A method for simultaneous localization and mapping based on fractional order gauss field

Abstract

The application provides a synchronization positioning and mapping method based on a fractional order Gaussian field, and belongs to the field of image processing. A parameterized model composed of anisotropic three-dimensional Gaussian primitives is used for representation, three-dimensional covariance is decomposed into rotation and scaling, and is mapped to an image plane to form a two-dimensional Gaussian through a differentiable projection to realize rasterization coding. Camera pose is estimated through minimization of projection error and loss iteration, and key frames are inserted or updated according to pose and Gaussian visibility determination; non-key frames are only updated locally. A mapping target function is coupled with luminosity, depth and isotropic regularization, and parameters are updated through gradient with an adaptive learning rate. A fractional order kernel controls power law attenuation, zero order degenerates into an ordinary Gaussian, and the fractional order is dynamically adjusted to balance robustness and structural consistency.

Description

Technical Field

[0001] This invention belongs to the field of image processing, and more specifically relates to a method for simultaneous localization and mapping based on a fractional-order Gaussian field. Background Technology

[0002] With the continuous development of cutting-edge technologies such as intelligent robots, autonomous driving, and augmented reality, Visual Simultaneous Localization and Mapping (VSLAM) has gradually become one of the key technologies for intelligent perception and environmental understanding. Through VSLAM technology, devices can achieve precise localization in unknown environments while continuously improving the scene map, thus laying the foundation for higher-order tasks such as path planning, object detection, and environmental interaction. Traditional SLAM methods mainly include sparse methods based on feature point matching and dense reconstruction methods based on pixels. Although these methods achieve good localization and mapping results in simple structures or stable environments, they often exhibit optimization bottlenecks and insufficient robustness when facing scenes with complex lighting, dynamic occlusion, missing textures, and extremely disordered geometric structures.

[0003] In recent years, implicit scene representation methods, especially 3D visual modeling techniques based on Gaussian fields, have become a new research hotspot in the SLAM field due to their inherent continuity, differentiability, and efficient fusion of geometric and photometric information. Gaussian field methods parameterize 3D spatial information using a Gaussian distribution, offering greater flexibility to adapt to scenes with rich details and complex spatial structures compared to previous grid- or voxel-based map representations. However, existing Gaussian field modeling is generally limited by the isotropic assumption, resulting in limited adaptability to discrete sampling and boundary region processing. This manifests as a significant decrease in map reconstruction quality in certain visually degraded scenes (such as extremely strong or weak illumination, local occlusion, etc.), and the inability to meet practical application requirements for detail and structural fidelity. Furthermore, traditional image processing methods within the integer-order calculus framework, while easy to implement in engineering, exhibit slow convergence speeds and insensitivity to illumination changes in key areas such as multi-scale scene representation, capturing subtle structural changes, and edge sensitivity.

[0004] Therefore, there is an urgent need to propose a novel simultaneous localization and mapping method based on fractional-order Gaussian field theory to promote the application and development of SLAM technology in high-precision and high-robust environmental perception scenarios. Summary of the Invention

[0005] This invention aims to address the shortcomings in robustness and accuracy of existing Visual Simultaneous Localization and Mapping (VSLAM) technologies in complex lighting, variable environments, preservation of structural details, and boundary handling. Specifically, this invention addresses the difficulty of achieving high-quality, real-time, and highly adaptable scene reconstruction and online camera localization under the constraints of traditional Gaussian field representations, such as isotropy assumptions, insufficient power-law decay, and discrete boundary effects. It proposes a simultaneous localization and mapping method combining fractional calculus and Gaussian field theory, significantly improving the system's generalization ability to complex environments, detail restoration capabilities, and lighting adaptability. This, in turn, promotes the practicality and stability of SLAM technology in real-world applications such as intelligent transportation and autonomous driving.

[0006] To achieve the above objectives, the present invention employs the following technical solution: the method comprises: S1. Combining the idea of ​​the Caputo derivative with the Gaussian weight kernel, a fractional-order Gaussian field method is proposed. S2. Represent the 3D Gaussian scene by formalizing the 3D scene representation into a parameterized model composed of anisotropic 3D Gaussian elements. The Gaussian elements are defined by their position in the world coordinate system, the 3D covariance matrix, and the opacity parameter. S3. Perform camera pose tracking. By optimizing the projection error relationship between the current frame and the existing Gaussian map, the six-degree-of-freedom pose estimation of the camera in the current frame is achieved. S4. Perform keyframe management, using inter-frame pose estimation and determining whether the current frame is inserted into the map as a keyframe. If it is a keyframe, update and maintain the 3D Gaussian primitives. If it is not a keyframe, do not update the global map, but only update the current frame state estimation and local parameters to ensure computational efficiency and structural information integrity. S5. Perform mapping and optimization. Apply isotropic constraints to the 3D Gaussian distribution through regularization terms. Combine photometric loss, depth loss, and isotropic regularization terms to form a complete mapping objective function. Use type-specific learning rates for gradient descent optimization for each parameter type.

