Unsupervised sparse CT image reconstruction method based on dual-frequency refinement diffusion prior

Abstract

The invention discloses an unsupervised sparse CT (Computed Tomography) image reconstruction method based on double-frequency refinement diffusion prior, which mainly comprises the following steps of: in a training stage, executing a forward noise adding process to obtain a diffusion model s theta; in the reconstruction stage, firstly, a pre-trained diffusion model s theta is adopted to optimize low-frequency information of an image, and an initialized CT image is obtained; and data consistency operation is applied to the reconstruction result of the diffusion model. And then, optimizing high-frequency information of the image by adopting a double-frequency refining module, and further improving the quality of the CT image. Through a double-frequency refining mechanism, a high-frequency prior image is embedded into an implicit neural network (INR) to learn implicit neural representation of a target image, so that the target of further optimizing high-frequency information of the image is achieved. And finally, combining a high-frequency optimization result with the low-frequency part to finally realize the purpose of double-frequency refinement. And the steps are repeatedly iterated until the circulation is finished, and a final reconstructed image is obtained. According to the method disclosed by the invention, the edges and details of the image can be better reserved while the artifacts of the CT reconstructed image are removed, and meanwhile, the calculation cost is saved.

Description

Technical Field

[0001] The present invention mainly relates to an unsupervised sparse CT image reconstruction method based on dual-frequency refined diffusion prior, belonging to the fields of medical artificial intelligence and computer technology. Background Art

[0002] X-ray computed tomography (CT), as a non-invasive imaging method, can achieve accurate reconstruction of the internal structure of the target object and is widely used in many fields such as industrial inspection, non-destructive testing, security inspection, clinical medical diagnosis and treatment. The health risks of ionizing radiation have prompted minimizing the CT scan dose to become the research focus. As an effective dose control strategy, sparse CT (Sparse-view CT, SVCT) reduces radiation by reducing the number of measurement projections. However, restricted by the Nyquist sampling theorem, traditional reconstruction methods face challenges under under-sampling conditions. For example, algorithms based on analytical reconstruction take the Radon transform and its inverse transform as the theoretical framework. The Radon transform converts the distribution of an object in the spatial domain into line integral data in the projection domain, and the analytical reconstruction algorithm recovers the original image from the projection data through the inverse Radon transform, but it will produce serious streak artifacts, such as the classic Filtered Back Projection (FBP) reconstruction method. Iterative reconstruction algorithms construct an objective function by combining the observed data and prior knowledge of CT images, and then obtain the reconstructed image through a regularization method. This process starts from an initial image estimate and, under the joint guidance of the projection measurement data and prior constraints, continuously adjusts the image estimate through iterative calculations to ensure that the loss between the simulated projection data and the actual measurement data gradually decreases, while satisfying the constraints of prior knowledge (such as smoothness, sparsity or edge characteristics, etc.) on the image characteristics, so as to obtain a high-quality reconstruction result. Nevertheless, common iterative methods have problems of slow convergence and poor stability. In addition, setting the regularization parameter is also very challenging. These limitations have promoted the development of new reconstruction algorithms to achieve a balance between low dose and high-quality reconstruction.

[0003] In recent years, the application of deep learning in the field of medical image imaging has been developing rapidly, and important progress has been made in sparse image reconstruction. Reconstruction methods based on deep learning can be divided into three categories: image-to-image reconstruction, data-to-image reconstruction, and single / double-domain iterative mapping reconstruction. Image-to-image reconstruction methods directly convert low-quality input images into high-quality reconstruction results through pre-trained models. Different from traditional reconstruction methods, this technology does not require access to the original measurement data, effectively reducing the dependence on the acquisition protocols of specific imaging devices. Currently, typical implementation schemes include various network architectures such as FBPConvNet, deconvolution network with dense connections (DD-Net), and residual encoder-decoder CNN (RED-CNN). However, it should be noted that such direct mapping methods usually lack explicit constraints on the consistency of measurement data, which may affect the physical accuracy of the reconstruction results. Data-to-image reconstruction methods directly establish the conversion relationship from the measurement domain to the image domain through deep neural network architectures, achieving the reconstruction from original acquisition data to high-quality medical images. Representative technical solutions include the iRadonMap system, which specifically designs a deep reconstruction network from projection data to tomographic images according to the physical characteristics of CT imaging. However, there are obvious engineering bottlenecks in this technical route: when dealing with high-dimensional measurement data generated by ultra-high-resolution or dynamic scans, the computational complexity of the network model will increase non-linearly, resulting in a sharp increase in GPU video memory occupancy and reconstruction latency. Deep reconstruction networks based on unfolded iterative optimization have made significant progress in recent years. These methods have achieved significant performance improvements by unfolding traditional iterative reconstruction algorithms into deep learning frameworks. For example, the Learning from Experts for Reconstruction Network (LEARN). These unfolded iterative reconstruction algorithms integrate the theoretical advantages of iterative optimization and the expressive power of deep learning to improve the reconstruction performance. Summary of the Invention

