A dual-wavelength body additive manufacturing optimization method based on a statistical model

Abstract

This invention discloses a statistical model-based optimization method for dual-wavelength bulk additive manufacturing. The dual-wavelength bulk additive manufacturing method uses two light sources of different wavelengths to project corresponding images onto a binary photoresist containing an initiator and an inhibitor. The spatial distribution of light energy is reconstructed through rotational exposure to achieve the printing of a three-dimensional structure. The initiation and inhibition components of the photoresist in dual-wavelength bulk additive manufacturing are controlled by two separate projection systems of different wavelengths, offering advantages such as high precision, high contrast, and high adaptability. This invention utilizes statistical principles to generate the optimal projection set for dual-wavelength bulk additive manufacturing. A maximum likelihood estimation dual-wavelength projection optimization model is established based on the Poisson distribution assumption of the photopolymerization reaction, and an iterative optimization algorithm for the projection is derived using the expectation-maximization algorithm. The computational model and algorithm of this invention have strong adaptability to both the projection system structure and photoresist parameters, and can be applied to various dual-wavelength bulk additive manufacturing apparatus modes.

Description

Technical Field

[0001] This invention relates to the field of bulk additive manufacturing based on the principle of computational axial lithography, and particularly to an optimization method for dual-wavelength bulk additive manufacturing based on a statistical model. Background Technology

[0002] Additive manufacturing (AM), also known as 3D printing or rapid prototyping, is a technology that processes and manufactures materials using an integral process. Unlike traditional machining-based manufacturing, this "bottom-up" approach has wide applications in industrial design, biomedicine, automotive, aerospace, and education. Currently, additive manufacturing primarily uses a layer-by-layer method to process target geometries, processing one slice of the 3D structure at a time, and then continuing to process layer by layer until the 3D object is complete. Mainstream additive manufacturing technologies include Fused Deposition Modeling (FDM), Stereolithography (SLA), Projection Laser Light (PSL), and Two-Photon Polymerization (2PP). However, current additive manufacturing technologies have some inherent drawbacks due to their layer-by-layer processing: firstly, the processing method typically results in lower vertical processing accuracy compared to horizontal processing accuracy; secondly, achieving higher interlayer resolution requires reducing layer thickness and increasing the number of layers, leading to increased processing times and longer manufacturing cycles. These disadvantages limit the application of additive manufacturing technology.

[0003] Volumetric additive manufacturing (VAM) is a novel additive manufacturing technology. This technology innovatively uses rotating volumetric exposure to directly create a stable light energy distribution within the photopolymer material, enabling rapid prototyping of three-dimensional structures. Thanks to this novel printing method, this technology overcomes many inherent shortcomings of traditional additive manufacturing techniques, offering high processing speeds. Currently, mainstream volumetric additive manufacturing technologies typically employ computed axial lithography (CAL), which uses 3D reconstruction theory to pre-calculate a set of projected images of the target 3D structure. Then, a projection optics system is used to back-project these images along the corresponding projection positions of each image. Various algorithms exist for calculating the projected image set, including Radon transform, gradient descent, and object space optimization. However, due to the absorption properties of the photopolymer material, CAL has certain limitations in terms of the size of the printed structure and the absorption of the photosensitive material. First, because only a single wavelength light source is used, the negative pixels in the projected image calculated by the algorithm cannot be physically replicated and require optimization. This optimization process incurs significant accuracy loss, thus reducing the expressiveness of the reconstructed structure. Second, for large-scale printing scenarios and with photoresists that absorb a lot of light, this method is difficult to manufacture structures of specified shapes due to the aggravated attenuation effect. To address this issue, the latest technology uses dual-wavelength projection combined with binary photoresist, providing a new approach for high-precision, rapid bulk additive manufacturing under large-scale conditions. This invention proposes an algorithm for this dual-wavelength projection mode bulk additive manufacturing system that can quickly and accurately calculate the optimal dual-wavelength projection set and has strong adaptability to different photoresist materials and projection optics systems. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of existing technologies by proposing a dual-wavelength bulk additive manufacturing optimization method based on a statistical model.

