Adaptive point cloud implicit curved surface reconstruction method based on truncation level B spline

Abstract

The invention discloses a self-adaptive point cloud implicit curved surface reconstruction method based on a truncation level B spline, and the method comprises the steps: constructing a first-layer truncation level B spline primary function space, and obtaining an initial implicit function through solving an optimization model; evaluating the fitting error of each point of the input point cloud, and determining the fitting error as a region which needs to be further refined; establishing a new truncation hierarchy B-spline primary function set, and expanding a current function space; performing truncation operation on the THB spline primary function which is completely positioned in the refined region; combining the truncated primary function with the primary function set generated by the new layer, and updating the whole primary function library; re-optimizing the model, and solving the updated implicit function; refining and optimizing layer by layer, and outputting a final implicit function and a zero level set thereof; and performing triangular meshing on the final zero level set to finish curved surface reconstruction. According to the method, an extra training process is not needed, excellent point-to-surface distance and curvature indexes can be kept at the noise level up to 10%, and the memory and calculation overhead can be greatly reduced.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of three-dimensional geometric reconstruction, and particularly relates to an adaptive point cloud implicit surface reconstruction method based on truncated hierarchical B-spline. BACKGROUND

[0002] With the rapid development of three-dimensional acquisition technologies such as laser scanning, structured light and multi-view stereo, a single scan can generate a million or even ten million discrete point clouds, greatly improving the convenience of three-dimensional data acquisition. However, the original point cloud often has problems such as uneven sampling, noise interference and local missing, and lacks necessary topological information and connectivity, making it difficult to be directly used for rendering, reverse engineering, simulation or medical diagnosis. How to convert this discrete and disordered point set into a continuous, smooth and topologically consistent surface model has become a key issue to be solved in the fields of computer graphics, computational geometry and computer vision.

[0003] Currently, surface reconstruction methods can be roughly divided into two categories: explicit and implicit. Explicit methods (such as Marching Cubes, Delaunay triangulation) approximate the target surface by directly generating triangular meshes, which has the advantages of simple implementation and fast calculation speed, but is extremely sensitive to abnormal topologies such as noise, cracks or false holes; when the point cloud size reaches more than a million, its memory overhead and mesh quality management problems are particularly prominent. In contrast, implicit methods (such as reconstruction based on Poisson equation, radial basis function interpolation) rely on solving implicit functions to describe the surface, which has good topological adaptability and noise resistance, and can automatically fill holes and smooth noise; but its core often depends on the construction and solution of large-scale sparse linear systems, and the computational complexity increases super-linearly with the size of the point cloud, making it difficult to meet the efficiency requirements of large-scale or real-time reconstruction.

[0004] In addition, most existing implicit reconstruction algorithms lack effective adaptive refinement mechanisms, and usually solve globally with fixed resolution, resulting in resource waste in smooth areas and insufficient sampling in local details, making it difficult to balance noise smoothing and detail preservation, and prone to overfitting or underfitting. Therefore, an adaptive refinement implicit surface reconstruction method that can efficiently process massive point clouds and dynamically allocate computing resources is needed to balance surface smoothness and local accuracy. SUMMARY

[0005] The application provides an adaptive point cloud implicit surface reconstruction method based on truncated hierarchical B-spline, which can realize high-precision restoration of noisy point cloud data, and balance computing resource consumption and resolution adaptation, overcoming the defects of low efficiency and insufficient accuracy in handling high noise or complex geometry in the prior art.

[0006] Technical solution: The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines described in this invention includes the following steps:

[0007] (1) Based on a globally uniform low-resolution grid, a first-layer truncated hierarchical B-spline basis function space is constructed. The initial implicit function is obtained by solving the optimization model, and its zero level set is extracted to generate the first-layer initial surface.

[0008] (2) For each point in the input point cloud, use the error function to evaluate its fitting error, and gather the points whose error is greater than the set threshold to determine the region that needs further refinement.

