CN108876922B - Grid repairing method based on internal dihedral angle compensation regularization - Google Patents

Grid repairing method based on internal dihedral angle compensation regularization Download PDF

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CN108876922B
CN108876922B CN201810601490.9A CN201810601490A CN108876922B CN 108876922 B CN108876922 B CN 108876922B CN 201810601490 A CN201810601490 A CN 201810601490A CN 108876922 B CN108876922 B CN 108876922B
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吴晓群
翟羽佳
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Beijing Technology and Business University
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Abstract

A grid repairing method based on internal dihedral angle compensation regularization belongs to the field of digital geometric processing and computer graphics. Firstly, initializing an input grid and reconstructing a connection relation of a grid missing region; then, according to the initial connection relation, a grid repairing model based on an L1 data item and an internal dihedral angle corner repairing regular item is established; and finally, iteratively solving the vertex position of the grid by using an Augmented Lagrangian Method (Augmented Lagrangian Method), thereby obtaining the reconstructed grid with the guaranteed characteristics. Compared with the existing method, the mesh repairing method based on the internal dihedral angle compensation regularization has obvious advantages in the aspect of keeping sharp characteristics such as creases, corner points, puncture points, sharp points and the like on the mesh curved surface, and has wide application prospects in mesh repairing.

Description

Grid repairing method based on internal dihedral angle corner-filling regularization
Technical Field
The invention relates to a grid repairing method based on internal dihedral angle corner-filling regularization, belonging to the field of digital geometric processing and computer graphics.
Background
The three-dimensional object surface reconstruction is widely applied to urban reconstruction, cultural relic restoration, medical image processing, movie and television games and the like. However, in the reconstruction process, many factors may cause the loss of information of a part of the surface of the three-dimensional model, so that the reconstructed grid has holes, for example, the limitation of measurement tools and measurement techniques, the occurrence of object occlusion in the measurement process, and the non-normative preprocessing process of the original data. The existence of the holes can influence a series of operations such as analysis and editing of a subsequent three-dimensional model, so that the three-dimensional model is limited to applications such as three-dimensional printing, shape retrieval, virtual reality and the like. Therefore, grid repair work is crucial.
For holes which are generally smoother, the existing method can better repair the holes. However, the prior art methods are often unsatisfactory for holes with sharp features. While sharp features of the grid include creases, corner points, prick points, cusps, and the like. Existing mesh patch algorithms are roughly divided into two categories: voxel-based mesh patch algorithms and surface-based mesh patch algorithms.
The grid surface to be repaired is firstly converted into a voxel representation form by a voxel-based grid repairing algorithm, and then grid repairing is carried out in a voxel space by different methods. The method has the defects that the mutual conversion of the mesh curved surface and the voxel space can damage the connectivity of the original mesh, the detailed characteristics of the mesh model are lost, and even when the processed mesh model is too complex, an error topological structure can be generated.
The mesh patch algorithm based on the curved surface generally detects a hole region, and then directly performs mesh patch on the hole region, which can be roughly classified into three types. (1) Grid patching algorithm based on interpolation: the interpolation grid may be generated with simple polynomial functions, triangular B-splines or radial basis functions, and typically has smooth and continuous boundaries. However, the method is only suitable for holes similar to a disc and is not suitable for repairing complicated holes with grooves, islands and the like. (2) Mesh repairing algorithm based on triangulation: finding out the missing area defined by the hole boundary, directly triangulating the polygonal holes to obtain an initial mesh, and then optimizing the mesh by adopting different methods to improve the fairness and consistency between the mesh and the surrounding shape. The method can quickly realize grid repair, but is difficult to accurately recover the detailed characteristics of the hole region (3) and a grid repair algorithm based on a template (exemplar-based): and according to the similarity measurement rule, searching a local grid similar to the characteristics of the hole to be repaired in the incomplete model, and repairing the hole by using the found similar local grid. Or searching an online three-dimensional model library for a similar grid model. However, the time complexity of such methods is often high, it is difficult to perform efficient repair on complex grids, and if the missing regions of the grids are unique, distorted or even erroneous repair results may be obtained.