[0007] In one approach, the fractional order of the Gaussian field kernel function controls the degree of power-law decay. When the fractional order is zero, the kernel function degenerates into a regular Gaussian kernel. The setting of the fractional order affects the decay behavior of the kernel function with distance and allows for flexible adjustment of the decay behavior parameters to suit the actual requirements of the mapping process for different image processing needs.

[0008] In one scheme, the three-dimensional covariance matrix of a three-dimensional Gaussian element is decomposed into the product of a rotation matrix and a scaling matrix. The rotation matrix is ​​used to describe the geometric orientation, and the scaling matrix is ​​used to describe the scaling ratio along different axes.

[0009] In one approach, three-dimensional Gaussian primitives are mapped to the image plane through a differentiable projection transformation. By utilizing the Jacobian matrix of the projection transformation, the view transformation matrix, and the transformation relationship from the three-dimensional covariance matrix in the world coordinate system to the two-dimensional covariance matrix in the image plane, specific settings are made for the view transformation, scale parameters, skip null amount, absolute translation threshold, and scale factor in the differentiable rendering process.

[0010] In one approach, the camera pose tracking process includes: rendering a 3D Gaussian field to the current frame image plane to generate a projection, calculating the photometric difference and depth consistency error between the rendered image and the observed image, solving for the current camera pose parameters, and updating the state for each frame during tracking.

[0011] In one approach, keyframe management utilizes inter-frame pose estimation and determines whether to insert a keyframe based on motion and overlap criteria. If a keyframe is identified, the parameters of the 3D Gaussian elements in the global map are updated and maintained. If a frame is identified as a normal frame, only the state of the current frame and local parameters are estimated and adjusted.

[0012] In one scheme, the mapping objective function is a weighted sum of photometric loss, depth loss, and isotropic regularization term, and structural similarity evaluation components are introduced into the objective function. The fractional order is dynamically adjusted based on absolute trajectory error monitoring, and each component in the objective function, namely photometric, depth, and regularization term, has a corresponding weight coefficient setting to support the expression of structural features.

[0013] In one approach, the parameters of the three-dimensional Gaussian trigonometric unit include: spatial location, three-dimensional covariance matrix parameter, and opacity parameter; the update is optimized by gradient descent with a type-specific learning rate, the learning rate is set according to the parameter type, the gradient of the mapping loss with respect to the parameters is obtained based on the fractional Gaussian field kernel and the chain rule, and iterative processing is performed using stepwise unified constraints.

[0014] Beneficial effects of this invention: This invention combines fractional-order calculus with implicit Gaussian field scene modeling, introducing a fractional-order Gaussian field to achieve a comprehensive expression of the global-local structure of spatial information. It also addresses the shortcomings of traditional Gaussian fields, such as isotropy assumptions, boundary discontinuities, and insufficient adaptability to complex lighting and details. This technical solution significantly improves the robustness and map reconstruction accuracy of visual simultaneous localization and mapping (VSLAM) systems in dynamic and diverse environments, while enhancing the faithful representation of detailed structures and boundary information. Furthermore, the power-law decay characteristic brought by the fractional-order modeling mechanism facilitates smoother and more natural map stitching and transitions at different spatial scales, improving the overall scene adaptability and real-time performance. In summary, this invention effectively enhances the practical performance of VSLAM technology in complex and dynamic scenes, providing key technical support for practical applications such as intelligent transportation, autonomous driving, and 3D environment understanding. Attached Figure Description

[0015] Figure 1 This is a flowchart of the method of the present invention; Figure 2 The figure shows a comparative analysis of the quality of images generated from new perspectives after 3D reconstruction using different 3DGS-based methods. Detailed Implementation

[0016] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Typical embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete.

[0017] Unless otherwise defined, all technical and scientific terms used in this invention have the same meaning as understood by one of ordinary skill in the art to which this invention pertains. The terminology used in this specification is for the purpose of describing particular embodiments only and is not intended to limit the invention. To facilitate understanding, the invention will now be described more fully with reference to the accompanying drawings. Typical embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to make the disclosure of the invention more thorough and complete.

