A conformal period flattening method for genus surfaces

Abstract

The present invention discloses a conformal periodic flattening method for a surface with genus, and relates to the field of surface flattening. The steps are as follows: for an input triangular mesh surface, first check the genus number of the surface. If the genus number is greater than 1, cut the surface into multiple sub-surfaces with genus 1, otherwise record the original surface as a sub-surface; secondly, search for the ring handle ring and tunnel ring of each sub-surface and cut them; then establish a periodic condition for the ring slit boundary inside each sub-surface, establish a flipping condition for the slit boundary between two sub-surfaces, construct a coupled conformal energy minimization model and solve it, obtain the vertex mapping coordinates, periodic translation and flip center; finally, construct a conformal periodic flattened mesh for each sub-surface according to the results. The method of the present invention constructs a periodic condition and flipping condition based on the slit, calculates the conformal periodic flattening mapping by conformal energy minimization, ensures the high conformality of the mapping at the slit, and is applicable to any surface with genus.

Description

Technical Field

[0001] The present invention relates to the field of surface flattening, and in particular to a conformal periodic flattening method for a genus surface. Background Art

[0002] In computer graphics and geometry processing, surface flattening is an important technique for mapping three-dimensional surfaces onto two-dimensional planes. Conventional surface flattening techniques prevent the surface from aligning at the cuts after cutting and flattening, or make it difficult to maintain low deformation errors at the cuts. Conformal periodic flattening compensates for these deficiencies by simultaneously aligning the cuts during flattening. This makes the flattened deformation error insensitive to the choice of cuts, making it crucial for applications such as mesh generation and texture mapping, where cut alignment is crucial. Summary of the Invention

[0003] In view of the deficiencies of the prior art, the present invention aims to provide a conformal periodic flattening method for a genus surface.

[0004] In order to achieve the above object, the present invention adopts the following technical solutions:

[0005] S1: Check input surface Genus ,like , cut the surface into pieces subsurfaces with genus 1 ,remember For subsurface and The gap between Article; if , then the subsurface Original surface ;

[0006] S2: Search each subsurface A pair of ring handle circles and tunnel circles are cut along them to obtain the cut subsurface , each sub-surface contains two circle cut boundaries , while inheriting the subsurface The gap between ;

[0007] S3: For each sub-surface Internal circle cut boundary Establish periodic conditions for any two subsurfaces The slit boundary between Establish the flipping condition, build the coupled conformal energy minimization model, and then use the optimization method to solve the model to obtain each subsurface Vertex mapping coordinates , periodic shift and flip center ;

[0008] S3.1: For each subsurface Internal circle cut boundary Establish a periodic condition so that the one-to-one corresponding points on each circle slit boundary are only shifted after flattening. The amount of translation is the quantity to be determined. Since there are two circle slit boundaries, there are two translations, which are recorded as , and thus construct the matrix:

[0009] ;

[0010] in, It is composed of subsurfaces The constructed cotangent Laplace matrix is ​​defined as:

[0011] ;

[0012] in, It's the edge The corresponding cotangent Laplace matrix edge weights, and It's the edge The two opposite angles; It's the edge The weight of and It's the edge The two opposite angles; expressed The relationship with adjacent mesh vertices, whose elements are The sum of the negative weights of the adjacent edges of the newly added cut corresponding to the vertex, expressed The relationship between them is the sum of the negative weights of all vertex adjacent edges corresponding to the boundaries of the two circle cuts, Representation matrix The transpose of

[0013] S3.2: Searching for Subsurfaces The boundary points are constructed according to the following antisymmetric matrix ,in

[0014] ;

[0015] Construct each subsurface Conformal energy function on :

[0016] ;

[0017] in They are matrices The first and second columns of , Is a subsurface The coordinates of the mesh vertices obtained after mapping;

[0018] S3.3: For any two subsurfaces The slit boundary between Establish the flipping condition so that the sub-surface The slit vertices on the periodically flattened plane With the subsurface The corresponding slit vertex on the periodically flattened plane satisfy ,in Slit exist The center, because there is shard cuts, so there are The center of the slit is recorded as ; The flipping condition maps all sub-surfaces Coupled together, a coupled conformal energy minimization model is constructed:

[0019] ;

[0020] S3.4: Use the optimization method to solve the coupled conformal energy minimization model constructed in S3.3 and obtain each subsurface Vertex mapping coordinates , periodic shift and flip center ;

[0021] S4: Vertex mapping coordinates as requested in S3 , periodic shift and flip center Construct each subsurface A conformal periodic flattened mesh.