[0005] The objective of this invention is achieved through the following technical solution: a dual-wavelength bulk additive manufacturing optimization method based on a statistical model, comprising two parts: a forward transmission model and a backward transmission model. The forward transmission model uses a set of projected images as input and the distribution of photopolymerization reaction in three-dimensional space as output, used to predict the printing result given the projection data. The backward transmission model uses the target three-dimensional geometry as input and the set of projected images as output, used to solve for the optimal projection data that yields the best printing result using the expectation-maximization (EM) algorithm combined with the Gaussian distribution approximation assumption. Based on the forward and backward transmission models, for any given parameters of the binary photopolymer material and the target three-dimensional geometry to be manufactured, two sets of optimal projection image sets are generated, and the binary photopolymer material is subjected to bulk additive manufacturing to obtain the optimal three-dimensional printed structure.

[0006] Furthermore, the binary photopolymer material is composed of prepolymer and monomer, and is a liquid substance containing two photosensitive components: a photoinitiator and a photoinhibitor.

[0007] Furthermore, the photoinitiator in the binary photopolymer material will generate initiating active species under light radiation of a certain wavelength, which will initiate the photopolymerization reaction of the prepolymer and monomer in the system; the photoinhibitor will generate inert active species under light radiation of a different wavelength, which will consume the initiating active species and thus inhibit the photopolymerization reaction; for any point in the binary photopolymer material, when the number of photopolymerization reactions increases to exceed the curing threshold of the binary photopolymer material, that point will be cured.

[0008] Furthermore, the target three-dimensional geometric structure is a volumetric three-dimensional model in the form of a three-dimensional matrix. Each voxel in the volumetric three-dimensional model represents a point in space, and the value of the voxel indicates whether the point is part of the final structure to be manufactured.

[0009] Furthermore, the optimal projection image set consists of two sets of projection image sets, one set of images corresponding to each projection position of the initiating light projection system and the other set of images corresponding to each projection position of the suppressing light projection system; the two sets of projection image sets may have different numbers of images and projection positions.

[0010] Furthermore, the optimal projection image set is obtained by using two different wavelengths of light as light sources, projecting the corresponding images onto the interior of a binary photoresist containing an initiator and an inhibitor, and generating a spatial light aggregation distribution with the highest similarity to the target three-dimensional geometry through rotational exposure, thereby obtaining a three-dimensional structure with the highest printing quality.

[0011] Furthermore, the statistical principle is based on the Poisson distribution assumption of photopolymerization reaction. This assumption holds that the number of initiating or inhibiting active species generated by each projected pixel exciting each spatial point follows a Poisson distribution, and the number of active species at all spatial points is independent of each other. The sum of the equivalent initiating active species received by each spatial point from all projected pixels, i.e., the sum of the number of initiating active species minus the sum of the number of inhibiting active species, is positively correlated with the final photopolymerization reaction degree of that spatial point. The photopolymerization reaction degree of all spatial points constitutes the final three-dimensional spatial total distribution.

[0012] Furthermore, the Expectation-Maximization (EM) algorithm is used to solve for the maximum likelihood estimate containing latent variables. It uses two steps to perform alternating iterative optimization until the optimal solution is found. The iterative process includes the following two steps:

[0013] 1) Expectation step E: Given a set of parameters to be estimated, calculate the conditional expectation of the log-likelihood function of the complete data containing observed and latent variables with respect to the given observed data;

[0014] 2) Maximize step M: Solve for the maximum value of the expected value under the above conditions to obtain a new set of parameter values ​​to be estimated, and use them as known parameters for the next round of iteration.

[0015] Furthermore, the Gaussian distribution approximation assumption approximates the Poisson distribution as a Gaussian distribution in the step of solving the expectation, and derives the general expression for the expectation calculation by using the conditional distribution function of the Gaussian distribution and the correlation between the inducing and inhibiting variables.

[0016] The beneficial effects of this invention are as follows: This invention realizes an optimal projection array generation algorithm for dual-wavelength bulk additive manufacturing based on statistical principles. A mathematical model for the three-dimensional reconstruction process of bulk additive manufacturing is established based on the Poisson distribution assumption of photopolymerization reactions. A dual-wavelength optimal projection calculation model based on maximum likelihood estimation is established by equating the target geometry to the three-dimensional spatial distribution of photopolymerization reactions. An iterative optimization algorithm for projection generation is derived by applying the expectation-maximization principle and the Gaussian distribution assumption. This invention can calculate the corresponding optimal dual-wavelength projection image set for any dual-wavelength photosensitive material under given relevant quantum parameters. Furthermore, the model of this invention has strong adaptability to projection system structures, making it applicable to various projection exposure modes in dual-wavelength bulk additive manufacturing. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0018] Figure 1 This is a schematic diagram illustrating the principle of the forward transmission model.

