measurements from a set of sinograms. Reconstructing the spatial distribution of radioactive tracer uptake Given the randomness of the photon detection process, PET data acquisition can be described by the following Poisson statistical model:

[0019] ; in, This is the system matrix, used for modeling the imaging system; Let the expected background event be (e.g., scattering and random coincidence). Assuming that the number of photons collected on different sinusoidal bins is independent, the following likelihood function can be obtained:

[0020] in, , and Representing vectors respectively , , The One element; Represents non-negative integers The factorial of the factorial; based on the likelihood function derived from the statistical modeling of the imaging system and noise, maximum likelihood (ML) reconstruction can be performed. However, due to the ill-conditioned nature of the inverse problem, ML reconstruction often suffers from severe noise amplification, especially under low-dose conditions. To achieve high-quality reconstruction from noisy sine wave data, it is necessary to introduce a factorial of the factorial of the factorial of the noise. Prior information.

[0021] like Figure 4 As shown, this embodiment of a PET image reconstruction method based on manifold prior constraints includes: Step 1: Obtain the PET historical image dataset (training set), the PET images to be reconstructed, and the PET measurement data used for training the prior. The preferred PET measurement data is projection data acquired by a PET detector, and the flow matching generation model is trained based on the training set; or, a pre-trained flow matching generation model is obtained.

[0022] Step 2: Learn the manifold prior for radioactive tracer activity images from the PET historical image dataset, and obtain the PET image reconstruction problem that satisfies the Poisson statistical model based on the relationship between PET measurement data and the PET image to be reconstructed. Introduce the manifold prior as a constraint in the PET image reconstruction problem to obtain an optimization model with manifold prior constraints.

[0023] A deterministic differentiable mapping from latent variables to PET images is established based on the flow matching model, and the PET image manifold prior is defined accordingly. The image manifold prior is introduced as a constraint into the PET image reconstruction optimization model, resulting in the PET image reconstruction problem with manifold constraints.

[0024] In this work, a flow matching generative model is used to learn from a set of existing PET images. The prior distribution is then incorporated into the reconstruction. Specifically, flow matching learns a deterministic differentiable mapping. It is obtained by solving ordinary differential equations from the flow matching generation model to define a deterministic differentiable mapping from latent space variables to PET images. Specifically, it is obtained by sampling from simple basis distributions (such as Gaussian distributions). The following ordinary differential equation (ODE) is used to transform the image to conform to the target image distribution. of The specific expression is:

[0025] (3); in , Indicates the intermediate image state. Indicates that the parameter is The velocity field is represented by a neural network. This invention uses a conditional flow matching loss to train the velocity field in the flow matching generation model, wherein the velocity field is determined by minimizing the following conditional flow matching loss. The specific expression is:

[0026] (4); in, This represents the target velocity field, i.e., along the conditional probability path. The conditional velocity field of the designed flow; the expectation of the random variable in the equation. Simultaneously satisfy ,as well as This is obtained. Furthermore, conditional flow matching with linear probability paths is employed: ,in, These are samples from the PET image training set. For samples from the basic distribution, As the final sample, and The difference is taken as the target difference value. Because and , can be obtained ,and:

[0027] ; Therefore, with training The square of the 2-norm between the difference between the lower velocity field and the target constructs the conditional flow matching loss function, and equation (4) can be simplified to: ; The expectation is as well as Taken.

[0028] After training the neural network, from arrive The specific expression for solving ODE in equation (3) is as follows: ; Clearly, the above equation defines the potential noise vector. To reconstruct the image A definite and differentiable mapping, i.e. In practice, the ODE in equation (7) can be approximated using numerical methods, and the trained flow matching generation model can be obtained by minimizing the conditional flow matching loss function. Forward Euler method, where Therefore, the following iterative update is obtained: .

[0029] Based on the Poisson statistical model, the relationship between PET measurement data, the PET image to be reconstructed, and background events is obtained, and the relationship is used as the PET image reconstruction problem that satisfies the Poisson statistical model.

