Reflective Fluorescence Molecular Tomography Method and System Based on ADMM Network

Abstract

The present invention discloses a method and system for reflectance fluorescence molecular tomography based on an ADMM network, which relates to the field of molecular imaging technology. The method includes: constructing a forward matrix and a training data set by using a decoupled fluorescence Monte Carlo model according to the reflectance fluorescence distribution to be reconstructed; constructing an objective function to be optimized by using the forward matrix; performing L1 regularization processing on the objective function to be optimized; decomposing the processed objective function to be optimized into three sub-problems by using the alternating direction method of multipliers; parameterizing the three sub-problems to obtain a parameterized ADMM iterative network; training and optimizing the parameterized ADMM iterative network by using the training data set and a loss function to obtain an ADMM network; and inputting the reflectance fluorescence distribution to be reconstructed into the ADMM network to obtain a reconstructed fluorophore distribution. The present invention can improve the quality of reflectance FMT image reconstruction.

Description

Technical Field

[0001] The present invention relates to the field of molecular imaging technology, and in particular, to a reflection fluorescence molecular tomography method and system based on an ADMM network. Background Art

[0002] Fluorescence Molecular Tomography (FMT) is an imaging technology based on diffused photons. Traditional FMT is usually transmission imaging. Therefore, when the excitation light or fluorescence cannot penetrate the entire sample, it is difficult for traditional transmission FMT to image the sample. Reflection FMT only uses reflected photons for imaging and is basically not limited by the sample size. However, due to the large number of source-detector pairs in reflection FMT and the single scanning angle and detection angle, the fluorescence distributions corresponding to different light sources are highly similar. Therefore, the inverse problem of reflection FMT is highly ill-posed, which means that the reconstruction results of traditional reconstruction methods based on mathematical iteration have many artifacts and poor imaging quality. Therefore, how to improve the image reconstruction quality of reflection FMT is of great significance for reflection FMT. Summary of the Invention

[0003] The purpose of the present invention is to provide a reflection fluorescence molecular tomography method and system based on an ADMM network, which can improve the image reconstruction quality of reflection FMT.

[0004] To achieve the above purpose, the present invention provides the following solutions:

[0005] A reflection fluorescence molecular tomography method based on an ADMM network, the method comprising:

[0006] Obtaining the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed;

[0007] According to the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed, using a decoupled fluorescence Monte Carlo model to construct a forward matrix and a training data set corresponding to the forward matrix; the training data set includes the true fluorophore distributions at different positions and the reflected fluorescence distributions of the excitation light source scanning plane corresponding to the true fluorophore distributions at different positions;

[0008] Constructing an objective function to be optimized using the forward matrix;

[0009] Performing L1 regularization processing on the objective function to be optimized to obtain a processed objective function to be optimized;

[0010] Using the alternating direction multiplier method to decompose the processed objective function to be optimized into three sub-problems; the three sub-problems include a first sub-problem, a second sub-problem, and a third sub-problem;

[0011] Parameterize the three sub - problems to obtain a parameterized ADMM iterative network;

[0012] Use the training data set and the loss function to train and optimize the parameterized ADMM iterative network to obtain an ADMM network;

[0013] Input the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed into the ADMM network to obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed.

[0014] Optionally, the processed objective function to be optimized is:

[0015]

[0016] where Y is the reflected fluorescence distribution of the excitation light source scanning surface; A is the forward matrix; x is the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

[0017] Optionally, the parameterizing the three sub - problems to obtain a parameterized ADMM iterative network specifically includes:

[0018] Use the first three - dimensional convolutional neural network as the initialization operator for the first sub - problem, the second three - dimensional convolutional neural network as the solution operator for the second sub - problem, and the third three - dimensional convolutional neural network as the update operator for the third sub - problem to obtain a parameterized ADMM iterative network.

[0019] Optionally, the parameterized ADMM iterative network is:

[0020]

[0021] z k+1 =F(W k x k+1 +u k )+(W k x k+1 +u k )

[0022] u k+1 =u k +W k x k+1 -z k+1

[0023] W k+1 =P(x k+1 )

[0024] In the formula, x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iterative network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iterative network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; u k+1 is the second auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network, and k represents the serial number of the iteration times.

[0025] Optionally, the loss function is:

[0026]

[0027] In the formula, loss represents the loss function; is the fluorophore distribution output after the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset; n is the total number of reflected fluorescence distributions of the excitation light source scanning planes in the training dataset; i is the serial number of the reflected fluorescence distribution of the excitation light source scanning plane; is the square of the L2 norm of; μ is a preset coefficient; M is the dimension of x.