[0019] Figure 2 Schematic diagram of two dual-wavelength volume printing exposure modes;

[0020] Figure 3 Schematic diagrams of optical paths using refractive index matching fluid and non-refractive matching fluid;

[0021] Figure 4 This is a schematic diagram illustrating the precise calculation of the actual beam.

[0022] Figure 5 A schematic diagram of the number distribution of active species particles induced by the target three-dimensional structure. Detailed Implementation

[0023] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings.

[0024] like Figure 1 As shown, this invention provides an optimization method for dual-wavelength bulk additive manufacturing based on a statistical model.

[0025] 1. Based on statistical parameter estimation theory, and combining the physicochemical mechanism of photoresist photopolymerization and the basic model of bulk additive manufacturing, this invention proposes a mathematical model for dual-wavelength bulk additive manufacturing and an optimized method for generating projection image sets. The dual-wavelength bulk additive manufacturing system uses two different wavelengths of light as light sources to project corresponding images onto the interior of a binary photoresist containing initiators and inhibitors. A stable spatial distribution of the photopolymerization reaction is formed through rotational exposure to achieve the printing of three-dimensional structures. In the dual-wavelength bulk additive manufacturing system, the initiating and inhibiting components of the binary photoresist are respectively controlled by a projection system using two different wavelengths of light. Due to the introduction of binary control, the dual-wavelength bulk additive manufacturing system has advantages over single-beam systems, including higher printing reconstruction accuracy, higher contrast, and greater adaptability to large-size and highly attenuated lithography systems.

[0026] 2. The mathematical model of the method of this invention is divided into two parts: a forward transmission model and a backward transmission model. The forward transmission model uses the projected image set as the model input and the three-dimensional photopolymerization reaction distribution as the system output, and is used to predict the printing result given the projection data; the backward transmission model uses the target three-dimensional geometry as the system input and the projected image set as the system output, and is used to solve for the optimal projection data that makes the printing result the best.

[0027] 3. Since bulk additive manufacturing technology, especially dual-wavelength systems, has a variety of possible printing modes, this invention uses a general approach to represent the mathematical model of bulk additive manufacturing.

[0028] 4. For the forward transmission model, see [link / reference]. Figure 1 First, for the projection data that triggers the light, the projection image set is represented as a vector P consisting of all n projected pixels; the three-dimensional space containing the photoresist region is represented as a spatial matrix V consisting of all m voxels. Consider any projection pixel p. j For any spatial voxel v i The contribution of photopolymerization to the reaction can be categorized into two cases:

[0029] 5. First case: Projected pixel p j The emitted beam of light passes through the spatial voxel v i At this point, the degree of photopolymerization of the spatial voxel is directly proportional to the actual light energy absorbed by the voxel, denoted as v. i =a ij p j , where a ij The scaling factor; second case: projected pixel p j The emitted beam of light did not pass through the spatial voxel v i At this time, the projected pixel corresponds to voxel v i The photopolymerization reaction does not contribute, and is equivalent to the aforementioned proportionality coefficient a. ij It is 0.

[0030] 6. Therefore, considering all projected pixels P for a given spatial voxel v i The total contribution of photopolymerization reaction can reduce voxel v i Represented as Considering all space voxels, this relationship can be expressed in matrix form as V = AP, where A is an m×n order photopolymerization contribution coefficient matrix. This is the forward transport model under the single-initiation lithography system.

[0031] 7. For a dual-wavelength binary lithography system, the above forward propagation model can be extended to V = AP - BQ, where P and Q are the projection vectors of the initiating light and the suppressing light, respectively, and A and B are the coefficient matrices of the initiating light and the suppressing light, respectively. This is the forward propagation model under the dual-wavelength binary lithography system.

