CN112258655A - Three-dimensional grid simplification method applied to VR virtual bank - Google Patents
Three-dimensional grid simplification method applied to VR virtual bank Download PDFInfo
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Abstract
The invention discloses a three-dimensional grid simplification method applied to a VR virtual bank, which comprises the following steps of: calculating the number of simplified theoretical triangular surfaces; secondly, calculating an error matrix of each original triangle contained in each triangular mesh; thirdly, obtaining a geometric error and a texture error of the three-dimensional grid model; fourthly, calculating the edge collapse error of each edge; fifthly, constructing a minimum stack; sixthly, selecting the edge with the minimum collapse error for collapse simplification; seventhly, performing side collapsing operation; eighthly, obtaining the number of the simplified triangular plates; and ninthly, if the number of the simplified triangular faces is larger than the number of the simplified theoretical triangular faces, continuing to simplify the process by turning to the sixth step, and otherwise, ending the simplified process. The invention can realize the purposes of simplifying the three-dimensional grid and simultaneously ensuring the quality of the display model without seriously influencing the operation of the system, thereby better realizing the online virtual bank system based on VR. The invention belongs to the technical field of image processing, and is used for three-dimensional grid simplification.
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 exhibition, engineering design, assembly and maintenance, medical research, military training and the like at present. The advent of virtual reality technology has enabled many things to appear in new forms to people, gradually changing their traditional lifestyle. The method depends on virtual online banking of VR technology. The virtual bank refers to a three-dimensional electronic bank created by using a virtual information processing technology. Because all the businesses of the virtual bank are handled through the network, the business efficiency is higher, the service and operation rate is lower, and the virtual bank has the advantages of low operation cost, high loan efficiency and the like. The virtual bank is released, which is an important ring in intelligent bank measures, financial innovation is promoted, better user experience is brought, and realization of general finance is further promoted.
Virtual banks have been currently tried in many countries such as the united kingdom, germany, singapore, etc. At present, the hong Kong has introduced virtual bank license plates in China, and 8 virtual bank license plates have been issued by the hong Kong administration. The companies for obtaining virtual bank license plates include China Bank hong Kong, Jingdong science and technology, Paocha bank, ant golden clothes, Tengchong, industry and commerce bank, millet and peace and safety group and many powerful financial and scientific companies and statues.
At present, virtual banking based on a VR system is mainly realized by a Unity 3D technology and a WebVR technology. The 3D model is the basis for VR system construction. In order for a user to have 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 technologies such as VR technology and computer graphics, the three-dimensional model constructed by the triangular mesh is gradually refined, which makes the morphological characteristics of the model very complex and results in an excessively large data volume of the three-dimensional model file, so that the real-time response mechanism of the virtual reality system is slow and unsmooth due to heavy memory burden, and is not beneficial to performing real-time human-computer interaction. And because the existing WebVR technology is influenced by network conditions and the computing capability of a browser, the WebVR system does not support loading of a model with overlarge data amount into a scene. Therefore, in order to optimize a virtual reality system and enhance the response speed and the interaction fluency of the virtual reality system, the model simplification method for VR virtual bank construction is invented, the aim is to reduce the data volume of a three-dimensional model, ensure the quality of the simplified model and spend as little time as possible to complete the process.
The existing grid simplification methods mainly include Quadratic Error Measure (QEM) algorithm proposed by Garland and progressive grid (PM) algorithm proposed by Hoppe and its improved algorithm.
The Progressive Mesh (PM) algorithm proposed by Hoppe is mesh simplification using the idea of edge collapse. As shown in fig. 1, a triangle side uv in the triangular mesh, u and v are two end points of the triangle side, and a and B are two triangles with uv as a common side; when the side uv is a collapsed side, the following operations are performed: vertex u is replaced with vertex v and other neighbor vertices are connected to the new vertex, and then vertex u and related information are deleted. This completes one collapse of the triangular edges of the target, where v is referred to as the collapsed target of u. It is known from fig. 1 that for a solid model, one triangle side collapse is performed, and one triangle vertex, three triangle sides, and two triangle patches can be deleted. By repeatedly selecting and deleting, a simplified grid with certain requirements is finally obtained.
