A three-dimensional point cloud generation method and system based on a learnable heat dissipation operator

Abstract

This application provides a method and system for generating 3D point clouds based on learnable heat dissipation operators. The method includes: mapping the local neighborhood of each 3D point cloud and constructing a discrete graph Laplacian operator; sequentially applying an anisotropic diffusion coefficient matrix to the point cloud, blurring it, adding Gaussian noise, establishing a probability distribution transition process, and obtaining a noisy heat dissipation state; inputting a frequency-aware model, achieving signal frequency decomposition through multi-directional window attention, and limiting high-frequency energy amplification through adaptive spectral filtering to obtain stable spatial state features; constructing an operator learning module based on frequency domain branching to fit the mapping from stable spatial state features to the evolution trajectory of the inverse heat dissipation equation; and decoding in parallel through the operator learning module, simultaneously sampling the point cloud state at multiple time steps, and generating the target 3D point cloud based on multiple intermediate diffusion states. This application improves the structural fidelity and local detail representation capability of 3D point cloud generation, enhances the stability of few-step sampling, and improves the generation quality.

Description

Technical Field

[0001] This application relates to the fields of computer vision and 3D geometric generation model technology, specifically to a method and system for generating 3D point clouds based on learnable heat dissipation operators. Background Technology

[0002] 3D point clouds represent objects by means of discrete data points acquired in 3D space. They inherently contain the geometric properties and topological structure of 3D entities and convey rich, fine-grained semantics in local details. High-quality 3D point cloud generation and completion are fundamental to achieving high-precision reconstruction and synthesis of real-world scenes, and are widely used in downstream tasks such as autonomous driving, robot navigation, 3D animation production, and industrial design.

[0003] Currently, most existing point cloud generation models directly transplant generation techniques that have achieved success in the 2D image domain, most typically diffusion models. Traditional point cloud diffusion models, such as PVD, LION, and DiT-3D, usually utilize end-to-end normal Gaussian Markov chains to synthesize 3D shapes in latent space or coordinate space. However, these methods that directly transplant traditional generation mechanisms treat point clouds as static sets of coordinates, failing to model the dynamic evolution of the underlying manifold, and thus struggling to capture complex geometric topologies. This results in a lack of robustness in structural fidelity in existing 3D generation processes, especially in scenarios with few inference steps, where point clouds are highly susceptible to structural collapse.

[0004] Recently, some studies have attempted to introduce physical heat conduction equations to simulate the "fuzzy" effect in the forward degradation process. However, existing heat equation methods mainly use isotropic heat kernel operators, which can only perform radially symmetric filtering, ignoring the directionality of local features and failing to effectively preserve the local gradients and fine semantics of three-dimensional geometry. Furthermore, solving the inverse heat dissipation equation is mathematically a typical ill-conditioned problem, with its spectral energy exponentially amplified in high-frequency components, leading to extreme instability in the inverse generation process and requiring extremely long iterations to converge.

[0005] A search revealed a patent application (application number 202311572637.3) that discloses a method for generating 3D point clouds based on a diffusion model. This method employs a spatial domain network to progressively add noise and sequentially predict the noise to complete the generation. However, its reverse reconstruction process lacks frequency domain regularization constraints, leading to the exponential amplification of high-frequency components during inverse diffusion, causing distortion of the point cloud structure. Furthermore, the sequential iteration mechanism results in layer-by-layer error accumulation and low inference efficiency, making it difficult to achieve stable, high-fidelity reconstruction with few sampling steps.

[0006] Therefore, how to break the limitation of isotropic diffusion and overcome the ill-conditioned instability of the inverse heat dissipation equation to achieve high-fidelity and robust 3D point cloud generation and completion with fewer steps has become a technical problem that urgently needs to be solved in this field. Summary of the Invention

[0007] To address one of the shortcomings in the existing technology, the purpose of this application is to provide a method and system for generating three-dimensional point clouds based on learnable heat dissipation operators.

[0008] A first aspect of this application provides a method for generating 3D point clouds based on learnable heat dissipation operators, comprising: Acquire 3D point cloud data; A discrete graph Laplacian operator is constructed by mapping the local neighborhood of each of the three-dimensional point clouds. The anisotropic diffusion coefficient matrix is ​​learned using an adaptive network; The point cloud is sequentially blurred using the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator, and Gaussian noise is added to establish a probability distribution transfer process to obtain the noisy heat dissipation state. The noisy heat dissipation state is input into the frequency sensing model, and the signal frequency is decomposed through a multi-directional window attention module. Then, the high-frequency energy amplification is limited by adaptive spectrum filtering to obtain stable spatial state characteristics. Based on the frequency-aware model, an operator learning module is constructed to fit the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation; Based on the mapping, the operator learning module is used to perform parallel decoding and simultaneously sample the point cloud state at multiple time steps to obtain the target 3D point cloud.

[0009] Optionally, the step of constructing a discrete graph Laplacian operator by mapping the local neighborhood of each of the three-dimensional point clouds includes: Using points in a 3D point cloud as graph nodes, and taking any one of the graph nodes as the center node, the spatial distance between the center node and each adjacent point is calculated based on geometric features. The k nearest neighbor points are selected in ascending order of distance to construct a k nearest neighbor graph, where k is a positive integer. The edge weights between the central node and each of the nearest neighbors are calculated using Gaussian radial basis functions to characterize local geometric similarity; Based on the k-nearest neighbor graph and the edge weights, a discrete graph Laplacian operator is constructed to describe the geometric smoothness of point clouds within their local neighborhoods.

[0010] Optionally, the anisotropic diffusion coefficient matrix is ​​a positive semi-definite tensor. ; The half-point positive definite point-level tensor Parameterized by Cholesky decomposition ,in, It is a lower triangular matrix. for The transpose of the matrix; the lower triangular matrix The diagonal elements are used to control the independent heat dissipation rates in different spatial directions, while the off-diagonal elements are used to control the heat dissipation coupling relationship between different spatial directions. The diagonal elements of the lower triangular matrix L are composed of fixed positive constants superimposed with the prediction parameters of the point-level multilayer perceptron based on the local geometric features of the three-dimensional point cloud, to ensure that the diagonal elements are always positive; the off-diagonal elements of the lower triangular matrix L are composed of the corresponding prediction parameters directly assigned.

