CN112258655B - Three-dimensional grid simplifying method applied to VR virtual bank - Google Patents
Three-dimensional grid simplifying method applied to VR virtual bank Download PDFInfo
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Abstract
The invention discloses a three-dimensional grid simplifying method applied to a VR virtual bank, which comprises the following steps of: 1. calculating the number of theoretical triangular faces after simplification; 2. calculating an error matrix of each original triangle contained in each triangular mesh; 3. obtaining geometric errors and texture errors of the three-dimensional grid model; 4. calculating an edge collapse error of each edge; 5. constructing a minimum stack; 6. selecting the side with the smallest collapse error to carry out collapse simplification; 7. performing side collapse operation; 8. the number of triangle facets after simplification is obtained; 9. if the number of the triangular patches after simplification is larger than the number of the theoretical triangular patches after simplification, continuing to simplify the step six, and otherwise ending the simplification flow. The invention can realize the purpose of simplifying the three-dimensional grid and simultaneously guaranteeing the quality of the display model without seriously affecting the operation of the system, thereby better realizing the VR-based online virtual banking system. The invention belongs to the technical field of image processing and is used for simplifying three-dimensional grids.
Description
Technical Field
The invention belongs to the technical field of image processing, and relates to a three-dimensional grid simplifying method, in particular to a three-dimensional grid simplifying method applied to a VR virtual bank.
Background
Virtual Reality (VR) technology is rapidly developed and is widely applied to the fields of travel exhibitions, engineering design, assembly and maintenance, medical research, military training and the like at present. The advent of virtual reality technology has led to the advent of many things in new forms in front of people, gradually changing the traditional lifestyle of people. Virtual online banking by VR technology has been developed. Virtual banking refers to three-dimensional electronic banking created using virtual information processing technology. Because all the services of the virtual bank are transacted through the network, the virtual bank has the advantages of faster office efficiency, lower service and operation rate, low operation cost, high lending efficiency and the like. The virtual bank is pushed out, which is an important ring in the intelligent bank measures, promotes the financial innovation, brings better user experience and further promotes the realization of the universal finance.
Virtual banks have been tried out in many countries such as the united kingdom, germany, singapore, etc. At present, hong Kong has first introduced virtual bank license plates and hong Kong gold administration has issued 8 virtual bank license plates. The companies for obtaining the virtual bank license plate include hong Kong, jingdong technology, slag banking, ant gold clothing, tencent, commercial banking, millet and peace officials.
At present, virtual banking implementation based on a VR system is mainly realized through a Unity 3D technology and a WebVR technology. The 3D model is the basis for VR system construction. In order for a user to obtain a realistic immersive interactive experience, the construction of scenes and models in the system must be as realistic as possible. However, with the rapid development of VR technology, computer graphics and other technologies, the three-dimensional model constructed by the triangular mesh is gradually refined, which makes the morphological characteristics of the model very complex and causes the data volume of the three-dimensional model file to be too large, so that the real-time response mechanism of the virtual reality system is slow and blocked due to heavy memory burden, and is not beneficial to real-time man-machine interaction. And because the existing WebVR technology is influenced by network conditions and browser operation capability, the WebVR system does not support loading of a model with excessive data volume into a scene. Therefore, in order to optimize the virtual reality system and enhance the response speed and interaction fluency, a model simplifying method for VR virtual bank construction is invented, and the purpose is to ensure the quality of a simplified model while reducing the data volume of a three-dimensional model, and to take as little time as possible to complete the process.
The prior grid simplifying method mainly comprises a Quadratic Error Measure (QEM) algorithm proposed by Garland and a progressive grid (PM) algorithm proposed by Hoppe and an improved algorithm thereof.
The Progressive Mesh (PM) algorithm proposed by Hoppe is mesh reduction using the idea of edge collapse. As shown in fig. 1, a triangle side uv is arranged in the triangular mesh, u and v are respectively two endpoints of the triangle side, and a and B are two triangles with uv as a common side; when the edge uv is a collapsed edge, the following operations are performed: the vertex u is replaced with the vertex v, and other neighbor vertices are connected to the new vertex, and then the vertex u and related information are deleted. This completes a target triangular edge collapse, where v is referred to as the collapsed target of u. It is known from fig. 1 that for a solid model, one triangle vertex, three triangle sides, and two triangle patches can be deleted by performing one triangle side collapse. By repeatedly selecting the pruning, a certain required simplified grid is finally obtained.