[0004] Object of the Invention: The present invention proposes an unsupervised sparse CT image reconstruction method based on a dual-frequency refined diffusion prior, which uses technologies such as generative deep learning models, wavelet transforms, and implicit neural networks to address the problems of low quality and unstable performance in current CT image reconstruction, restore more complete image edges and details, remove artifacts, and improve the quality of the reconstructed images.

[0005] To achieve the above object, the present invention discloses an unsupervised sparse CT image reconstruction method based on a dual-frequency refined diffusion prior, which realizes the refinement of image edges and details and reduces the phenomenon of artifacts. The implementation scheme includes the following:

[0006] Solving the reconstruction of SVCT using a diffusion model in the image domain, which can be mathematically expressed as:

[0007] Y = Ax + n (1)

[0008] Where Y represents the collected projection data, x represents the reconstructed image, and A is the sparse measurement matrix. To reconstruct high-quality CT images from sparse measurements, the reconstruction model proposed by the present invention is as follows:

[0009]

[0010] The objective function consists of three terms. The first term represents data fidelity, ensuring that the sparse measurements are consistent with the estimated measurements obtained through the undersampling mask; the second term is the implicit neural network consistency term, ensuring that the signal f learned through the implicit neural network φ transformed by the measurement matrix A can be as close as possible to the original sampled data Y; the third term represents the diffusion prior term; where the parameters η and λ help to maintain an appropriate balance between data fidelity and the regularization term.

[0011] To solve the above equation, x and φ can be updated separately by using an iterative strategy:.

[0012]

[0013] In the process of solving the x sub-problem of the above equation using the image domain diffusion model, we adopt the Split-Bregman strategy and introduce an intermediate variable u, then the solution of the x sub-problem can be expressed as follows:

[0014]

[0015] Where: σ(t)>0 is a function that monotonically increases with t, representing the noise scale at time step t.

[0016] To maintain consistency between variables in the subsequent text, we use to replace u. Then the x sub-problem in (3) can be solved by optimizing the following two separate sub-problems:

[0017]

[0018] For equation (5). We choose This is a choice that can improve the generation ability. The training process is as follows:

[0019]

[0020] Where σ(t)>0 is a function that monotonically increases with t, representing the noise scale at time step t.

[0021] The reconstruction process can be defined as the solution of the following reverse SDE: Where, represents the standard Wiener process, represents the data gradient direction at time t. Although unknown, it can be estimated through a neural network to optimize the following objective to train the neural network:

[0022]

[0023] Here, ζ(t) represents the weight parameter, and t is sampled from a continuous uniform distribution [0, T]. p t (x(t)|x(0)) represents the conditional distribution, and s θ is the score estimated by the network to approximate the gradient both When the model is sufficiently trained, s θ (x(t), t) can be calculated at any time t thus providing prior information for image reconstruction.

[0024] We adopt the Simultaneous Iterative Reconstruction Technique (SIRT) to obtain ensuring that the intermediate results are consistent with the initial sparse projection Y. The iterative process is outlined as follows:

[0025]

[0026] where, represents a diagonal matrix, and a q represents the q-th row vector of matrix A, and q is a hyperparameter for adjusting the step size.