[0011] Optionally, the step of sequentially applying the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator to the point cloud for blurring and adding Gaussian noise to establish a probability distribution transition process and obtain a noisy heat dissipation state includes: Based on the anisotropic diffusion coefficient matrix and the discrete graph Laplace operator Establish an anisotropic heat dissipation equation, in time At that time, three-dimensional coordinates The anisotropic heat dissipation equation at a specific location is expressed as follows:

[0012] in, Indicates time Continuous geometric features of the 3D point cloud at the location; The anisotropic heat dissipation equation is used to analyze the input point cloud features. Dynamic blur degradation is performed to gradually smooth high-frequency local details in the point cloud during the forward process. The anisotropic heat dissipation equation based on the frequency variable is obtained through Fourier transform. Frequency domain analytical solution:

[0013] in, and These represent the solutions for heat dissipation at the initial time and the solutions at the final time, respectively. The frequency domain spectrum characterization at that location shows a characteristic of exponential smooth decay in the high-frequency band within the frequency domain space; Let the discrete-time series of the positive probability distribution transition process be... , , , ,in, Indicates the initial time. Indicates the final heat dissipation time; In the i-th forward transition step, according to the time interval The anisotropic diffusion coefficient matrix and the discrete graph Laplace operator Based on the exponential heat kernel in the analytical solution Constructing a linear heat dissipation operator under a first-order implicit backward difference scheme for:

[0014] The point cloud state at time step i-1 Input the linear heat dissipation operator, and obtain the average heat dissipation state at the i-th time step by solving the following linear equation. :

[0015] in, This represents the mean state of the point cloud after only thermal dissipation blurring. In the mean state of heat dissipation Gaussian random perturbation is superimposed on the data to obtain the point cloud state at the i-th time step. The variance of the Gaussian random perturbation is determined by... control; The probability distribution transition process between adjacent discrete time steps is represented as follows:

[0016] Substituting into the mean state expression, we get:

[0017] in, Indicates the first Point cloud state at each time step; According to the described forward transition process, from the initial point cloud state Calculate in sequence , , , This causes the point cloud state to be gradually subjected to the combined effects of thermal dissipation ambiguity and Gaussian random perturbation during the forward process, and the final time step obtained As a state of heat dissipation with added noise.

[0018] Optionally, the frequency-aware model includes an embedding module, a multi-directional window attention module, a feedforward network module, and an adaptive spectrum filtering module; The noisy heat dissipation state is input into the frequency sensing model, and the following operation is performed: The noisy heat dissipation state is encoded into a voxelized three-dimensional feature sequence; The embedding module performs conditional embedding and time-step embedding on the three-dimensional feature sequence to obtain the input features. The input features are input into the multi-directional window attention module, and the input features are modeled at multiple scales using three-dimensional windows with different axial sizes to obtain decoupled multi-band spatial topological features. The multi-band spatial topology features output by the multi-directional window attention module are input into the feedforward network module to obtain spatial state features. The spatial state features are input into the adaptive spectrum filtering module, and the filtered spatial state features are obtained through Fourier transform, frequency domain filtering, high-frequency component suppression and inverse Fourier transform. The multi-directional window attention module and the adaptive spectrum filtering module are used together to limit the amplification of high-frequency components during the inverse reconstruction process, thereby achieving frequency domain constraints on the inverse heat dissipation process.

[0019] Optionally, the step of achieving signal frequency decomposition through a multi-directional window attention module to obtain decoupled multi-band spatial topological features includes: The input feature sequence is projected along the channel dimension to multiple attention heads, and the multi-head attention mechanism is divided into 8 parallel feature processing groups. Each feature processing group is independently configured with a three-dimensional window size, so that the eight groups correspond to different frequency perception scales and orientation preferences. Each feature group performs attention calculations in parallel within its corresponding window size, extracting low-frequency global structural features, mid-frequency directional structural features, or high-frequency local detail features, respectively. The attention feature vectors output by the parallel computation of the eight feature groups are concatenated and aggregated to obtain decoupled multi-band spatial topological features.

[0020] Optionally, limiting high-frequency energy amplification through adaptive spectral filtering includes: The spatial state features output by the feedforward network module are obtained, and they are mapped to the frequency domain using the fast Fourier transform to obtain the complex frequency domain feature matrix. A learnable spectral filter matrix is ​​constructed and multiplied element-wise with the complex frequency domain feature matrix to apply adaptive weight constraints in the frequency domain. Based on the complex frequency domain feature matrix after weight constraints, adaptive truncation or energy suppression is performed on the high-frequency components of the complex frequency domain feature matrix that exceed the preset hard cutoff frequency threshold, thereby blocking the high-frequency noise amplification path in the reverse reconstruction process. The filtered frequency domain features are restored to the spatial domain through inverse fast Fourier transform, and the filtered spatial state features are output.

[0021] Optionally, the step of constructing an operator learning module based on the frequency-aware model to fit the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation includes: Based on the spatial and frequency domain processing mechanisms of the frequency-aware model, an operator learning module, namely the Fourier neural operator framework, is constructed. The framework The inter-layer update of layer l is represented as:

[0022] in, Indicates the first Layer features, Indicates the first +1 layer features; Represents a non-linear activation function. Represents spatial transformation operators, This represents a learnable spectral filtering matrix. and The spatial transformation operator represents the Fourier transform and inverse transform, respectively. It includes a multi-directional window attention module and a feedforward network module for extracting multi-scale spatial structure features; the learnable spectral filtering matrix Used to approximate the integral operator in the frequency domain to enhance the operator learning module's ability to approximate continuous-time inverse heat dissipation trajectories; The noisy heat dissipation state at the final time step Input the Fourier neural operator framework From the noisy heat dissipation state Gradually restore to the target point cloud state The sequence is defined as the inverse heat dissipation evolution trajectory g; The inverse heat dissipation evolution trajectory g at M target time steps , , , Discrete sampling is performed to obtain the target point cloud state corresponding to M target time steps; Using the Fourier neural operator framework Fit from Mapping to the states of the M target point clouds; The training phase of the operator learning module specifically includes: Calculate the error between the predicted point cloud state and the target point cloud state at the M target time steps, introduce a weighting coefficient that varies with time step to combine the errors at each time step, and construct and optimize the diffusion loss target. Specifically: ; The inference phase of the operator learning module specifically includes: The trained operator learning module is used for actual generation, and the noisy heat dissipation state is then applied. After performing reverse prediction and inference of a set number of steps, the final reconstruction result is output. .