The quadratic error measure method is inferior to the collapse cost control method in grid simplification quality, but has higher calculation speed and lower memory consumption. Let side e be the target collapsed side, u and v be the two end points of triangle side e, p (e) be the union of plane set formed by the planes of the related triangles of vertex u and v, when collapsing side e to a new position point v bar ＝∣x bar ,y bar ,z bar When 1|, the error Q generated is defined as the collapsed vertex v bar The sum of squares of the distances to the individual planes in p (e), i.e
D in formula (1) p For a new vertex v bar Distance to plane pAnd (5) separating. Let v= |x bar ,y bar ,z bar 1 is v bar P= [ a, b, c, d ]] T Plane equation representing the plane of the triangle: ax+by+cz+d=0 (a 2 +b 2 +c 2 =1), then the expression (1) can be expressed as
M in formula (2) p Error matrix called triangle
However, these mesh simplification methods are all simplified based on the original model of the non-texture map, and the model displayed in the virtual bank system is a realistic model with the texture map, so that the model simplified by these methods cannot be directly applied to the virtual bank. And the calculated amount of the existing algorithm is larger, so that the running time of the algorithm is longer, and the algorithm cannot be well transplanted into a program running in a virtual bank.
Disclosure of Invention
The invention aims to provide a three-dimensional grid simplifying method applied to a VR virtual bank, which can realize the purpose that the quality of a display model is ensured while three-dimensional grid simplification is carried out and the operation of a system is not seriously influenced, thereby better realizing an online virtual banking system based on VR.
The technical scheme adopted by the invention for realizing the purposes is as follows:
a three-dimensional grid simplifying method applied to VR virtual banks comprises the following steps in sequence:
1. reading in three-dimensional grid model related information, and calculating the theoretical triangular face number after simplification according to the set model simplification precision epsilon;
wherein epsilon is more than 0 and less than 100%, and the three-dimensional grid model related information comprises vertex information and texture mapping information;
2. calculating an error matrix M of each original triangle contained in each triangular mesh through read-in three-dimensional mesh model related information p ；
3. Obtaining geometric Error of three-dimensional grid model v And texture error Er uv ；
4. Calculating an edge collapse error of each edge;
5. constructing a minimum stack using the edge collapse error;
6. selecting an edge with the minimum collapse cost for collapse simplification;
7. performing edge collapse operation, selecting a new vertex, and recording the corresponding relation after collapse;
8. deleting the edge related to the collapsed edge from the heap, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum heap to obtain the number of triangle facets after simplification;
9. if the number of the triangular patches after simplification is larger than the number of the theoretical triangular patches after simplification, continuing to simplify the step six, and otherwise ending the simplification flow.
As a limitation, the step three obtains the geometric Error of the three-dimensional grid model v The process of (2) is carried out according to the following steps:
the degree of regularity of the original triangle is expressed as
re＝2(cosα+cosβ+cosγ-1) ④
Wherein alpha, beta and gamma are the degrees of the inner angle of the triangle respectively;
setting the error adjustment factor as the average value of the regularities of all generated triangles in the selected new vertex neighborhood after the edge collapse, namely
Third, a new error calculation method is determined based on the quadratic error measure and triangle regularization degree judgment, namely
(IV) useRepresenting mesh reduction weights
Wherein f is a single triangular grid in the three-dimensional grid model, s f Is the area s of the triangular mesh f texf The area of the texture triangle of the triangular grid f on the projection surface is set as the area of the texture triangle;
(V) defining the texture error adjustment factor as
(VI) introducing a texture error adjustment factor, redefining a secondary error matrix Q in the prior art as
Q v ＝P(α)Q ⑧
(seventh) Q in formula (6) is changed to Q v Obtaining a new geometric error calculation formula as
As a further limitation, the third step obtains texture error Er of the three-dimensional grid model uv The process of (2) is carried out according to the following steps:
s1, weighting two vertexes of one e in the three-dimensional grid model in the color space distance to obtain
Wherein D is RGB (e) Representing vertex P 1 、P 2 Distance, w, of colors in RGB space r 、w g 、w b Is a weighting coefficient;
s2, prescribing folding edge P 1 、P 2 The texture error caused is
Wherein D (e) represents vertex P 1 、P 2 Distance in three-dimensional geometric space.