[0018] The overall concept of this invention is to construct a simultaneous localization and mapping (SLAM) method based on fractional-order Gaussian fields (FGF-SLAM). The FGF-SLAM algorithm supports both RGB-D and monocular input data streams. The processing flow begins in a 3D Gaussian scene representation module, where the input data is initialized and transformed into an anisotropic 3D Gaussian set through a structured parameterization process.

[0019] Through a differentiable projection transformation, each 3D Gaussian is projected onto a 2D Gaussian on the image plane to achieve efficient rasterization operations. These Gaussian primitives collectively encode the geometric and photometric properties of the environment.

[0020] Meanwhile, the camera pose tracking module coordinates the relationship between pose estimation and rendering optimization. It employs a strategy combining the "L1+FGF-SSIM" loss function to iteratively optimize the camera pose, achieving accurate online pose estimation for sequential frames. The keyframe management module then retrieves, selects, and crops keyframes based on pose constraints and Gaussian visibility metrics, significantly improving the computational efficiency and reconstruction quality of the SLAM system.

[0021] In mapping and optimization, the simultaneous localization and mapping algorithm based on fractional-order Gaussian fields utilizes the pose information optimized in the preceding stages to optimize Gaussian insertion and parameter updates. This optimization includes isotropic regularization to prevent excessive anisotropic deformation in under-observed regions, and employs an "L1+FGF-SSIM" strategy to construct a similar loss function for the mapping process. The FGF-SSIM part uses an adaptive fractional order, and its parameters are dynamically adjusted through ATE monitoring, thereby ensuring robustness under different illumination conditions while maintaining structural consistency and contrast sensitivity.

[0022] like Figure 1 As shown, a synchronous localization and mapping method based on a fractional-order Gaussian field specifically includes the following steps: S1. Combining the idea of ​​the Caputo derivative with a Gaussian weighted kernel, a novel fractional-order Gaussian field (FGF) method is proposed. This overcomes the boundary effects and isotropic requirements in discrete image processing. Fractional-order Gaussian field kernel function. We can obtain:

[0023] When calculating fractional-order local statistics, the convolution kernel in the discrete case is normalized to maintain numerical stability. The normalized convolution kernel used for calculating fractional-order local statistics is defined. as follows:

[0024] in, Indicates the fractional order, controlling the degree of power-law decay, when When the value is 0, this kernel degenerates into a regular Gaussian kernel. The Gamma function is expressed as a fractional normalization factor. This is the standard deviation of the Gaussian term, which controls the intensity of Gaussian decay. (m,n) represents the pixel. p Relative to the center pixel Integer displacement coordinates. Expressed in standard deviation A two-dimensional Gaussian function with parameters.

[0025] S2. Represent the 3D Gaussian scene by formalizing the 3D scene representation into a parameterized model composed of anisotropic 3D Gaussian primitives.

[0026] To construct a Visual Simultaneous Localization and Mapping (VSLAM) system based on dense 3D Gaussian scenes (3DGS) capable of high-fidelity geometric reconstruction, the 3D scene representation is formalized into a parametric model with adaptive control capabilities composed of anisotropic 3D Gaussian primitives:

[0027] Each three-dimensional Gaussian element Defined by the following parameter: position in the world coordinate system , representing the three-dimensional covariance matrix of anisotropic geometry And the opacity parameter that controls transparency. To ensure numerical stability and optimization efficiency, the covariance matrix is ​​decomposed into... ,in Represents the rotation matrix. Represents the scaling matrix. This represents the transpose of the matrix. Such decomposition facilitates efficient optimization in gradient-based methods using PyTorch's automatic differentiation framework.

[0028] When transforming from the world coordinate system to the camera coordinate system, the 3D Gaussian... It will be projected onto the image plane as a two-dimensional Gaussian through a differentiable projection transformation. :

[0029] in, The Jacobian matrix represents the projection transformation. Represents the view transformation matrix. The original three-dimensional covariance matrix in the world coordinate system. Let be the two-dimensional covariance matrix projected onto the image plane. This representation allows for efficient rasterization operations that skip blank areas while maintaining differentiability throughout the rendering pipeline.

[0030] S3, Camera pose tracking, projects 3D Gaussian elements from the world coordinate system to a 2D Gaussian on the image plane through a differentiable projection transformation. The projection process includes the Jacobian matrix of the projection transformation, the view transformation matrix, and the 2D covariance matrix, achieving differentiability.

