A method for twisting deformation of a three-dimensional mesh model

Abstract

The application provides a three-dimensional mesh model torsion deformation method, which comprises the following steps: constructing a Laplacian coordinate through a Cartesian coordinate on a network model; performing a translation and rotation deformation operation on the network model to generate a deformed Laplacian coordinate; combining a control point set by a user and a deformation constraint applied to complete reconstruction of the Cartesian coordinate; improving mesh torsion deformation precision, improving mesh quality, and realizing personalized torsion deformation.

Description

Technical Field

[0001] This invention relates to three-dimensional mesh models, and more particularly to a method for torsional deformation of three-dimensional mesh models. Background Technology

[0002] In fields such as industry, medicine, and art design, including film production, mesh editing, and game engines, users often need to twist and deform models to simulate real-world effects due to personalized requirements. In the field of medical external fixation devices, doctors frequently need to twist and deform patient areas to obtain models that conform to clinical significance. The biggest challenge in torsional deformation is preserving the local geometric features of the deformed area and avoiding distortion of details. Laplacian coordinates, due to their unique definition, can well preserve detailed features, but they are prone to rotational instability during torsional deformation. This paper addresses the issue of rotational instability during torsional deformation by adding affine matrices to the model vertices, correcting the Laplacian coordinates, and building upon the Laplacian coordinate deformation method. Summary of the Invention

[0003] The purpose of this invention is to address the shortcomings of the prior art by providing a method for torsional deformation of a three-dimensional mesh model, thereby improving the accuracy of mesh torsional deformation and enabling personalized torsional deformation.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] This invention provides a method for torsional deformation of a three-dimensional mesh model, comprising the following steps:

[0006] S1. Construct Laplacian coordinates using Cartesian coordinates on the network model;

[0007] S2. Perform translation, rotation, and deformation operations on the network model to generate deformed Laplacian coordinates;

[0008] S3. Combine the control points set by the user and the applied deformation constraints to complete the reconstruction of the Cartesian coordinates.

[0009] Furthermore, in S1, constructing the Laplacian coordinates includes:

[0010] S101. Describe the mesh model M using (P,S,C), where P is the set of vertices in the mesh model, S is the set of edges in the mesh model, and C is the set of faces in the mesh model.

[0011] S102, Let p1, p2, p3...p n Let n be a point in the point set p, where n is the number of points in the point set.

[0012] S103. Let N(i) = {j|(i,j)∈S} represent p. i The set consisting of the number of neighboring points of a point, in p i The Laplacian coordinates δ at point i for:

[0013]

[0014] Where L is the matrix for transforming between Cartesian and Laplacian coordinates, called the Laplacian coefficient matrix; ω ij It is related to vertex p i The relevant weighting coefficients; i and j are the indexes; N is the number of neighboring points of the i-th point.

[0015] Furthermore, S2 includes the following steps:

[0016] S201. Pre-deformation is achieved by using Laplacian coordinate deformation on the network model in S1.

[0017] S202, For each vertex p after model deformation i Estimate an affine matrix Q i ;

[0018]

[0019] Where, p ij Represents vertex p i The j-th neighbor;

[0020] To maintain the linearity of equation (2), Q is... i Approximately expressed as a linear function of P′, with matrix Q i Written as:

[0021]

[0022] The simulated rotation coordinates U after adding the affine matrix are obtained. i (s i ,r i ) T and the coordinates v of the initial deformation i Find the minimum solution to equation (4):

[0023] ||U i (s i ,r i ) T -v i || 2 (4)

[0024] S203. Simulate the transformation of the vertices, update the error function, and solve for P″.

[0025]

[0026] H is the set of control points.

[0027] Furthermore, S3 includes the following steps:

[0028] S301. Calculate the initial Laplacian coordinates based on the Laplacian coefficient matrix. Then, through the estimated local transformation Q i Update Laplacian coordinates

[0029] S302, calculate respectively and The modulus lengths norm0 and norm1, where,

[0030]

[0031] S303. Calculate the scaling factor (scale) of the Laplacian coordinates under large-angle torsion:

[0032]

[0033] S304. Use the scaling factor to correct the Laplacian coordinates to complete the adjustment under large-angle torsion.