[0022] A conformal periodic flattening method for surfaces with genus is applied in the field of computer-aided engineering: after the surface is conformally flattened, surface processing is performed on the flattened plane, and then the processing effect is restored to the surface, such as mesh generation, texture mapping, etc.

[0023] Beneficial effects:

[0024] 1. Conformal periodic flattening designs periodic and flipping conditions for the kerf, thereby ensuring both flattening conformality and kerf alignment. It can be directly applied to scenarios such as mesh generation and texture mapping.

[0025] 2. The conformal period flattening method is universal and can be applied to any surface with defect.

[0026] 3. The model of the conformal energy minimization method is simple and has strict theoretical guarantees. It can be solved efficiently, gives high-precision conformal mapping, and is robust. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 It is a flow chart of the algorithm described in the present invention.

[0028] Figure 2 It is the model in the specific implementation case 1 and the divided ring handle circle and tunnel circle.

[0029] Figure 3 It is the conformal flattening result achieved by the algorithm in specific implementation case 1.

[0030] Figure 4 It is the model in the specific implementation case 2 and the segmentation cut line, ring handle circle and tunnel circle.

[0031] Figure 5 This is the conformal flattening result of the algorithm implemented in the specific implementation case 2. The left picture is Figure 4 The conformal flattened plane of the left half of the subsurface is Figure 4 The conformally flattened plane of the right half of the subsurface. DETAILED DESCRIPTION

[0032] To make the purpose, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions of the embodiments of the present invention are clearly and completely described below in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Therefore, the detailed description of the embodiments of the present invention provided in the accompanying drawings below is not intended to limit the scope of the claimed invention.

[0033] The present invention provides a conformal periodic flattening method for a genus surface, the specific steps of which are:

[0034] S1: Check input surface Genus ,like , cut the surface into pieces subsurfaces with genus 1 ,remember For subsurface and The gap between Article; if , then the subsurface Original surface ;

[0035] S2: Search each subsurface A pair of ring handle circles and tunnel circles are cut along them to obtain the cut subsurface , each sub-surface contains two circle cut boundaries , while inheriting the subsurface The gap between ;

[0036] S3: For each sub-surface Internal circle cut boundary Establish periodic conditions for any two subsurfaces The slit boundary between Establish the flipping condition, build the coupled conformal energy minimization model, and then use the optimization method to solve the model to obtain each subsurface Vertex mapping coordinates , periodic shift and flip center ;

[0037] S3.1: For each subsurface Internal circle cut boundary Establish a periodic condition so that the one-to-one corresponding points on each circle slit boundary are only shifted after flattening. The amount of translation is the quantity to be determined. Since there are two circle slit boundaries, there are two translations, which are recorded as , and thus construct the matrix:

[0038] ;

[0039] in, It is composed of subsurfaces The constructed cotangent Laplace matrix is ​​defined as:

[0040] ;

[0041] in, It's the edge The corresponding cotangent Laplace matrix edge weights, and It's the edge The two opposite angles; It's the edge The weight of and It's the edge The two opposite angles; expressed The relationship with adjacent mesh vertices, whose elements are The sum of the negative weights of the adjacent edges of the newly added cut corresponding to the vertex, expressed The relationship between them is the sum of the negative weights of all vertex adjacent edges corresponding to the boundaries of the two circle cuts, Representation matrix The transpose of

[0042] S3.2: Searching for Subsurfaces The boundary points are constructed according to the following antisymmetric matrix ,in

[0043] ;

[0044] Construct each subsurface Conformal energy function on :

[0045] ;

[0046] in They are matrices The first and second columns of , Is a subsurface The coordinates of the mesh vertices obtained after mapping;

[0047] S3.3: For any two subsurfaces The slit boundary between Establish the flipping condition so that the sub-surface The slit vertices on the periodically flattened plane With the subsurface The corresponding slit vertex on the periodically flattened plane satisfy ,in Slit exist The center, because there is shard cuts, so there are The center of the slit is recorded as ; The flipping condition maps all sub-surfaces Coupled together, a coupled conformal energy minimization model is constructed:

[0048] ;

[0049] S3.4: Use the optimization method to solve the coupled conformal energy minimization model constructed in S3.3 and obtain each subsurface Vertex mapping coordinates , periodic shift and flip center ;

[0050] S4: Vertex mapping coordinates as requested in S3 , periodic shift and flip center Construct each subsurface A conformal periodic flattened mesh.