[0023] Optionally, the three-dimensional point cloud data includes complete data and incomplete data; When complete data is input, the method performs unconditional or class-conditional point cloud generation based on sampled noise; When inputting incomplete data that is partially missing, sparse, or partially observed, the method performs 3D point cloud completion, specifically as follows: A point-voxel convolutional neural network was used to extract multi-scale geometric features and missing context features of the observable region. The multi-scale geometric features and missing context features are dynamically injected into the frequency-aware model through a three-dimensional adaptive layer normalization mechanism, guiding the inverse heat dissipation evolution trajectory to converge toward the target missing manifold and generating a complete point cloud.

[0024] A second aspect of this application provides a 3D point cloud generation system based on a learnable heat dissipation operator, comprising: The data acquisition module acquires 3D point cloud data; The local graph construction module constructs a discrete graph Laplacian operator by constructing a graph in the local neighborhood of each of the three-dimensional point clouds. The diffusion coefficient learning module uses an adaptive network to learn the anisotropic diffusion coefficient matrix. The probability transfer module sequentially applies the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator to the point cloud, blurs it, adds Gaussian noise, establishes a probability distribution transfer process, and obtains the noisy heat dissipation state. The frequency-aware reconstruction module inputs the noisy heat dissipation state into the frequency-aware model, decomposes the signal frequency through the multi-directional window attention module, and limits high-frequency energy amplification through adaptive spectrum filtering to obtain stable spatial state characteristics. The operator learning module is constructed based on the frequency-aware model to fit the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation; The point cloud output module, based on the mapping, uses the operator learning module to perform parallel decoding and simultaneously samples the point cloud state at multiple time steps to obtain the target 3D point cloud.

[0025] The 3D point cloud generation method based on learnable heat dissipation operators provided in this application improves the structural fidelity and local detail representation ability of 3D point cloud generation, while effectively enhancing the stability and generation quality of low-step sampling. It can be applied to tasks such as 3D point cloud generation and completion.

[0026] Other technical effects resulting from the additional features will be further illustrated in the corresponding embodiments. Attached Figure Description

[0027] Other features, objects, and advantages of this application will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings: Figure 1 This is a flowchart illustrating a method for generating 3D point clouds based on a learnable heat dissipation operator according to an exemplary embodiment; Figure 2 This is a schematic diagram illustrating the process of constructing a local neighborhood graph, establishing a discrete graph Laplacian operator, and parameterizing the anisotropic diffusion coefficient matrix, as an exemplary embodiment. Figure 3 This is a schematic diagram illustrating the positive probability distribution transfer process of fusing a heat dissipation model with a Gaussian noise addition process, as shown in an exemplary embodiment. Figure 4 A framework diagram of a frequency-aware inverse reconstruction model is shown as an exemplary embodiment, including a multi-directional window attention module, an adaptive spectral filtering module, and an operator learning module; Figure 5 This is a structural diagram of a 3D point cloud generation system based on a learnable heat dissipation operator, as shown in an exemplary embodiment. Detailed Implementation

[0028] The present application will now be described in detail with reference to specific embodiments. These embodiments will help those skilled in the art to further understand the present application, but do not limit the present application in any way. It should be noted that those skilled in the art can make several modifications and improvements without departing from the concept of the present application, and these all fall within the protection scope of the present application. Parts not described in detail in the following embodiments can be implemented using existing technology.

[0029] In existing technologies, it is impossible to overcome the ill-conditioned instability of the inverse heat dissipation equation while breaking the limitation of isotropic diffusion. Based on the above problems, embodiments of this application provide a three-dimensional point cloud generation method based on a learnable heat dissipation operator to solve the aforementioned issues.

[0030] Reference Figure 1 As shown in one embodiment of this application, a method for generating three-dimensional point clouds based on learnable heat dissipation operators is provided, including the following steps: S100, acquire 3D point cloud data; S200 constructs a discrete graph Laplacian operator by mapping the local neighborhood of each 3D point cloud. S300 utilizes an adaptive network to learn the anisotropic diffusion coefficient matrix; S400, the point cloud is successively blurred by applying the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator, and Gaussian noise is added to establish the probability distribution transfer process and obtain the noisy heat dissipation state. S500 inputs the noisy heat dissipation state into the frequency sensing model, decomposes the signal frequency through a multi-directional window attention module, and limits high-frequency energy amplification through adaptive spectrum filtering to obtain stable spatial state characteristics. S600, based on the frequency-aware model, constructs an operator learning module to fit the mapping from steady-state features to the evolution trajectory of the inverse heat dissipation equation; The S700, based on mapping, uses an operator learning module to perform parallel decoding, simultaneously sampling the point cloud state at multiple time steps to obtain the target 3D point cloud.

[0031] The above embodiments, through the processing flow from S100 to S700, combine local neighborhood graph modeling, anisotropic heat dissipation modeling, forward probability distribution transfer, frequency-aware reverse reconstruction, and operator learning parallel sampling, enabling the point cloud generation process to simultaneously possess local geometric structure constraints, directional adaptive diffusion capabilities, and frequency domain stable reconstruction capabilities. Therefore, in the forward degradation stage of the point cloud, it can more accurately simulate the smooth evolution of local geometric details, and in the reverse reconstruction stage, it can suppress high-frequency noise amplification and reduce the accumulation of step-by-step sampling errors, thereby improving the structural fidelity, local detail representation ability, and few-step sampling stability of the generated 3D point cloud.

[0032] In some specific embodiments of this application, the three-dimensional point cloud data obtained in S100 can be complete three-dimensional point cloud data, partially missing three-dimensional point cloud data, sparse three-dimensional point cloud data, or partially observed three-dimensional point cloud data.

[0033] 3D point cloud data can be represented as ,in, Let represent the i-th point, and n represent the number of points in the point cloud. Each point It can include three-dimensional coordinates, and may further include normal vectors, color, intensity, category labels or other point-level attributes.

[0034] The above embodiments provide the data foundation for 3D point cloud processing.

[0035] In order to enable explicit modeling of local geometric neighborhood relationships in point clouds, in some specific embodiments of this application, S200, a graph is constructed in the local neighborhood of each 3D point cloud, and a discrete graph Laplacian operator is constructed, such as... Figure 2 As shown, the following steps can be taken: S201: Using points in the 3D point cloud as graph nodes, for any point as the center node, calculate the spatial distance between the geometric features of the points and the adjacent points, sort them in ascending order of spatial distance, and select k neighboring points corresponding to the center node, where k is a preset positive integer.

[0036] Specifically, geometric features can be three-dimensional spatial coordinates or features extracted from three-dimensional spatial coordinates; Specifically, a k-nearest neighbor graph can be constructed by selecting the k nearest points based on the Euclidean distance between points. In other implementations, feature distance, or a fusion of spatial distance and feature distance, can also be used to determine neighborhood associations, thereby constructing a local neighborhood graph.