[0027] The output result with data consistency imposed is decomposed into four subbands using the Discrete Wavelet Transform (DWT) This wavelet transform can be expressed as:

[0028]

[0029] where W(·) represents the wavelet transform operation. and represent the low-frequency component and high-frequency elements respectively. To simplify our expression, we introduce the shorthand notation

[0030] In the present invention, in order to further enhance the complex details of the reconstructed image, an Implicit Neural Representation (INR) network model is designed to optimize the high-frequency components of the image. In this model, we specifically select three particular high-frequency subbands, namely To refine the image edges and features in the high-frequency band. To achieve this, first, perform a zeroing operation on all elements of the low-frequency subband matrix \(x'-E(x')\). Here, \(E(\cdot)\) represents the extractor of high-frequency components in the wavelet domain, where \(E\) T \(E(x') = x'\). This process can be represented by the following equation:

[0031]

[0032] where \(W\) T (·) is the inverse wavelet transform operation, which converts data from the wavelet domain to the image domain.

[0033] Next, we process the sub-problem \(\varphi\) * . In implicit neural representation learning, an image is represented as a continuous function by a neural network. It can be defined as:

[0034]

[0035] where \(\varphi\) represents the trainable parameters (weights and biases) of the network, \(p=(x,y)\in\mathbb{R}\) 2 is any 2D spatial coordinate in the imaging plane, is the corresponding image intensity at position \(p\) in the image.

[0036] Since neural networks (especially MLPs) are prone to spectral bias when dealing with high-frequency signals or complex textures, that is, they tend to learn low-frequency information more and are difficult to express high-frequency patterns with rich details. Therefore, the input coordinate \(p\) is encoded by Fourier feature mapping \(\gamma(p)\) and used as the network input based on coordinates. Specifically, the encoding form is as follows:

[0037] \(\gamma(p)=[\cos(2\pi Bp),\sin(2\pi Bp)] T ,(10)

[0038] where the matrix \(B\) represents the coefficients of the Fourier transform, and its elements are sampled from the Gaussian distribution . The hyperparameter \(\sigma\) controls the standard deviation of this distribution and determines the scale of the feature transformation. After Fourier feature mapping, the network \(f\) φ uses the encoded coordinate \(\gamma(p)\) as the input, thereby enhancing its ability to express high-frequency information.

[0039] High-frequency prior embedding: Embed the processed high-frequency prior image into the coordinate-based MLP \(f\) φ to map the spatial coordinates to the corresponding intensity values in the prior image . That is, \(f\) φ : where \(i\) represents the coordinate index in the image spatial domain, and the image has a total of \(N\) pixels. The randomly initialized MLP is trained by optimizing the following objective function:

[0040]

[0041] Training the network: After optimization, the internal information of the high-frequency prior image is encoded into the MLP network to establish a feasible starting point for the implicit neural network. Since it is generated by unconditional random sampling. Therefore, we need to make the image generated by combining the CT imaging model \(A\) such that \(f\) φ approximate the measurement data \(Y\). Training is performed by minimizing the L2 norm loss, and this process can be expressed as follows:

[0042]

[0043] After training the network, the final reconstructed image Specifically, the reconstructed image is represented by the mapping relationship of all coordinate points .

[0044] Then, by applying the wavelet transform on the optimized high-frequency subband \(x''\) is obtained h .

[0045] Combining \(x' - E(x')\) and \(x''\) h through the inverse wavelet transform operation, the reconstruction result can be obtained:

[0046] \(x(t - 1)=W\) T \(\{(x' - E(x')), x''\) h \}, (13)

[0047] The beneficial effects achieved by the present invention: The unsupervised sparse CT image reconstruction method based on dual-frequency refined diffusion prior proposed by the present invention takes into account that the image reconstructed from the sparse sinogram has high-frequency and low-frequency artifacts. Among them, the low frequency represents the global structure and background, while the high frequency represents details and edges. The present invention uses the wavelet frame transform to decompose the image reconstructed by the diffusion model into low-frequency and high-frequency parts. Since the high-frequency structure is used to capture detail information in medical images and is crucial for accurate analysis, we use an implicit neural network to further refine the high-frequency components. Before inputting into the network, the image coordinates are embedded with Fourier features so that the network can better perceive high-frequency details.