[0028] The present invention also provides the following solution:

[0029] A reflectance fluorescence molecular tomography system based on an ADMM network, the system comprising:

[0030] A to-be-reconstructed reflected fluorescence distribution acquisition module, configured to acquire the reflected fluorescence distribution of an excitation light source scanning plane to be reconstructed;

[0031] A forward matrix and training dataset construction module, which is used to construct a forward matrix and a training dataset corresponding to the forward matrix according to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed, by using a decoupled fluorescence Monte Carlo model; the training dataset includes the true fluorophore distributions at different positions and the reflected fluorescence distributions of the excitation light source scanning surface corresponding to the true fluorophore distributions at different positions;

[0032] An objective function to be optimized construction module, which is used to construct an objective function to be optimized by using the forward matrix;

[0033] An L1 regularization processing module, which is used to perform L1 regularization processing on the objective function to be optimized, and obtain the processed objective function to be optimized;

[0034] A three-subproblem decomposition module, which is used to decompose the processed objective function to be optimized into three subproblems by using the alternating direction method of multipliers; the three subproblems include a first subproblem, a second subproblem, and a third subproblem;

[0035] A three-subproblem parameterization module, which is used to parameterize the three subproblems to obtain a parameterized ADMM iterative network;

[0036] A parameterized ADMM iterative network training and optimization module, which is used to train and optimize the parameterized ADMM iterative network by using the training dataset and a loss function, and obtain an ADMM network;

[0037] A fluorophore distribution reconstruction module, which is used to input the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed into the ADMM network, and obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed.

[0038] Optionally, the processed objective function to be optimized is:

[0039]

[0040] where Y is the reflected fluorescence distribution of the excitation light source scanning surface; A is the forward matrix; x is the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

[0041] Optionally, the three-subproblem parameterization module specifically includes:

[0042] Three sub-problem parameterization units are used to obtain a parameterized ADMM iterative network by using a first three-dimensional convolutional neural network as the initialization operator for the first sub-problem, a second three-dimensional convolutional neural network as the solution operator for the second sub-problem, and a third three-dimensional convolutional neural network as the update operator for the third sub-problem.

[0043] Optionally, the parameterized ADMM iterative network is:

[0044]

[0045] z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k )

[0046] u k+1 = u k + W k x k+1 - z k+1

[0047] W k+1 = P(x k+1 )

[0048] where x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iterative network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iterative network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; u k+1 is the second auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network, and k represents the sequence number of the iteration times.

[0049] Optionally, the loss function is:

[0050]

[0051] In the formula, loss represents the loss function; is the fluorophore distribution output after the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset; n is the total number of reflected fluorescence distributions of the excitation light source scanning plane in the training dataset; i is the serial number of the reflected fluorescence distribution of the excitation light source scanning plane; is the square of the L2 norm of; μ is a preset coefficient; M is the dimension of x.

[0052] According to the specific embodiments provided by the present invention, the following technical effects are disclosed:

[0053] The reflection-type fluorescence molecular tomography method and system based on the ADMM network disclosed by the present invention use the alternating direction method of multipliers (ADMM) to decompose the objective function to be optimized after L1 regularization into an iterative form of three sub-problems, and then parameterize the three sub-problems respectively to obtain a parameterized ADMM iterative network. The parameterized ADMM iterative network is trained using the training dataset, and the parameterized ADMM iterative network is optimized using the loss function to obtain an ADMM network for reconstructing the fluorophore distribution. The ADMM network establishes a mapping between the reflected fluorescence distribution of the excitation light source scanning plane and the fluorophore distribution to be reconstructed, reduces the ill-posedness of image reconstruction, and improves the quality of reflection-type fluorescence molecular tomography (FMT) image reconstruction. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0055] Figure 1 is a flowchart of an embodiment of the reflection-type fluorescence molecular tomography method based on the ADMM network of the present invention;

[0056] Figure 2 is a three-dimensional view of the simulation cube model of the present invention;

[0057] Figure 3 is a tomogram of the xy plane of the simulation cube model of the present invention;

[0058] Figure 4It is the light source distribution diagram of the simulation cube of the present invention;

[0059] Figure 5 It is the detector distribution diagram of the simulation cube of the present invention;

[0060] Figure 6 It is the three-dimensional diagram of the reconstruction result of the present invention;

[0061] Figure 7 It is the tomogram of the xy plane of the reconstruction result of the present invention;

[0062] Figure 8 It is the structural diagram of the embodiment of the reflection-type fluorescence molecular tomography system based on the ADMM network of the present invention. Detailed implementation manners

[0063] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0064] The purpose of the present invention is to provide a reflection-type fluorescence molecular tomography method and system based on the ADMM network, which can improve the quality of reflection-type FMT image reconstruction.

[0065] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below in conjunction with the accompanying drawings and specific implementation manners.

[0066] Figure 1 It is the flowchart of the embodiment of the reflection-type fluorescence molecular tomography method based on the ADMM network of the present invention. Refer to Figure 1 The reflection-type fluorescence molecular tomography method based on the ADMM network includes:

[0067] Step 101: Obtain the reflection fluorescence distribution of the excitation light source scanning surface to be reconstructed.

[0068] Step 102: According to the reflection fluorescence distribution of the excitation light source scanning surface to be reconstructed, use the decoupled fluorescence Monte Carlo model to construct a forward matrix and a training data set corresponding to the forward matrix; the training data set includes the true fluorophore distributions at different positions and the reflection fluorescence distributions of the excitation light source scanning surface corresponding to the true fluorophore distributions at different positions.