[0051] Example 1: This example takes the Kitten model with a genus of 1 as an example to further explain the present invention in detail. The specific steps are as follows:

[0052] S1: Surface The genus number is 1, and there is no need to split it into multiple sub-surfaces. The original surface ;

[0053] S2: Search Surface A pair of ring handle rings and tunnel rings are cut along them to obtain the cut surface , including 2 circle cutting boundaries ,like Figure 2 As shown in Line 1;

[0054] S3: For curved surfaces Internal circle cut boundary Establish periodic conditions. Since there is no sub-surface division, there is no need to establish flipping conditions. The conformal energy minimization model constructed is only on the surface. Then, the optimization method is used to solve the model and obtain the vertex mapping coordinates and periodic shift ;

[0055] S3.1: For curved surfaces The circle cut boundary Establish a periodic condition so that the corresponding points on each circle slit boundary only differ in translation after flattening, where the translation amount is is the quantity to be determined. Since there are two circle cut boundaries, there are two translations, which are recorded as , and thus construct the matrix:

[0056] ;

[0057] in, It is made of surface The constructed cotangent Laplace matrix is ​​defined as:

[0058] ;

[0059] in, It's the edge The corresponding cotangent Laplace matrix edge weights, and It's the edge The two opposite angles; It's the edge The weight of and It's the edge Two opposite angles. expressed The relationship with adjacent mesh vertices, whose elements are The sum of the negative weights of the adjacent edges of the newly added cut corresponding to the vertex, expressed The relationship between them is the sum of the negative weights of all vertex adjacent edges corresponding to the boundaries of the two circle cuts, Representation matrix The transpose of

[0060] S3.2: Search Surface The boundary points are constructed according to the following antisymmetric matrix ,in

[0061] ;

[0062] Construction Surface Conformal energy function on :

[0063] ;

[0064] among them They are matrices The first and second columns of , yes The coordinates of the mesh vertices obtained after mapping;

[0065] S3.3: Since there is no sub-surface splitting, there is no need to construct a flipping condition. The final conformal energy minimization model is:

[0066] ;

[0067] S3.4: Use optimization methods to solve the conformal energy minimization model constructed in S3.3 and obtain the vertex mapping coordinates and periodic shift ;

[0068] S4: Vertex coordinates obtained in S3 and periodic shift Construction Surface The conformal period flattened grid of Figure 3 shown.

[0069] Example 2: This example takes the Filterclip model with a genus of 2 as an example to further explain the present invention in detail. The specific steps are as follows:

[0070] S1: Surface The genus of is 2, so the surface is cut into two subsurfaces with edges of genus 1. ,in and The gap between ,like Figure 4 As shown in Line 2;

[0071] S2: Search each subsurface A pair of ring handle circles and tunnel circles are cut along them to obtain the cut subsurface , each containing 2 circle cut boundaries as well as ,like Figure 4 As shown in Line 1, Inherited The gap between ;

[0072] S3: For each sub-surface Internal circle cut boundary Establish periodic conditions for the two subsurfaces The slit boundary between Establish the flipping condition, construct the coupled conformal energy minimization model, and then use the optimization method to solve the model to obtain all subsurfaces Vertex mapping coordinates , periodic shift and flip center ;

[0073] S3.1: For each subsurface Internal circle cut boundary Establish periodic conditions so that the one-to-one corresponding points on each circle slit boundary are only translated after flattening. The translation amount is the quantity to be determined. Since there are two circle slit boundaries, there are two translation amounts, that is, , and thus construct the matrix:

[0074] ;

[0075] in, is composed of subsurface meshes The constructed cotangent Laplace matrix is ​​defined as:

[0076] ;

[0077] in, It's the edge The corresponding cotangent Laplace matrix edge weights, and It's the edge The two opposite angles; It's the edge The weight of and It's the edge The two opposite angles; expressed The relationship with adjacent mesh vertices, whose elements are The sum of the negative weights of the adjacent edges of the newly added cut corresponding to the vertex, expressed The relationship between them is the sum of the negative weights of all vertex adjacent edges corresponding to the boundaries of the two circle cuts, Representation matrix The transpose of

[0078] S3.2: Searching for Subsurfaces The boundary points are constructed according to the following antisymmetric matrix ,in

[0079] ;

[0080] Construct each subsurface Conformal energy function on :

