Cotangent angle Laplace grid smoothing method and system based on multiple control points

Abstract

The present invention relates to a cotangent angle Laplace grid smoothing method based on multiple control points: S1: obtaining and processing the original unsmoothed grid data after grid reconstruction; S2: constructing the original grid matrix; S3: solving the cotangent angle element values ​​corresponding one-to-one to the grid vertices; S4: constructing the row and column positioning number container of the sparse matrix; S5: selecting the number and position of the control anchor points, and expanding the row and column number container, and finally generating the matrix; S6: solving the equation of the multi-objective optimization problem and updating the vertices of the original grid. The method and system for grid smoothing based on control point generation and multi-objective optimization of the present invention converts the Cartesian coordinates of a three-dimensional model into Laplace coordinates through a discrete Laplace operator. Compared with directly performing addition and subtraction smoothing of the Cartesian coordinates, the method and system of the present invention are more realistic and stable. The present invention selects the optimal number of control points and weight coefficients for tooth and skull grids to achieve the best efficiency and grid quality.

Description

Technical Field

[0001] The present invention relates to the technical field of medical image processing, and in particular to a cotangent angle Laplace grid smoothing method and system based on multiple control points. Background Art

[0002] CBCT data that has not been meshed has problems such as rough surfaces and multi-angle errors. Currently, there is no good way to deal with the rough surfaces, and the real CBCT data cannot be restored, resulting in the generated mesh having a weird shape, which is not conducive to doctors' viewing. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to design a cotangent angle Laplace grid smoothing method and system based on multiple control points to solve the existing technical problems.

[0004] To solve the above technical problems, the cotangent angle Laplace grid smoothing method based on multiple control points of the present invention comprises the following steps:

[0005] Step S1: Acquire and process the original unsmoothed mesh data after mesh reconstruction to obtain a valid mesh that can be processed later;

[0006] Step S2: Construct the original grid matrix. The steps are as follows:

[0007] Step S21: Use Eigen to initialize a 1*3 matrix a, and assign all elements to 0, a=[0, 0, 0];

[0008] Step S22: traverse each grid vertex coordinate, and construct the x, y, z coordinate point values ​​of each vertex coordinate into a 1*3 matrix m=[x, y, z];

[0009] Step S23: Increase the number of rows of the initialized 1*3 matrix with 0 elements in the row direction, and continuously insert the 1*3 matrix constructed in step S22 into the last row. After all the vertices of the mesh are traversed, an (n+1)*3 original mesh vertex matrix M is generated. n : Where n is the number of vertices, and the rows are arranged in the order of vertex serial numbers 1, 2, 3…, n;

[0010] Step S24: Delete the first row of the (n+1)*3 matrix to form an n*3 matrix, and save it for subsequent processing;

[0011] Step S3: solving the cotangent angle element values ​​corresponding to the mesh vertices one by one;

[0012] Step S4: constructing a row and column positioning number container of a sparse matrix, i.e., a Laplace cotangent angle matrix;

[0013] Step S5: Select the number and position of the control anchor points, and based on this, expand the row and column number container, and finally generate a matrix;

[0014] Step S6: Solve the equations of the multi-objective optimization problem and update the vertices of the original mesh.

[0015] Furthermore, step S1 includes the following steps:

[0016] Step S11: Check the quality of the mesh to check the polygonal soup of the mesh to eliminate errors;

[0017] Step S12: Check and repair the direction of the mesh to make it consistent, so that the reversed facets are forward facing, and output a valid mesh that can be processed later.

[0018] Furthermore, step S3 includes the following steps:

[0019] Step S31: traverse each vertex of the original mesh, find all half-edge structures adjacent to each vertex, and find the number of all adjacent vertices corresponding to each vertex and the corresponding index through the half-edge structure;

[0020] Step S32: traverse all adjacent vertices of each vertex to find adjacent facets between the vertex and the adjacent vertex, mainly by finding common facets through all adjacent faces of two vertices to obtain two adjacent facets, and find the common vertices and independent points of the adjacent facets;

[0021] Step S33: construct a vector using the left and right independent points between the two vertices found in step S32, calculate the sum of the cotangent angles using dot product and cross product operations of the vectors, and store the values ​​in the weight container in the order of the original mesh vertices.

[0022] Furthermore, step S5 includes the following steps:

[0023] Step S51: randomly selecting 20% ​​of all mesh vertices as anchor control points in a scattered manner;

[0024] Step S52: using all mesh vertices as control anchor points as method 2;

[0025] Step S53: Based on step S4, the row and column positioning containers are correspondingly expanded by using the first and second methods respectively;

[0026] Step S54: based on step S33, the weight container is augmented with the anchor point weight coefficient;

[0027] Step S55: The final construction of the Laplace cotangent angle coefficient matrix is ​​performed using the row and column positioning number containers and weight coefficient containers formed in steps S4 and S5 to form a sparse matrix A of (N+K)*N.