Aiming at the defects of the existing method, the invention provides a mesh repairing method for keeping a sharp characteristic, namely an L1 data item is used for reducing the dependence of an internal dihedral angle corner-filling regular item on mesh missing information. Firstly, initializing input incomplete grids, reconstructing the connection relation of grid missing regions, then searching the optimal missing vertex position through internal dihedral angle corner-filling regularization, and finally reconstructing a new complete grid. The method provides how to carry out grid repairing of the feature protection by using the inner dihedral angle compensation angle, and the grid information is considered globally aiming at the input incomplete grid instead of only aiming at the missing area, so that the sharp feature of the grid can be maintained while the grid repairing is finished.
Disclosure of Invention
The invention aims to solve the problems that sharp features cannot be effectively kept and an optimization algorithm is complex in the conventional mesh repairing technology, and provides a mesh repairing method based on internal dihedral angle compensation regularization.
The core idea of the invention is as follows: initializing the input incomplete grids, and reconstructing the connection relation of the grid missing areas; reducing the dependence of an internal dihedral angle supplementary angle regular term on grid missing information by using an L1 data item, and then searching an optimal vertex position through internal dihedral angle supplementary angle regularization; globally considering mesh information for the input mesh; firstly, initializing an input grid and reconstructing a connection relation of a grid missing region; then, according to the initial connection relation, a grid repairing model based on an L1 data item and an internal dihedral angle corner repairing regular item is established; and finally, iteratively solving the vertex position of the grid by using an Augmented Lagrangian Method (Augmented Lagrangian Method), thereby obtaining the reconstructed grid with the guaranteed characteristics.
The meshes mentioned in the present invention are all represented by triangular patches.
A grid repairing method based on internal dihedral angle corner-filling regularization comprises the following specific steps:
the method comprises the following steps: preprocessing an input grid, and reconstructing a connection relation of a grid missing region to obtain an initial grid with a complete connection relation;
the input grid is marked as M0, the initial grid is marked as M, and the preprocessing of the input grid specifically comprises:
the input grid being a defective grid M0For the incomplete mesh, a triangulation method based on minimum weight proposed in section 4 of literature 1 is adopted to triangulate the hole areaSubdividing, reconstructing a grid missing region, namely the connection relation of a hole region, and outputting an initial grid M with a complete connection relation;
document 1: liepa, Peter. filling holes in disks, Proceedings of the 2003 Europaphic/ACM SIGGRAPH symposium on Geometry processing, Europaphic Association,2003.
Step two, establishing an energy function of grid repairing based on an L1 data item and an internal dihedral angle corner repairing regular item according to the initial grid M with the complete connection relation obtained in the step one:
incomplete grid M for memory input0The set of vertices is
Figure BDA0001693178320000031
Where M is the input grid M0The number of the middle vertexes; the vertex set of the initial mesh M is v ═ v1,v2,...,vnH, the edge set is e ═ e1,e2,...,edThe length of the side is set as l ═ l1,l2,...,ldThe set of internal dihedral angles is theta ═ theta1,θ2,...,θdN is the number of top points in the initial grid M, and n is more than M; d is the number of edges in the initial mesh M, e in the set of edgesiIs the length of l in the length set liAngle theta in the set of internal angles thetaiRepresenting a shared edge eiThe internal dihedral angle between the two triangular surface patches;
keeping the known vertex positions in the initial mesh M as close as possible to the original positions after the mesh inpainting process, the mesh features are maintained by an internal dihedral regularization constraint, while the mesh features can be kept by minimizing