[0031] Estimating camera pose for each frame of input image while freezing Gaussian parameters. The optimization problem is formulated as a composite loss function to enhance robustness under different lighting conditions:

[0032] in, Represents the photometric loss component. This represents the geometric loss component when depth information is available. and These are the weighting coefficients of the two loss terms, used to balance the contributions of the different terms to the total loss. This separation allows the tracking phase to focus on camera motion estimation, avoiding modifications to the environment model.

[0033] For monocular scenes, a fractal Gaussian field is introduced into the SSIM loss function to obtain the fractional-order structural similarity loss (FGF-SSIM), which combines the traditional L1 norm (absolute value loss function) with the fractional-order structural similarity loss (FGF-SSIM):

[0034] in, Indicates the rendered image. Represents a real image. It is a valid RGB mask region. Represents pixel coordinates, and These are the weighting parameters for the L1 and FGF-SSIM components, respectively. The FGF-SSIM index can better preserve edge information and has multi-scale analysis capabilities, which is crucial for improving the accuracy of SLAM.

[0035] When depth information is available, depth residuals are also introduced to further constrain camera pose optimization:

[0036] in, For depth-based loss functions, This represents the rendered depth map. Represents the true depth map. It is an effective depth mask region. Represents pixel coordinates. This joint method utilizes both photometric and geometric constraints to achieve robust pose estimation.

[0037] S4. Perform keyframe management, using inter-frame pose estimation and determining whether the current frame is inserted into the map as a keyframe. If it is a keyframe, update and maintain the 3D Gaussian primitives. If it is not a keyframe, do not update the global map, but only update the current frame state estimation and local parameters to ensure computational efficiency and structural information integrity.

[0038] Keyframe selection employs a co-visibility analysis strategy based on a Gaussian distribution, aiming to maximize information diversity while minimizing computational overhead. During rasterization, the Gaussian distribution... Visibility is quantitatively assessed in the following ways:

[0039] in, For Gauss Visibility. Indicates an indicator function, Representing Gauss Opacity It is a Gaussian probability density function. The integral term represents the accumulation of rays on Gaussian elements. This is the visibility threshold. This visibility assessment is a preprocessing step before keyframe management.

[0040] Keyframe selection is based on the current frame. With the latest keyframe This can be accomplished through a rigorous co-visibility analysis between them:

[0041] in and Representing a frame and The set of visibility Represents the cardinality of a set. It is the intersection-union ratio threshold. Furthermore, relative motion constraints are employed to ensure sufficient baseline for the triangulation process:

[0042] in It is a frame and The relative translation vector between them It is the absolute translation threshold. It is a scaling factor. Representing a frame The median depth.

[0043] S5. Perform mapping and optimization. Apply isotropic constraints to the 3D Gaussian distribution through regularization terms. Combine photometric loss, depth loss, and isotropic regularization terms to form a complete mapping objective function. Use type-specific learning rates for gradient descent optimization for each parameter type.

[0044] To prevent excessive anisotropic deformation of the Gaussian distribution in regions with insufficient observation, isotropic constraints are imposed on it using regularization terms:

[0045] in Represents the i-th Gaussian The scaling parameters correspond to the scale of the Gaussian distribution along the x, y, and z axes; Scale to the mean of the scaling parameter. It is a unit vector. This represents the total number of Gaussian distributions. This regularization mechanism effectively constrains the spatial expansion of the Gaussian distribution.

[0046] The complete mapping objective function incorporates FGF-SSIM to better preserve structural features:

[0047] in These are the weighting coefficients for photometric loss, depth loss, and isotropic regularization, respectively. The FGF-SSIM components employ an adaptive fractional order, which is dynamically adjusted based on ATE (absolute trajectory error) monitoring.

[0048] To optimize efficiency, the gradient of the FGF-SSIM loss with respect to the rendered image was derived. The fractional order in FGF-SSIM... and It will be dynamically adjusted based on ATE monitoring to maintain robustness.

[0049]

[0050] in, This represents the pixel values ​​of the currently rendered image (an optimization variable). Indicates the pixel value of the reference image. Represents the total loss function For variables gradient, and These are weight parameters. It is a symbolic function. and The relative contributions of the brightness and contrast-structure components are controlled, and the partial derivatives are calculated using a fractional Gaussian field kernel (FGF) and the chain rule. It is a fractional-order brightness statistic. Comparison - structural statistics. The ATE (Absolute Trajectory Error) monitoring mechanism will dynamically adjust. and The value is adjusted to optimize performance under different environmental conditions.