[0028] Furthermore, step S6 includes the following steps:

[0029] Step S61: Use the sparse matrix A in step S55 as the left side of the equation;

[0030] Step S62: Repeat steps S2 to S5, and change the off-diagonal elements of the weight item to the inverse of the number of adjacent points to form a new (N+K)*N uniform weight matrix Au. Extract each column of the original mesh vertex matrix and multiply it with Au to obtain the right side of the equation, that is, the Laplace coordinate value L x , L y , L z ;

[0031] Step S63: Using the uniform weight Au and the updated Laplace coordinate L x , L y , L z The constructed equation is solved to obtain the updated mesh vertex values, which is the new smoothed mesh.

[0032] Furthermore, step S4 includes the following steps:

[0033] Step S41: construct a vertex number positioning container in the row direction and add the row numbers on the diagonal line;

[0034] Step S42: construct an adjacent point location serial number container in the column direction and add the column numbers on the diagonal line.

[0035] The present invention also provides a cotangent angle Laplace grid smoothing system based on multiple control points, comprising:

[0036] at least one processor; and

[0037] at least one memory in communication with the processor;

[0038] The memory stores instructions that can be executed by the processor, and the instructions are executed by the processor to enable the electronic device to perform the aforementioned cotangent angle Laplace grid smoothing method based on multiple control points.

[0039] Beneficial Effects of the Invention: The present invention's method and system for mesh smoothing based on control point generation and multi-objective optimization primarily utilizes a discrete Laplace operator to convert the Cartesian coordinates of a three-dimensional model into Laplace coordinates. Laplace coordinates carry many detailed features of local surfaces, such as curvature and normals. Compared to directly smoothing Cartesian coordinates through addition and subtraction, the present invention's method and system are more realistic and stable. Considering the selection of the position and number of control points and the Lagrangian factor weights during the smoothing process, the present invention selects the optimal number of control points and weight coefficients for tooth and skull meshes to achieve optimal efficiency and mesh quality. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The specific embodiments of the present invention will be further explained below with reference to the accompanying drawings.

[0041] Figure 1 is a flow chart of the present invention;

[0042] Figure 2(a), Figure 2(b), and Figure 2(c) are comparisons of skull mesh quality under different weight coefficients;

[0043] Figure 3(a), Figure 3(b) and Figure 3(c) are comparison diagrams of tooth mesh quality using different control point selection methods. DETAILED DESCRIPTION

[0044] Example 1

[0045] Combine Figure 1 The cotangent angle Laplace grid smoothing method based on multiple control points in this embodiment includes the following steps:

[0046] Step S1: Acquire and process the original unsmoothed mesh data after mesh reconstruction to obtain a valid mesh that can be processed later. Step S1 includes the following steps:

[0047] Step S11: Check the quality of the mesh to check the polygonal soup of the mesh to eliminate errors;

[0048] Step S12: Check and repair the direction of the mesh to make it consistent, so that the reversed facets are in the positive direction, and output a valid mesh that can be processed later;

[0049] Step S2: Construct the original grid matrix. The steps are as follows:

[0050] Step S21: Use Eigen to initialize a 1*3 matrix a, and assign all elements to 0, a=[0, 0, 0];

[0051] Step S22: traverse each grid vertex coordinate, and construct the x, y, z coordinate point values ​​of each vertex coordinate into a 1*3 matrix m=[x, y, z];

[0052] Step S23: Increase the number of rows of the initialized 1*3 matrix with 0 elements in the row direction, and continuously insert the 1*3 matrix constructed in step S22 into the last row. After all the vertices of the mesh are traversed, an (n+1)*3 original mesh vertex matrix M is generated. n : Where n is the number of vertices, and the rows are arranged in the order of vertex serial numbers 1, 2, 3…, n;

[0053] Step S24: Delete the first row of the (n+1)*3 matrix to form an n*3 matrix, and save it for subsequent processing;

[0054] Step S3: Calculate the cotangent angle element value corresponding to each mesh vertex. Step S3 includes the following steps:

[0055] Step S31: traverse each vertex of the original mesh, find all half-edge structures adjacent to each vertex, and find the number of all adjacent vertices corresponding to each vertex and the corresponding index through the half-edge structure;

[0056] Step S32: traverse all adjacent vertices of each vertex to find adjacent facets between the vertex and the adjacent vertex, mainly by finding common facets through all adjacent faces of two vertices to obtain two adjacent facets, and find the common vertices and independent points of the adjacent facets;

[0057] Step S33: Construct a vector from the left and right independent points between the two vertices found in step S32, calculate the sum of the cotangent angles by vector dot product and cross product operations, and store the values ​​in the weight container in the order of the original mesh vertices.