[0081] ;

[0082] in They are matrices The first and second columns of , Is a subsurface The coordinates of the mesh vertices obtained after mapping;

[0083] S3.3: For any two subsurfaces The slit boundary between Establish the flipping condition so that the sub-surface The slit vertices on the periodically flattened plane With the subsurface The corresponding slit vertex on the periodically flattened plane satisfy ,in Slit exist The center of all slits is . Thus the mapping of all sub-surfaces Coupled together to build a coupled conformal energy minimization model:

[0084] ;

[0085] S3.4: Use the optimization method to solve the coupled conformal energy minimization model constructed in S3.3 and obtain all subsurfaces Vertex mapping coordinates , periodic shift and flip center ;

[0086] S4: Vertex mapping coordinates as requested in S3 , periodic shift and flip center Construct each subsurface The conformal period flattened grid of Figure 5 As shown, due to the surface The genus of is 2, so there are two periodic flattened planes, the outer boundary of each plane satisfies the periodicity condition, and the cavity boundaries inside the two planes satisfy the flipping condition.

[0087] The technical means disclosed in the solutions of the present invention are not limited to those disclosed in the above-mentioned embodiments, but also include technical solutions composed of any combination of the above-mentioned technical features. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications are also considered to be within the scope of protection of the present invention.

Claims

1. A conformal periodic flattening method for a genus surface, characterized in that: Applied in the field of texture mapping, it is used to map a three-dimensional surface onto a two-dimensional plane. Specifically, after the surface slices are conformally flattened, the surface processing is performed on the plane, and then the processing effect is restored back to the surface. The following steps are included: S1: Check input surface Genus ,like , cut the surface into pieces subsurfaces with genus 1 ,remember For subsurface and The gap between Article; if , then the subsurface Original surface ; S2: Search each subsurface A pair of ring handle rings and tunnel rings are cut along them to obtain the cut subsurface , each sub-surface contains two circle cut boundaries , while inheriting the subsurface The gap between ; S3: For each sub-surface Internal circle cut boundary Establish periodic conditions for any two subsurfaces The slit boundary between Establish the flipping condition, build the coupled conformal energy minimization model, and then use the optimization method to solve the model to obtain each subsurface Vertex mapping coordinates , periodic shift and flip center ; S4: Vertex mapping coordinates according to S3 , periodic shift and flip center Construct each subsurface A conformal periodic flattened mesh.

2. The conformal period flattening method for a genus surface according to claim 1, wherein: Step S3 specifically includes the following steps: S3.1: For each subsurface Internal circle cut boundary Establish a periodic condition so that the one-to-one corresponding points on each circle slit boundary are only shifted after flattening. The amount of translation is the quantity to be determined. Since there are two circle slit boundaries, there are two translations, which are recorded as , thus constructing the matrix ； in, It is composed of subsurfaces The constructed cotangent Laplace matrix is ​​defined as: ； in, It's the edge The corresponding cotangent Laplace matrix edge weights, and It's the edge The two opposite angles; It's the edge The weight of and It's the edge The two opposite angles; expressed The relationship with adjacent mesh vertices, whose elements are The sum of the negative weights of the adjacent edges of the newly added cut corresponding to the vertex, expressed The relationship between them is the sum of the negative weights of all vertex adjacent edges corresponding to the boundaries of the two circle cuts, Representation matrix The transpose of S3.2: Searching for Subsurfaces The boundary points are constructed according to the following antisymmetric matrix ,in ； Construct each subsurface Conformal energy function on : ； in They are matrices The first and second columns of , Is a subsurface The coordinates of the mesh vertices obtained after mapping; S3.3: For any two subsurfaces The slit boundary between Establish the flipping condition so that the sub-surface The slit vertices on the periodically flattened plane With the subsurface The corresponding slit vertex on the periodically flattened plane satisfy ,in Slit exist The center, because there is shard cuts, so there are The center of the slit is recorded as ; The flipping condition maps all sub-surfaces Coupled together to build a coupled conformal energy minimization model ； S3.4: Use the optimization method to solve the coupled conformal energy minimization model constructed in S3.3 and obtain each subsurface Vertex mapping coordinates , periodic shift and flip center .

3. The conformal period flattening method for a genus surface according to claim 1, characterized in that: The application in the field of computer-aided engineering is: after the surface is conformally flattened, surface processing is performed on the flattened plane, and then the processing effect is restored back to the surface.

Images