[0037] S202, construct a k-nearest neighbor graph, where the central node is... It is connected to all k neighboring points by edges, and the weight of each edge is calculated using the Gaussian radial basis function of spatial distance, with the specific formula as follows:

[0038] in, Indicates the central node Geometric features, Representing neighborhood points Geometric features, Indicates the central node The local neighborhood of , where c represents the Gaussian filter scaling parameter.

[0039] S203, a discrete graph Laplacian operator is established based on the k-nearest neighbor graph to describe the smooth transformation relationship of the point cloud in the local neighborhood. The specific formula is as follows:

[0040] In the above embodiments, the local neighborhood of each central node is determined by a k-nearest neighbor graph, and weights are assigned to the neighborhood edges using Gaussian radial basis functions. This allows neighboring points that are closer or have more similar geometric features to contribute more to local smooth propagation. A discrete graph Laplacian operator is constructed based on the k-nearest neighbor graph and edge weights, which can explicitly express the local geometric adjacency relationships of the point cloud, providing a discrete geometric propagation structure for the subsequent heat dissipation process. This improves the adaptability of heat dissipation modeling to local surfaces, edges, and detailed structures.

[0041] To enable the heat dissipation process to adaptively adjust the diffusion direction and intensity based on the local geometry of the point cloud, this application utilizes an adaptive network model to predict the anisotropic diffusion coefficient matrix K. In some specific embodiments of this application, S300, the anisotropic diffusion coefficient matrix is ​​learned using an adaptive network, such as... Figure 2 As shown, the following steps can be taken: S301 clarifies that the anisotropic diffusion coefficient matrix K is a point-level tensor matrix used to characterize the thermal diffusion coupling relationship between different directions in three-dimensional space. K is the diffusion coefficient matrix in the heat dissipation equation, distinct from the exponential form of the heat core. .

[0042] S302 extracts local geometric features of each point through a point-level multilayer perceptron (the adaptive network model is a point-level multilayer perceptron) and predicts a six-dimensional parameter vector. The vector is mapped to the three diagonal elements and three off-diagonal elements of the lower triangular matrix L.

[0043] S303 applies the Sigmoid activation function to the parameter vector and superimposes a fixed positive constant on the diagonal element parameters to ensure that the diagonal elements of L are always positive; S304, the lower triangular matrix L is assembled based on the constrained parameters and calculated using Cholesky parametric methods. Construct an anisotropic diffusion coefficient matrix K that satisfies the positive semidefinite constraint.

[0044] Specifically, matrix K is used to substitute into the heat dissipation control equation. The diagonal elements of L control the heat dissipation rate in independent spatial directions, while the off-diagonal elements control the coupling between directions, thereby improving the diffusion stability and geometric adaptability of the heat dissipation model in different spatial directions.

[0045] In the above embodiments, the lower triangular matrix L is adaptively predicted using a point-level multilayer perceptron, and then... An anisotropic diffusion coefficient matrix K is constructed, satisfying a positive semi-definite constraint to avoid anomalous diffusion caused by instability of the diffusion coefficient matrix. The diagonal elements in the lower triangular matrix L control the independent diffusion rates in different spatial directions, while the off-diagonal elements control the coupled diffusion relationships between different directions. This enables the heat dissipation model to perform directional adaptive diffusion based on the local geometry of the point cloud, improving its ability to represent the local directional structure and complex geometric details of 3D point clouds.

[0046] To construct a smooth degradation path that conforms to the local geometric topology, thereby improving the fidelity of the reverse reconstruction process in restoring the global structure and local details of the 3D point cloud, in some specific embodiments of this application, in S400, the point cloud is sequentially blurred using an anisotropic diffusion coefficient matrix and a discrete graph Laplacian operator, and Gaussian noise is added to establish a probability distribution transfer process, thereby obtaining a noisy heat dissipation state, such as... Figure 2 and Figure 3 As shown, the following steps can be taken: S401, based on the anisotropic diffusion coefficient matrix and discrete graph Laplace operator Establish an anisotropic heat dissipation equation, in time At that time, three-dimensional coordinates The anisotropic heat dissipation equation at a specific location is expressed as follows:

[0047] in, Indicates time The continuous geometric features of the three-dimensional point cloud at that location.

[0048] S402 utilizes anisotropic heat dissipation equations to analyze the input point cloud features. Dynamic blur degradation is performed to gradually smooth high-frequency local details in the point cloud during the forward process. An anisotropic heat dissipation equation based on the frequency variable is obtained through Fourier transform. Frequency domain analytical solution:

[0049] in, and These represent the solutions for heat dissipation at the initial time and the solutions at the final time, respectively. The frequency domain spectrum characterization at that location shows a characteristic of exponential smooth decay in the high-frequency band within the frequency domain space; S403, assuming the discrete-time series of the positive probability distribution transition process is... , , , ,in, Indicates the initial time. Indicates the final heat dissipation time; S404, in the i-th forward transition step, according to the time interval Anisotropic diffusion coefficient matrix and discrete graph Laplace operator Based on the exponential heat kernel in the analytical solution Constructing a linear heat dissipation operator under a first-order implicit backward difference scheme for:

[0050] The point cloud state at time step i-1 Input a linear heat dissipation operator, and obtain the average heat dissipation state at the i-th time step by solving the following linear equation. :

[0051] in, This represents the mean state of the point cloud after only thermal dissipation blurring. S405, under average heat dissipation state Gaussian random perturbation is superimposed on the data to obtain the point cloud state at the i-th time step. The variance of the Gaussian random perturbation is determined by... control; The probability distribution transition process between adjacent discrete time steps is represented as follows:

[0052] Substituting into the mean state expression, we get:

[0053] in, Indicates the first The point cloud state at each time step is transitioned from the initial point cloud state according to the forward transition process. Calculate in sequence , , , This causes the point cloud state to be gradually subjected to the combined effects of thermal dissipation ambiguity and Gaussian random perturbation during the forward process, and the final time step obtained As a state of heat dissipation with added noise.

[0054] In the above embodiments, a linear heat dissipation operator is constructed using an implicit backward difference scheme, and this linear heat dissipation operator is used to solve for the heat dissipation mean state at each time step, giving the point cloud blurring degradation in the forward process a clear geometric evolution constraint. Superimposing Gaussian random perturbations on the heat dissipation mean state combines the deterministic geometric smoothing process with the random noise modeling process, ensuring that the obtained noisy heat dissipation state simultaneously contains local geometric degradation information and the random perturbation information required by the diffusion model. This improves the generative model's ability to express the global structure and local details of the 3D point cloud.