[0069] In step 102, the forward matrix is the forward matrix constructed for the object scanned by the excitation light source. The forward matrix describes the influence of the fluorophore concentration at different positions inside the object scanned by the excitation light source (the fluorophore distribution at different positions) on the reflected fluorescence distribution of the excitation light source scanning plane. The reflection-type fluorescence molecular tomography method based on the ADMM network in the present invention is a process of reconstructing the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed on the surface of the object to obtain the fluorophore distribution at a certain position inside the object. Therefore, it is necessary to use the true fluorophore distribution at different positions inside the object and the reflected fluorescence distribution of the excitation light source scanning plane corresponding to the true fluorophore distribution at different positions inside the object to train the parameterized ADMM iterative network.

[0070] Step 103: Construct an objective function to be optimized by using the forward matrix.

[0071] Step 104: Perform L1 regularization on the objective function to be optimized to obtain the processed objective function to be optimized.

[0072] In this step 104, the processed objective function to be optimized is:

[0073]

[0074] In the formula, Y is the reflected fluorescence distribution of the excitation light source scanning plane; A is the forward matrix; x is the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning plane; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

[0075] Step 105: Use the alternating direction multiplier method to decompose the processed objective function to be optimized into three sub-problems; the three sub-problems include the first sub-problem, the second sub-problem, and the third sub-problem.

[0076] Step 106: Parameterize the three sub-problems to obtain a parameterized ADMM iterative network.

[0077] This step 106 specifically includes:

[0078] Use the first three-dimensional convolutional neural network as the initialization operator for the first sub-problem, use the second three-dimensional convolutional neural network as the solution operator for the second sub-problem, and use the third three-dimensional convolutional neural network as the update operator for the third sub-problem to obtain a parameterized ADMM iterative network.

[0079] Among them, the parameterized ADMM iterative network is:

[0080]

[0081] z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k )

[0082] u k+1 = u k + W k x k+1 - z k+1

[0083] W k+1 = P(x k+1 )

[0084] where x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iterative network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iterative network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network, and k represents the sequence number of the iteration times.

[0085] Step 107: Train and optimize the parameterized ADMM iterative network using the training dataset and the loss function to obtain the ADMM network.

[0086] In this step 107, the loss function is:

[0087]

[0088] where loss represents the loss function; is the fluorophore distribution output after the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflected fluorescence distribution of the i-th excitation light source scanning plane in the training dataset; n is the total number of reflected fluorescence distributions of the excitation light source scanning planes in the training dataset; i is the serial number of the reflected fluorescence distribution of the excitation light source scanning plane; is the square of the L2 norm of; μ is a preset coefficient; M is the dimension of x.

[0089] Step 108: Input the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed into the ADMM network to obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed.

[0090] In this step 108, the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning plane to be reconstructed is the reconstructed fluorophore distribution.

[0091] The following uses a specific embodiment to illustrate the technical solution of the present invention:

[0092] The method for reflectance fluorescence molecular tomography based on the ADMM (alternating direction method of multipliers) network of the present invention specifically includes the following steps:

[0093] Step 1: Obtain the reflected fluorescence data on the sample surface, that is, obtain the reflected fluorescence distribution of the excitation light source scanning plane.

[0094] Step 2: Use the decoupled fluorescence Monte Carlo model to construct (calculate) the forward matrix; use the forward matrix to construct the objective function to be optimized; the objective function to be optimized introduces a sparse prior by adding L1 regularization.

[0095] The objective function to be optimized is:

[0096]

[0097] In the formula, E(x) is the objective function to be optimized during the reconstruction of the reflectance fluorescence molecular tomography image, Y is the detected surface reflected fluorescence distribution, A is the forward matrix obtained by using the decoupled fluorescence Monte Carlo model, x is the fluorophore distribution, λ is the regularization parameter, W is the weight matrix and is a diagonal matrix, is the square of the L2 norm of ·, ||·||1 is the L1 norm of ·.

[0098] The reconstruction of the reflectance fluorescence molecular tomography image is to use the objective function to be optimized Optimized to the minimum value. In the formula, A is the forward matrix, Y is the reflected fluorescence distribution of the excitation light scanning surface (excitation light source scanning surface), x is the fluorophore distribution. When E0(x) is minimized, x is the reconstruction result. However, due to the usually large dimension of A and the noise in Y as the detected data, the above optimization problem (the objective function to be optimized) has severe ill-posedness (very poor solution accuracy). Therefore, a regularization term is usually added to E0(x), and the added regularization term often varies according to different actual problems. L1 regularization changes the objective function to be optimized into Originally, it was to optimize E0(x), and after L1 regularization, it is to optimize E(x). L1 regularization can make the optimization result x of E(x) sparser, thus making the reconstruction effect better than the case without L1 regularization. L1 regularization is a common method for FMT image reconstruction.

[0099] Step 3: Use ADMM to decompose the objective function to be optimized into an iterative form of three sub-problems; then use a three-dimensional convolutional neural network as the initialization operator, solution operator, and update operator for the three sub-problems respectively to obtain a parameterized ADMM iterative network, that is, parameterize the three sub-problems respectively to obtain a parameterized ADMM iterative network.