[0030] The aforementioned flow matching generation model effectively constrains the solution to a "reasonable PET image" manifold characterized by the training data. After training the neural network, a latent space vector derived from a Gaussian distribution is defined. To the image domain A deterministic and differentiable mapping is introduced into the PET image reconstruction problem, employing manifold prior constraints. Under these constraints, the PET image to be reconstructed is used as the optimization object, and the objective function is the minimization of the combination of linear terms and the logarithmic terms related to PET measurement data. This update the image reconstruction problem, resulting in an optimization model with manifold prior constraints, specifically:

[0031] ; (8); Step 3: Introduce scaled Lagrange multipliers to the optimization model with manifold prior constraints, construct an augmented Lagrange function, and solve the optimization model using an alternating iterative method based on variable splitting. The reconstructed PET image is obtained when the stopping condition is met. The iterative process is as follows: Data Consistency Update: Under the condition of fixed manifold variables and Lagrange multipliers, obtain a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term, and perform one or more iterative updates on the image variables to obtain the updated image variables; Manifold Projection Update: Project the intermediate variables formed by combining the updated image variables and Lagrange multipliers onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints; Lagrange Multiplier Update: Update the Lagrange multipliers according to the constraint residuals between the updated image variables and the manifold variables; Repeat the data consistency update step, the manifold projection update step, and the Lagrange multiplier update step until the stopping condition is met, and finally output the reconstructed PET image.

[0032] The optimization model is solved using an alternating iterative approach based on variable splitting (preferably ADMM). Scaled Lagrange multipliers are introduced into the image reconstruction problem to construct an augmented Lagrange function. The first and second sub-optimization problems in the augmented Lagrange function are solved alternately, and the scaled Lagrange multipliers are updated to iteratively reconstruct the PET image. The first sub-problem includes a Poisson data fidelity term, and the second sub-optimization problem is a nonlinear optimization problem.

[0033] The expression in equation (8) forms a constrained optimization problem. Here, we describe an ADMM-based algorithm to solve this problem. First, the augmented Lagrangian function corresponding to equation (8) is constructed as follows:

[0034] (9); in, For Lagrange multipliers, For penalty parameters, To augment the Lagrange function, For the system matrix, For deterministic differentiable mappings, For background noise, Here, M represents the projected data, and M represents the dimension of the projected data. Indicates the first A background event, Denotes the square of the 2-norm. For the inner product of the Lagrange multipliers, Indicates the first PET measurement data.

[0035] Based on the augmented Lagrangian function, by adjusting image variables Latent space variables The optimization model is solved by alternating updates of Lagrange multipliers.

[0036] Update the Lagrange multipliers: (10); Specifically, the alternating solution of the first and second sub-optimization problems is as follows: Step 31: In the first sub-optimization problem, the negative Poisson log-likelihood term is subjected to upper bounding of the EM substitution function for expectation maximization. The negative Poisson log-likelihood term after upper bounding is then used to... The update is performed, and based on the image voxel separability condition of the objective function during the update process, a closed-loop update is performed on the image voxels to obtain the update result.

[0037] Among them, under the condition of fixed manifold variables and Lagrange multipliers, the subproblem containing the Poisson negative log-likelihood term and the augmented Lagrange quadratic penalty term is obtained, specifically: Applying the ADMM algorithm, the image variable subproblem (the first sub-optimization problem) can be written as: (11); (12); Among them, parameters This subproblem is an ML reconstruction problem with a penalty term. Utilizing the properties of the Poisson log-likelihood function, it employs either the EM surrogate function or a majorization-minimization algorithm for efficient solution. Specifically, it constructs an EM surrogate function that maximizes the expectation of the negative Poisson log-likelihood term and updates it using inner iterations. The optimized estimate for the subproblem in the current iteration. It is the Lagrange multiplier from the previous iteration.

[0038] Introducing the inner EM index and initialize Constructing substitution functions The negative Poisson log-likelihood term is principalized by substitution function; the specific expression is: (13); (14); in, , It is a vector of all 1s. This indicates element-wise multiplication; division is also performed element-wise.

[0039] The objective function of equation (16) is convex and differentiable, and separable across individual image voxels. Therefore, it has the following closed-form update per voxel: [The text abruptly ends here, likely due to an incomplete translation or missing information.] To perform an update, the specific expression is:

[0040] (15); in, For the updated .