[0009] (3) Subdivide the grid cells within the refinement region to generate a higher resolution grid, and on this basis, establish a new set of truncated hierarchical B-spline basis functions to expand the current function space;

[0010] (5) Perform a truncation operation on the THB spline basis functions that are completely located in the refinement region in the previous layer to remove redundant effects; merge the truncated basis functions with the basis function set generated in the new layer and update the entire basis function library;

[0011] (5) Optimize the model again and solve the updated implicit function;

[0012] (6) Repeat steps (2) to (5) to refine and optimize layer by layer until the fitting error of all points is lower than the set threshold or the preset maximum refinement level is reached, and output the final implicit function and its zero level set.

[0013] (7) Triangulate the final zero-level set to complete the surface reconstruction; smooth or simplify the reconstructed surface as needed to obtain the final triangular mesh model.

[0014] Furthermore, the optimization model described in step (1) is as follows:

[0015] min f(T1,T2,T3)=T1+λ1T2+λ2T3

[0016] Where p is defined i For point cloud data, Ensure that the curved surface closely fits the point cloud data. By using the constraint surface gradient, make it parallel to the normal vector n i Maintain consistency to improve normal accuracy; Regularization terms ensure surface smoothness and reduce the impact of noise. λ1 is the normal uniformity coefficient, λ2 is the regularization coefficient, and f... xx f yy f zz It is the second-order partial derivative of the function with respect to x, y, and z, f xy fyz f xz It is the mixed second-order partial derivative of the function; the normal uniformity coefficient is 1, and the regularization term coefficient is 0.1.

[0017] Furthermore, the process of constructing the first-level truncated hierarchical B-spline basis function space described in step (1) is as follows:

[0018] Starting with an initial level m=0, construct a globally uniform low-resolution grid with N control points. x N y N z Based on this grid, construct the set of truncated hierarchical B-spline basis functions for layer 0. in The complete set of all basis functions constructed for layer 0 (initial layer), suppφ is the support set of function φ, and this set of functions constitutes the truncated hierarchical B-spline basis function space of layer 0, with domain D. 0 The bounding box for the entire point cloud:

[0019] D 0 =[x min ,x max ]×[y min ,y max ]×[z min , z max ]

[0020] Among them, (x min y min , z min ) and (x max y max , z max ) represents the minimum and maximum values ​​of the point cloud coordinates; the size of each grid cell is:

[0021]

[0022] Based on the initial truncation level B-spline basis function space The control coefficient c is optimized by solving an energy optimization model with normal fitting and regularization terms. φ Thus, the initial implicit function is obtained:

[0023]

[0024] And extract the zero-level set f 0 (x) = 0, thus obtaining the initial reconstructed surface of the first layer.

[0025] Furthermore, the implementation process of step (2) is as follows:

[0026] The current refinement level is l, and its implicit function representation is:

[0027]

[0028] in, B is the control coefficient for the l-th level. i (u) B j (v) B k (w) is a cubic uniform B-spline basis function;

[0029] For each p in the point cloud i (x m y m , z m (m = 1, 2, ..., M), the error ε is defined as the implicit function value f(x) m y m , z m Normalized value relative to the gradient magnitude:

[0030]

[0031] It is the gradient magnitude, used for normalization to avoid the absolute influence of the gradient magnitude on the error, and to give the error a relative meaning;

[0032] For a point cloud Q = {q1, q2, ..., q...} M Each point q in} i =(x i y i , z i ), calculate the fitting error ε of the current level m. m If ε m If the value is relatively large, it means that the current resolution cannot accurately approximate that point;

[0033] Based on the error threshold ε e To identify points {q} with larger errors i |ε i >ε e}; For each error point q i =(x i y i , z i ), calculate the parametric coordinates of these points in the current mesh; map these points to the cells of the current mesh, calculate their corresponding mesh index range, and determine the refinement region D by merging overlapping cells. m+1 .