[0100] Among them, using ADMM to decompose the objective function to be optimized into an iteration of three sub-problems (the iterative form of the three sub-problems into which the objective function to be optimized is decomposed), the expression is as follows:

[0101]

[0102] z k+1 = soft λ / ρ (W k x k+1 + u k ),

[0103] u k+1 = u k + W k x k+1 - z k+1 ),

[0104]

[0105] When k < N, it can also be expressed as

[0106] The solution objectives of the three sub-problems are x, z, and W respectively. Among them, N is the maximum number of iteration rounds, ρ is the regularization parameter, z and u are auxiliary variables, δ is a positive number greater than 0 to prevent the denominator from approaching 0. CG[Β,Γ,Λ] is the conjugate gradient operator, which can solve the equation ΒΛ = Γ, and soft λ / ρ (·) is with is the soft thresholding function with the threshold, diag(·) is the diagonal matrix generated with the vector · as the diagonal elements, x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration, x k is the fluorophore distribution obtained after the k-th iteration, z k+1 , u k+1 are the auxiliary variables obtained after the (k + 1)-th iteration respectively, z k , u k are the auxiliary variables obtained after the k-th iteration respectively, W k+1 is the regularization matrix obtained after the (k + 1)-th iteration.

[0107] Among them, CG[Β,Γ,Λ] is used to explain the function of the CG operator. Suppose there is a linear equation system ΒΛ = Γ, where B and Γ are known and Λ is unknown. Then Λ res = CG[Β,Γ,Λ0], Λ res is the result of solving Λ by CG, and Λ0 is the preset initialized Λ. ΒΛ = Γ is only used to describe the role of the CG operator.

[0108] In the ADMM iteration, the steps: That is, solve the equation What needs to be solved is x, and the solution result is denoted as x k+1 , x k is the x solved by CG in the previous iteration and is used as the initialization of x in this round. CG represents the conjugate gradient method, and the method of solving the linear equation system by the conjugate gradient method is well-known.

[0109] Step 4: Generate a simulation training dataset, and use the simulation training dataset and the loss function to train the parameterized ADMM iteration network to obtain the ADMM network.

[0110] Among them, the determination method of the ADMM network is:

[0111] Obtain the dataset to be trained; the dataset to be trained includes the surface reflection fluorescence distribution to be trained and the corresponding true fluorophore distribution;

[0112] Use the decoupled fluorescence Monte Carlo model to calculate the forward matrix; use the forward matrix to construct the objective function to be optimized; the objective function to be optimized introduces a sparse prior by adding L1 regularization.

[0113] The alternating direction method of multipliers (ADMM) is used to decompose the objective function to be optimized into an iterative form of three sub-problems; the three sub-problems are parameterized to obtain a parameterized ADMM iterative network. The parameterization methods for the three sub-problems are as follows: three-dimensional convolutional neural networks are used as the initialization operator, the solution operator, and the update operator for the three sub-problems respectively to obtain a parameterized ADMM iterative network. The generation method of the parameterized ADMM iterative network is as follows: the three sub-problems in each layer of iteration are parameterized to obtain a parameterized ADMM iterative network; the parameterization method is: a three-dimensional convolutional neural network (denoted as O) is used as the initialization operator for the first sub-problem; another three-dimensional convolutional neural network (denoted as F) is used as the solution operator for the second sub-problem; another three-dimensional convolutional neural network (denoted as P) is used as the update operator for the third sub-problem; the three sub-problems in each layer of iteration are parameterized to obtain a parameterized ADMM iterative network.

[0114] The computational expression of the parameterized ADMM iterative network is as follows:

[0115]

[0116] z k+1 = F(W k x k+1 + u k )+(W k x k+1 + u k ),

[0117] u k+1 = u k + W k x k+1 - z k+1 ),

[0118] W k+1 = P(x k+1 ),

[0119] where It can also be expressed as

[0120] In the formula, the three operators O, F, and P are three three-dimensional convolutional neural networks with the same structure and non-shared parameters; the three-dimensional convolutional neural network consists of a convolutional layer and an activation function. The convolutional layer includes a convolutional kernel and a bias. The activation function is set to ReLU, ReLU(·)= max(·, 0). The dot in the parentheses is only used to describe the ReLU function and does not correspond to any of the above variables.

[0121] The convolutional kernel parameters and bias parameters of the parameterized ADMM iterative network are optimized by calculating the gradient of the loss function.

[0122] Input the training dataset into the parameterized ADMM iterative network and train it in combination with the loss function to obtain the ADMM network. That is, generate the training dataset through simulation, use the training dataset to train the parameterized ADMM iterative network to obtain the ADMM network; calculate the gradient of the loss function to optimize the convolution kernel parameters and bias parameters of the parameterized ADMM iterative network; when the number of training rounds reaches the preset number of training rounds, terminate the training to obtain the ADMM network. The method process is as follows: L1-regularize the objective function to be optimized - decompose ADMM into iterative forms of several sub-problems - parameterize the ADMM iterative architecture using a neural network (operator) to obtain the parameterized ADMM iterative network - generate the training dataset through simulation, train the parameterized ADMM iterative network to obtain the ADMM network - use ADMM for image reconstruction. Therefore, the traditional ADMM method is to solve for x given Y and A. The inputs of the ADMM network are A (forward matrix) and Y (reflected fluorescence distribution), and the output is x (fluorophore distribution).