[0032] 8. Based on the above mathematical model, the method of this invention is applicable to various different volume printing exposure conditions, including... Figure 2 The modes shown, including rotating exposure after the two projected beams are combined by a dichroic mirror and rotating exposure when the optical axes of the two projected beams are separated by an arbitrary angle, are also applicable to... Figure 3 As shown, the optical path modes can be used with or without a refractive index matching fluid.

[0033] 9. Furthermore, the method of the present invention is also applicable to precise calculation models that take into account the shape parameters of the projected beam, see [link to relevant documentation]. Figure 4For the various modes described above, the method of this invention only needs to calculate the respective coefficient matrices A and B according to the optical path shape parameters for each case.

[0034] 10. The following section details the backpropagation model for calculating the optimal projected image set. The backpropagation model applies statistical parameter estimation theory to solve the projection optimization problem. For this purpose, the projection vectors P and Q are considered as parameters to be estimated, and the spatial matrix V is considered as the observation result, with each voxel storing the value of the number of initiating active species particles. Based on the physicochemical properties of photoresist and photopolymerization reactions, the number of active species particles generated at any spatial location under a certain light energy irradiation can be considered to follow a Poisson distribution, and the degree of photopolymerization is proportional to the number of initiating active species particles. Therefore, the target spatial matrix V can be seen as the spatial distribution of the number of initiating active species particles in three-dimensional space; for the maximum likelihood estimation model, it is equivalent to the observation data.

[0035] 11. For a given target three-dimensional geometry, the number distribution matrix of initiating active species particles with the same geometric characteristics can be established as known observation data for the parameter estimation system. See [link to relevant documentation]. Figure 5 .

[0036] 12. Let p be the pixel that triggers the projection. j For space voxel v i The number of contributing initiating particles is a random variable X. ij Suppress pixel q k For space voxel v i The number of priming particles consumed is a random variable Y. ik Both of them obey parameters λ. ij =a ij p j and μ ik =b ik q k Poisson distribution:

[0037]

[0038] Let the space v be a vole. i The equivalent sum of the number of induced particles is a random variable V. i According to the forward transmission model, the relationship between the random variables of the particle number satisfies

[0039] 13. Since the form of solving the parameter matrices P and Q directly using maximum likelihood estimation contains latent variables, the idea of ​​expectation maximization (EM) algorithm is used for iterative parameter optimization.

[0040] 14. The objective function of the EM algorithm is the log-likelihood function of the complete data:

[0041]

[0042] Due to lnX ij ! Item and lnY ik The ! term is irrelevant to the parameter being estimated and can be ignored. Therefore, the final objective function to be optimized is:

[0043]

[0044] 15. The E-step in the Expectation Maximization (EM) algorithm requires given parameters P. current and Q current Calculate the conditional expectation of the above objective function with respect to the observed data using the log-likelihood function of the complete data:

[0045]

[0046] 16. In order to investigate X ij and Y ik To calculate the conditional expectation, considering that using the Poisson distribution to calculate the conditional expectation is quite complex and cannot be given an analytical expression, a Gaussian approximation is introduced. For cases where the Poisson distribution parameter is large (typically λ > 8), the Poisson distribution can be approximated as a Gaussian distribution with both expectation and variance λ, and the approximation error is extremely small and negligible. In this invention, the number of active species particles is relatively large, satisfying the Gaussian approximation condition, and therefore can be approximated as a Gaussian distribution.

[0047] 17. The Gaussian distribution has the property of invariance of sum and difference distributions, meaning that the sum and difference of any number of random variables following a Gaussian distribution still follow a Gaussian distribution; furthermore, the conditional distributions of two random variables are also Gaussian distributions. Let random variables Z1 and Z2 follow a Gaussian distribution: Then there is The covariance matrices of Z1 and Z2 are Then the conditional expectation of Z1 with respect to Z2 is:

[0048]

[0049] 18. For random variable X ij and Y ik Applying the Gaussian approximation, we have X ij ~N(λ) ij ,λ ij ), Y ik ~N(μ) ik ,μ ik Based on the forward transport model and the properties of the Gaussian distribution, we have V i ~N(λ) i -μ i ,λ i +μ i ),in

[0050] 18. To calculate the number of initiating active species particles X. ij Conditional expectation E(X) ij |V i =v i X needs to be calculated. ij and V i The correlation coefficient. Introducing the intermediate variable T = V. i -X ij To remove the light-inducing pixel X ij The sum of the number of active species particles generated by all pixels other than the one that has this effect is X. ij and V i The covariance is:

[0051] Cov(X ij V i )=E(X ij V i )-E(X ij )E(V i )=E[X ij (X ij +T)]-E(X ij )E(X ij +T).