[0041] All operations are performed element-wise. After the EM layer iteration, we have:

[0042] (16); (17); Note that the above update guarantees that the solution naturally satisfies the nonnegativity constraint. Furthermore, to improve computational efficiency, an inexact ADMM strategy is employed: this subproblem can be solved with only a small number of EM-style iterations.

[0043] Step 32: In the second sub-optimization problem, by solving a nonlinear optimization problem, the noisy intermediate solution is projected onto the manifold constraints learned by the flow model to obtain the updated result with prior constraints.

[0044] The intermediate variables formed by combining the updated image variables with Lagrange multipliers are projected onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints, specifically: The ADMM algorithm is applied to project the updated image variables and the intermediate variables formed by combining Lagrange multipliers onto the image manifold prior. In solving the nonlinear optimization problem, manifold variables satisfying the manifold prior constraints are obtained. Specifically, the nonlinear optimization problem involves determining the relationship between the deterministically differentiable output and the current estimated value. Add standardized Lagrange multipliers When the square of the 2-norm is minimized, we obtain the first... The estimated value from the 1st iteration is used as the second sub-optimization problem, and its specific expression is: (18); The subproblem in equation (11) can be rewritten as: (19); in, This is the updated result for the second sub-optimization problem.

[0045] This is a differentiable nonlinear optimization problem, corresponding to projecting onto a learned image manifold prior. This invention uses an efficient quasi-Newton algorithm to solve it, namely the finite-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm; where the objective function is related to... The gradient is calculated through automatic differentiation. This invention also employs an inaccurate ADMM strategy, using only a small number of L-BFGS iterations to solve this subproblem.

[0046] To accelerate convergence, a hot-start strategy is adopted for both subproblems in equations (10) and (11), that is, the solution of the previous iteration is used as the initialization of the current iteration. In addition, since the proposed formulation is non-convex, a multi-start strategy may be helpful in practical applications.

[0047] Step 6: When the stopping condition is met, output the reconstructed PET image.

[0048] The following describes an experimental procedure: Dataset: In the numerical simulation, a realistic 3D brain phantom (voxel dimensions 128×128×60; isotropic voxel size 2mm) derived from the Zubal human phantom was used. Three spherical lesions (radius 1.5–2.5 voxels) were further added to this phantom, exhibiting higher radiotracer uptake. A compartmental model was used to model biological variation, with kinetic parameters sampled from a normal distribution (coefficient of variation CV = 30%). A spatially invariant Gaussian point spread function (PSF) with a full width at half maximum (FWHM) of 4.5mm was applied. Reference full-dose acquisition (including…) The true coincidence count was scaled to simulate a low-count scenario at 15%. An additional 30% of uniformly distributed random and scattering events were added to all simulations. For the deep learning-based approach, the EM algorithm was used to reconstruct the full dose data, generating 12,000 high-quality 2D slices for training. Testing was conducted on a separate set of 2,000 2D slices simulated using the same acquisition settings.

[0049] Clinical PET data were obtained from the MICCAI Ultra-Low Dose PET Imaging Challenge dataset. This dataset contains acquisition data from two whole-body PET / CT systems: United Imaging uEXPLORER and Siemens Biograph Vision Quadra. The uEXPLORER dataset contains 159 individuals with 18F-FDG; the Quadra dataset contains 115 individuals with 18F-FDG, each acquired statically for 360 seconds. Axial head slices were extracted from both datasets, resulting in 4,979 slices from uEXPLORER and 6,307 slices from Quadra (a total of 11,286 slices). 11,000 slices were used for training, and the remaining 286 slices were used for testing. In the experiments, the scanner geometry was modeled after Siemens Biograph mCT, and the sine curve contained 256 radial bins and 224 projection angles. Attenuation and detector sensitivity effects were incorporated into the model using pre-calculated correction factors for each response line.