As a further limitation, in the fourth step, the calculation formula of the edge collapse error is as follows
Cost(e)＝ω(Error v +λ*Er uv ) Ⅰ
Wherein Cost represents the edge collapse error; omega is a threshold value when processing boundary edges, lambda is a scale factor which represents the importance of texture relative to the importance of geometry according to the specific conditions of geometry and texture of a certain network model, and when a certain triangular edge is inside a grid, the omega value is set to be 1; let ω be a real number greater than 1 when the triangle edge is the boundary edge.
Compared with the prior art, the technical proposal adopted by the invention has the following technical progress:
(1) The prior three-dimensional grid simplification technology basically only considers the geometric feature simplification of the 3D model, but in order to show higher object reduction degree, the three-dimensional model to be imported is often provided with a texture map, and the simplification of the model texture map features is added in addition to the full consideration of the geometric feature simplification of the model, so that the model after grid simplification can fully reserve the geometric features and the texture features of the model, and the fidelity degree of the model is better ensured, so that the model is more suitable for a virtual reality system than the model after the simplification of the prior art;
(2) For the simplification of the geometric features of the grid, one of a QEM algorithm and a progressive grid algorithm is selected in the prior art, but the QEM algorithm technology is old, the quality of a simplified model is low, the progressive grid algorithm has higher simplified quality, but the calculated amount is huge, the simplified time is overlong, the QEM algorithm and the progressive grid algorithm are fused, the simplification thought of the progressive grid algorithm is utilized, the error control matrix of the QEM algorithm is added, the advantages of the QEM algorithm and the progressive grid algorithm are taken, the simplified time is short, and the quality of the simplified model can be better ensured;
(3) On the basis of utilizing a QEM algorithm secondary error matrix, the invention introduces triangle regularities re=2 (cosalpha+cosbeta+cosgamma-1) as an adjusting factor, thereby generating a new error control formulaThe geometric error control of grid simplification is carried out by utilizing the formula, so that the occurrence of an elongated triangle in the three-dimensional grid is avoided, and the grid simplification quality is optimized;
(4) For error control of texture map portion in mesh simplification, the invention first uses the ratio of texture area to mesh areaTo calculate the texture adjustment factor +.>And utilize Q v The (alpha) Q is converted into a secondary error matrix Q, so that texture simplification factors are considered in grid simplification errors, the phenomena of texture stretching, texture blurring and the like of a three-dimensional grid in the mapping process are reduced, and the degree of fit between the geometric appearance of the model and the texture mapping is ensured; then, the texture error of the model is controlled by utilizing the color difference between the triangle vertexes, and the RGB color space is introduced to realize the texture error Er uv Using RGB weighted color difference formula to calculate the color difference between two points, finally passing through the formula Cost (e) =ω (Error) v +λ*Er uv ) Calculating the final simplification error of the weighting of the geometric error and the texture error of the grid model;
(5) In the prior art, only the simplified quantity of the whole model is considered, and the simplified quality of the boundary features of the model is ignored, so that the conditions of unobvious features, feature loss and the like of the simplified model are caused, the boundary processing of the model is considered, the boundary points and boundary edges are processed, the boundary points are directly processed to select the boundary points as new vertices for edge collapse, an error threshold omega is added to the boundary edges, so that the simplified priority is reduced, the geometric features of the edges of the model are reserved, and particularly, the detail features of the complex model with obvious features can be better ensured by the boundary processing;
(6) The model simplification is carried out by using the technology of the invention, and the model with the texture map can be simplified, so that the range of model simplification is wider;
(7) By using the technology of the invention to simplify the model, the simplified model not only can keep better geometric appearance characteristics, but also can keep texture characteristics, and boundary characteristics of the simplified model can be kept, so that the fidelity of the three-dimensional model can be well kept;
(8) The invention optimizes the operand and the data structure, ensures that the grid simplification process is completed in a shorter time, thereby ensuring the fluency of system operation, and is more suitable for the virtual reality technology; the invention can be used in the construction of a virtual reality system, so that a complex model can be simplified and loaded quickly, the quality of the model can be ensured, the interactive experience of a user can be improved, and the development of a virtual reality technology can be further promoted;
(9) Based on the existing advantages of the technology and the characteristic of being suitable for a virtual reality system, and combining the heat development of the emerging concept of the virtual bank nowadays, the technology can be applied to the VR virtual bank, so that the key problem of constructing the virtual bank system, namely the problem of loading a complex model with large data volume, is solved, and the development of the whole emerging industry of the virtual bank can be further promoted.