[0052] According to the forward transmission model, X... ij Since T and T are independent, the above equation becomes:

[0053]

[0054] The correlation coefficient is thus obtained:

[0055]

[0056] 18. To calculate the number of active species particles Y consumed. ik Conditional expectation E(Y) ik |V i =v i ), need to calculate Y ik and V i The correlation coefficient. Introducing the intermediate variable W = V. i +Y ik To suppress light pixel Y ik The sum of the number of active species particles generated by all pixels other than those involved in the action, then Y ik and V i The covariance is:

[0057] Cov(Y ik V i )=E(Y ik V i )-E(Yik )E(V i )=E[Y ik (WY ik )]-E(Y ik )E(WY ik ).

[0058] Similarly, Y ik If W and W are independent, then the above equation becomes:

[0059]

[0060] The correlation coefficient is calculated as follows:

[0061]

[0062] 19. Combining the above derivation with the conditional expectation expression of the Gaussian distribution, X can be finally determined. ij and Y ik Conditional expectation:

[0063]

[0064]

[0065] 20. In the E-step of the Expectation Maximization (EM) algorithm, given a set of currently known parameters P... current and Q current It can calculate the current and Based on the above conclusions regarding conditional expectations, the current conditional expectation value can be calculated, and then the current optimization objective function E(L(X,Y;P)) can be calculated. current Q current )|V=V).

[0066] 21. In the Expectation-Maximization (EM) algorithm, the M-step calculates the extreme points of the above objective function as the parameters for the next iteration, i.e., the new optimization parameters are:

[0067] P next Q current =argmaxE(L(X,Y;P) current Q current )|V=V).

[0068] 22. For the first round of optimization, P and Q can be initialized to arbitrary values. In subsequent iterations, the above process is repeated to optimize the parameter matrices P and Q until convergence to the optimal solution. This constitutes the entire process of the backpropagation model.

[0069] For any given binary photopolymer material and the target three-dimensional geometry to be manufactured, the forward and backward transmission models of this invention can generate two sets of optimal projection images, so that applying these image sets to the binary photopolymer material for bulk additive manufacturing can obtain the best three-dimensional printed structure.

[0070] 23. The above algorithm is applicable to any type of dual-wavelength bulk additive manufacturing exposure system. For example... Figure 2 As shown, two basic exposure system layouts are illustrated. Projector 1 is a projection optics system using an induction wavelength light source, and projector 2 is a projection optics system using a suppression wavelength light source. The binary photoresist 3 is located within a cylindrical space. Figure 2 In the left-hand diagram, the dichroic mirror 4 functions to combine the colors of the dual-wavelength projection beams. In the right-hand diagram, the optical axes of the two projectors are at any angle. Figure 3 The diagram illustrates the optical path scenarios with and without refractive index matching. When the refractive index matching liquid 5 is placed in the square container, its refractive index is equal to that of the binary photoresist, maintaining the conical shape of the projected beam, making the geometric calculation relatively simple. Without the refractive index matching liquid, the path of each projected ray needs to be calculated based on the shape parameters of the photoresist container, making the geometric calculation relatively complex. Adjusting the projection image plane 6 to the center plane of the photoresist container maximizes the resolution. In all the above cases, the projected beam is calculated as an equivalent straight line. To further improve accuracy, a precise model of the beam can be established based on the optical parameters of the projection system, such as... Figure 4 As shown. In this case, the light emitted by each projected pixel is no longer a straight line, but a cone-shaped beam.

[0071] 24. For any of the above hardware system scenarios, the mathematical model and algorithm flow remain unchanged. Only the corresponding contribution matrices A and B need to be calculated based on the geometric relationships of the different hardware system structures. The algorithm is implemented using matrices A and B as known parameters, which are pre-calculated structural parameters for any hardware system and are independent of the 3D model to be printed. Therefore, this algorithm is universally applicable to dual-wavelength bulk additive manufacturing systems.