[0034] Furthermore, the implementation process of step (3) is as follows:

[0035] In region D that needs further refinement m+1Inside, based on the refinement markers of the m-th layer mesh, the truncated hierarchical B-spline basis function space W of the next layer is constructed. m+1 A higher resolution mesh is generated by uniformly subdividing the mesh cells and increasing the number of control points; a new set of basis functions is then constructed based on the refined mesh. in, Let φ be the complete set of all basis functions constructed at layer (m+1), and let suppφ be the support set of function φ. This refers to the set of basis functions at the (m+1)th layer that still contribute to the refined region after truncation; correspondingly, the size of each mesh cell becomes... If the current refinement region has a grid index range of [i] in the x direction... min i max If ], then the corresponding number of control points is: Similarly, we can obtain This generates a new set of truncated hierarchical B-spline basis functions. Used to expand the entire function space.

[0036] Furthermore, the implementation process of step (4) is as follows:

[0037] For the set of B-spline basis functions of the truncated level above. The support domain is identified as being completely contained within the current refined region D. m+1 The basis functions within are truncated, and a truncation operation is performed on them to remove redundant effects;

[0038] Specifically, for If its support satisfies Its contribution has been covered by the next layer of basis functions, so it should be marked as inactive, that is, truncated and no longer participates in the subsequent function expression;

[0039] The adjustment operation definition corresponding to the truncation mechanism is as follows:

[0040]

[0041] Where φ′ is a sub-basis function in the new layer whose support domain completely covers φ, a φ′ For the reconstructed coefficients, the new set of basis functions obtained after truncation is: in, This represents the residual part obtained after performing a cut operation on the truncated basis function of the previous layer. This represents the (m+1)th layer of basis functions newly introduced within the refined region, which ultimately updates the entire basis function library, forming the expanded truncated hierarchical B-spline basis function system.

[0042] Furthermore, the implementation process of steps (5) and (6) is as follows:

[0043] After obtaining the updated set of B-spline basis functions at the truncation level Then, a new basis function space is constructed, and an optimization model is re-established based on this space. By solving the energy function that minimizes the normal and regularization terms, new control coefficients are obtained. Then, the implicit function is calculated:

[0044]

[0045] Furthermore, the implementation process of step (6) is as follows:

[0046] At each level, construct an updated truncated hierarchical B-spline basis function space and the corresponding implicit function, until the error at all points is below the tolerance ε. e Alternatively, the maximum refinement level N can be reached, and the final implicit function f(q) and its zero level set f(q) = 0 can be output as the result of point cloud reconstruction surface.

[0047] Furthermore, the implicit function calculation process is as follows:

[0048] After substituting the three-dimensional tensor product B-spline into the optimization model, the tensor product of the basis functions is defined as... The three-dimensional tensor product spline function is then written as:

[0049] In p i =(x i ,y i ,z i The data fitting matrix A at point ) is Then the first term in the optimization model min f(T1,T2,T3)=T1+λ1T2+λ2T Transformed into: Define the normal constraint matrix B: Define matrix d:

[0050] Then the second item Transformed into:

[0051]

[0052] Define matrix H xx H xy H yy H yz H xz H zz :

[0053]

[0054] The smoothness regularization H is then expressed as: H = H xx+2H xy +H yy +2H yz +2H xz +H zz ;

[0055] Third item Transformed into:

[0056]

[0057] Putting the above three items together, we get:

[0058]

[0059] Then, by merging the three terms, the optimization problem is reduced to a quadratic form:

[0060] J(C)=C T MC-2b T C+c

[0061] Where M = A + λ1B + λ2H, λ1 is the normal uniformity system, λ2 is the regularization coefficient, and b = λ1d is the linear term of the normal constraint; Taking the derivative with respect to C: The optimal solution is C = M -1 b, written as C=(A+λ1B+λ2H) -1 (λ1d); The constant term c originates from the irrelevant terms of the normal constraint:

[0062] Beneficial Effects: Compared with existing technologies, the present invention offers the following advantages: By utilizing the prior smoothing properties of tensor product B-splines without external training during implicit surface reconstruction, the present invention effectively suppresses noisy point cloud data, significantly improving reconstruction accuracy; it innovatively introduces a truncated hierarchical B-spline (THB-spline) adaptive refinement mechanism for hierarchical refinement, achieving adaptive adjustment of local resolution in complex regions, effectively reducing memory and computational consumption while ensuring the restoration of model details; the present invention can be directly applied to various 3D scanning scenarios and has good promotional value and practical application prospects in fields such as industrial design, cultural heritage digitization, and virtual reality modeling. Attached Figure Description

[0063] Figure 1 This is a flowchart of the present invention;

[0064] Figure 2 The image shows the reconstruction effect achieved using this invention. Detailed Implementation

[0065] The present invention will now be described in further detail with reference to the accompanying drawings.