[0058] Step S4: constructing a row and column positioning number container of a sparse matrix, namely a Laplace cotangent angle matrix; Step S4 includes the following steps:

[0059] Step S41: construct a vertex number positioning container in the row direction and add the row numbers on the diagonal line;

[0060] Step S42: constructing a column-wise adjacent point location serial number container and adding the column numbers on the diagonal line;

[0061] Step S5: Select the number and position of the control anchor points, and based on this, expand the row and column number containers, and finally generate a matrix; Step S5 includes the following steps:

[0062] Step S51: randomly selecting 20% ​​of all mesh vertices as anchor control points in a scattered manner;

[0063] Step S52: using all mesh vertices as control anchor points as method 2;

[0064] Step S53: Based on step S4, the row and column positioning containers are correspondingly expanded by using the first and second methods respectively;

[0065] Step S54: based on step S33, the weight container is augmented with the anchor point weight coefficient;

[0066] Step S55: The Laplace cotangent angle coefficient matrix is ​​finally constructed using the row and column positioning number containers and weight coefficient containers formed in steps S4 and S5 to form a (N+K)*N sparse matrix A;

[0067] Step S6: solving the multi-objective optimization problem equations and updating the vertices of the original mesh; Step S6 includes the following steps:

[0068] Step S61: Use the sparse matrix A in step S55 as the left side of the equation;

[0069] Step S62: Repeat steps S2 to S5, and change the off-diagonal elements of the weight item to the inverse of the number of adjacent points to form a new (N+K)*N uniform weight matrix Au. Extract each column of the original mesh vertex matrix and multiply it with Au to obtain the right side of the equation, that is, the Laplace coordinate value L x , L y , L z ;

[0070] Step S63: Using the uniform weight Au and the updated Laplace coordinate L x , L y , L z The constructed equation is solved to obtain the updated mesh vertex values, which is the new smoothed mesh.

[0071] Considering the location and number of control points and the Lagrangian factor weights that need to be selected during the smoothing process, the present invention selects the optimal number of control points and weight coefficients for the tooth and skull meshes to achieve the best efficiency and mesh quality. Specifically,

[0072] The original mesh output after segmentation is shown in Figure 2(a). The mesh is relatively rough, which affects the aesthetics of the skull mesh. Therefore, the cotangent angle Laplace smoothing method based on multiple control points in this embodiment is used to make the mesh smoother and more visually aesthetically pleasing. The same number of control points and different Lagrangian factor coefficients are used for Laplace smoothing. Experiments are conducted using different Lagrangian factors, with the Lagrangian factor ranging from 0.0 to 0.5. Figures 2(b) and 2(c) are skull mesh images with Lagrangian factor weights of 0.18 and 0.2, respectively. In this embodiment, preferably, the skull mesh selects a Lagrangian factor of 0.2, as shown in Figure 2(c) for display.

[0073] The segmented tooth mesh, shown in Figure 3(a), is rough, affecting both aesthetics and the subsequent algorithm flow. Therefore, the mesh also needs to be smoothed. Figures 3(b) and 3(c) use the vertices of the tooth mesh as control points and smooth the mesh using different Lagrangian factors. Figures 3(b) and 3(c) show tooth meshes with Lagrangian factor weights of 0.1 and 0.15, respectively. In this embodiment, a Lagrangian factor of 0.15 is preferably selected for the tooth mesh, as shown in Figure 3(c).

[0074] The position and number of control points also have a certain impact on the smoothing effect. The position of the control points affects the smoothing range, with vertices closer to the control points having a more pronounced smoothing effect. The same applies to the number of control points. If there are a sufficient number of control points, the smoothing range of surrounding vertices will be greater. This embodiment uses all vertices of the mesh as control points and performs Laplacian smoothing on the vertices of the mesh, making the mesh smoother.

[0075] Example 2

[0076] The cotangent angle Laplace grid smoothing system based on multiple control points of this embodiment includes:

[0077] at least one processor; and

[0078] at least one memory in communication with the processor;

[0079] The memory stores instructions that can be executed by the processor, and the instructions are executed by the processor to enable the electronic device to perform the cotangent angle Laplace grid smoothing method based on multiple control points in embodiment 1.

[0080] In the above description, many specific details are set forth in order to fully understand the present invention. However, the above description is only a preferred embodiment of the present invention. The present invention can be implemented in many other ways different from those described herein, so the present invention is not limited to the specific implementation disclosed above. At the same time, any person skilled in the art can make many possible changes and modifications to the technical solution of the present invention using the methods and technical contents disclosed above without departing from the scope of the technical solution of the present invention, or modify it into an equivalent embodiment of equivalent changes. Any simple modification, equivalent change and modification made to the above embodiment based on the technical essence of the present invention without departing from the content of the technical solution of the present invention still falls within the scope of protection of the technical solution of the present invention.