[0050] Implementation details: The proposed method and all comparison methods were implemented on a Linux server using PyTorch 2.8.0 and Python 3.10. The server was equipped with two NVIDIA H100 NVL GPUs interconnected via NVLink (each with 96GB of HBM3 memory) and 1.5TB of DDR5 memory. The velocity field for flow matching was represented by a four-layer U-Net and trained using the Adam optimizer with an initial learning rate of 1×10⁻⁶. -4In the image reconstruction stage, ODE is numerically solved using a forward Euler method with 20 discrete steps. ADMM penalty parameters... Set to 100. In the proposed imprecise ADMM algorithm, the first subproblem undergoes 4 EM-class iterations, and the second subproblem undergoes 1 L-BFGS iteration. The proposed algorithm is initialized as follows: ,and These hyperparameters are selected empirically to ensure good reconstruction performance.

[0051] Method comparison: The proposed method is compared with the following methods: 1) ML reconstruction; 2) Total Variation (TV) reconstruction; 3) Depth Unfolded Reconstruction (FBSEM); 4) Score-based Diffusion Posteriori Sampling Reconstruction (DPS). To ensure a fair comparison, the score-based diffusion model is trained using the same U-Net architecture as the proposed method. During the inference phase, the total reconstruction time of the proposed method (26.58 s for 35 ADMM iterations) is made approximately equivalent to the time required for a single sampling iteration of DPS (27.80 s). All methods are evaluated using three quantitative metrics: Normalized Root Mean Square Error (NRMSE), Peak Signal-to-Noise Ratio (PSNR), and Structural Similarity Index (SSIM).

[0052] (2) Results: Figure 1 The reconstructed images and corresponding error maps obtained using ML, TV, FBSEM, DPS, and the proposed method under a 15% dose level simulation are shown. It can be seen that ML reconstruction is severely contaminated by noise and exhibits artifacts. Although TV is an improvement over ML, it leads to over-smoothing and loss of anatomical details. Supervised learning methods (i.e., FBSEM) and unsupervised learning methods (i.e., DPS) both achieve significant performance improvements compared to traditional methods. However, the proposed method still has significant advantages over these two deep learning methods. Notably, the proposed method and the diffusion method (i.e., DPS) use the same training dataset and U-Net architecture, and the training and inference times are comparable. Furthermore, as... Figure 2 As shown, the proposed ADMM–EM-based algorithm exhibits excellent convergence behavior. Table 1 summarizes the quantitative results, further confirming the above observations. From Figure 2 It is evident from this that, although the problem itself is non-convex, the proposed ADMM-EM-based algorithm, when using a given initialization scheme (i.e., assuming...), can achieve good results. ,and Even under these conditions, it still exhibits excellent convergence characteristics.

[0053] Table 1. Quantitative comparison of reconstruction methods based on simulation results at a 15% dose level The optimal result is marked in bold.

[0054] Figure 3 The reconstruction results are presented using real data under a 20% dose condition, and Table 2 provides the corresponding quantitative comparisons. Consistent with simulation results, the proposed method outperforms the traditional baseline method and two deep learning-based methods.

[0055] Table 2. Quantitative comparison of reconstruction methods based on real data at 20% dose level In summary, the present invention brings the following effects: 1. By using a flow matching generative model to learn the statistical structure of high-quality PET images and representing it as a deterministic differentiable mapping from the latent space to the image space, we can construct a manifold prior that conforms to the distribution of PET images and improve the prior representation ability.

[0056] 2. By unifying Poisson statistical PET data consistency and manifold prior constraints into the same optimization framework, data-driven priors can be introduced while maintaining physical consistency. This achieves synergistic optimization of data fidelity and anatomical priors, which is beneficial for improving the quality and detail retention of low-dose PET reconstruction.

[0057] 3. By employing an alternating iterative algorithm based on variable splitting, data consistency updates and manifold projection updates are processed separately, which facilitates solving complex constrained optimization problems and is expected to provide an efficient, robust, and interpretable advanced solution for clinical PET imaging.

[0058] 4. In a preferred embodiment, data consistency updates can be efficiently solved by combining EM-type substitution functions, and a single model can be flexibly adapted to multi-dose scenarios without the need for retraining for different noise levels.

[0059] Based on the above method, this invention proposes a PET image reconstruction system based on manifold prior constraints, including: a data acquisition module, a problem construction module, and an iterative reconstruction module.