[0048] We adopted an innovative optimization scheme in the SVCT reconstruction research, conducted a detailed analysis and extensive research on the regularization term, which ensured the practicality and usability value of our method. The data consistency term directly reflects the physical characteristics of the imaging system. The addition of the data term not only ensures that the reconstructed image is consistent with the original acquired data, but also makes the reconstruction result conform to physical laws, thereby improving the reliability and stability of the reconstruction. During the Step 3 of the reconstruction, we introduced data consistency to further correct the reconstruction result and reduce the cumulative error during the reconstruction process, thus enhancing the accuracy and credibility of the reconstructed image.

[0049] Throughout the process, we selected the image optimized by the wavelet decomposition diffusion model to save time cost and make up for the disadvantage that the diffusion model tends to generate smooth low-frequency structures in image reconstruction and is difficult to restore sharp edges and texture details. Through wavelet decomposition, we can process high-frequency information and enhance the detailed part of the image without affecting the global structure. At the same time, we adopted an image domain optimization strategy, that is, directly adjusting in the image domain instead of relying on the processing in the projection domain. Compared with the projection domain processing method, this method can more effectively retain the detailed information of the image during the reconstruction process and avoid additional errors (such as artifacts, noise amplification, etc.) that may be introduced during the reconstruction of projection data. This strategy fully retains the image details and provides a more efficient and reliable solution for sparse CT reconstruction.

[0050] In summary, through the organic combination of multiple strategies such as regularization analysis, data consistency correction, wavelet decomposition combined with the diffusion model, refinement of high frequencies by the implicit neural network, and image domain optimization, our method realizes efficient and high-quality sparse CT image reconstruction while ensuring physical rationality. This method not only improves the accuracy and detailed performance of the reconstruction, but also reduces the computational cost, making it more feasible in practical applications. Brief Description of the Drawings

[0051] Figure 1 It is the flowchart of the sparse CT reconstruction method of the technical solution of the present invention.

[0052] Figure 2 It is the structural diagram of the sparse CT reconstruction method of the present invention.

[0053] Figure 3 It is the algorithm diagram in the reconstruction stage of the present invention.

[0054] Figure 4 It is the reconstruction result (20view) of the present invention on the public test set. Specific Embodiment

[0055] The following further explains and illustrates the specific embodiment and effect of the present invention in conjunction with the drawings:

[0056] The reconstruction of solving SVCT using a diffusion model in the image domain can be mathematically expressed as:

[0057] Y = Ax + n (1)

[0058] Where Y represents the acquired projection data, x represents the reconstructed image, and A is the sparse measurement matrix. To reconstruct a high-quality CT image from sparse measurements, the reconstruction model proposed in the present invention is as follows:

[0059]

[0060] The objective function consists of three terms. The first term represents data fidelity, ensuring that the sparse measurement is consistent with the estimated measurement obtained through the undersampling mask; the second term is the implicit neural network consistency term, ensuring that the signal f learned through the implicit neural network φ transformed by the measurement matrix A can be as close as possible to the original sampled data Y; the third term represents the diffusion prior term; where the parameters η and λ help maintain an appropriate balance between data fidelity and the regularization term.

[0061] To solve the above equation, x and φ can be updated separately by using an iterative strategy:

[0062]

[0063] In the process of using the image domain diffusion model to solve the x sub-problem of the above equation, we adopt the Split-Bregman strategy and introduce an intermediate variable u, then the solution of the u sub-problem can be expressed as follows:

[0064]

[0065] Where: σ(t)>0 is a function that monotonically increases with t, representing the noise scale at time step t.

[0066] To maintain consistency between variables in the subsequent text, we use to replace u. Then the x sub-problem in (3) can be solved by optimizing the following two separate sub-problems:

[0067]

[0068] For formula (5). We choose This is a choice that can improve the generation ability. The training process is as follows:

[0069]

[0070] where σ(t)>0 is a function that monotonically increases with t, representing the noise scale at time step t.

[0071] The reconstruction process can be defined as the solution to the following reverse SDE: where, represents the standard Wiener process, represents the data gradient direction at time t. Although is unknown, it can be estimated by the neural network Therefore, we can optimize the following objective to train the neural network:

[0072]

[0073] Here, ζ(t) represents the weight parameter, and t is sampled from the continuous uniform distribution [0, T]. p t (x(t)|x(0)) represents the conditional distribution, s θ is the score estimated by the network to approximate the gradient both When the model is sufficiently trained, the θ at any time t can be calculated by s thus providing prior information for image reconstruction.