Claims

1. A cotangent angle Laplace grid smoothing method based on multiple control points, characterized by: The mesh is a tooth and skull mesh in the medical imaging field, and includes the following steps: Step S1: Acquire and process the original unsmoothed mesh data after mesh reconstruction to obtain a valid mesh that can be processed later; Step S2: Construct the original grid matrix. The steps are as follows: Step S21: Use Eigen to initialize a 1*3 matrix a and assign all elements to 0. ; Step S22: traverse each grid vertex coordinate and construct the x, y, z coordinate point values ​​of each vertex coordinate into a 1*3 matrix ; Step S23: Increase the number of rows of the initialized 1*3 matrix with 0 elements in the row direction, and continuously insert the 1*3 matrix constructed in step S22 into the last row. After all the vertices of the mesh are traversed, an (n+1)*3 original mesh vertex matrix M is generated. n : ; Where n is the number of vertices, and the rows are arranged in the order of vertex serial numbers 1, 2, 3..., n; Step S24: Delete the first row of the (n+1)*3 matrix to form an n*3 matrix, and save it for subsequent processing; Step S3: solving the cotangent angle element values ​​corresponding to the mesh vertices one by one; Step S4: constructing a row and column positioning number container of a sparse matrix, i.e., a Laplace cotangent angle matrix; Step S5: Select the number and position of the control anchor points, and based on this, expand the row and column number container, and finally generate a matrix; Step S5 includes the following steps: Step S51: randomly selecting 20% ​​of all mesh vertices as anchor control points in a scattered manner; Step S52: using all mesh vertices as control anchor points as method 2; Step S53: Based on step S4, the row and column positioning containers are correspondingly expanded by using the first and second methods respectively; Step S54: based on step S33, the weight container is augmented with the anchor point weight coefficient; Step S55: Final construction of the Laplace cotangent angle coefficient matrix is ​​performed using the row and column positioning number containers and weight coefficient containers formed in steps S4 and S5 to form a (N+K)*N sparse matrix A; Step S6: solving the multi-objective optimization problem equations and updating the vertices of the original mesh; Step S6 includes the following steps: Step S61: Use the sparse matrix A in step S55 as the left side of the equation; Step S62: Repeat steps S2 to S5, and change the off-diagonal elements of the weight item to the inverse of the number of adjacent points to form a new (N+K)*N uniform weight matrix Au. Extract each column of the original mesh vertex matrix and multiply it with Au to obtain the right side of the equation, that is, the Laplace coordinate value L. x , L y , L z ; Step S63: Using the uniform weight Au and the updated Laplace coordinate L x , L y , L z The constructed equation is solved to obtain the updated mesh vertex values, which is the new smoothed mesh.

2. The cotangent angle Laplace grid smoothing method based on multiple control points according to claim 1, characterized in that: Step S1 includes the following steps: Step S11: Perform a quality check on the mesh to check the polygonal soup of the mesh to eliminate errors; Step S12: Check and repair the direction of the mesh to make it consistent, so that the reversed facets are forward facing, and output a valid mesh that can be processed later.

3. The cotangent angle Laplace grid smoothing method based on multiple control points according to claim 1, characterized in that: Step S3 includes the following steps: Step S31: traverse each vertex of the original mesh, find all half-edge structures adjacent to each vertex, and find the number of all adjacent vertices corresponding to each vertex and the corresponding index through the half-edge structure; Step S32: traverse all adjacent vertices of each vertex to find adjacent facets between the vertex and the adjacent vertex, mainly by finding common facets through all adjacent faces of two vertices to obtain two adjacent facets, and find the common vertices and independent points of the adjacent facets; Step S33: construct a vector using the left and right independent points between the two vertices found in step S32, calculate the sum of the cotangent angles using the dot product and cross product operations of the vectors, and store the values ​​in the weight container in the order of the original mesh vertices.

4. The cotangent angle Laplace grid smoothing method based on multiple control points according to claim 1, characterized in that: Step S4 includes the following steps: Step S41: construct a vertex number positioning container in the row direction and add the row numbers on the diagonal line; Step S42: construct an adjacent point location serial number container in the column direction and add the column numbers on the diagonal line.

5. A cotangent angle Laplace grid smoothing system based on multiple control points, characterized by: include: at least one processor; as well as at least one memory in communication with the processor; The memory stores instructions that can be executed by the processor, and the instructions are executed by the processor to enable the electronic device to perform the cotangent angle Laplace grid smoothing method based on multiple control points according to any one of claims 1 to 4.

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