[0123] Among them, the training dataset is obtained through simulation, that is, constructed using the decoupled fluorescence Monte Carlo model. The training dataset includes: the reflected fluorescence distribution on the sample surface and the corresponding true fluorophore distribution. During the training process, the reflected fluorescence distribution on the sample surface is used as the input of the parameterized ADMM iterative network to obtain the fluorophore distribution output by the parameterized ADMM iterative network. Calculate the loss function using the fluorophore distribution output by the parameterized ADMM iterative network and the corresponding true fluorophore distribution, and then optimize the trainable parameters in the parameterized ADMM iterative network, such as convolution kernel parameters and bias parameters, by calculating the gradient of the loss function. When the preset number of training rounds is reached, stop the training, and the parameters in the parameterized ADMM iterative network are optimized to obtain the ADMM network.

[0124] The expression of the loss function is as follows:

[0125]

[0126] In the formula, is the fluorophore distribution output by the parameterized ADMM iterative network corresponding to the input of the i-th reflected fluorescence distribution in the training dataset during the training process; is the true fluorophore distribution corresponding to the i-th reflected fluorescence distribution in the training dataset; M is the dimension of the vector x, and n is the number of data pairs in the training dataset; μ is a preset coefficient used to balance the values of the two terms before and after. In the example, μ = 0.05; when the number of training rounds reaches the preset number of training rounds, terminate the training to obtain the ADMM network.

[0127] Step 5: Use the reflected fluorescence distribution on the excitation light source scanning surface as the input of the ADMM network to obtain the reconstructed fluorophore distribution, that is, input the fluorescence distribution into the ADMM network to reconstruct the fluorophore distribution.

[0128] Mathematically, Y = Ax, where Y is the reflected fluorescence distribution that can be detected by the detector, A is the forward matrix that can be calculated by methods such as the decoupled fluorescence Monte Carlo model, and x is the fluorophore distribution that cannot be directly detected and is also the target to be solved in the present invention.

[0129] The reflection-type fluorescence molecular tomography method based on the ADMM network disclosed in the present invention obtains the reflected fluorescence distribution on the sample surface; constructs the forward matrix using the decoupled fluorescence Monte Carlo model; constructs the objective function to be optimized using the forward matrix; introduces sparse prior by adding L1 regularization to the objective function to be optimized; uses ADMM to decompose the objective function to be optimized into an iterative form of three sub-problems; then uses a three-dimensional convolutional neural network as the initialization operator, solution operator, and update operator for the three sub-problems respectively to obtain a parameterized ADMM iterative network; generates a simulation training dataset, trains the parameterized ADMM iterative network using the simulation dataset to obtain the ADMM network; uses the reflected fluorescence distribution on the sample surface as the input of the ADMM network to obtain the reconstructed fluorophore distribution. By decomposing the objective function to be optimized into an iterative form of three sub-problems using ADMM and then parameterizing the three sub-problems respectively, and training the network using the training dataset, the present invention establishes a mapping between the surface reflected fluorescence distribution and the fluorophore distribution to be reconstructed, reduces the ill-posedness of traditional image reconstruction methods, and improves the quality of reflection-type fluorescence molecular tomography image reconstruction.

[0130] The reasons why the present invention can improve the reconstruction quality compared with the existing methods are as follows:

[0131] Iterative form of the traditional ADMM method:

[0132]

[0133] z k+1 = soft λ / ρ (W k x k+1 + u k ),

[0134] u k+1 = u k + W k x k+1 - z k+1 ,

[0135]

[0136] Each step of the calculation is deterministic. Given a set of Y, A, and the initial W, x can be calculated. However, there are non-negligible errors when solving or updating x, z, and W. λ / ρ (·) and When solving or updating x, z, and W, there are non-negligible errors.

[0137] The iterative form of the present invention is:

[0138]

[0139] z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k ),

[0140] u k+1 = u k + W k x k+1 - z k+1 ,

[0141] W k+1 = P(x k+1 ),

[0142] In the iterative steps of x, z, and W, three parameterized operators O, F, and P are added. The parameter values in these three parameterized operators are optimized through the training dataset and the defined loss function. O optimizes the input of x in CG, F replaces the soft threshold function soft λ / ρ (·) to solve for z, and P replaces for updating W. These three operators are designed as (but not limited to) a neural network structure of convolution + ReLU. After training, the parameters in the operators are optimized to obtain the ADMM network. At this time, O can well initialize x, making the result of CG solving more accurate. Compared with soft λ / ρ (·) and , F and P can solve and update z and W more accurately, further improving the solving accuracy of CG. Thus, while the ADMM network retains the iterative form of the traditional ADMM method, the accuracy of solving x is much higher than that of the traditional ADMM iteration.

[0143] Next, the actual effect of the reflection-type fluorescence molecular tomography method based on the ADMM network of the present invention is described:

[0144] First, a homogeneous cube model is constructed, such as Figure 2As shown in the figure. The function of constructing the homogeneous cube model is to obtain the reflected fluorescence distribution on the excitation light source scanning plane and calculate the forward matrix of the model; when the model changes, the reflected fluorescence distribution Y on the excitation light source scanning plane will change, and the forward matrix A will also change; the reflected fluorescence distribution Y on the excitation light source scanning plane is directly detected by the detector, and the forward matrix A is calculated by solving the coupled fluorescence Monte Carlo model.