[0074] We adopt the Simultaneous Iterative Reconstruction Technique (SIRT) to obtain ensuring that the intermediate result is consistent with the initial sparse projection Y. The iterative process is outlined as follows:

[0075]

[0076] where, represents the diagonal matrix, and a q represents the q-th row vector of matrix A, where q is a hyperparameter for adjusting the step size.

[0077] The output result with data consistency imposed is decomposed into four sub-bands using the Discrete Wavelet Transform (DWT) This wavelet transform can be expressed as:

[0078]

[0079] where W(·) represents the wavelet transform operation. and They are respectively represented as low-frequency components and high-frequency elements. To simplify our expression, we introduce the shorthand notations

[0080] In the present invention, in order to further enhance the complex details of the reconstructed image, an implicit neural representation (INR) network model is designed to optimize the high-frequency components of the image. In this model, we specifically select three particular high-frequency subbands, namely to refine the image edges and features within the high-frequency band. To achieve this, first, the entire zeroing operation of the low-frequency subband matrix x′ - E(x′) is performed. Among them, E(·) represents the extractor of high-frequency components in the wavelet domain, where E T E(x′) = x′. This process can be represented by the following equation:

[0081]

[0082] where, W T (·) is the inverse wavelet transform operation, which transforms the data from the wavelet domain to the image domain.

[0083] Next, we deal with the sub-problem φ * . In implicit neural representation learning, an image is represented as a continuous function by a neural network. It can be defined as:

[0084]

[0085] where φ represents the trainable parameters (weights and biases) of the network, p = (x, y) ∈ R 2 is any 2D spatial coordinate in the imaging plane, is the corresponding image intensity at position p in the image.

[0086] Since neural networks (especially MLPs) are prone to spectral bias when dealing with high-frequency signals or complex textures, that is, they tend to learn low-frequency information more and are difficult to express high-frequency patterns with rich details. Therefore, the input coordinate p is encoded by Fourier feature mapping γ(p) and used as the coordinate-based network input. Specifically, the encoding form is as follows:

[0087] γ(p) = [cos2πBp sin2πBp] T ,(10)

[0088] where, the matrix B represents the coefficients of the Fourier transform, and its elements are sampled from the Gaussian distribution The hyperparameter σ controls the standard deviation of this distribution and determines the scale of the feature transformation. After Fourier feature mapping, the network f φ uses the encoded coordinate γ(p) as the input, thereby enhancing its ability to express high-frequency information.

[0089] High-frequency prior embedding: Embed the processed high-frequency prior image into the coordinate-based MLPf φ to map the spatial coordinates in the prior image to the corresponding intensity values. That is, f φ : where i represents the coordinate index in the image spatial domain, and the image has a total of N pixels. The randomly initialized MLP is trained by optimizing the following objective function:

[0090]

[0091] Training the network: After optimization, the internal information of the high-frequency prior image is encoded into the MLP network to establish a feasible starting point for the implicit neural network. Since it is generated by unconditional random sampling. Therefore, we need to make the image generated by f φ approximate the measured data Y by combining the CT imaging model A. The training is performed by minimizing the L2 norm loss, and this process can be expressed as follows:

[0092]

[0093] After training the network, the final reconstructed image Specifically, the reconstructed image is represented by the mapping relationship of all coordinate points .

[0094] Then, by applying the wavelet transform to to obtain the optimized high-frequency subband x″ h .

[0095] Combining x′ - E(x′) and x″ h through the inverse wavelet transform operation, the reconstruction result can be obtained:

[0096] x(t - 1) = W T {(x' - E(x')), x″ h}, (13)

[0097] As Figure 2 shown, continuously repeat the iterative optimization process until the step size t of the diffusion model is 0, and the loop ends. The output at this time is the high-quality CT image we need.

[0098] In this experiment, the peak signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) were used as quantitative evaluation indicators. We randomly selected 7 test sets from the publicly available dataset AAPM, and tested the average values of the indicators under 20 views as shown in Table 1. The experimental results show that the algorithm proposed in this paper has achieved significant performance improvement in the sparse CT image reconstruction task.