The quadratic error measure method is second to the collapse cost control method in grid simplification quality, but the calculation speed is high, and the memory consumption is low. Let e be the target collapsed side, u and v be the two endpoints of the triangle side e, p (e) be the union of the plane sets formed by the planes of the related triangles with the vertexes u and v, when collapsing the side e to a certain oneNew position point vbar＝∣xbar,ybar,zbar1 | the error Q produced is defined as the collapsed vertex vbarThe sum of the squares of the distances to the respective planes in p (e), i.e.
D in formula (1)pIs a new vertex vbarDistance to plane p. Let v | xbar,ybar,zbar1 | is vbarWith the homogeneous coordinate formula of (a), (b), (c), (d)]TThe plane equation representing the plane in which the triangle lies: ax + by + cz + d is 0 (a)2+b2+c21), then the formula (1) can be represented as
M in the formula (2)pError matrix called triangle
However, these mesh simplification methods are based on the original model of texture-free maps for simplification, and since the models displayed in the virtual bank system are all realistic models with texture maps, the simplified models of these methods cannot be directly applied to virtual banks. And because the calculation amount of the existing algorithm is large, the running time of the algorithm is long, and the algorithm cannot be well transplanted to a program run by a virtual bank.
Disclosure of Invention
The invention aims to provide a three-dimensional grid simplification method applied to VR virtual banks, which can realize the purposes of simplifying the three-dimensional grid, ensuring the quality of a display model and not seriously influencing the operation of a system, thereby better realizing the VR-based online virtual bank system.
In order to achieve the purpose, the invention adopts the following technical scheme:
a three-dimensional grid simplification method applied to a VR virtual bank comprises the following steps in sequence:
reading related information of a three-dimensional grid model, and calculating the number of simplified theoretical triangular surfaces according to a set model simplification precision epsilon;
wherein, the epsilon is more than 0 and less than 100 percent, and the related information of the three-dimensional mesh model comprises vertex information and texture mapping information;
secondly, calculating an error matrix M of each original triangle contained in each triangular mesh through the read-in related information of the three-dimensional mesh modelp；
Thirdly, obtaining the geometric Error of the three-dimensional grid modelvAnd texture error Eruv；
Fourthly, calculating the edge collapse error of each edge;
fifthly, constructing a minimum stack by using the edge collapse error;
sixthly, selecting the edge with the minimum collapse cost to perform collapse simplification;
seventhly, performing side collapse operation, selecting a new vertex, and recording a corresponding relation after collapse;
eighthly, deleting the edge associated with the collapsed edge from the stack, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum stack to obtain the number of faces of the simplified triangular plate;
and ninthly, if the number of the simplified triangular faces is larger than the number of the simplified theoretical triangular faces, continuing to simplify the process by turning to the sixth step, and otherwise, ending the simplified process.
As a limitation, the third step obtains the geometric Error of the three-dimensional mesh modelvThe process of (2) is carried out according to the following steps:
the degree of regularity of (a) the original triangle is expressed as
re＝2(cosα+cosβ+cosγ-1) ④
Wherein alpha, beta and gamma are degrees of the inner angle of the triangle respectively;
secondly, setting the error adjustment factor as the average value of the regularization degrees of all the generated triangles in the neighborhood of the new vertex selected after the edges are collapsed, namely
(III) determining a new error calculation method based on the quadratic error measure and the triangle regularization judgment, namely
(IV) use
Representing mesh reduction weights
Wherein f is a single triangular mesh in the three-dimensional mesh model, sfIs the area, s, of the triangular mesh ftexfThe area of the texture triangle of the triangular mesh f on the projection plane is shown;
(V) defining the texture error adjustment factor as
Sixthly, texture error adjustment factors are introduced, and a quadratic error matrix Q in the prior art is redefined into
Qv＝P(α)Q ⑧
(VII) changing Q in the formula (III) to QvObtaining a new geometric error calculation formula as
As a further limitation, the third step obtains the texture error Er of the three-dimensional mesh modeluvThe process of (2) is carried out according to the following steps:
s1, weighting the distance of two vertexes of one e in the three-dimensional mesh model in the color space to obtain
Wherein D isRGB(e) Represents a vertex P1、P2Distance of the color of (1) in RGB space, wr、wg、wbIs a weighting coefficient;
s2, defining folding edge P1、P2The resulting texture error is
Wherein D (e) represents a vertex P1、P2Distance in three-dimensional geometric space.