[0055] The forward heat dissipation process has a smoothing effect on high-frequency local details in the frequency domain, while the reverse reconstruction process needs to recover the initial point cloud structure from the noisy heat dissipation state. Therefore, there is a risk of high-frequency components being amplified during the inverse heat dissipation solution process. To address this issue, this application further introduces Fourier regularization and adaptive spectral filtering in the reverse reconstruction process to improve the stability of the reverse process. Some specific embodiments of this application, such as... Figure 4 As shown, the noisy heat dissipation state is input into the frequency sensing model. The frequency sensing model can adopt a three-dimensional diffusion Transformer structure, or other neural network structures capable of processing three-dimensional feature sequences or three-dimensional feature maps.

[0056] In this embodiment, the frequency-aware model includes a point cloud embedding module, a time step embedding module, a multi-directional window attention module, a feedforward network module, an adaptive spectrum filtering module, and an output module. The output module is used to output the point cloud state corresponding to the current time step or multiple target time steps.

[0057] Furthermore, based on the structure of the frequency sensing model described above, the S500 inputs the noisy heat dissipation state into the frequency sensing model, performs signal frequency decomposition through a multi-directional window attention module, and limits high-frequency energy amplification through adaptive spectral filtering to obtain stable spatial state characteristics. This can be achieved through the following steps: S501, the input point cloud state input point cloud embedding module and the time step embedding module are jointly encoded, and spatial geometric features and diffusion time step information are fused to generate an initial three-dimensional feature map; S502, normalizes and represents the initial 3D feature map as a feature map. Where C represents the number of feature channels, and D, H, and W represent spatial depth, height, and width, respectively.

[0058] S503 inputs the feature map F into the multi-directional window attention module, and performs multi-frequency spatial modeling of the 3D features based on the multi-scale window receptive field, thereby achieving explicit decoupling between the global structure and local details.

[0059] Specifically, the input feature map F is projected along the channel dimension onto multiple attention heads, and the multi-head mechanism is divided into eight parallel multi-scale feature groups. Each feature group uses a three-dimensional window size for attention computation. The three-dimensional window size includes: ; ; ; ; ; ; ; Where s represents the preset base window size, for example, s can be set to 4.

[0060] In the above window naming, L and H represent low-frequency perception tendency (corresponding to a large-size window 4s) and high-frequency perception tendency (corresponding to a small-size window s), respectively; the three letters correspond to the D / H / W axes in three-dimensional space, with each letter independently setting the window size and frequency band preference along that axis; WA is an abbreviation for Window Attention. When the three letters are all L (LLL-WA), an isotropic large window is constructed to extract low-frequency global structure; when they are all H (HHH-WA), an isotropic small window is constructed to extract high-frequency local details; when they are a mixture of L and H (such as HLL-WA, HLH-WA, etc.), an anisotropic window with asymmetric axial size is constructed to specifically capture mid-frequency directional structural features that vary along a specific direction. Thus, the eight window modes work in parallel, achieving explicit decoupling of three-dimensional spatial frequency bands and directional topology.

[0061] Each feature group performs attention calculations in parallel within its corresponding window size, extracting low-frequency global structural features, mid-frequency directional structural features, or high-frequency local detail features, respectively. The attention feature vectors output from the parallel computation of the eight feature groups are concatenated and aggregated to obtain decoupled multi-band spatial topological features.

[0062] Larger 3D windows are used to extract low-frequency global structural features, while smaller 3D windows are used to extract high-frequency local detail features. Windows with different axial proportions are used to extract mid-frequency structural features that vary along different spatial directions. Through the aforementioned multi-directional window attention module, multi-band spatial topological features can be obtained, enabling the reverse reconstruction process to simultaneously ensure global shape consistency and local detail recovery capabilities.

[0063] S504 inputs the multi-band spatial topology features output by the multi-directional window attention module into the feedforward network module to obtain spatial state features; Specifically, the feedforward network module performs channel transformation and nonlinear mapping on the features output by the multi-directional window attention module. The feedforward network module may include linear mapping layers, nonlinear activation layers, and residual connection structures, which are used to expand the feature channel dimension, enhance feature expressiveness, and preserve input features while stabilizing network training, respectively. After processing by the feedforward network module, spatial state features are obtained for subsequent adaptive spectral filtering.

[0064] The S505 adaptive spectrum filter module is used to limit high-frequency energy amplification.

[0065] Specifically, the spatial state features output by the feedforward network module are obtained, and mapped to the frequency domain using a fast Fourier transform to obtain a complex frequency domain feature matrix. A learnable spectral filter matrix is ​​constructed and multiplied element-wise with the complex frequency domain feature matrix to apply adaptive weight constraints in the frequency domain. Based on the complex frequency domain feature matrix after weight constraints, adaptive truncation or energy suppression is performed on the high-frequency components of the complex frequency domain feature matrix that exceed a preset hard cutoff frequency threshold to block the high-frequency noise amplification path in the inverse reconstruction process. The filtered frequency domain features are restored to the spatial domain through an inverse fast Fourier transform to output the filtered spatial state features.

[0066] The preset threshold in the aforementioned adaptive spectral filtering process can be set according to the spatial resolution of the spatial state features, the frequency domain distribution range, and the task's requirement for preserving local details. It can also appropriately truncate or suppress the energy of the frequency domain features based on the sampling frequency. For example, the size of the spatial state features output by the feedforward network module can be set to 128×32×32×32, where 128 represents the number of feature channels and 32×32×32 represents the three-dimensional axial resolution. The size of the learnable spectral filtering matrix should be set to the same size of 128×32×32×32 to facilitate element-wise multiplication and adaptive adjustment of the response intensity for different channels and frequency positions. The hard truncation range can be set to retain 16×16×16 frequency domain components in each feature channel, and high-frequency positions outside the range of these components can be zeroed, truncated, or have their energy attenuated. In practical applications, this can be flexibly adjusted according to the task's requirement for preserving details. The core principle is to accurately truncate high-frequency noise that is easily amplified exponentially during the reverse reconstruction process, while fully preserving the mid-to-low-frequency structural features that determine the object's contour and topological coherence.

[0067] The frequency domain analytical form of the reverse heat dissipation process can be expressed as:

[0068] Where ω represents the frequency variable, Indicates the final heat dissipation time. The frequency domain representation of the final noisy heat dissipation state. The frequency domain representation represents the initial state for reconstruction. As can be seen from the above form, high-frequency terms may be exponentially amplified during the inverse solution process. Therefore, the above embodiment uses Fourier regularization and adaptive spectral filtering to suppress the amplification of high-frequency noise, thereby improving the stability of the inverse reconstruction process.