[0066] likeFigure 1 The present invention proposes an adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines, the specific implementation process of which is as follows:

[0067] Step 1: Based on a globally uniform low-resolution grid, construct the first-layer truncated hierarchical B-spline basis function space, obtain the initial implicit functions by solving the optimization model, extract their zero level sets, and generate the first-layer initial surface.

[0068] The optimized model is:

[0069] min f(T1,T2,T3)=T1+λ1T2+λ2T3

[0070] Where p is defined i For point cloud data, Ensure that the curved surface closely fits the point cloud data. By using the constraint surface gradient, make it parallel to the normal vector n i Maintain consistency to improve normal accuracy; The regularization term ensures surface smoothness and reduces the impact of noise. λ1 is the normal uniformity coefficient, set to 1, and λ2 is the regularization coefficient, set to 0.1. xx f yy f zz It is the second-order partial derivative of the function with respect to x, y, and z, f xy f yz f xz It is the mixed second-order partial derivative of the function.

[0071] Initialize the basis function space: Starting with an initial level m = 0, construct a globally uniform low-resolution grid with a control point count of (N). x N y N z Based on this grid, an initial set of basis functions is defined. in The complete set of all basis functions constructed for layer 0 (initial layer), suppφ is the support set of function φ, and the domain is D. 0 The bounding box for the entire point cloud:

[0072] D 0 =[x min x max ]×[y min ,y max ]×[z min , z max ]

[0073] Among them, (x min y min , z min ) and (x maxy max , z max ) represents the minimum and maximum values ​​of the point cloud coordinates; the size of each grid cell is:

[0074]

[0075] Based on the initial truncation level B-spline basis function space The control coefficient c is optimized by solving an energy optimization model with normal fitting and regularization terms. φ Thus, the initial implicit function is obtained:

[0076]

[0077] And extract the zero-level set f 0 (x) = 0, thus obtaining the initial reconstructed surface of the first layer.

[0078] Step 2: For each point in the input point cloud, use the error function to evaluate its fitting error, and cluster the points whose errors are greater than a set threshold to determine the regions that need further refinement.

[0079] To achieve adaptive refinement, it is necessary to first accurately locate regions with high reconstruction errors and then perform directional mesh refinement in these regions. This invention introduces an error function to evaluate the reconstruction quality of each data point. Specifically, by calculating the distance error from each data point to the reconstructed surface, if the error at a certain point exceeds a pre-set threshold, it indicates that the geometry at that location is complex and requires refinement to improve fitting accuracy. This error-based evaluation strategy makes the refinement process more targeted, thereby achieving optimized allocation of computational resources while ensuring reconstruction quality. Let the current refinement level be l, its implicit function expression is:

[0080]

[0081] in, B is the control coefficient for the l-th level. i (u) B j (v) B k (w) is a cubic uniform B-spline basis function.

[0082] For each p in the point cloud i (x m y m , z m (m = 1, 2, ..., M), the error ε is defined as the implicit function value f(x) m y m , z m Normalized value relative to the gradient magnitude:

[0083]

[0084] It is the gradient magnitude. It is used for normalization to avoid the absolute influence of the gradient magnitude on the error, thus giving the error a relative meaning.

[0085] For a point cloud Q = {q1, q2, ..., q...} M Each point q in} i =(x i y i , z i ), calculate the fitting error ε of the current level m. m If ε m If the value is too large, it means that the current resolution cannot accurately approximate that point.