[0099] Table 1 Comparison table of evaluation criteria

[0100] Method PSNR (dB) SSIM (100%) FBP 10.312 20.003 Ours 36.651 93.259

[0101] Finally, it should be noted that the above description is only a specific example of the present invention and does not constitute any limitation to the present invention. Obviously, for professionals in this field, after understanding the content and principle of the present invention, various modifications and changes in form and details may be made without departing from the principle and structure of the present invention. However, these modifications and changes based on the idea of the present invention are still within the scope of protection of the claims of the present invention.

Claims

1. An unsupervised sparse CT image reconstruction method based on a dual-frequency refined diffusion prior, characterized in that It includes the following steps: Step 1: Training phase: Prepare the training dataset required for the diffusion model and perform the forward noise addition process. The L2 norm is used as the loss function to train the image-domain diffusion model s θ . Step 2: Start reconstruction: Sample the random variable x(T). Input the randomly sampled variable into the trained diffusion model s θ to obtain the model output result Step 3: Apply data consistency to the results output by the diffusion model to obtain Step 4: Decompose the result with data consistency applied into a low-frequency sub-band and a high-frequency sub-band by using the discrete wavelet technique. Reset the low-frequency part to a matrix of all zeros, and use the inverse wavelet technique to transform the reset low-frequency part and the high-frequency part from the wavelet domain to the image domain, denoted as . Step 5: High-frequency prior embedding: Fourier feature embedding is performed on the coordinates of . The high-frequency prior image after Fourier feature embedding is embedded into the implicit neural network. Step 6: Training the network: Based on the MLP with high-frequency prior embedding and the original measurement data Y, we train the network to learn the neural representation of the target image. The aim is to optimize the high-frequency part. Combine the high-frequency part optimized by the implicit neural network and the low-frequency part in the optimization result of the wavelet initial decomposition diffusion model, and use the inverse wavelet technique to transform it into the image domain to obtain the next input result x(t-1). Step 7: Repeat steps 2-6 until the loop ends to obtain the final reconstructed image.

2. A sparse CT image reconstruction method based on wavelet implicit neural network refinement for the diffusion model according to claim 1, characterized in that: The training phase includes the following steps: The diffusion model mainly consists of two processes: the forward process is to gradually transform the data into a Gaussian distribution to generate training samples. The reverse process is to train the neural network s θ (x(t), t) to learn denoising and optimize the following objective to train the neural network: Among them, ζ(t) represents the weight parameter to adjust the contribution of different t to the loss. In VE-SDE, ζ(t) = σ 2 (t), where σ(t)>0 is a monotonically increasing function of t, representing the noise scale at time step t. Secondly, x(T) ~ N(0, σ 2 (T)²I) in the reverse process indicates that the random variable x(T) follows a multivariate normal distribution with a mean of 0 and a variance of σ 2 (T)².

3. The specific implementation of step 2 in claim 1 is as follows: Use the trained score function s θ (x(t), t) to approximate the actual gradient And obtain the result optimized by the diffusion model, as shown below: Here, x(t) is the image before optimization, ∈ represents the added random noise, which increases the randomness of sampling and ensures the diversity of the generated data.

4. The specific implementation of step 3 in claim 1 is as follows: To ensure that the diffusion reconstruction result is consistent with the original projection data and make the finally generated image conform to physical meaning both in terms of overall structure and details, the introduction of data consistency is essential. As follows: Among them, Y is the projection data of the original sampling, represents a diagonal matrix, and q is a hyperparameter for adjusting the step size.

5. The specific implementation of step 4 in claim 1 is as follows: The image is decomposed into different frequency components using 2D discrete wavelet transform while retaining the time information. Haar wavelet is selected as the wavelet basis function in this method. The specific decomposition process includes filtering and downsampling of the signal. First, the image Using 2D DWT with a row-column decomposition method, first perform 1D DWT on each row to obtain low-frequency and high-frequency components. When performing wavelet transform on each row of the signal x[m, n], convolution operations are carried out using a low-pass filter g[n] and a high-pass filter h[n] and downsampling is performed: Among them, L(m,k) represents the low-frequency row component, H(m,k) represents the high-frequency row component, m is the row index, and k is the column index after downsampling. Then perform 1D DWT on L(m,k) and H(m,k) to obtain: where n is the column index and j is the row index after downsampling. The signal after performing the wavelet transform is denoted as: Set the low-frequency subband to a matrix of all zeros, and then use the inverse wavelet transform (W T ) to transform it into the image domain using the following formula:

6. The specific implementation of step 5 in claim 1 is as follows: First, perform Fourier feature embedding on the coordinate p of : γ(p) = [cos2πBpsin2πBp] T . Then, perform high-frequency prior embedding: Embed the processed high-frequency prior image into the coordinate-based MLP f φ to map the spatial coordinates to the corresponding intensity values in the prior image . That is, where i represents the coordinate index in the image spatial domain, and the image has a total of N pixels. The randomly initialized MLP is trained by optimizing the following objective function:

7. The specific implementation of step 6 in claim 1 is as follows: Training network: After optimization, the internal information of the high-frequency prior image is encoded into the MLP network to establish a feasible starting point for the implicit neural network. Since it is generated by unconditional random sampling, we need to approximate the measured data Y by combining the CT imaging model A so that the image generated by f φ is approximated to the measured data Y. Training is performed by minimizing the L2 norm loss, and this process can be expressed as follows: Finally, after training the network, the final reconstructed image Specifically, the reconstructed image is represented by the mapping relationships of all coordinate points The optimized decomposed low-frequency subbands of the combined diffusion model and the high-frequency subbands optimized by INR are combined, and then transformed to the image domain using the inverse wavelet transform to obtain the next input result. E is the high-frequency extractor, : x(t - 1) = W T {(x' - E(x')), x'' h} Repeat steps 2-step 6 until t = 0, end the loop, and output the final reconstruction result.

8. A sparse CT image reconstruction method based on wavelet implicit neural network refinement for diffusion models, and its innovations include the following: 8(a). During the training phase, we use the dataset from AAPM to train the diffusion model s in the image domain θ . In the reconstruction phase, we take the randomly sampled pure Gaussian noise as the initial input of the diffusion model, and then impose data consistency on it to correct the reconstruction result. Then, we use wavelets to decompose the image processed with data consistency into four sub-bands (aLL(x), dLH(x), dHL(x), dHH(x)), and the high-frequency sub-bands (dLH(x), dHL(x), dHH(x)) are optimized using an implicit neural network. Finally, the optimized high-frequency sub-bands and the low-frequency sub-band processed by diffusion (aLL(x)) are transformed back to the image domain using the inverse wavelet technique (W T ) as the input for the next iteration of the diffusion model. The loop continues until the final reconstruction result is obtained 8(b). In recent years, generative models have made significant progress in the field of image generation and have been widely applied to computer tomography (CT) reconstruction tasks. In sparse CT (SVCT) reconstruction, diffusion models can be used for image domain data completion to generate clearer images. However, diffusion models still have the following limitations in this task: ① High computational cost: The forward process of the diffusion model relies on multiple noise additions and reverse denoising, resulting in a long training time. For example, in this study, we used a dataset containing 1200 520×520 CT images and performed 300,000 iterations of training on an NVIDIA RTX 4090 device, which took about 35 hours. ② High-frequency information loss: The step-by-step denoising process of the diffusion model depends on the estimation of the noise distribution, which may cause the image to be over-smoothed, thus losing high-frequency details. In SVCT reconstruction, high-resolution structures (such as lesion areas, bone edges, etc.) are crucial, and the diffusion model may generate blurred reconstruction results, making it difficult to accurately recover texture information, thereby affecting the accuracy of medical diagnosis. 8(c). To address the above issues, this study introduces wavelet decomposition technology and divides the undersampled image into four subbands. Since the low-frequency components mainly represent the global structure of the image, we use a diffusion model to reconstruct the low-frequency components. For the high-frequency components, we use an implicit neural network (INR) embedded with Fourier features to recover the details. To meet the input requirements of this network, we first replace the low-frequency components with a matrix of all zeros, and then perform the inverse wavelet transform to return to the image domain for processing. The diffusion model only solves the prior problem. We propose to optimize the prior embedding based on physical consistency to ensure that the reconstruction result is consistent with the initial sparse projection Y. Through the above improvements, we effectively alleviate the computational burden of the diffusion model in the SVCT task and enhance the high-frequency detail fidelity of the reconstruction result, thereby improving the quality of the final reconstructed image.

Images