(1) Obtaining the vertex position of the optimal mesh by the energy function of the formula;
Figure BDA0001693178320000041
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0001693178320000042
is to satisfy λ Ef(v,v0)+Er(l, θ) the minimum vertex position, i.e., v; ef(v,v0) For data items, Er(l, θ) is a regularization term; lambda is a data item parameter, and the data item is used for keeping the original characteristics of the grid and reducing the dependence of internal dihedral angle regularization on missing vertex information;
step 2.1, calculating a data item, specifically by the following formula (2):
Figure BDA0001693178320000043
wherein v isiSet of vertices representing the initial mesh, v ═ v1,v2,...,vnItem i in (j);
Figure BDA0001693178320000044
a set of vertices representing the input incomplete mesh,
Figure BDA0001693178320000045
item i in (1); l Fv-v0||1Represents Fv-v0L1 regularization; f represents a m × n projection matrix, defined as (3):
Figure BDA0001693178320000046
step 2.2, calculating a regular term, specifically calculating by a formula (4):
Figure BDA0001693178320000047
wherein liRepresenting the edge e in the initial gridiI.e. the set of side lengths l ═ l1,l2,...,ldItem i in (j); theta.theta.iRepresenting a shared edge eiThe internal dihedral angle (pi-theta) between two triangular patchesi) Refers to the complement of the internal dihedral angle; simplified dihedral corner knotTwo half-planes of the structure are Deltav1v3v4And Δ v1v2v3The shared edge of the two half-planes is v1v3V is corresponding to the third vertex2And v4(ii) a Definition of T1,T2Is two lengths | | v1v3Vector of | |, T1Is plane Δ v1v3v4Inner normal direction of, T2Is plane Δ v1v2v3Outer normal direction of (T)1And T2The included angle between them is pi-theta, then | v |1v3Is the vector T1And T2The arc length of the arc sandwiched therebetween; the normal vector formula is calculated based on the edges and included angles of the triangular patch as follows (5):
Figure BDA0001693178320000051
wherein cot represents an inverse cotangent function, which is equal to a cosine function divided by a sine function; theta4,1,3Is side v1v4And the side v1v3Angle of (a) of1,3,4Is side v1v3And the side v3v4Angle of (a) of2,3,1Is side v2v3And the side v1v3Angle of (a) theta3,1,2Is side v1v2And the side v1v3The included angle of (A); according to T1,T2The arc length, i.e./is determinedi(π-θi) Expressed as (6):
Figure BDA0001693178320000052
wherein the content of the first and second substances,
Figure BDA0001693178320000053
therefore, the regularization term is specifically calculated as the following equation (7):
Figure BDA0001693178320000054
wherein, K1Represents a matrix in formula (6)
Figure BDA0001693178320000055
||K1v||1Represents K1L1 regularization of v;
in conjunction with equations (2) and (7), the energy function (1) of mesh patch can be written as equation (8) below:
Figure BDA0001693178320000056
wherein the content of the first and second substances,
Figure BDA0001693178320000057
is to satisfy lambda | Fv-v0||1+||K1v||1The minimum vertex position, i.e., v;
step three, solving the energy function of grid repair in the step two by applying a value-added Lagrange method, which specifically comprises the following steps:
step 3.1, solving equation (8) is converted into solving the optimization problem with the constraint by the following formula (9):
Figure BDA0001693178320000061
wherein z-Fv-v0，p＝K1v，||z||1L1 regularization representing z, | p | | luminance1L1 regularization, which represents p;
Figure BDA0001693178320000062
solving the problem of satisfying lambda | z | | non-woven phosphor1+||p||1The smallest z, p;
the constraint problem in (9) above can be transformed to solve the functional saddle point problem of the following formula (10) according to the incremental lagrangian method:
Figure BDA0001693178320000063
wherein λ iszAnd λpIs a lagrange multiplier; < lambdaz,z-(Fv-v0) Is > represents lambdazAnd z- (Fv-v)0) Inner product of < lambda >p,p-K1v > represents lambdapAnd p-K1The inner product of v;
Figure BDA0001693178320000064
represents z- (Fv-v)0) The regularization of (a) L2,
Figure BDA0001693178320000065