By way of further limitation, in the fourth step, the edge collapse error is calculated by the formula
Cost(e)＝ω(Errorv+λ*Eruv) Ⅰ
Wherein Cost represents the edge collapse error; omega is a threshold value when processing boundary edges, lambda is a scale factor which expresses the importance of the texture relative to the geometrical importance according to the specific situation of the geometry and the texture of a certain network model, and when a certain triangular edge is in the interior of a mesh, the value of omega is set to 1; when the triangle edge is a boundary edge, let ω be a real number greater than 1.
Due to the adoption of the technical scheme, compared with the prior art, the invention has the technical progress that:
(1) the three-dimensional mesh simplification technology basically only considers the geometric feature simplification of a 3D model, but in order to show higher object reduction degree, the virtual reality technology needs the imported three-dimensional model with a texture mapping, and the method also adds the simplification of the texture mapping feature of the model besides fully considering the geometric feature simplification of the model, so that the model after mesh simplification can fully retain the geometric feature and the texture feature of the model, and better ensures the fidelity of the model, therefore, the model simplified by the prior art is more suitable for a virtual reality system;
(2) for simplification of grid geometric features, one of a QEM algorithm or a progressive grid algorithm is selected in the prior art, but the QEM algorithm technology is old and the quality of a simplified model is not high, while the progressive grid algorithm has high simplified quality, but the calculation amount is huge and the simplified time is too long, the QEM algorithm and the progressive grid algorithm are fused by the technology, the advantages of the QEM algorithm and the progressive grid algorithm are taken by utilizing the simplified thought of the progressive grid algorithm and the error control matrix of the QEM algorithm, the simplified time can be short, and the quality of the simplified model can be better ensured;
(3) on the basis of utilizing a QEM algorithm quadratic error matrix, the invention introduces a triangle regularization re-2 (cos alpha + cos beta + cos gamma-1) as an adjusting factor, thereby generating a new error control formula
The grid simplification geometric error control is carried out by using the formula, so that the appearance of long and narrow triangles in the three-dimensional grid is avoided, and the grid simplification quality is optimized;
(4) for error control of the texture mapping part in mesh simplification, the invention first uses the ratio of the texture area to the mesh area
To calculate the texture adjustment factor
And use of QvConverting a quadratic error matrix Q into P (alpha) Q, thereby considering texture simplification factors into grid simplification errors, reducing phenomena of texture stretching, texture blurring and the like of a three-dimensional grid in a mapping process, and ensuring the conformity of the geometric appearance of the model and a texture mapping; and then, controlling the texture error of the model by using the color difference between the vertexes of the triangle, and introducing an RGB color space to realize the texture error EruvThe color difference between two points is calculated by using an RGB weighted color difference formula, and finally, the formula cost (e) is ω (Error)v+λ*Eruv) Calculating a final simplified error weighted by 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 characteristics of the model is ignored, so that the simplified model has the conditions of unobvious characteristics or lost characteristics and the like;
(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 the model simplification range is wider;
(7) the method has the advantages that the model is simplified by the technology, the simplified model not only can keep better geometric appearance characteristics, but also can keep texture characteristics, the boundary characteristics of the simplified model are kept, and the fidelity of the three-dimensional model can be well kept;
(8) the invention optimizes the operation quantity and the data structure, and ensures that the grid simplification process is completed in a shorter time, thereby ensuring the smooth degree of system operation, and being more suitable for the virtual reality technology; the method is used in the construction of the virtual reality system, so that the complex model can be quickly simplified and loaded, the quality of the model is ensured, the interactive experience of a user can be improved, and the development of the virtual reality technology is further promoted;
(9) based on the existing advantages of the technology and the characteristics of being suitable for a virtual reality system, the technology of the invention is combined with the fire-heat development of the emerging concept of the current virtual bank, so that the technology of the invention can be applied to the VR virtual bank, the key problem of constructing the virtual bank system, namely the loading problem of 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, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention.
In the drawings:
FIG. 1 is a schematic view of a prior art edge collapse of the present invention;
FIG. 2 is a schematic diagram of a grid projection according to 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 an overall flow diagram of an embodiment of the present invention;
FIG. 5 is a simplified error calculation flow diagram according to an embodiment of the present invention;
FIG. 6 is a flowchart illustrating a boundary processing 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 for purposes of illustration and explanation only and are not intended to limit the present invention.