[0086] Relying on setting an error threshold ε e To identify points {q} with larger errors i |ε i >ε e For each error point q i =(x i y i , z i The algorithm calculates the parametric coordinates of these points within the current mesh. It maps these points to cells in the current mesh, calculates their corresponding mesh index range, and determines the refinement region D by merging overlapping cells. m+1 .

[0087] Step 3: Subdivide the mesh cells within the refinement region to generate a higher resolution mesh, and on this basis, establish a new set of truncated hierarchical B-spline basis functions to expand the current function space.

[0088] In region D that needs further refinement m+1 Inside, based on the refinement markers of the m-th layer mesh, the truncated hierarchical B-spline basis function space W of the next layer is constructed. m+1 A higher resolution mesh is generated by uniformly subdividing the mesh cells and increasing the number of control points. A new set of basis functions is then constructed based on the refined mesh. in Let suppφ be the complete set of all basis functions constructed for the (m+1)th layer (initial layer), and let suppφ be the support set of function φ. This refers to the set of basis functions at the (m+1)th layer that still contribute to the refined region after truncation. Correspondingly, the size of each mesh cell becomes [size missing]. If the current refinement region has a grid index range of [i] in the x direction... min i max If ], then the corresponding number of control points is: Similarly, we can obtain This generates a new set of truncated hierarchical B-spline basis functions. Used to expand the entire function space.

[0089] Step 4: Perform a truncation operation on the THB spline basis functions that are completely located within the refinement region in the previous layer to eliminate redundant effects; merge the truncated basis functions with the basis function set generated in the new layer, and update the entire basis function library.

[0090] For the set of B-spline basis functions of the truncated level above. The support domain is identified as being completely contained within the current refined region D. m+1 The basis functions within are truncated, and a truncation operation is performed on them to remove redundant effects.

[0091] Specifically, for If its support satisfies If its contribution has been covered by the next layer of basis functions, it should be marked as inactive (i.e. truncated) and no longer participate in subsequent function expressions.

[0092] The adjustment operation definition corresponding to the truncation mechanism is as follows:

[0093]

[0094] Where φ′ is a sub-basis function in the new layer whose support domain completely covers φ, a φ′ For the reconstructed coefficients, the new set of basis functions obtained after truncation is: in, This represents the residual part obtained after performing a cut operation on the truncated basis function of the previous layer. This represents the (m+1)th layer of basis functions newly introduced within the refined region, ultimately updating the entire basis function library to form the expanded truncated hierarchical B-spline basis function system.

[0095] Step 5: Reconstruct the optimization model in the new THB spline function space and solve the updated implicit function; repeat the above steps to refine and optimize layer by layer until the fitting error of all points is lower than the set threshold or the preset maximum refinement level is reached, and output the final implicit function and its zero level set.

[0096] After obtaining the updated set of B-spline basis functions at the truncation level Then, a new basis function space is constructed, and an optimization model is re-established based on this space. By solving the energy function that minimizes the normal and regularization terms, new control coefficients are obtained. Then, the implicit function is calculated:

[0097]

[0098] Subsequently, an error assessment was performed on all point cloud points to determine whether their fitting error met the error tolerance ε. e If there is a point cloud error greater than ε e Then, the refined region is identified and a new round of mesh refinement, truncation, and optimization steps are performed.

[0099] At each level, construct an updated truncated hierarchical B-spline basis function space and the corresponding implicit function, until one of the following termination conditions is met: the error at all points is less than the tolerance ε. e Alternatively, the maximum refinement level N can be reached, and the final implicit function f(q) and its zero level set f(q) = 0 can be output as the result of point cloud reconstruction surface.

[0100] The implicit function calculation process in the above steps is as follows:

[0101] After substituting the three-dimensional tensor product B-spline into the optimization model, the tensor product of the basis functions is defined as... The three-dimensional tensor product spline function is then written as:

[0102] In p i =(x i y i , z i The data fitting matrix A at point ) is Then the first term in the optimization model min f(T1, T2, T3) = T1 + λ2T2 + λ2T Transformed into: Define the normal constraint matrix B: Define matrix d:

[0103] Then the second item Transformed into:

[0104]

[0105] Define matrix H xx H xy H yy H yz H xz H zz :

[0106]

[0107] The smoothness regularization H is then expressed as: H = H xx +2H xy +H yy +2H yz +2H xz +H zz .