[0034] The beneficial effects of this invention are: improving the accuracy of mesh torsional deformation, improving mesh quality, and achieving personalized torsional deformation. Attached Figure Description

[0035] Figure 1 This is a flowchart of a three-dimensional mesh model torsional deformation method according to the present invention;

[0036] Figure 2 This is a schematic diagram of the model deformation partitioning;

[0037] Figure 3 A diagram showing the display of the bounding box and the selection of fixed anchor points;

[0038] Figure 4 Diagram showing the selection of the torsion shaft;

[0039] Figure 5 This is a diagram showing the deformation effect of the finger model;

[0040] Figure 6 This is a local mesh quality map of the finger model;

[0041] Figure 7 This is a rendering of the Wrist model after deformation.

[0042] Figure 8 This is a local mesh quality map of the Wrist model. Detailed Implementation

[0043] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0044] Please see Figure 1 A method for torsional deformation of a three-dimensional mesh model includes the following steps:

[0045] S1. Construct Laplacian coordinates using Cartesian coordinates on the network model;

[0046] S2. Perform translation, rotation, and deformation operations on the network model to generate deformed Laplacian coordinates;

[0047] S3. Combine the control points set by the user and the applied deformation constraints to complete the reconstruction of the Cartesian coordinates.

[0048] In S1, constructing the Laplacian coordinates includes:

[0049] S101. Describe a mesh model M using (P,S,C), where P is the set of vertices in the mesh model; S is the set of edges in the mesh model; and C is the set of faces in the mesh model.

[0050] S102, Let p1, p2, p3...p n Let n be a point in the point set p, where n is the number of points in the point set.

[0051] S103. Let N(i) = {j|(i,j)∈S} represent p. i The set consisting of the number of neighboring points of a point, in p i The Laplacian coordinates δ at point i for:

[0052]

[0053] Where L is the matrix for converting between Cartesian coordinates and Laplacian coordinates, called the Laplacian coefficient matrix; i and j are the indices; N is the number of neighborhood points of the i-th point.

[0054] ω ij It is related to vertex p i The relevant weighting coefficients. Specifically, this method possesses translation invariance.

[0055]

[0056] Where T is the translation matrix; Tp i =p i +d;δ i The initial grid is at point p. i Laplacian coordinates at the location.

[0057] The Laplacian coordinates remain unchanged before and after the translation, meaning that the translation does not change the Laplacian coordinates.

[0058] Specifically, this method is sensitive to rotation and scaling.

[0059]

[0060] Where R is the rotation or scaling matrix; δ i The initial grid is at point p. i Laplacian coordinates at the location.

[0061] Laplacian coordinates are invariant to translation, but not to rotation and scaling. Rotation or scaling operations applied to deformed models will affect the deformed model. Rotation is essential for torsional deformation of mesh models using Laplacian coordinates, and large-angle twists can also produce scaling effects. Therefore, considering the sensitivity of Laplacian coordinates to rotation and scaling, the main method of this invention is to first pre-deform the model using Laplacian coordinates, and then perform deformation on each deformed vertex p. i Estimate an affine matrix Q i Simulate the transformation of the vertices, update the error function, and solve for P″; matrix Q. i It should include rotation and scaling; an approximate solution is obtained using the idea of ​​least squares.

[0062] S2 includes the following steps:

[0063] S201. Pre-deformation is achieved by using Laplacian coordinate deformation on the network model in S1.

[0064] S202, For each vertex p after model deformation i Estimate an affine matrix Q i ;

[0065]

[0066] Where, p ij Represents vertex p i The j-th neighbor;

[0067] To maintain the linearity of equation (2), Q is... iApproximately expressed as a linear function of P′, with matrix Q i Written as:

[0068]

[0069] The simulated rotation coordinates U after adding the affine matrix are obtained. i (s i ,r i ) T and the coordinates v of the initial deformation i Find the minimum solution to equation (4):

[0070] ||U i (s i ,r i ) T -v i || 2 (4)

[0071] S203. Simulate the transformation of the vertices, update the error function, and solve for P″.