represents p-K1L2 regularization of v; r is a radical of hydrogenz，rpIs a penalty factor, and rz＞0，rpIs greater than 0; the optimization problem translates into a saddle point problem of equation (11) as follows:
Figure BDA0001693178320000066
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0001693178320000067
is to solve the equation satisfying the variational equation L (v, z, p; lambda)z,λp) The smallest v, z, p;
step 3.2, solving an optimization problem (11); specifically, the problem (11) is converted into 3 subproblems which are solved in sequence, and then the Lagrange multiplier is updated in an iteration mode, and the method is realized through the following substeps:
step 3.2A fixes p, z, solves v, i.e. solves the v sub-problem, which can be transformed into a quadratic form of the following equation (12):
Figure BDA0001693178320000068
wherein the content of the first and second substances,
Figure BDA0001693178320000071
is to be satisfied withFoot
Figure BDA0001693178320000072
A minimum v;
the subproblems can be converted into linear equations for solution;
step 3.2B fixes v, p, and solves z, namely solves z subproblem which can be converted into the form of the following formula (13):
Figure BDA0001693178320000073
wherein the content of the first and second substances,
Figure BDA0001693178320000074
is to satisfy
Figure BDA0001693178320000075
A minimum z;
the problem (13) can be decomposed and has a closed form solution of the following equation (14):
Figure BDA0001693178320000076
wherein the content of the first and second substances,
Figure BDA0001693178320000077
is taken to be 0 and
Figure BDA0001693178320000078
maximum value of (1);
step 3.2C fixes v, z, solves p, i.e. solves p subproblems, which can be converted into the form of the following formula (15):
Figure BDA0001693178320000079
wherein the content of the first and second substances,
Figure BDA00016931783200000710
is to obtainSatisfy the requirement of
Figure BDA00016931783200000711
A minimum of p;
then all problems like the equations (13), (15) have a closed form solution of the following equation (16):
Figure BDA0001693178320000081
wherein the content of the first and second substances,
Figure BDA0001693178320000082
is taken to be 0 and
Figure BDA0001693178320000083
the maximum value of (a);
step 3.3, updating Lagrange multipliers, wherein the relationship between the l +1 iteration and the l is as follows (17):
Figure BDA0001693178320000084
step 3.4, iterative solution;
initial value of order
Figure BDA0001693178320000085
Iteratively solving equations (12), (13), (15) in sequence, updating the lagrange multiplier (17) until a termination condition is satisfied;
wherein the termination condition is as follows: assuming two consecutive iterations, e.g., l +1 iterations, the distance of the control vertex is recorded
Figure BDA0001693178320000086
When ε is less than a given threshold ε0When so, the iteration stops.
Advantageous effects
The invention provides a mesh repairing method based on internal dihedral angle regularization, which has the following beneficial effects compared with the existing mesh repairing method:
1. the method provided by the invention has obvious advantages in the aspects of maintaining sharp characteristics such as grid creases, corner points, puncture points, sharp points and the like in the repairing process;
2. the method provided by the invention has wide application prospects in the fields of digital entertainment, virtual reality, industrial manufacturing and the like.
Drawings
FIG. 1 is a frame diagram of a mesh repairing method based on inner dihedral angle corner repairing according to the present invention;
fig. 2 is a schematic diagram of a mesh repairing method based on internal dihedral angle corner compensation and the mesh dihedral angle corner compensation in embodiment 1 of the present invention;
fig. 3 is a mesh repairing method based on inner dihedral angle corner filling and an algorithm diagram in embodiment 1.
Detailed Description
The method of the present invention is described in detail below with reference to the accompanying drawings and examples.
As shown in fig. 1, it is a mesh repairing method based on inner dihedral angle corner filling and the algorithm frame diagram of this embodiment 1.