[0069] In the above embodiments, multi-scale frequency modeling of 3D features is achieved through a multi-directional window attention module, enabling the model to capture low-frequency global shape, mid-frequency directional structure, and high-frequency local details, thereby enhancing its ability to express complex 3D shapes and multi-scale geometric structures. Furthermore, the multi-band spatial topology features output by the multi-directional window attention module are processed by a feedforward network module and then input into an adaptive spectral filtering module. This allows subsequent frequency domain filtering to suppress anomalous high-frequency noise while preserving effective geometric details, thereby improving the stability of the reverse reconstruction process.

[0070] In some specific embodiments of this application, the operator learning module constructed in S600 is also referred to as the Fourier neural operator framework. Fourier neural operator framework It includes a spatial transformation branch and a frequency domain transformation branch. The spatial transformation branch is used to extract multi-scale spatial structure features of the point cloud state, while the frequency domain transformation branch is used to filter the point cloud state features in the frequency domain and approximate the integral operator in the inverse reconstruction process.

[0071] Reference Figure 4 As shown, the spatial transformation branch and frequency domain processing branch in the frequency-aware model are jointly constructed into a Fourier neural operator form. The inter-layer update in the Fourier neural operator form is expressed as:

[0072] in, Indicates the first Layer features, Represents a non-linear activation function. Represents spatial transformation operators, This represents a learnable spectral filtering matrix. and These represent the Fourier transform and inverse Fourier transform, respectively. Spatial transformation operator. It can include a multi-directional window attention module and a feedforward network module for extracting multi-scale spatial structure features; it can learn a spectral filtering matrix. Used to approximate integral operators in the frequency domain to enhance the operator learning module's ability to approximate continuous-time inverse heat dissipation trajectories.

[0073] In one specific implementation, an operator learning module is provided. The training process.

[0074] Operator Learning Module In the final noise state As input, the output is the point cloud state corresponding to multiple target time steps. The target inverse thermal dissipation trajectory is denoted as... By optimizing the error between the predicted trajectory and the target trajectory, The mapping relationship from a noise state to a continuous-time inverse heat dissipation trajectory is learned. In some specific embodiments of this application, the diffusion loss target... Represented as:

[0075] in, This represents the m-th target time step, where M represents the number of target time steps. This represents the operator learning module to be trained. This represents the target inverse heat dissipation trajectory mapping.

[0076] The above embodiments construct an operator learning module based on Fourier neural operators to predict the point cloud state at multiple time steps in parallel, which can improve sampling efficiency and maintain good generation stability and reconstruction quality under the condition of few inference steps.

[0077] Based on the mapping and operator learning module obtained above, in a specific embodiment of this application, in step S700, parallel decoding is performed using the operator learning module based on the mapping, and the point cloud state at multiple time steps is sampled simultaneously to obtain the target 3D point cloud. This can be achieved by the following steps: S701, set the target number of time steps M for parallel decoding, where M is an integer greater than or equal to 2; Here, M can be 4. In other specific implementations, M can also be adaptively configured according to the model size, number of sampling steps and computing resources. S702 inputs the noisy heat dissipation state into the operator learning module, and performs parallel decoding operation based on the decoupled temporal dependency provided by the continuous mapping, independently obtains and synchronously outputs the intermediate diffusion states corresponding to the M target time steps; This mapping abstracts the discrete inverse process into a functional relationship in the continuous time domain, enabling the operator module to directly predict the state at any specified time step without relying on the intermediate results of adjacent time steps. S703 performs reverse reconstruction based on M intermediate diffusion states to generate a target 3D point cloud.

[0078] The parallel decoding mechanism described above replaces the time-step serial iteration of the traditional diffusion model, which can effectively reduce the computational overhead in the inference process and block the cumulative effect of single-step prediction error, thereby simultaneously improving generation stability and inference efficiency under the condition of fewer sampling steps.

[0079] Based on the two core requirements of 3D point cloud generation and completion, in one specific embodiment of this application, an adaptive condition guidance mechanism is provided according to the difference in the completeness of the input data.

[0080] 3D point cloud data includes complete 3D point cloud data, partially missing 3D point cloud data, sparse 3D point cloud data, or partially observed 3D point cloud data; When the 3D point cloud data is complete 3D point cloud data, the method is used for unconditional point cloud generation or category-conditional point cloud generation based on sampling noise. When the 3D point cloud data is partially missing, sparse, or only partially observed, the method is used for 3D point cloud completion. The specific operation is as follows: The point cloud conditional feature extraction network extracts local geometric features, global shape features, and contextual structure features of the missing regions from the observable regions. The point cloud conditional feature extractor uses a point-voxel convolutional neural network (PVCNN), which is composed of alternating combinations of point convolution and voxel convolution, as the conditional feature extraction network to explicitly extract multi-scale local geometric structure control features of the mask region. By utilizing a three-dimensional adaptive layer normalization mechanism, multi-scale local geometric control features are dynamically injected into each module of the frequency sensing model, guiding the inverse heat dissipation evolution trajectory to approximate the target missing manifold, and generating a high-fidelity completed three-dimensional point cloud of the target.

[0081] Specifically, the approximation of the target missing manifold to the inverse heat dissipation evolution trajectory here refers to the following: During the reverse reconstruction process, an adaptive layer normalization mechanism is used to transform the features of the known region into scaling and translation adjustment parameters for each layer of the network in real time, thereby dynamically calibrating the spatial distribution center and fluctuation range of the generated features. This adjustment process is equivalent to adding explicit geometric guidance constraints to the inverse heat dissipation process, effectively guiding the originally aimless free generation trajectory to always evolve along a reasonable form compatible with the known observed structure. As a result, the model can automatically maintain the coherence of the overall three-dimensional topology when reconstructing the missing region, avoiding structural breaks or curvature abrupt changes, and ultimately achieving a natural and smooth splicing and high-fidelity reconstruction of the missing part and the known part.

[0082] The above embodiments are applicable to tasks of generating complete 3D point clouds, generating category-conditional point clouds, and completing partially missing point clouds, and have good task adaptability and application scalability.