The method is suitable for the technical field of image processing and is used for simplifying the three-dimensional grid.
Drawings
The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate the invention and together with the embodiments of the invention, serve to explain the invention.
In the drawings:
FIG. 1 is a prior art side collapse schematic of the present invention;
FIG. 2 is a grid projection schematic diagram of an embodiment of the present invention;
FIG. 3 is a schematic diagram of an RGB color space according to an embodiment of the present invention;
FIG. 4 is a general flow chart of an embodiment of the present invention;
FIG. 5 is a simplified error calculation flow chart according to an embodiment of the present invention;
FIG. 6 is a flowchart illustrating a boundary process according to an embodiment of the present invention.
Detailed Description
Preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are presented for purposes of illustration and explanation only and are not intended to limit the present invention.
Embodiment three-dimensional grid simplifying method applied to VR virtual bank
As shown in fig. 4, this embodiment is performed in the following sequence of steps:
1. loading a three-dimensional grid model in an OBJ format by the system, performing simplified initialization operation, reading related information of the three-dimensional grid model, generating a list for storage, and calculating the simplified theoretical triangular surface number according to the set model simplified precision epsilon;
epsilon is the percentage set according to the actual needs, 0 < epsilon < 100%, epsilon=80% in the embodiment; the three-dimensional grid model related information comprises vertex information and texture mapping information;
2. calculating an error matrix M of each original triangle contained in each triangular mesh through read-in three-dimensional mesh model related information p ；
Calculating an error matrix M in this step p The method in the prior art is adopted;
3. obtaining geometric Error of three-dimensional grid model v And texture error Er uv ；
4. The edge collapse error of each edge is calculated,
5. constructing a minimum stack by utilizing the edge collapse error, and recording related data at the same time;
6. selecting an edge with the minimum collapse cost for collapse simplification;
7. performing edge collapse operation, selecting a new vertex, and recording the corresponding relation after collapse;
8. deleting the edge related to the collapsed edge from the heap, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum heap to obtain the number of triangle facets after simplification;
9. if the number of the triangular patches after simplification is larger than the number of the theoretical triangular patches after simplification, turning to the sixth step to continue simplification, otherwise ending the simplification process, and loading the simplification model into the VR scene.
In the present embodiment, as shown in fig. 5, in the process of obtaining the geometric Error of the three-dimensional grid model in step four, triangle regularities are introduced as adjustment factors on the basis of the quadratic Error measure to adjust the collapse Error. The degree of regularity of a triangle is expressed as
re＝2(cosα+cosβ+cosγ-1) ④
Wherein α, β, γ are the degrees of the interior angle of the triangle respectively, when re=1, the triangle is the best in the regularity, and the triangle is an equilateral triangle, and the best case is in the three-dimensional grid; when re=0, the regularity of the triangle is worst, and when a certain internal angle of the triangle is 180 °, the triangle is degraded into a straight line; according to the judgment of the triangle regularities, setting an error adjusting factor as the average value of the regularities of all the generated triangles in the selected new vertex neighborhood after the edge collapse, namely
Based on the secondary error measure and triangle regularization degree judgment, a new error calculation method is determined, namely
As shown in the formula (6), when the degree of regularization of the triangle generated after the edge collapse is larger, the simplification error is smaller, the priority of the triangle in the simplification process is higher, and otherwise, the simplification error is larger; in the embodiment, the formula (6) is adopted to avoid the occurrence of an elongated triangle after the side collapse, so that the simplified transition of the grid is ensured to be smoother, and the quality of the generated grid is ensured.