As can be seen from fig. 1, the present invention performs the following steps for inputting a defective grid:
step A, reconstructing the connection relation of grid holes;
triangulating the hole area of the input incomplete mesh by using a triangulation method based on minimum weight, reconstructing the connection relation of the hole area and obtaining an initial mesh with a complete connection relation;
b, based on the grid with the complete connection relation, establishing an energy function of grid repair based on an L1 data item and an internal dihedral angle corner-filling regular item;
incomplete grid M for memory input0The set of vertices of (1) is
Figure BDA0001693178320000091
Where M is the input grid M0The number of the middle vertexes; the vertex set of the initial mesh M is v ═ v1,v2,...,vnH, the edge set is e ═ e1,e2,...,edH, set of side lengths l ═ l1,l2,...,ldThe set of internal dihedral angles is theta ═ theta1,θ2,...,θdN (n > M) is the number of vertices in the initial mesh M, d is the number of edges in the initial mesh M, and e is an edge in the set of edges eiIs the length of the length set liAngle theta in the set of internal angles thetaiRepresenting a shared edge eiThe internal dihedral angle between the two triangular patches; keeping the known vertex position in the initial mesh M as close to the original position as possible after mesh repairing processing, keeping the mesh characteristics through internal dihedral angle regularization constraint, and simultaneously obtaining the optimal vertex position of the mesh through optimizing the following energy function;
Figure BDA0001693178320000092
wherein the content of the first and second substances,
Figure BDA0001693178320000093
is to satisfy λ Ef(v,v0)+Er(l, θ) the minimum vertex position, i.e., v; ef(v,v0) For data items, Er(l, θ) is a regularization term; lambda is a data item parameter, and the data item is used for keeping the original characteristics of the grid and reducing the dependence of internal dihedral angle regularization on missing vertex information;
step B.1, calculating a data item, and calculating the distance between the new peak after the mesh is repaired and the corresponding peak in the input mesh; the invention uses the new vertex Fv obtained by calculation to approach the corresponding vertex v0Thus the data item is composed of
Figure BDA0001693178320000101
Calculating to obtain;
wherein v isiSet of vertices representing the initial mesh, v ═ v1,v2,...,vnItem i in (j);
Figure BDA0001693178320000102
a set of vertices representing the input incomplete mesh,
Figure BDA0001693178320000103
item i in (1); i Fv-v0||1Represents Fv-v0L1 regularization; f is an m × n projection matrix defined as follows:
Figure BDA0001693178320000104
step B.2, calculating a regular term, and keeping the grid characteristics by using dihedral angle constraint, so that the regular term is expressed as:
Figure BDA0001693178320000105
wherein liRepresenting the edge e in the initial gridiI.e. the set of side lengths l ═ l1,l2,...,ldItem i of (9); theta.theta.iRepresenting a shared edge eiThe internal dihedral angle (pi-theta) between two triangular patchesi) Refers to the complement of the internal dihedral angle; two half-planes of simplified dihedral structure are Δ v1v3v4And Δ v1v2v3The shared edge of the two half-planes is v1v3Corresponding to the third vertex is v2And v4(ii) a Definition of T1,T2Is two lengths | | | v1v3Vector of | |, T1Is the plane Δ v1v3v4Inner normal direction of, T2Is plane Δ v1v2v3Outer normal direction of, T1And T2The included angle between them is pi-theta, then | v1v3Is the vector T1And T2The arc length of the arc sandwiched therebetween; the normal vector formula is calculated based on the edges and included angles of the triangular patch as follows:
Figure BDA0001693178320000106
wherein, theta4,1,3Is side v1v4And the side v1v3Angle of (a) theta1,3,4Is side v1v3And the side v3v4Angle of (a) of2,3,1Is side v2v3And the side v1v3Angle of (a) theta3,1,2Is side v1v2And the side v1v3The included angle of (A); according to T1,T2Determining the arc length, i.e. /)i(π-θi) Expressed as:
Figure BDA0001693178320000111
wherein
Figure BDA0001693178320000112
Therefore, the regularization term is specifically calculated as follows:
Figure BDA0001693178320000113
wherein, | | K1v||1Represents K1L1 regularization of v;
to sum up, the equation
Figure BDA0001693178320000114
Can be written as:
Figure BDA0001693178320000115
wherein the content of the first and second substances,
Figure BDA0001693178320000116
is to satisfy lambda | Fv-v0||1+||K1v||1The minimum vertex position, i.e., v;
step C, solving an energy function by applying a value-added Lagrange method to obtain an optimal vertex position;
step C.1, equation
Figure BDA0001693178320000119
Translating into the following optimization problem with constraints:
Figure BDA0001693178320000117
wherein, z is Fv-v0，p＝K1v，||z||1L1 regularization representing z, | | p | | computing luminance1L1 regularization, which represents p;
Figure BDA0001693178320000118
solving the problem of satisfying lambda | z | | non-woven phosphor1+||p||1The smallest z, p;
then the constraint problem can be converted into the following functional saddle point problem according to the value-added Lagrange method:
Figure BDA0001693178320000121
wherein λ iszAnd λpIs a lagrange multiplier; < lambdaz,z-(Fv-v0) Is > represents lambdazAnd z- (Fv-v)0) Inner product of < lambda >p,p-K1v > represents lambdapAnd p-K1The inner product of v;
Figure BDA0001693178320000122
represents z- (Fv-v)0) The regularization of (1) L2,
Figure BDA0001693178320000123
represents p-K1L2 regularization of v; r is a radical of hydrogenz，rpIs a penalty factor, and rz＞0，rpIs greater than 0; the optimization problem translates into the saddle point problem as follows:
Figure BDA0001693178320000124
wherein the content of the first and second substances,
Figure BDA0001693178320000125
is to solve the equation satisfying the variational equation L (v, z, p; lambda)z,λp) The smallest v, z, p;
step C.2, solving the optimization problem
Figure BDA0001693178320000126
And converting the problem into 3 subproblems which are solved in sequence, and then iteratively updating Lagrange multipliers, wherein the 3 subproblems are respectively as follows:
● fixing p, z, solving v, i.e. solving the v sub-problem, which can be transformed into the following quadratic form:
Figure BDA0001693178320000127
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0001693178320000128
is to satisfy
Figure BDA0001693178320000129
A minimum v;
the problem can be converted into a linear equation to be solved;
● fixing v, p, solving z, i.e. solving the z sub-problem, which can be converted to the following form:
Figure BDA00016931783200001210
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00016931783200001211
is to satisfy
Figure BDA00016931783200001212
A minimum z;
the problem can be broken down and has the following closed form solution:
Figure BDA0001693178320000131
wherein the content of the first and second substances,
Figure BDA0001693178320000132
is taken to be 0 and
Figure BDA0001693178320000133
maximum value of (1);
● fixing v, z, solving p, i.e. solving the p sub-problem, which can be converted to the following form:
Figure BDA0001693178320000134
wherein the content of the first and second substances,
Figure BDA0001693178320000135
is to satisfy
Figure BDA0001693178320000136
The smallest p;
then similarly, the problem has a closed form solution as follows:
Figure BDA0001693178320000137
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0001693178320000138
is taken to be 0 and
Figure BDA0001693178320000139
the maximum value of (a);
and C.3, updating the Lagrange multiplier, wherein the relationship between the l +1 iteration and the l is as follows:
Figure BDA00016931783200001310
Figure BDA00016931783200001311
step C.4, iterative solution;
starting value
Figure BDA00016931783200001312
Equation is solved by iteration in turn
Figure BDA00016931783200001313
Figure BDA00016931783200001314
Figure BDA00016931783200001315
Updating Lagrange multipliers
Figure BDA0001693178320000141
Until the termination condition is met, see the algorithm map of fig. 3.
While the foregoing is directed to the preferred embodiment of the present invention, it is not intended that the invention be limited to the embodiment and the drawings disclosed herein. Equivalents and modifications may be made without departing from the spirit of the disclosure, which is to be considered as within the scope of the invention.