[0072]

[0073] H is the set of control points.

[0074] That is, the minimum solution of equation (4) can be obtained, and p can be calculated. i The linear transformation matrix Q' i Converted to p via equation (5) i P″ is a quadratic function of ′, and the error function can be solved by using the least squares method.

[0075] The deformed mesh vertices are obtained, and the connection relationships between the vertices are restored to complete the mesh reconstruction, realizing the rotation-invariant mesh deformation process.

[0076] The Cartesian coordinates obtained by estimating the local transformation for each vertex of the mesh model are sufficient. However, in cases where the torsion angle is too large, the model will scale, and the Laplacian coordinates need to be adjusted.

[0077] S3 includes the following steps:

[0078] S301. Calculate the initial Laplacian coordinates based on the Laplacian coefficient matrix. Then, through the estimated local transformation Q i Update Laplacian coordinates

[0079] L is the matrix for converting between Cartesian and Laplacian coordinates, called the Laplacian coefficient matrix, as detailed in formula (1).

[0080] S302, calculate respectively and The modulus lengths norm0 and norm1, where,

[0081]

[0082] S303. Calculate the scaling factor (scale) of the Laplacian coordinates under large-angle torsion:

[0083]

[0084] S304. Use the scaling factor to correct the Laplacian coordinates to complete the adjustment under large-angle torsion.

[0085] Example 1

[0086] During the torsional deformation process, the vertex set of the mesh model is divided into three parts. Taking the Wrist model as an example, each part is as follows: Figure 2 The figures show the fixed anchor point, the torsion anchor point, and the position to be determined, respectively. The coordinates of the fixed anchor point remain unchanged before and after deformation, the coordinates of the torsion anchor point are user-defined, and the coordinates of the position to be determined will be obtained through the Laplacian deformation method.

[0087] Selection of fixed anchor points

[0088] In selecting fixed anchor points, the Axis Aligned Bounding Box (AABB) method was used. An AABB is a cuboid that encloses the model. It is simple to define; the eight vertices of the cuboid are composed of the maximum and minimum values ​​of the x, y, and z axes of the enclosed model, and its six faces are parallel to the coordinate axes. It is often used for collision detection.

[0089] Its definition is shown in equation (9):

[0090] box AABB ={(p min ,p max )|p min =(x min ,y min ,z min ),p max =(x max ,y max ,z max )} (9)

[0091] The process of building an AABB is as follows: traverse all vertices of the model, find the maximum and minimum values ​​of their x, y, and z coordinates respectively, and combine them to form the eight vertices of a cuboid. The resulting cuboid is the AABB of the current model.

[0092] After generating the AABB, users can select any plane of it to move it on the plane and adjust the size of the bounding box in that direction; they can select any vertex of it to scale and adjust the overall size of the bounding box; and they can rotate the entire bounding box along the center point of the AABB to adjust its orientation.

[0093] After customizing the bounding box interaction on the model, the vertices inside the bounding box and on its plane are defined as fixed anchor points for torsional deformation, and their positions do not change before and after deformation, such as... Figure 3 As shown.

[0094] Selection of torsion anchor points

[0095] In selecting torsion anchor points, the AABB bounding box method is continued to be used to determine the positions of the torsion anchor points. Then, the interactive position picking function is used to find a suitable torsion axis on the model. The torsion axis diagram is shown below. Figure 4 As shown, after the user sets the torsion angle, the coordinates of the torsion anchor point will be rotated around the selected torsion axis, and the new coordinates obtained are the position of the torsion anchor point after deformation, as well as the position of the index vertex after the model is deformed.