[0051] The Gaussian parameters are optimized using gradient descent with a type-specific learning rate:

[0052] in, This indicates the Gaussian parameters being optimized. Indicates the number of iterations. It's the learning rate. This is the gradient of the mapping loss with respect to the parameters. The learning rate for each parameter type is carefully tuned to ensure the accuracy of the geometry and the stability of the optimization convergence.

[0053] Through the synergistic integration of these modules, FGF-SLAM can achieve efficient and high-quality scene reconstruction and rendering in complex lighting environments. This architecture provides a useful reference for enhancing the robustness and practicality of VSLAM technology in real-world intelligent transportation systems.

[0054] Experimental verification of the method of this invention: like Figure 2 As shown, on the Replica dataset, a comparative analysis was conducted on the quality of new perspective images generated after 3D reconstruction using different 3DGS-based methods. To evaluate the advantages brought by the introduction of fractional Gaussian fields, two representative state-of-the-art models from CVPR 2024 (SplaTAM and MonoGS) were selected for performance comparison. All three methods demonstrated strong scene reconstruction capabilities when the input was RGB-D data; however, the proposed FGF-SLAM performed better in detail preservation and long-range dependency modeling.

[0055] In the Office1 scene, SplaTAM fills in occluded areas of the input image (such as the corner behind the pillow) using blank Gaussian units, resulting in noticeable artifacts. While MonoGS-SLAM attempts to reconstruct these areas, it introduces significant color distortion near the edges, with the whitish artifacts inconsistent with the original beige pillow and black sofa. In contrast, FGF-SLAM fully utilizes its long-range memory during the new perspective synthesis process, effectively handling these difficult-to-reconstruct areas and generating a more natural and coherent scene reconstruction. This advantage is even more pronounced in the sofa gap reconstruction in the Office3 scene: with only a frontal view input, SplaTAM reconstructs a relatively blurry gap structure, MonoGS-SLAM shows some edge structure, while FGF-SLAM accurately reconstructs a sharp and geometrically consistent gap.

[0056] Furthermore, FGF-SLAM excels in detail reconstruction. In an Office 4 scene with murals, SplaTAM can only capture a rough visual pattern, failing to reproduce the structural details of specific lines. While MonoGS-SLAM achieves reasonable reconstruction results, it still has limitations in reproducing delicate patterns of complex shapes (such as facial features and hat outlines). FGF-SLAM, on the other hand, accurately preserves all fine details and exhibits higher line accuracy and geometric fidelity. These results further confirm the effectiveness of fractional-order Gaussian fields in 3D reconstruction tasks, directly benefiting from the long-distance dependency modeling capabilities of fractional-order kernels, thereby enhancing the consistency of structures within the spatial domain.

[0057] As shown in Table 1, the performance of the SLAM framework was evaluated using various image quality metrics on the TUM monocular dataset. The results show that it achieves a consistent advantage across all evaluation metrics. Experimental results demonstrate that FGF-SLAM achieves superior performance across all evaluation dimensions, surpassing methods such as MonoGS-SLAM and UDGS-SLAM.

[0058] Under monocular input settings, FGF-SLAM exhibits superior performance, achieving a peak signal-to-noise ratio (PSNR) of 36.04 dB, a structural similarity (SSIM) of 0.95, a deep learning-based image quality assessment metric (LPIPS) of 0.07, and an absolute trajectory error (RMSE ATE) of 1.73. These results are +12.11 dB, +0.15, and -0.18 dB higher than the second-best method, respectively. The TUM Mono dataset, a widely adopted real-world benchmark in SLAM, suffers from severe motion blur due to rapid movement during data acquisition. Furthermore, traditional RGB sensors have limited sensitivity under short exposures, often resulting in underexposed and noisy images. The data sequence also contains numerous rapidly rotating, textureless regions and repetitive patterns. These factors collectively place extremely high demands on tracking robustness, pose estimation accuracy, and scene reconstruction capabilities under harsh visual conditions.

[0059] By incorporating fractional calculus into the 3DGS-based SLAM framework, the FGF-SSIM loss function enables fine-grained and independent tuning of the brightness and contrast-structure components. In TUM scenes, characterized by severe illumination variations and low image quality, the method adaptively reduces sensitivity to abrupt brightness changes while enhancing the preservation of edge and texture information, thereby improving the structural similarity and perceptual quality of the reconstructed scene. Most notably, the PSNR is improved by 72% compared to the second-best method. Therefore, FGF-SSIM not only retains the efficient rendering capabilities of 3DGS-SLAM but also significantly enhances the model's ability to reconstruct lighting, structural, and texture details through a robust fractional calculus mechanism, achieving industry-leading overall performance.