Claims (1)

1. A grid repairing method based on internal dihedral angle corner-filling regularization is characterized in that: initializing the input incomplete grids, and reconstructing the connection relation of the grid missing areas; reducing the dependence of an internal dihedral angle supplementary angle regular term on grid missing information by using an L1 data item, and then searching an optimal vertex position through internal dihedral angle supplementary angle regularization; globally considering mesh information for the input mesh; finally, the vertex position of the grid is iteratively solved by using an incremental Lagrangian Method (Augmented Lagrangian Method), so that the reconstructed grid which keeps the sharp features is obtained;
the mesh is represented by a triangular patch, and the specific steps are as follows:
the method comprises the following steps: preprocessing an input grid, reconstructing a connection relation of a grid missing region, and obtaining an initial grid with a complete connection relation;
wherein, the input grid is marked as M0The initial grid is marked as M, and the preprocessing of the input grid specifically comprises the following steps:
the input grid being a defective grid M0Triangulating the hole area of the incomplete mesh by adopting a triangulation method based on minimum weight proposed in section 4 of literature 1, reconstructing the connection relation of the missing mesh area, namely the hole area, and outputting an initial mesh M with a complete connection relation;
document 1: liepa, Peter. filling holes in disks, Proceedings of the 2003 Europaphic/ACM SIGGRAPH symposium on Geometry processing, Europaphic Association, 2003;
step two, establishing an energy function of grid repairing based on an L1 data item and an internal dihedral angle corner repairing regular item according to the initial grid M with the complete connection relation obtained in the step one:
incomplete grid M for memory input0The set of vertices is
Figure FDA0003572506640000011
Where M is the input grid M0The number of the middle vertexes; the set of vertices for the initial mesh M is v ═ v1,v2,...,vnH, set of edges e ═ e1,e2,...,edThe length of the side is set as l ═ l1,l2,...,ldA set of internal dihedral angles θ ═ θ1,θ2,...,θdN is the number of top points in the initial grid M, and n is more than M; d is the number of edges in the initial mesh M, and e is the edge in the edge set eiIs the length of l in the length set liAngles in the set of internal angles thetaθiRepresenting a shared edge eiThe internal dihedral angle between the two triangular surface patches;
keeping the known vertex position in the initial grid M close to the original position after the grid repairing treatment, keeping the grid characteristic through the internal dihedral angle regularization constraint, and meanwhile obtaining the optimal vertex position of the grid through minimizing the energy function of the formula (1);
Figure FDA0003572506640000021
wherein the content of the first and second substances,
Figure FDA0003572506640000022
is to satisfy λ Ef(v,v0)+Er(l, θ) the minimum vertex position, i.e., v; ef(v,v0) For data items, Er(l, θ) is a regularization term; lambda is a data item parameter, and the data item is used for keeping the original characteristics of the grid and reducing the dependence of internal dihedral angle regularization on missing vertex information;
step 2.1, calculating a data item, specifically by the following formula (2):
Figure FDA0003572506640000023
wherein v isiSet of vertices v ═ v { v } representing the initial mesh1,v2,...,vnItem i in (j);
Figure FDA0003572506640000024
set of vertices representing an input incomplete mesh
Figure FDA0003572506640000025
Item i in (1); l Fv-v0||1Represents Fv-v0L1 regularization; f represents a m × n projection matrix, defined as (3):
Figure FDA0003572506640000026
step 2.2, calculating a regular term, specifically calculating by a formula (4):
Figure FDA0003572506640000027
wherein liRepresenting the edge e in the initial gridiI.e. the set of side lengths l ═ l1,l2,...,ldItem i in (j); thetaiRepresenting a shared edge eiThe internal dihedral angle (pi-theta) between two triangular patchesi) Refers to the complement of the internal dihedral angle;
two half-planes of simplified dihedral structure are Δ v1v3v4And Δ v1v2v3The shared edge of the two half-planes is v1v3Corresponding to the third vertex is v2And v4(ii) a Definition of T1,T2Is two lengths | | | v1v3Vector of | |, T1Is plane Δ v1v3v4Inner normal direction of (T)2Is the plane Δ v1v2v3Outer normal direction of (T)1And T2The included angle between them is pi-theta, then | v1v3Is the vector T | | π - θ |1And T2The arc length of the arc sandwiched therebetween; the normal vector formula is calculated based on the edges and included angles of the triangular surface patch as follows (5):
Figure FDA0003572506640000031