[0060] The system comprises several modules: a data acquisition module for acquiring historical PET image datasets, PET images to be reconstructed, and PET measurement data; a problem construction module for learning manifold priors about radioactive tracer activity images from the historical PET image dataset and obtaining a PET image reconstruction problem satisfying a Poisson statistical model based on the relationship between PET measurement data and the PET images to be reconstructed, introducing manifold priors as constraints to obtain an optimization model with manifold prior constraints; and an iterative reconstruction module for introducing scaled Lagrange multipliers into the optimization model with manifold prior constraints, constructing an augmented Lagrange function, and employing a method based on... The optimization model is solved using an alternating iterative approach involving variable splitting. The reconstructed PET image is obtained when the stopping condition is met. The iterative process is as follows: under the condition of fixed manifold variables and Lagrange multipliers, a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term is obtained. The image variables are updated one or more times to obtain the updated image variables. The intermediate variables formed by combining the updated image variables and Lagrange multipliers are projected onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints. The Lagrange multipliers are updated according to the constraint residuals between the updated image variables and the manifold variables, and the reconstructed PET image is finally output.

[0061] The present invention also provides a computer device, including a memory and a processor, wherein the memory stores a program, and when the program is executed by the processor, the processor performs the steps of a PET image reconstruction method based on manifold prior constraints.

[0062] According to the disclosed embodiments, the computer device can communicate with one or more external devices (e.g., keyboard, pointing device, Bluetooth communication, etc.) or with any device that enables the computing device to communicate with one or more other computing devices (e.g., router, demodulator, etc.).

[0063] The present invention also provides a storage medium storing a computer program thereon, characterized in that the computer program, when executed by a processor, implements the steps of a PET image reconstruction method based on manifold prior constraints.

[0064] According to the disclosed embodiments, the storage medium can be a non-volatile computer-readable storage medium, such as, but not limited to: portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof. In this invention, the storage medium can be any tangible medium containing or storing a program that can be used by or in conjunction with an instruction execution system, apparatus, or device.

[0065] The above description, in conjunction with specific preferred embodiments, provides a more detailed explanation of the present invention. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such deductions or substitutions should be considered to fall within the scope of protection of the present invention.

Claims

1. A PET image reconstruction method based on manifold prior constraints, characterized in that, Includes the following steps: Acquire historical PET image datasets, PET images to be reconstructed, and PET measurement data; The manifold prior for radioactive tracer activity images is learned from the PET historical image dataset, and a PET image reconstruction problem satisfying the Poisson statistical model is obtained based on the relationship between PET measurement data and the PET image to be reconstructed. The manifold prior is introduced as a constraint in the PET image reconstruction problem to obtain an optimization model with manifold prior constraints. A scaled Lagrange multiplier is introduced into the optimization model with manifold prior constraints to construct an augmented Lagrange function. The optimization model is solved using an alternating iterative approach based on variable splitting. The reconstructed PET image is obtained when the stopping condition is met. The iterative process is as follows: under the condition of fixed manifold variables and Lagrange multipliers, a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term is obtained. The image variables are updated one or more times to obtain updated image variables. The intermediate variable formed by combining the updated image variables and the Lagrange multipliers is projected onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints. The Lagrange multipliers are updated according to the constraint residuals between the updated image variables and the manifold variables, and the reconstructed PET image is finally output.

2. The image reconstruction method as described in claim 1, characterized in that, The process involves obtaining a PET image reconstruction problem that satisfies a Poisson statistical model based on the relationship between PET measurement data and the PET image to be reconstructed. A manifold prior is introduced as a constraint into the PET image reconstruction problem, resulting in an optimization model with manifold prior constraints. Specifically: Based on the Poisson statistical model, the relationship between PET measurement data, the PET image to be reconstructed, and background events is obtained, and the relationship is used as the PET image reconstruction problem that satisfies the Poisson statistical model. In the PET image reconstruction problem, a manifold prior is introduced as a constraint. Under the manifold prior constraint, the PET image to be reconstructed is taken as the optimization object, and the objective function is the minimization of the combination of linear terms and the logarithmic terms related to PET measurement data. This yields an optimization model with manifold prior constraints.