For simplification of a three-dimensional grid model with texture, not only geometrical errors caused by edge folding are considered, but also texture errors are considered, so that the simplified model is guaranteed to be maximally faithful to an original model in both geometrical characteristics and texture characteristics, and for texture mapping characteristics, the embodiment firstly introduces the grid texture area as an error adjustment factor, and in the process of three-dimensional grid mapping, the geometrical area and the texture area of a triangular grid can cause larger deviation due to projection directions, and in order to meet a plump visual effect, a rendering system tends to stretch the texture, so that the texture can be guaranteed to be completely filled with the grid, and blurring of the texture mapping is caused; if the ratio of the texture area to the geometric area of the grid can be increased, distortion and blurring generated by the texture in the mapping process can be reduced, so that the quality of the texture mapping is ensured. FIG. 2 is a schematic view of triangular meshes projected onto a texture surface, in FIG. 2, f is an individual triangular mesh in a three-dimensional model, its view is v, and normal vectors of planes in which f and v lie are n, respectively f And n v The included angle of the two vectors is theta, the included angle of the vectors is inversely proportional to the quality of the texture mapping, and the area of the triangular mesh f is s f While the area of the texture triangle on the projection surface is s texf For the present embodimentThe larger the ratio, the lower the simplification priority, and conversely the higher the simplification priority, the larger the ratio that the part of texture information occupies. The present embodiment defines the texture error adjustment factor as
I.e. the average of the ratio of the texture area to the geometric area of all connected triangles around a certain triangle vertex, then the prior art quadratic error matrix Q is redefined as after the introduction of the texture error adjustment factor
Q v ＝P(α)Q ⑧
Since the possible adjustment effect of the adjustment factor is not obvious when the alpha value is small, the texture adjustment factor is not directly multiplied by Q here in order to maintain a good degree of differentiation; in the formula P (alpha) =T α T can be a positive integer greater than 1; q in formula (6) is changed to Q v Obtaining a new geometric error calculation formula as
As shown in fig. 5, a texture error Er of the three-dimensional mesh model is obtained uv In the process of (a), the texture error caused by folding the edge is defined as the texture color difference between the two vertices of the folded edge. Firstly, defining an RGB color space, taking RGB three values as coordinate axes of the color space respectively, and then each color corresponds to a point in the color space, as shown in figure 3. The greater the difference between the colors of two points, the greater the distance between their corresponding points in the color space and vice versa. And the distance between the black (0, 0) and white (255, 255, 255) corresponding points with the largest color difference is the farthest. For one e in the model, the present embodiment uses D (e) to represent the vertex P 1 、P 2 Distance in three-dimensional geometric space, D RGB (e) Representing vertex P 1 、P 2 The distance of the colors in RGB space, i.e. the color difference of two points. In addition, since the human eyes feel and acquire different primary colors of red Huang Lansan, in order to make the simplified texture picture more fit to human vision, the two-point color space distance is weighted as shown in the formula (9):
w in (9) r 、w g 、w b Is a weighting coefficient, where (w r ,w g ,w b ) The value of (2) is (3, 4). Then define the folded edge P 1 、P 2 The texture error caused is
Wherein, the liquid crystal display device comprises a liquid crystal display device,the two factors D (e) and phi are used to balance the geometric error and texture error so that the measurement values of the two errors are at the same level.
In the fifth step, the edge collapse error of each edge is calculated by adopting the formula I:
Cost(e)＝ω(Error v +λ*Er uv ) Ⅰ
cost represents edge collapse error; ω is a threshold value when processing boundary edges, λ is an artificially set parameter, which is a scale factor representing the texture importance relative to the geometric importance according to the geometry and texture specific situation of a certain network model, and the specific flow is shown in fig. 6.
The boundary points and boundary edges of the grid model should be preferentially ensured not to be deleted when the grid is simplified, because once the boundary of the model is changed, the edge characteristics of the model are likely to be affected; the traditional secondary error measurement method can have a larger influence on the boundary characteristics of the open boundary grid, so that for the boundary vertexes of the three-dimensional grid, the embodiment adopts a method of directly taking the boundary points as new vertex positions after edge collapse, and the integral quality of the simplified model can be well ensured; for boundary edges, if an unfolded method is adopted, for some grids with relatively more boundaries and obvious boundaries, the integral simplification degree of the grid model can be influenced, for the processing of the boundary edges, a threshold omega is added on the basis of collapse errors of the boundary edges, and when a certain triangular edge is in the grid, the omega value is set to be 1; when the triangle edge is the boundary edge, omega is a real number larger than 1, so that the collapse cost of the boundary edge is increased, and the collapse priority is reduced. By the processing mode, detail features at the boundary of the model can be conveniently and simply kept.