[0108] Third item Transformed into:

[0109]

[0110] Putting the above three items together, we get:

[0111]

[0112] Then, by merging the three terms, the optimization problem is reduced to a quadratic form:

[0113] J(C)=C T MC-2b T C+c

[0114] Where M = A + λ1B + λ2H, λ1 is the normal uniformity system, λ2 is the regularization coefficient, and b = λ1d is the linear term of the normal constraint; Taking the derivative with respect to C: The optimal solution is C = M -1 b, written as C=(A+λ1B+λ2H) -1 (λ1d); The constant term c originates from the irrelevant terms of the normal constraint:

[0115] Step 6: Triangulate the final zero-level set to complete the surface reconstruction; post-processing such as smoothing or simplification can be performed on the reconstructed surface as needed to obtain the final triangular mesh model.

[0116] The zero-level set is meshed using Marching Cubes isosurface extraction technology to generate a triangular mesh model. Subsequently, surface post-processing techniques can be used to smooth or simplify the surface, resulting in a high-quality surface model that can be used for visualization or subsequent analysis.

[0117] Figure 2 Several sets of different input point cloud data and their corresponding surface reconstruction results are shown. It can be seen that by using different original point clouds, the reconstructed surface has a high degree of fit with the original point cloud, small error, and accurate reconstruction effect, which shows that the invention is highly effective.

[0118] To comprehensively assess the reconstruction performance of this invention, the following evaluation dimensions are used: 1) Average and maximum distances from points to the surface: These measure the average and worst-case fitting errors between the implicit surface and the original point cloud, respectively. Smaller values ​​indicate better fitting results. 2) Average curvature: This measures the overall curvature of the surface. Lower average curvature typically corresponds to a smoother surface. 3) Standard deviation of curvature: This describes the magnitude of local curvature changes on the surface. A smaller standard deviation indicates stable local changes. 4) Memory usage: This records the memory resources used during algorithm execution. Lower memory requirements indicate higher computational efficiency and better adaptability to large-scale point clouds.

[0119] To verify the robustness of the method of the present invention under actual noise conditions, Gaussian noise of different intensities was added to the model, as shown in Table 1:

[0120] Table 1 Noise Levels

[0121]

[0122] Experimental results show that as the noise level of the input point cloud increases, both the average and maximum distances from points to the surface increase, indicating that noise interference leads to an increase in fitting error. Simultaneously, the average curvature and standard deviation of curvature decrease, suggesting that noise weakens local details to some extent, making the surface smoother overall. Furthermore, memory usage also increases. Although the increase in noise affects accuracy and detail, the proposed method still maintains relatively good overall reconstruction quality, verifying the robustness and adaptability of this invention to noise.

[0123] An adaptive hierarchical refinement method based on truncated hierarchical B-splines was added, and its surface reconstruction results were compared with those using the original method (three-dimensional tensor product spline function of one layer of basis functions), as shown in Table 2.

[0124] Table 2 Comparison of evaluation index values ​​under the new method (truncated hierarchical B-spline) and the old method

[0125]

[0126] After enabling the adaptive resolution refinement module (i.e., the truncated hierarchical B-splines of this invention), compared with the unrefined control, the average and maximum distances from points to the surface increased slightly, but the overall error remained within acceptable limits. Simultaneously, the control mesh size was significantly reduced, and the memory consumption of the Fish, Deer, and Camel models decreased by approximately 26%, 27%, and 28%, respectively, demonstrating that the local refinement strategy significantly improved computational efficiency while maintaining fitting accuracy. For the Deer and Camel models, the standard deviation of curvature decreased from 1.040656 to 0.883453 and from 5.050540 to 4.434555, respectively, indicating that locally complex structural regions were sampled more evenly, with a more concentrated curvature distribution and smoother surfaces; the moderate increase in average curvature reflects better detail recovery. Overall, the refinement method of this invention achieves a better balance between accuracy and resource consumption, making it suitable for handling large-scale or high-noise point cloud reconstruction tasks.