[0096] Selection of torsion angle

[0097] like Figure 5 As shown, Figure 5 (a) is the left and front view of the undeformed finger model. Figure 5 (b) is the model obtained by deformation using the Laplacian coordinate mesh torsion deformation algorithm proposed in this invention. The fixed anchor point is selected from the bottom of the finger model to a vertex 15mm from the bottom, and the torsion anchor point is selected from the top of the finger model to a vertex 25mm from the top. The central axis of the thumb is the torsion axis, and the torsion angle is set to -25°. The Tangent weight is selected for the Laplacian coordinate weights. The finger model has 5700 vertices and 10671 elements, and the deformation time is 0.734s, meeting the real-time deformation requirements.

[0098] like Figure 6 As shown, the deformation method of the present invention generates a significantly improved mesh quality, essentially eliminating distorted triangular facets and achieving a more natural deformation.

[0099] like Figure 7 As shown, Figure 7 (a) is the front and rear view of the undeformed wrist model. Figure 7 (b) The model obtained after torsional deformation using the algorithm proposed in this invention. The bottom of the wrist model, extending to a vertex 35mm from the bottom, is selected as the fixed anchor point. The top of the wrist model, extending to a vertex 30mm from the top, is selected as the torsion anchor point. The central axis of the wrist model is used as the torsion axis, the torsion angle is set to -20°, and the weight is set to Tangent weight. The deformation time is 7.647s, meeting the real-time requirements, and the deformation is natural. The tortuous part of the wrist has been adjusted, generating a wrist model that conforms to clinical significance, achieving the expected results.

[0100] like Figure 8 As shown, the part of the Wrist model with the largest deformation amplitude was magnified, which solved the problem of detail distortion during deformation, and the deformation quality was significantly improved, making the deformation more natural.

[0101] To reduce the number of degenerate meshes after torsional deformation, the mesh quality of the deformed finger and wrist models was optimized. After applying the mesh quality optimization algorithm, the average metric values ​​improved, the overall quality of the triangular facets was enhanced, and the overall mesh quality was improved.

[0102] The embodiments described above are merely illustrative of implementation methods of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be defined by the appended claims.

Claims

1. A method for torsional deformation of a three-dimensional mesh model, characterized in that, The 3D mesh model is either a finger model or a wrist model, and includes the following steps: S1. Construct Laplacian coordinates using Cartesian coordinates on the network model; S2. Perform translation, rotation, and deformation operations on the network model to generate deformed Laplacian coordinates; S3. Combine the control points set by the user and the applied deformation constraints to complete the reconstruction of the Cartesian coordinates; The user-defined control points include the selection of fixed anchor points, torsion anchor points, and torsion angles; In S1, constructing the Laplacian coordinates includes: S101, Use Describe the mesh model ,in, It is the set of vertices in a mesh model; It is the set of edges in the mesh model; It is a collection of faces in a mesh model; S102, Let , for vertex set The points in, where, The number of vertices in the vertex set; S103, Use express The set consisting of the number of neighboring points of a point, in Laplacian coordinates of the point for: （1）； in, It is a matrix that transforms between Cartesian coordinates and Laplacian coordinates, and is called the Laplacian coefficient matrix; Is with vertex The relevant weighting coefficients; and For serial numbers; For the first The number of points in the neighborhood of a given point; S2 includes the following steps: S201. Pre-deformation is achieved by using Laplacian coordinate deformation on the network model in S1. S202, For each vertex after model deformation Estimate an affine matrix ; （2）； in, Represents vertices The 1 neighbor; in order to maintain the linearity of equation (2), Approximate expression linear functions, matrices Written as: （3）； Obtain the simulated rotated coordinates after adding the affine matrix. and the coordinates of the initial deformation Find the minimum solution to equation (4): （4）； S203. Simulate the transformation of the vertices, update the error function, and solve. ; （5）； The set of control points; S3 includes the following steps: S301. Calculate the initial Laplacian coordinates based on the Laplacian coefficient matrix. Then through the estimated local transformation Update Laplacian coordinates ; S302, calculate respectively and Length of the module and ,in, （6）； S303. Calculate the scaling factor of the Laplacian coordinates under large-angle torsion. for: ； S304, using the scaling factor Correct the Laplacian coordinates to complete the adjustment under large-angle torsion.

Images