wherein cot is an inverse cotangent function, equal to cosine divided by sine function; theta.theta.4,1,3Is side v1v4And the side v1v3Angle of (a) theta1,3,4Is side v1v3And the side v3v4Angle of (a) of2,3,1Is side v2v3And the side v1v3Angle of (a) theta3,1,2Is side v1v2And the side v1v3The included angle of (c); according to T1,T2The arc length, i.e./is determinedi(π-θi) Expressed as:
Figure FDA0003572506640000032
wherein the content of the first and second substances,
Figure FDA0003572506640000033
therefore, the regularization term is specifically calculated as the following equation (7):
Figure FDA0003572506640000034
wherein, K1Represents the matrix in formula (6)
Figure FDA0003572506640000035
||K1v||1Represents K1L1 regularization of v;
in conjunction with equations (2) and (7), the energy function (1) of the mesh patch can be written as the following equation (8):
Figure FDA0003572506640000036
wherein the content of the first and second substances,
Figure FDA0003572506640000037
is to satisfy lambda | Fv-v0||1+||K1v||1The minimum vertex position, i.e., v;
step three, solving the energy function of grid repair in the step two by applying a value-added Lagrange method, which specifically comprises the following steps:
step 3.1, solving equation (8) is converted into solving the optimization problem with the constraint according to the following formula (9):
Figure FDA0003572506640000041
wherein, z is Fv-v0，p＝K1v，||z||1L1 regularization representing z, | p | | luminance1L1 regularization, which represents p;
Figure FDA0003572506640000042
solving the problem of satisfying lambda | z | | non-woven phosphor1+||p||1The smallest z, p;
the constraint problem of the above equation (9) can be converted into a functional saddle point problem solving the following equation (10) according to the incremental lagrange method:
Figure FDA0003572506640000043
wherein λ iszAnd λpIs a lagrange multiplier; < lambdaz,z-(Fv-v0) Is > represents lambdazAnd z- (Fv-v)0) Inner product of, < lambdap,p-K1v > represents lambdapAnd p-K1The inner product of v;
Figure FDA0003572506640000044
represents z- (Fv-v)0) The regularization of (1) L2,
Figure FDA0003572506640000045
represents p-K1L2 regularization of v; r is a radical of hydrogenz，rpIs a penalty factor, and rz＞0，rpIs greater than 0; the optimization problem translates into a saddle point problem of equation (11) as follows:
Figure FDA0003572506640000046
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003572506640000047
is to solve the equation satisfying the variational equation L (v, z, p; lambda)z,λp) Minimum v, z, p;
step 3.2, solving an optimization problem (11); specifically, the problem (11) is converted into 3 subproblems which are solved in sequence, and then the Lagrange multiplier is updated in an iteration mode, and the method is realized through the following substeps:
step 3.2A fixes p, z, solves v, i.e. solves the v subproblem, which can be transformed into a quadratic form of the following equation (12):
Figure FDA0003572506640000048
wherein the content of the first and second substances,
Figure FDA0003572506640000049
is to satisfy
Figure FDA0003572506640000051
A minimum v;
the formula (12) can be converted into a linear equation to be solved;
step 3.2B fixes v, p, and solves z, namely solves z subproblem which can be converted into the form of the following formula (13):
Figure FDA0003572506640000052
wherein the content of the first and second substances,
Figure FDA0003572506640000053
is to satisfy
Figure FDA0003572506640000054
A minimum z;
this equation (13) can be decomposed and has a closed form solution of equation (14) as follows:
Figure FDA0003572506640000055
wherein the content of the first and second substances,
Figure FDA0003572506640000056
Figure FDA0003572506640000057
is taken to be 0 and
Figure FDA0003572506640000058
maximum value of (2);
step 3.2C, fixing v, z, and solving p, namely solving a p subproblem which can be converted into a form of the following formula (15):
Figure FDA0003572506640000059
wherein the content of the first and second substances,
Figure FDA00035725066400000510
is to satisfy
Figure FDA00035725066400000511
A minimum of p;
then, similarly to (13), the equation (15) has a closed form solution of the following equation (16):
Figure FDA00035725066400000512
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003572506640000061
Figure FDA0003572506640000062
is taken to be 0 and
Figure FDA0003572506640000063
maximum value of (d);
and 3.3, updating the Lagrangian multiplier, wherein the relationship between the l +1 iteration and the l is as follows (17):
Figure FDA0003572506640000064
step 3.4, iterative solution;
initial value of order
Figure FDA0003572506640000065
Solving equations (12), (13) and (15) iteratively in sequence, and updating a Lagrange multiplier (17) until a termination condition is met;
wherein the termination condition is as follows: assuming two consecutive iterations, e.g., l +1 iterations, the distance of the control vertex is recorded
Figure FDA0003572506640000066
When ε is less than a given threshold ε0When so, the iteration stops.
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