[0145] The absorption coefficient is set to 0.03 cm -1 , and the reduced scattering coefficient is set to 5 cm -1 . The positions of the cylindrical fluorophores in the x-y plane are as Figure 3 shown in the circular area in the figure. The diameter of each fluorophore is 0.2 mm, the length is 1.2 mm, and the concentration of the fluorophores is the same. There are 100 light sources in total, evenly distributed on the y = 0 plane, corresponding to the x coordinates from x = 1.1 mm - 2.9 mm, with a spacing of 0.2 mm, and the corresponding z coordinates from z = 1.1 mm - 2.9 mm, with a spacing of 0.2 mm, as Figure 4 shown by the triangular points in the figure. There are 400 detectors in total, evenly distributed on the y = 0 plane, corresponding to the x coordinates from x = 0.1 mm - 3.9 mm, with a spacing of 0.2 mm, and the corresponding z coordinates from z = 0.1 mm - 3.9 mm, with a spacing of 0.2 mm, as Figure 5 shown by the dot points in the figure. As Figure 4 and Figure 5 shown, both the light sources and detectors are distributed on the y = 0 plane, that is, the reflectance FMT only detects the reflected fluorescence distribution, rather than detecting the fluorescence distribution at various angles of the sample. The total number of source-detector pairs is 100 × 400 = 40000, and the reconstruction model is voxelized by (0.2 mm) 3 , with a total of 8000 voxels. The dimension of the forward matrix A obtained by solving the coupled fluorescence Monte Carlo model is 40000 × 8000. It can be seen that different from traditional fluorescence molecular tomography, the light source settings and detector positions of reflectance fluorescence molecular tomography are both located on the y = 0 plane. Therefore, when the light source cannot penetrate the entire sample, traditional fluorescence molecular tomography cannot detect the fluorescence signal, while reflectance fluorescence molecular tomography can detect the fluorescence signal of fluorophores at a certain depth.

[0146] The detected reflected fluorescence distribution is input into the ADMM network to obtain the image reconstruction result, as Figure 6 and Figure 7 shown in the figure. By comparing Figure 6 and Figure 2 , and comparing Figure 7 and Figure 3 , it can be found that the reconstruction result is in good agreement with the real result, and it can simultaneously locate fluorophores at different depths with high reconstruction quality.

[0147] Compared with the prior art, the advantages of the present invention are as follows:

[0148] 1. Compared with the traditional fluorescence molecular tomography that requires collecting the 360° fluorescence distribution on the surface, the present invention only needs to collect the reflected fluorescence signal to complete the three-dimensional image reconstruction.

[0149] 2. Compared with the traditional reflectance fluorescence molecular tomography algorithm, the present invention can improve the quality of image reconstruction.

[0150] 3. Compared with the existing data-driven deep learning methods, the present invention has strong interpretability.

[0151] The existing data-driven deep learning methods can directly construct the non-linear mapping between the detected fluorescence and the fluorophore, avoiding the modeling error and the ill-posedness of the inverse problem. Currently, many diffusion optical imaging methods based on deep learning have been proposed. An FMT network based on the improved U-Net is used to improve the reconstruction quality and resolution of FMT. In addition, a multi-layer perceptron is used to improve the image reconstruction quality of bioluminescence tomography. These data-driven deep learning methods, although they can greatly improve the quality of image reconstruction, have problems such as poor generalization, poor interpretability, and the need for a large amount of training data sets. These problems have restricted the application of deep learning in FMT image reconstruction. To address these problems, a model-driven deep learning method based on the projected gradient descent method is used for FMT image reconstruction. This method uses the residual block structure of the multi-layer three-dimensional convolutional neural network to fit the gradient of the regularization term, improving the interpretability of the network and the imaging quality of transmissive FMT. However, this method is only applicable to transmissive FMT and cannot be applied to reflectance FMT imaging. Because reflectance FMT needs to introduce a sparse prior through L1 regularization, and the L1 regularization term is non-differentiable, the above method cannot solve the optimization objective function of the reflectance FMT in the present invention, that is, it cannot reconstruct the reflectance FMT.

[0152] The present invention first decomposes the optimization objective function by using ADMM, separates each term into an iterative form of sub-problems, and uses different operators to initialize, solve, and update different sub-problems respectively. Then, the three operators are parameterized, which is applicable to solving almost any form of optimization objective function. From the process, the present invention does not fit a certain part of the optimization objective function, but first decomposes the optimization objective function, and then performs different processes such as initializing, solving, and updating different sub-problems; from the result, the present invention is applicable to reflectance FMT imaging with a sparse prior introduced, while all the above methods are not applicable.

[0153] Figure 8 This is the structural diagram of the embodiment of the reflectance fluorescence molecular tomography system based on the ADMM network of the present invention. See Figure 8, the reflection fluorescence molecular tomography system based on the ADMM network includes:

[0154] A reflection fluorescence distribution acquisition module 801 to be reconstructed, configured to acquire the reflection fluorescence distribution of the excitation light source scanning plane to be reconstructed.