Claims (4)
1. The three-dimensional grid simplifying method applied to the VR virtual bank is characterized by comprising the following steps of:
1. reading in three-dimensional grid model related information, and calculating the theoretical triangular face number after simplification according to the set model simplification precision epsilon;
wherein epsilon is more than 0 and less than 100%, and the three-dimensional grid model related information comprises vertex information and texture mapping information;
2. calculating an error matrix M of each original triangle contained in each triangular mesh through read-in three-dimensional mesh model related information p ；
3. Obtaining texture error Er of three-dimensional grid model uv Obtaining geometric Error of the three-dimensional grid model according to the Error matrix Mp v ；
4. Combining texture error Er uv And geometric Error v Calculating an edge collapse error of each edge;
5. constructing a minimum stack using the edge collapse error;
6. selecting an edge with the minimum collapse cost for collapse simplification;
7. performing edge collapse operation, selecting a new vertex, and recording the corresponding relation after collapse;
8. deleting the edge related to the collapsed edge from the heap, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum heap to obtain the number of triangle facets after simplification;
9. if the number of the triangle facets after simplification is larger than the number of the theoretical triangle facets after simplification, the step six is continued to be simplified, and otherwise, the simplification flow is ended.
2. The method for three-dimensional mesh simplification applied to VR virtual banking according to claim 1, wherein said step three obtains a geometric Error of a three-dimensional mesh model v The process of (2) is carried out according to the following steps:
the degree of regularity of the original triangle is expressed as
re＝2(cosα+cosβ+cosγ-1) ④
Wherein alpha, beta and gamma are the degrees of the inner angle of the triangle respectively;
setting the error adjustment factor as the average value of the regularities of all generated triangles in the selected new vertex neighborhood after the edge collapse, namely
p (e) represents the union of plane sets formed by planes of related triangles of triangle vertexes u and v, wherein u and v are two endpoints of triangle edge e respectively;
third, a new error calculation method is determined based on the quadratic error measure and triangle regularization degree judgment, namely
Q represents the collapsed vertex v bar Sum of squares of distances to respective planes in p (e), where v bar Representing a new point to which edge e collapses, v bar ＝∣x bar ,y bar ,z bar ,1∣；
(IV) useRepresenting mesh reduction weights
Wherein f is a single triangular grid in the three-dimensional grid model, s f Is the area s of the triangular mesh f texf The area of the texture triangle of the triangular grid f on the projection surface is set as the area of the texture triangle;
(V) defining the texture error adjustment factor as
(VI) introducing a texture error adjustment factor, redefining the secondary error matrix Q in the prior art as Q v ＝P(α)Q ⑧
Wherein P (α) =t α T is greater than 1A positive integer;
(seventh) Q in formula (6) is changed to Q v Obtaining a new geometric error calculation formula as
3. The three-dimensional grid simplifying method for VR virtual bank as set forth in claim 2, wherein said step three obtains texture error Er of three-dimensional grid model uv The process of (2) is carried out according to the following steps: s1, weighting two vertexes of one edge e in the three-dimensional grid model in the color space distance to obtain
Wherein D is RGB (e) Representing vertex P 1 、P 2 Distance, w, of colors in RGB space r 、w g 、w b Is a weighting coefficient; s2, prescribing folding edge P 1 、P 2 The texture error caused is
Wherein D (e) represents vertex P 1 、P 2 Distance in three-dimensional geometric space;representing factors for balancing geometry errors with texture errors.
4. The three-dimensional grid simplifying method for VR virtual bank as in claim 3, wherein in said step four, the calculation formula of the edge collapse error is
Cost(e)＝ω(Error v +λ*Er uv )
Wherein Cost represents the edge collapse error; omega is a threshold value when processing boundary edges, lambda is a scale factor which represents the importance of texture relative to the importance of geometry according to the specific conditions of geometry and texture of a certain network model, and when a certain triangular edge is inside a grid, the omega value is set to be 1; let ω be a real number greater than 1 when the triangle edge is the boundary edge.
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