3. The image reconstruction method as described in claim 1, characterized in that, The optimization model is solved using an alternating iterative approach based on variable splitting, specifically as follows: By introducing Lagrange multipliers and penalty parameters Construct the augmented Lagrange function, the specific expression of which is: ； in, This represents the augmented Lagrange function. Indicates the dimension of the projected data. Represents the system matrix. For deterministic differentiable mappings, Indicates the first A background event, Denotes the square of the 2-norm. Indicates the first One PET measurement data; Based on the augmented Lagrangian function, by adjusting image variables Latent space variables The optimization model is solved by alternating updates of the Lagrange multipliers.

4. The image reconstruction method as described in claim 3, characterized in that, The subproblem of obtaining the Poisson negative log-likelihood term and the augmented Lagrange quadratic penalty term under the condition of fixed manifold variables and Lagrange multipliers is as follows: The image variable subproblem can be written as: ； Among them, parameters We construct an expectation-maximizing EM substitution function for the negative Poisson log-likelihood term and update it using inner iterations. This represents the optimized estimate of the subproblem in the current iteration. It is the Lagrange multiplier from the previous iteration.

5. The image reconstruction method as described in claim 3, characterized in that, The intermediate variable formed by combining the updated image variable with the Lagrange multiplier is projected onto the image manifold prior to obtain a manifold variable that satisfies the manifold prior constraint, specifically as follows: The intermediate variables formed by combining the updated image variables with the Lagrange multipliers are projected onto the image manifold prior, and when solving the nonlinear optimization problem, manifold variables that satisfy the manifold prior constraints are obtained. The nonlinear optimization problem specifically involves finding the deterministic differentiable map output and the current estimate. Add standardized Lagrange multipliers When the square of the 2-norm is minimized, we obtain the first... The estimated value after one iteration.

6. The image reconstruction method as described in claim 1, characterized in that, The deterministic differentiable mapping The flow matching generation model defines a deterministic differentiable mapping from latent space variables to PET images by solving ordinary differential equations.

7. The image reconstruction method as described in claim 6, characterized in that, It also includes training the flow matching generation model, specifically: The velocity field in the flow matching generation model is trained using conditional flow matching loss; Using a linear probability path ,in, These are samples from the PET image training set. For samples from the basic distribution, As the final sample, and The difference is taken as the target difference value; After training The conditional flow matching loss function is constructed by the square of the 2-norm between the difference between the lower velocity field and the target, and the trained flow matching generation model is obtained by minimizing the conditional flow matching loss function.

8. A PET image reconstruction system based on manifold prior constraints, characterized in that, include: The data acquisition module is used to acquire historical positron emission tomography (PET) image datasets, PET images to be reconstructed, and PET measurement data. The problem construction module is used to learn the manifold prior about the activity images of radioactive tracers from the PET historical image dataset, and obtain the PET image reconstruction problem that satisfies the Poisson statistical model based on the relationship between PET measurement data and the PET image to be reconstructed. The manifold prior is introduced as a constraint in the PET image reconstruction problem to obtain an optimization model with manifold prior constraints. The iterative reconstruction module introduces scaled Lagrange multipliers into the optimization model with manifold prior constraints, constructs an augmented Lagrange function, and solves the optimization model using an alternating iterative approach based on variable splitting. The reconstructed PET image is obtained when the stopping condition is met. Specifically, the iterative process involves: under fixed manifold variables and Lagrange multipliers, obtaining a subproblem containing a Poisson negative log-likelihood term and an augmented Lagrange quadratic penalty term; updating the image variables one or more times to obtain updated image variables; projecting the intermediate variable formed by combining the updated image variables and the Lagrange multipliers onto the image manifold prior to obtain manifold variables that satisfy the manifold prior constraints; updating the Lagrange multipliers based on the constraint residuals between the updated image variables and the manifold variables; and finally outputting the reconstructed PET image.

9. A computer device, characterized in that, The device includes a memory and a processor, wherein the memory stores a computer program that, when executed by the processor, causes the processor to perform the PET image reconstruction method based on manifold prior constraints as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the PET image reconstruction method based on manifold prior constraints as described in any one of claims 1 to 7.

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