[0155] A forward matrix and training data set construction module 802, configured to construct a forward matrix and a training data set corresponding to the forward matrix according to the reflection fluorescence distribution of the excitation light source scanning plane to be reconstructed; the training data set includes the true fluorophore distributions at different positions and the reflection fluorescence distributions of the excitation light source scanning plane corresponding to the true fluorophore distributions at different positions.

[0156] An objective function to be optimized construction module 803, configured to construct an objective function to be optimized by using the forward matrix.

[0157] An L1 regularization processing module 804, configured to perform L1 regularization processing on the objective function to be optimized to obtain a processed objective function to be optimized.

[0158] Specifically, the processed objective function to be optimized is:

[0159]

[0160] In the formula, Y is the reflection fluorescence distribution of the excitation light source scanning plane; A is the forward matrix; x is the fluorophore distribution corresponding to the reflection fluorescence distribution of the excitation light source scanning plane; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

[0161] A three sub - problem decomposition module 805, configured to decompose the processed objective function to be optimized into three sub - problems by using the alternating direction method of multipliers; the three sub - problems include a first sub - problem, a second sub - problem, and a third sub - problem.

[0162] A three sub - problem parameterization module 806, configured to parameterize the three sub - problems to obtain a parameterized ADMM iterative network.

[0163] The three sub - problem parameterization module 806 specifically includes:

[0164] A three sub - problem parameterization unit, configured to use a first three - dimensional convolutional neural network as the initialization operator for the first sub - problem, use a second three - dimensional convolutional neural network as the solution operator for the second sub - problem, and use a third three - dimensional convolutional neural network as the update operator for the third sub - problem to obtain a parameterized ADMM iterative network.

[0165] Specifically, the parameterized ADMM iterative network is as follows:

[0166]

[0167] z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k )

[0168] u k+1 = u k + W k x k+1 - z k+1

[0169] W k+1 = P(x k+1 )

[0170] In the formula, x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iterative network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iterative network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iterative network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; u k+1 is the second auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iterative network, and k represents the serial number of the iteration times.

[0171] The parameterized ADMM iterative network training and optimization module 807 is used to train and optimize the parameterized ADMM iterative network by using the training data set and the loss function to obtain the ADMM network.

[0172] Specifically, the loss function is as follows:

[0173]

[0174] In the formula, loss represents the loss function; is the fluorophore distribution output after the reflected fluorescence distribution of the i-th excitation light source scanning surface in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflected fluorescence distribution of the i-th excitation light source scanning surface in the training dataset; n is the total number of reflected fluorescence distributions of the excitation light source scanning surfaces in the training dataset; i is the serial number of the reflected fluorescence distribution of the excitation light source scanning surface; is the square of the L2 norm of ; μ is a preset coefficient; M is the dimension of x.

[0175] The fluorophore distribution reconstruction module 808 is configured to input the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed into the ADMM network to obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed.

[0176] In this specification, each embodiment is described in a progressive manner. The key point of each embodiment is to illustrate the differences from other embodiments. The same or similar parts among the embodiments can be referred to each other. For the system disclosed in the embodiments, since it corresponds to the method disclosed in the embodiments, the description is relatively simple. For the relevant parts, please refer to the description in the method section.

[0177] In this article, specific examples are used to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core idea; at the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manner and application scope. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A reflection fluorescence molecular tomography method based on the ADMM network, characterized in that The method includes: Obtaining the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed; According to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed, using the decoupled fluorescence Monte Carlo model to construct a forward matrix and a training data set corresponding to the forward matrix; the training data set includes the true fluorophore distributions at different positions and the reflected fluorescence distributions of the excitation light source scanning surface corresponding to the true fluorophore distributions at different positions; Constructing an objective function to be optimized using the forward matrix; Performing L1 regularization on the objective function to be optimized to obtain the processed objective function to be optimized; Using the alternating direction method of multipliers to decompose the processed objective function to be optimized into three sub-problems; the three sub-problems include a first sub-problem, a second sub-problem, and a third sub-problem; Parameterizing the three sub-problems to obtain a parameterized ADMM iterative network; Training and optimizing the parameterized ADMM iterative network using the training data set and a loss function to obtain an ADMM network; Inputting the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed into the ADMM network to obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed.

2. The method for reflectance fluorescence molecular tomography based on the ADMM network according to claim 1, wherein The processed objective function to be optimized is: Wherein, Y is the reflected fluorescence distribution of the excitation light source scanning surface; A is the forward matrix; x is the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

3. The method for reflectance fluorescence molecular tomography based on the ADMM network according to claim 2, wherein The parameterizing the three sub-problems to obtain a parameterized ADMM iterative network specifically includes: Using a first three-dimensional convolutional neural network as the initialization operator for the first sub-problem, using a second three-dimensional convolutional neural network as the solution operator for the second sub-problem, and using a third three-dimensional convolutional neural network as the update operator for the third sub-problem to obtain a parameterized ADMM iterative network.