[0083] Based on the same technical concept, other embodiments of this application provide a three-dimensional point cloud generation system 100 based on a learnable heat dissipation operator, such as... Figure 5 As shown, it includes: Data acquisition module 110 acquires 3D point cloud data; The local graph construction module 120 constructs a discrete graph Laplacian operator by constructing a graph in the local neighborhood of each 3D point cloud. Diffusion coefficient learning module 130 uses an adaptive network to learn the anisotropic diffusion coefficient matrix; The probability transfer module 140 sequentially applies anisotropic diffusion coefficient matrix and discrete graph Laplacian operator to the point cloud to blur it and add Gaussian noise, establishes the probability distribution transfer process, and obtains the noisy heat dissipation state. The frequency sensing reconstruction module 150 inputs the noisy heat dissipation state into the frequency sensing model, realizes signal frequency decomposition through the multi-directional window attention module, and limits high-frequency energy amplification through adaptive spectrum filtering to obtain stable spatial state characteristics. Operator learning module 160 is constructed based on the frequency-aware model to fit the mapping from steady-space state features to the evolution trajectory of the inverse heat dissipation equation. The point cloud output module 170, based on mapping, uses the operator learning module to perform parallel decoding and simultaneously samples the point cloud state at multiple time steps to obtain the target 3D point cloud.

[0084] The specific implementation techniques of each module / unit in the above examples of this application can be referred to the steps of the three-dimensional point cloud generation method based on learnable heat dissipation operators in the above embodiments, and will not be repeated here.

[0085] The preferred features in the above embodiments can be used individually in any embodiment, or in any combination thereof, provided they do not conflict with each other. Furthermore, parts not described in detail in the embodiments can be implemented using existing technologies.

[0086] The foregoing has described some specific embodiments of this application. It should be understood that this application is not limited to the specific embodiments described above, and those skilled in the art can make various modifications or variations within the scope of the claims, which do not affect the substantive content of this application. The above-described preferred features can be used in any combination without conflict.

Claims

1. A method for generating 3D point clouds based on learnable heat dissipation operators, characterized in that, include: Acquire 3D point cloud data; A discrete graph Laplacian operator is constructed by mapping the local neighborhood of each of the three-dimensional point clouds. The anisotropic diffusion coefficient matrix is ​​learned using an adaptive network; The point cloud is sequentially blurred using the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator, and Gaussian noise is added to establish a probability distribution transfer process to obtain the noisy heat dissipation state. The noisy heat dissipation state is input into the frequency sensing model, and the signal frequency is decomposed through a multi-directional window attention module. Then, the high-frequency energy amplification is limited by adaptive spectrum filtering to obtain stable spatial state characteristics. Based on the frequency-aware model, an operator learning module is constructed to fit the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation; Based on the mapping, the operator learning module is used to perform parallel decoding and simultaneously sample the point cloud state at multiple time steps to obtain the target 3D point cloud.

2. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 1, characterized in that, The step of constructing a discrete graph Laplacian operator by mapping the local neighborhood of each of the three-dimensional point clouds includes: Using points in a 3D point cloud as graph nodes, and taking any one of the graph nodes as the center node, the spatial distance between the center node and each adjacent point is calculated based on geometric features. The k nearest neighbor points are selected in ascending order of distance to construct a k nearest neighbor graph, where k is a positive integer. The edge weights between the central node and each of the nearest neighbors are calculated using Gaussian radial basis functions to characterize local geometric similarity; Based on the k-nearest neighbor graph and the edge weights, a discrete graph Laplacian operator is constructed to describe the geometric smoothness of point clouds within their local neighborhoods.

3. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 1, characterized in that, The anisotropic diffusion coefficient matrix is ​​a semi-positive definite-point tensor. ; The half-point positive definite point-level tensor Parameterized by Cholesky decomposition ,in, It is a lower triangular matrix. for The transpose of the matrix; the lower triangular matrix The diagonal elements are used to control the independent heat dissipation rates in different spatial directions, while the off-diagonal elements are used to control the heat dissipation coupling relationship between different spatial directions. The diagonal elements of the lower triangular matrix L are composed of fixed positive constants superimposed with the prediction parameters of the point-level multilayer perceptron based on the local geometric features of the three-dimensional point cloud, to ensure that the diagonal elements are always positive; the off-diagonal elements of the lower triangular matrix L are composed of the corresponding prediction parameters directly assigned.

4. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 1, characterized in that, The process of sequentially applying the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator to the point cloud, blurring it, adding Gaussian noise, establishing a probability distribution transition process, and obtaining the noisy heat dissipation state includes: Based on the anisotropic diffusion coefficient matrix and the discrete graph Laplace operator Establish an anisotropic heat dissipation equation, in time At that time, three-dimensional coordinates The anisotropic heat dissipation equation at a specific location is expressed as follows: ； in, Indicates time Continuous geometric features of the 3D point cloud at the location; The anisotropic heat dissipation equation is used to analyze the input point cloud features. Dynamic blur degradation is performed to gradually smooth high-frequency local details in the point cloud during the forward process. The anisotropic heat dissipation equation based on the frequency variable is obtained through Fourier transform. Frequency domain analytical solution: ； in, and These represent the solutions for heat dissipation at the initial time and the solutions at the final time, respectively. The frequency domain spectrum characterization at that location shows a characteristic of exponential smooth decay in the high-frequency band within the frequency domain space; Let the discrete-time series of the positive probability distribution transition process be... , , , ,in, Indicates the initial time. Indicates the final heat dissipation time; In the i-th forward transition step, according to the time interval The anisotropic diffusion coefficient matrix and the discrete graph Laplace operator Based on the exponential heat kernel in the analytical solution Constructing a linear heat dissipation operator under a first-order implicit backward difference scheme for: ； The point cloud state at time step i-1 Input the linear heat dissipation operator, and obtain the average heat dissipation state at the i-th time step by solving the following linear equation. : ； in, This represents the mean state of the point cloud after only thermal dissipation blurring. In the mean state of heat dissipation Gaussian random perturbation is superimposed on the data to obtain the point cloud state at the i-th time step. The variance of the Gaussian random perturbation is determined by... control; The probability distribution transition process between adjacent discrete time steps is represented as follows: ； Substituting into the mean state expression, we get: ； in, Indicates the first Point cloud state at each time step; According to the described forward transition process, from the initial point cloud state Calculate in sequence , , , This causes the point cloud state to be gradually subjected to the combined effects of thermal dissipation ambiguity and Gaussian random perturbation during the forward process, and the final time step obtained As a state of heat dissipation with added noise.

5. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 1, characterized in that, The frequency sensing model includes an embedding module, a multi-directional window attention module, a feedforward network module, and an adaptive spectrum filtering module; The noisy heat dissipation state is input into the frequency sensing model, and the following operation is performed: The noisy heat dissipation state is encoded into a voxelized three-dimensional feature sequence; The embedding module performs conditional embedding and time-step embedding on the three-dimensional feature sequence to obtain the input features. The input features are input into the multi-directional window attention module, and the input features are modeled at multiple scales using three-dimensional windows with different axial sizes to obtain decoupled multi-band spatial topological features. The multi-band spatial topology features output by the multi-directional window attention module are input into the feedforward network module to obtain spatial state features. The spatial state features are input into the adaptive spectrum filtering module, and the filtered spatial state features are obtained through Fourier transform, frequency domain filtering, high-frequency component suppression and inverse Fourier transform. The multi-directional window attention module and the adaptive spectrum filtering module are used together to limit the amplification of high-frequency components during the inverse reconstruction process, thereby achieving frequency domain constraints on the inverse heat dissipation process.

6. The method for generating three-dimensional point clouds based on learnable heat dissipation operators according to claim 5, characterized in that, The process of achieving signal frequency decomposition through a multi-directional window attention module to obtain decoupled multi-band spatial topological features includes: The input feature sequence is projected along the channel dimension to multiple attention heads, and the multiple attention heads are divided into 8 parallel feature processing groups; Each feature processing group is independently configured with a three-dimensional window size, so that the eight groups correspond to different frequency perception scales and orientation preferences. The window size includes: ； ； ； ； ； ； ； ； Where s represents the preset base window size, L and H represent low-frequency perception tendency and high-frequency perception tendency, respectively; D, H and W correspond to the depth, height and axial width of the three-dimensional feature tensor, respectively, and D, H and W at each position independently set the window size and frequency band preference on that axis; WA represents window attention; Each feature group performs attention calculations in parallel within its corresponding window size, extracting low-frequency global structural features, mid-frequency directional structural features, or high-frequency local detail features, respectively. The attention feature vectors output by the eight feature processing groups in parallel are concatenated and aggregated to obtain decoupled multi-band spatial topological features.

7. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 5, characterized in that, The method of limiting high-frequency energy amplification through adaptive spectral filtering includes: The spatial state features output by the feedforward network module are obtained, and they are mapped to the frequency domain using the fast Fourier transform to obtain the complex frequency domain feature matrix. A learnable spectral filter matrix is ​​constructed and multiplied element-wise with the complex frequency domain feature matrix to apply adaptive weight constraints in the frequency domain. Based on the complex frequency domain feature matrix after weight constraints, adaptive truncation or energy suppression is performed on the high-frequency components of the complex frequency domain feature matrix that exceed the preset hard cutoff frequency threshold, thereby blocking the high-frequency noise amplification path in the reverse reconstruction process. The filtered frequency domain features are restored to the spatial domain through inverse fast Fourier transform, and the filtered spatial state features are output.

8. The method for generating 3D point clouds based on learnable heat dissipation operators according to claim 7, characterized in that, The operator learning module constructed based on the frequency-aware model, which fits the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation, includes: Based on the spatial and frequency domain processing mechanisms of the frequency-aware model, an operator learning module, namely the Fourier neural operator framework, is constructed. The Fourier neural operator framework The inter-layer update of layer l is represented as: ； in, Indicates the first Layer features, Indicates the first +1 layer features; Represents a non-linear activation function. Represents spatial transformation operators, This represents a learnable spectral filtering matrix. and The spatial transformation operator represents the Fourier transform and the inverse transform, respectively. It includes a multi-directional window attention module and a feedforward network module for extracting multi-scale spatial structure features; the learnable spectral filtering matrix Used to approximate the integral operator in the frequency domain, thereby enhancing the operator learning module's ability to approximate continuous-time inverse heat dissipation trajectories; The noisy heat dissipation state at the final time step Input the Fourier neural operator framework From the noisy heat dissipation state Gradually restore to the target point cloud state The sequence is defined as the inverse heat dissipation evolution trajectory g; The inverse heat dissipation evolution trajectory g at M target time steps , , , Discrete sampling is performed to obtain the target point cloud state corresponding to M target time steps; Using the Fourier neural operator framework Fit from Mapping to the states of the M target point clouds; The training phase of the operator learning module specifically includes: Calculate the error difference between the predicted point cloud state and the target point cloud state at the M target time steps, introduce a weighting coefficient that varies with time step to combine the errors at each time step, and construct and optimize the diffusion loss target. Specifically: ； The inference phase of the operator learning module specifically includes: The trained operator learning module is used for actual generation, and the noisy heat dissipation state is then applied. Perform reverse prediction, and after a set number of inference steps, output the final reconstruction result. .

9. The method for generating three-dimensional point clouds based on learnable heat dissipation operators according to claim 1, characterized in that, The three-dimensional point cloud data includes complete data and incomplete data; When complete data is input, the method performs unconditional or class-conditional point cloud generation based on sampled noise; When inputting incomplete data that is partially missing, sparse, or partially observed, the method performs 3D point cloud completion, specifically as follows: A point-voxel convolutional neural network was used to extract multi-scale geometric features and missing context features of the observable region. The multi-scale geometric features and missing context features are dynamically injected into the frequency-aware model through a three-dimensional adaptive layer normalization mechanism, guiding the inverse heat dissipation evolution trajectory to converge toward the target missing manifold and generating a complete point cloud.

10. A three-dimensional point cloud generation system based on a learnable heat dissipation operator, characterized in that, include: The data acquisition module acquires 3D point cloud data; The local graph construction module constructs a graph in the local neighborhood of each of the three-dimensional point clouds and builds a discrete graph Laplacian operator. The diffusion coefficient learning module uses an adaptive network to learn the anisotropic diffusion coefficient matrix. The probability transfer module sequentially applies the anisotropic diffusion coefficient matrix and the discrete graph Laplacian operator to the point cloud, blurs it, adds Gaussian noise, establishes a probability distribution transfer process, and obtains the noisy heat dissipation state. The frequency-aware reconstruction module inputs the noisy heat dissipation state into the frequency-aware model, decomposes the signal frequency through the multi-directional window attention module, and limits high-frequency energy amplification through adaptive spectrum filtering to obtain stable spatial state characteristics. The operator learning module is constructed based on the frequency-aware model to fit the mapping from the stable space state features to the evolution trajectory of the inverse heat dissipation equation; The point cloud output module, based on the mapping, uses the operator learning module to perform parallel decoding and simultaneously samples the point cloud state at multiple time steps to obtain the target 3D point cloud.

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