[0127] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these all fall within the protection scope of this application. Therefore, the protection scope of this patent application should be determined by the appended claims.

Claims

1. An adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines, characterized in that, Includes the following steps: (1) Based on a globally uniform low-resolution grid, a first-layer truncated hierarchical B-spline basis function space is constructed. The initial implicit function is obtained by solving the optimization model, and its zero level set is extracted to generate the first-layer initial surface. (2) For each point in the input point cloud, use the error function to evaluate its fitting error, and gather the points whose error is greater than the set threshold to determine the region that needs further refinement. (3) Subdivide the grid cells within the refinement region to generate a higher resolution grid, and on this basis, establish a new set of truncated hierarchical B-spline basis functions to expand the current function space; (4) Perform a truncation operation on the THB spline basis functions that are completely located in the refinement region in the previous layer to remove redundant effects; merge the truncated basis functions with the basis function set generated in the new layer and update the entire basis function library; (5) Optimize the model again and solve the updated implicit function; (6) Repeat steps (2) to (5) to refine and optimize layer by layer until the fitting error of all points is lower than the set threshold or the preset maximum refinement level is reached, and output the final implicit function and its zero level set. (7) Triangulate the final zero-level set to complete the surface reconstruction; smooth or simplify the reconstructed surface as needed to obtain the final triangular mesh model.

2. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The optimization model described in step (1) is: min f(T1,T2,T3)=T1+λ1T2+λ2T3 Where p is defined i For point cloud data, Ensure that the curved surface closely fits the point cloud data. By using the constraint surface gradient, make it parallel to the normal vector n i Maintain consistency to improve normal accuracy; Regularization terms ensure surface smoothness and reduce the impact of noise. λ1 is the normal uniformity coefficient, λ2 is the regularization coefficient, and f... xx f yy f zz It is the second-order partial derivative of the function with respect to x, y, and z, f xy f yz f xz It is the mixed second-order partial derivative of the function.

3. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The process of constructing the first-level truncated hierarchical B-spline basis function space in step (1) is as follows: Starting with an initial level m=0, construct a globally uniform low-resolution grid with N control points. x N y N z Based on this grid, construct the set of truncated hierarchical B-spline basis functions for layer 0. in The complete set of all basis functions constructed for layer 0 (initial layer), suppφ is the support set of function φ, and this set of functions constitutes the truncated hierarchical B-spline basis function space of layer 0, with domain D. 0 The bounding box for the entire point cloud: D 0 =[x min ,x max ]×[y min ,y max ]×[z min ,z max ] Among them, (x min y min , z min ) and (x max y max , z max ) represents the minimum and maximum values ​​of the point cloud coordinates; the size of each grid cell is: Based on the initial truncation level B-spline basis function space The control coefficient c is optimized by solving an energy optimization model with normal fitting and regularization terms. φ Thus, the initial implicit function is obtained: And extract the zero-level set f 0 (x) = 0, thus obtaining the initial reconstructed surface of the first layer.

4. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implementation process of step (2) is as follows: The current refinement level is l, and its implicit function representation is: in, B is the control coefficient for the l-th level. i (u)B j (v)B k (w) is a cubic uniform B-spline basis function; For each p in the point cloud i (x m y m , z m (m = 1, 2, ..., M), the error ε is defined as the implicit function value f(x) m y m , z m Normalized value relative to the gradient magnitude: It is the gradient magnitude, used for normalization to avoid the absolute influence of the gradient magnitude on the error, and to give the error a relative meaning; For a point cloud Q = {q1, q2, ..., q...} M Each point q in} i =(x i y i , z i ), calculate the fitting error ε of the current level m. m If ε m If the value is relatively large, it means that the current resolution cannot accurately approximate that point; Based on the error threshold ε e To identify points {q} with larger errors i |ε i >ε e }; For each error point q i =(x i y i , z i ), calculate the parametric coordinates of these points in the current mesh; map these points to the cells of the current mesh, calculate their corresponding mesh index range, and determine the refinement region D by merging overlapping cells. m+1 .

5. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implementation process of step (3) is as follows: In region D that needs further refinement m+1 Inside, based on the refinement markers of the m-th layer mesh, the truncated hierarchical B-spline basis function space W of the next layer is constructed. m+1 A higher resolution mesh is generated by uniformly subdividing the mesh cells and increasing the number of control points; Construct a new set of basis functions based on the refined mesh: in, Let φ be the complete set of all basis functions constructed at layer (m+1), and let suppφ be the support set of function φ. This refers to the set of basis functions at the (m+1)th layer that still contribute to the refined region after truncation; correspondingly, the size of each mesh cell becomes... If the current refinement region has a grid index range of [i] in the x direction... min i max If ], then the corresponding number of control points is: Similarly, we can obtain This generates a new set of truncated hierarchical B-spline basis functions. Used to expand the entire function space.

6. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implementation process of step (4) is as follows: For the set of B-spline basis functions of the truncated level above. The support domain is identified as being completely contained within the current refined region D. m+1 The basis functions within are truncated, and a truncation operation is performed on them to remove redundant effects; Specifically, for If its support satisfies Its contribution has been covered by the next layer of basis functions, so it should be marked as inactive, that is, truncated and no longer participates in the subsequent function expression; The adjustment operation definition corresponding to the truncation mechanism is as follows: Where φ′ is a sub-basis function in the new layer whose support domain completely covers φ, a φ′ For the reconstructed coefficients, the new set of basis functions obtained after truncation is: in, This represents the residual part obtained after performing a cut operation on the truncated basis function of the previous layer. This represents the (m+1)th layer of basis functions newly introduced within the refined region, which ultimately updates the entire basis function library, forming the expanded truncated hierarchical B-spline basis function system.

7. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implementation process of steps (5) and (6) is as follows: After obtaining the updated set of B-spline basis functions at the truncation level Then, a new basis function space is constructed, and an optimization model is re-established based on this space. By solving the energy function that minimizes the normal and regularization terms, new control coefficients are obtained. Then, the implicit function is calculated:

8. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implementation process of step (6) is as follows: At each level, construct an updated truncated hierarchical B-spline basis function space and the corresponding implicit function, until the error at all points is below the tolerance ε. e Alternatively, the maximum refinement level N can be reached, and the final implicit function f(q) and its zero level set f(q) = 0 can be output as the result of point cloud reconstruction surface.

9. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 1, characterized in that, The implicit function calculation process is as follows: After substituting the three-dimensional tensor product B-spline into the optimization model, the tensor product of the basis functions is defined as... The three-dimensional tensor product spline function is then written as: In p i =(x i y i , z i The data fitting matrix A at point ) is Then the first term in the optimization model min f(T1, T2, T3) = T1 + λ1T2 + λ2T Transformed into: Define the normal constraint matrix B: Define matrix d: Then the second item Transformed into: Define matrix H xx H xy H yy H yz H xz H zz : The smoothness regularization H is then expressed as: H = H xx +2H xy +H yy +2H yz +2H xz +H zz ; Third item Transformed into: Combining the above three items, we get: Then, by merging the three terms, the optimization problem is reduced to a quadratic form: J(C)=C T MC-2b T C+c Where M = A + λ1B + λ2H, λ1 is the normal uniformity system, λ2 is the regularization coefficient, and b = λ1b is the linear term of the normal constraint; Taking the derivative with respect to C: The optimal solution is C = M -1 b, written as C=(A+λ1B+λ2H) -1 (λ1d); The constant term c originates from the irrelevant terms of the normal constraint:

10. The adaptive implicit surface reconstruction method for point clouds based on truncated hierarchical B-splines according to claim 2, characterized in that, The normal consistency coefficient is 1, and the regularization term coefficient is 0.1.

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