4. The method for reflectance fluorescence molecular tomography based on the ADMM network according to claim 3, wherein The parameterized ADMM iterative network is: z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k ) u k+1 = u k + W k x k+1 - z k+1 W k+1 = P(x k+1 ) where x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iteration network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iteration network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iteration network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iteration network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; u k+1 is the second auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network, where k represents the sequence number of the iteration times.

5. The method for reflectance fluorescence molecular tomography based on the ADMM network according to claim 4, wherein, The loss function is: where loss represents the loss function; is the fluorophore distribution output after the reflection fluorescence distribution of the i-th excitation light source scanning plane in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflection fluorescence distribution of the i-th excitation light source scanning plane in the training dataset; n is the total number of reflection fluorescence distributions of the excitation light source scanning planes in the training dataset; i is the serial number of the reflection fluorescence distribution of the excitation light source scanning plane; is the square of the L2 norm of; μ is a preset coefficient; M is the dimension of x.

6. A reflective fluorescence molecular tomography system based on the ADMM network, characterized in that The system includes: A reflected fluorescence distribution acquisition module to be reconstructed, configured to obtain the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed; A forward matrix and training data set construction module, configured to construct a forward matrix and a training data set corresponding to the forward matrix according to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed using the decoupled fluorescence Monte Carlo model; the training data set includes the true fluorophore distributions at different positions and the reflected fluorescence distributions of the excitation light source scanning surface corresponding to the true fluorophore distributions at different positions; An objective function construction module to be optimized, configured to construct an objective function to be optimized using the forward matrix; An L1 regularization processing module, configured to perform L1 regularization on the objective function to be optimized to obtain the processed objective function to be optimized; A three-sub-problem decomposition module, configured to decompose the processed objective function to be optimized into three sub-problems using the alternating direction method of multipliers; the three sub-problems include a first sub-problem, a second sub-problem, and a third sub-problem; A three-sub-problem parameterization module, configured to parameterize the three sub-problems to obtain a parameterized ADMM iterative network; A parameterized ADMM iterative network training and optimization module, configured to train and optimize the parameterized ADMM iterative network using the training data set and a loss function to obtain an ADMM network; The fluorophore distribution reconstruction module is configured to input the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed into the ADMM network, and obtain the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface to be reconstructed.

7. The reflectance fluorescence molecular tomography system based on the ADMM network according to claim 6, wherein The processed objective function to be optimized is: Wherein, Y is the reflected fluorescence distribution of the excitation light source scanning surface; A is the forward matrix; x is the fluorophore distribution corresponding to the reflected fluorescence distribution of the excitation light source scanning surface; λ is the regularization parameter; W is the weight matrix; is the square of the L2 norm of Ax - Y, ||Wx||1 is the L1 norm of Wx; E(x) is the objective function to be optimized, and the x corresponding to the minimum of E(x) is the reconstructed fluorophore distribution.

8. The reflectance fluorescence molecular tomography system based on the ADMM network according to claim 7, wherein The three sub-problem parameterization modules specifically include: Three sub-problem parameterization units, which are configured to use a first three-dimensional convolutional neural network as the initialization operator of the first sub-problem, use a second three-dimensional convolutional neural network as the solution operator of the second sub-problem, and use a third three-dimensional convolutional neural network as the update operator of the third sub-problem, to obtain a parameterized ADMM iterative network.

9. The reflectance fluorescence molecular tomography system based on the ADMM network according to claim 8, wherein The parameterized ADMM iterative network is: z k+1 = F(W k x k+1 + u k ) + (W k x k+1 + u k ) u k+1 = u k + W k x k+1 - z k+1 W k+1 = P(x k+1 ) where x k+1 is the fluorophore distribution obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; CG represents the conjugate gradient method; O represents the first three-dimensional convolutional neural network; F represents the second three-dimensional convolutional neural network; P represents the third three-dimensional convolutional neural network; T represents the transpose of a matrix; ρ is the regularization parameter; W k is the regularization matrix obtained after the k-th iteration of the parameterized ADMM iteration network; z k is the first auxiliary variable obtained after the k-th iteration of the parameterized ADMM iteration network; u k is the second auxiliary variable obtained after the k-th iteration of the parameterized ADMM iteration network; x k is the fluorophore distribution obtained after the k-th iteration of the parameterized ADMM iteration network; N is the maximum number of iteration rounds; z k+1 is the first auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; u k+1 is the second auxiliary variable obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network; W k+1 is the regularization matrix obtained after the (k + 1)-th iteration of the parameterized ADMM iteration network, and k represents the sequence number of the iteration times.

10. The reflectance fluorescence molecular tomography system based on the ADMM network according to claim 9, wherein, The loss function is: where loss represents the loss function; is the fluorophore distribution output after the reflection fluorescence distribution of the i-th excitation light source scanning surface in the training dataset is input into the parameterized ADMM iterative network; is the true fluorophore distribution corresponding to the reflection fluorescence distribution of the i-th excitation light source scanning surface in the training dataset; n is the total number of reflection fluorescence distributions of the excitation light source scanning surfaces in the training dataset; i is the serial number of the reflection fluorescence distribution of the excitation light source scanning surface; is the square of the L2 norm of; μ is a preset coefficient; M is the dimension of x.

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