[0072] The above embodiments are used to explain and illustrate the present invention, but not to limit the present invention. Any modifications and changes made to the present invention within the spirit and scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A statistical model-based optimization method for dual-wavelength bulk additive manufacturing, characterized in that, This method consists of two parts: a forward transmission model and a backward transmission model. The forward transmission model uses a set of projected images as input and the distribution of photopolymerization reaction in three-dimensional space as output, and is used to predict the printing results given the projection data. The backward transfer model uses the target 3D geometry as input and the projected image set as output. It is used to solve for the optimal projected data that makes the printing result best by combining the expectation-maximization (EM) algorithm with statistical parameter estimation theory and the Gaussian distribution approximation assumption. Based on the forward transfer model and the backward transfer model, for any given binary photopolymer material and the target 3D geometry to be manufactured, two sets of optimal projected image sets are generated. The binary photopolymer material is then subjected to bulk additive manufacturing to obtain the optimal 3D printed structure. The Expectation-Maximization (EM) algorithm is used to solve for maximum likelihood estimation containing latent variables. It uses two steps to perform alternating iterative optimization until the optimal solution is found. The iterative process includes the following two steps: 1) Expectation step E: Given a set of parameters to be estimated, calculate the conditional expectation of the log-likelihood function of the complete data containing observed and latent variables with respect to the given observed data; 2) Maximization step M: Solve for the maximum value of the expected value under the above conditions to obtain a new set of parameter values ​​to be estimated, and use them as known parameters for the next round of iteration; The binary photopolymer material is composed of prepolymers and monomers, and is a liquid substance containing two photosensitive components: a photoinitiator and a photoinhibitor. Under light radiation of a certain wavelength, the photoinitiator in the binary photopolymer material generates initiating active species, which initiate the photopolymerization reaction between the prepolymer and monomers in the system. Under light radiation of a different wavelength, the photoinhibitor generates inert active species, which consume the initiating active species and thus inhibit the photopolymerization reaction. For any point in the binary photopolymer material, when the number of photopolymerization reactions exceeds the curing threshold of the binary photopolymer material, that point will cure. The statistical parameter estimation theory is based on the Poisson distribution assumption of photopolymerization reaction. This assumption holds that the number of initiating or inhibiting active species generated by each projected pixel exciting each spatial point follows a Poisson distribution, and the number of active species at all spatial points is independent of each other. The sum of the equivalent initiating active species received by each spatial point from all projected pixels, that is, the sum of the number of initiating active species minus the sum of the number of inhibiting active species, is positively correlated with the final photopolymerization reaction degree of that spatial point. The photopolymerization reaction degree of all spatial points constitutes the final three-dimensional spatial total distribution.

2. The dual-wavelength bulk additive manufacturing optimization method based on a statistical model according to claim 1, characterized in that: The target three-dimensional geometric structure is a volumetric three-dimensional model in the form of a three-dimensional matrix. Each voxel in the volumetric three-dimensional model represents a point in space, and the value of the voxel indicates whether the point is part of the final structure to be manufactured.

3. The dual-wavelength bulk additive manufacturing optimization method based on a statistical model according to claim 1, characterized in that: The optimal projection image set consists of two sets of projection image sets. One set consists of images corresponding to each projection position of the initiating light projection system, and the other set consists of images corresponding to each projection position of the suppressing light projection system. The two sets of projection image sets can have different numbers of images and projection positions.

4. The dual-wavelength bulk additive manufacturing optimization method based on a statistical model according to claim 3, characterized in that: The optimal projection image set is obtained by using two different wavelengths of light as light sources, projecting the corresponding images onto the interior of a binary photoresist containing an initiator and an inhibitor, and generating a spatial light aggregation distribution with the highest similarity to the target three-dimensional geometry through rotational exposure, thereby obtaining a three-dimensional structure with the highest printing quality.

5. The dual-wavelength bulk additive manufacturing optimization method based on a statistical model according to claim 1, characterized in that: The Gaussian distribution approximation assumption approximates the Poisson distribution as a Gaussian distribution in the step of solving the expectation, and derives the expression for calculating the expectation by using the conditional distribution function of the Gaussian distribution and the correlation between the inducing and inhibiting variables.

Images