[0060] Table 1 Comparison of the performance of different 3DGS-based SLAM methods on the TUM Mono dataset.

[0061] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. The storage medium can be a magnetic disk, optical disk, read-only memory (ROM), or random access memory (RAM), etc.

[0062] It should be understood that the above detailed description of the technical solutions of the present invention with reference to preferred embodiments is illustrative and not restrictive. Those skilled in the art can modify the technical solutions described in the embodiments or make equivalent substitutions for some of the technical features based on reading this specification; however, these modifications or substitutions do not cause the essence of the corresponding technical solutions to depart from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A synchronous localization and mapping method based on a fractional-order Gaussian field, characterized in that: The method includes: S1. Combining the idea of ​​the Caputo derivative with the Gaussian weight kernel, a fractional-order Gaussian field method is proposed. S2. Represent the 3D Gaussian scene by formalizing the 3D scene representation into a parameterized model composed of anisotropic 3D Gaussian elements. The Gaussian elements are defined by their position in the world coordinate system, the 3D covariance matrix, and the opacity parameter. S3, Camera pose tracking: By optimizing the projection error relationship between the current frame and the existing Gaussian map, the six-degree-of-freedom pose estimation of the camera in the current frame is achieved. S4. Perform keyframe management, using inter-frame pose estimation and determining whether the current frame is inserted into the map as a keyframe. If it is a keyframe, update and maintain the 3D Gaussian primitives. If it is not a keyframe, do not update the global map, but only update the current frame state estimation and local parameters to ensure computational efficiency and structural information integrity. S5. Perform mapping and optimization. Apply isotropic constraints to the 3D Gaussian distribution through regularization terms. Combine photometric loss, depth loss, and isotropic regularization terms to form a complete mapping objective function. Use type-specific learning rates for gradient descent optimization for each parameter type. The three-dimensional Gaussian primitives are mapped to the image plane through a differentiable projection transformation. By utilizing the Jacobian matrix of the projection transformation, the view transformation matrix, and the transformation relationship from the three-dimensional covariance matrix in the world coordinate system to the two-dimensional covariance matrix in the image plane, specific settings are made for the view transformation, scale parameters, skip null amount, absolute translation threshold, and scale factor in the differentiable rendering process. The camera pose tracking process includes: rendering a three-dimensional Gaussian field to generate a projection onto the current frame image plane, calculating the luminance difference and depth consistency error between the rendered image and the observed image, solving the current camera pose parameters, and updating the state for each frame during tracking. The keyframe management utilizes inter-frame pose estimation and determines whether to insert a keyframe based on motion and overlapping area criteria. If it is determined to be a keyframe, the parameters of the three-dimensional Gaussian elements in the global map are updated and maintained. If it is a normal frame, only the state estimation of the current frame and the adjustment of local parameters are performed. The mapping objective function is a weighted sum of photometric loss, depth loss and isotropic regularization term, and structural similarity evaluation components are introduced into the objective function; The fractional order is dynamically adjusted based on absolute trajectory error monitoring, and each component of the objective function, including luminosity, depth, and regularization terms, has corresponding weight coefficient settings, supporting the expression of structural features.

2. The synchronous localization and mapping method based on a fractional-order Gaussian field according to claim 1, characterized in that: In the fractional-order Gaussian field kernel function of the fractional-order Gaussian field method, the fractional order controls the degree of power-law decay. When the fractional order is zero, the kernel function degenerates into an ordinary Gaussian kernel. The setting of the fractional order affects the decay behavior of the kernel function with distance.

3. The synchronous localization and mapping method based on a fractional-order Gaussian field according to claim 1, characterized in that: The three-dimensional covariance matrix is ​​decomposed into the product of a rotation matrix and a scaling matrix. The rotation matrix describes the geometric direction, and the scaling matrix describes the scaling ratio along different axes.

4. The synchronous localization and mapping method based on a fractional-order Gaussian field according to claim 1, characterized in that: The parameters of the three-dimensional Gaussian primitive include: spatial location, three-dimensional covariance matrix parameters, and opacity parameters; the primitive is updated by gradient descent with a specific learning rate, the learning rate is set according to the parameter type, the gradient of the mapping loss with respect to the parameters is obtained based on the fractional Gaussian field kernel and the chain rule, and it is iterated using step-by-step unified constraints.

Images