Embodiment of the invention provides a three-dimensional grid simplification method applied to VR virtual bank
As shown in fig. 4, the present embodiment is performed in the following order of steps:
the method comprises the steps that firstly, a three-dimensional grid model in an OBJ format is loaded by a system, simplified initialization operation is carried out, relevant information of the three-dimensional grid model is read in, a list is generated and stored, and the number of simplified theoretical triangular surfaces is calculated according to set model simplification precision epsilon;
epsilon is a percentage set according to actual needs, 0 < epsilon < 100%, and epsilon is 80% in the embodiment; the related information of the three-dimensional mesh model comprises vertex information and texture mapping information;
secondly, calculating an error matrix M of each original triangle contained in each triangular mesh through the read-in related information of the three-dimensional mesh modelp；
In this step, an error matrix M is calculatedpThe method of (1) adopts the method in the prior art;
thirdly, obtaining the geometric Error of the three-dimensional grid modelvAnd texture error Eruv；
Fourthly, calculating the edge collapse error of each edge,
constructing a minimum stack by using the edge collapse error, and simultaneously recording related data;
sixthly, selecting the edge with the minimum collapse cost to perform collapse simplification;
seventhly, performing side collapse operation, selecting a new vertex, and recording a corresponding relation after collapse;
eighthly, deleting the edge associated with the collapsed edge from the stack, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum stack to obtain the number of faces of the simplified triangular plate;
and ninthly, if the number of the simplified triangular faces is larger than the number of the simplified theoretical triangular faces, continuing to simplify the simplified step by turning to the step six, otherwise, ending the simplified process and loading the simplified model into the VR scene.
As shown in fig. 5, in the present embodiment, in the process of obtaining the geometric Error of the three-dimensional mesh model in step four, a triangular regularization is introduced as an adjustment factor 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 alpha, beta and gamma are degrees of inner angles of the triangle respectively, when re is 1, the regular degree of the triangle is the best, and the triangle is an equilateral triangle at the moment, and the condition is the best in a three-dimensional grid; when re is 0, the regularity of the triangle is the worst, and at this time, a certain internal angle of the triangle is 180 degrees, and the triangle is degenerated into a straight line; according to the judgment of the triangle regularity, the error adjusting factor is set as the average value of the regularity of all the generated triangles in the new vertex neighborhood selected after the edges are collapsed, namely
Based on the quadratic error measure and the triangle regularization decision, a new error calculation method is determined, i.e.
It is known from the formula, when the degree of regularity of the triangle generated after the edges are collapsed is larger, the smaller the simplification error is, the higher the priority of the triangle in the simplification process is, and otherwise, the larger the simplification error is; the embodiment adopts the formula (I) to avoid the appearance of long and narrow triangles after the edges are collapsed, and ensures that the simplified transition of the mesh is smoother, thereby ensuring the quality of the generated mesh.
For simplification of a three-dimensional mesh model with textures, not only a geometric error caused by edge folding but also a texture error of the three-dimensional mesh model are considered to ensure that the simplified model is faithful to an original model to the maximum extent in both geometric characteristics and texture characteristics, aiming at texture mapping characteristics, the embodiment firstly introduces a mesh texture area as an error adjusting factor, in the three-dimensional mesh mapping process, the geometric area and the texture area of a triangular mesh cause large deviation due to a projection direction, and in order to meet a fuller visual effect, a rendering system often stretches textures so as to ensure that the textures can be completely filled in the mesh, thereby causing blurring of the texture mapping; 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, and therefore texture mapping quality is guaranteed. FIG. 2 is a schematic diagram of a triangular mesh projected onto a texture surface, and in FIG. 2, f is a single triangular mesh in a three-dimensional model, and the view is that v, and the normal vectors of the plane where f and v are respectively nfAnd nvThe included angle between two vectors is theta, the included angle between the vectors is inversely proportional to the quality of the texture map, and the area of the triangular mesh f is sfAnd the area of the texture triangle on the projection surface is stexfFor the present embodiment
The larger the ratio is, the larger the ratio occupied by the part of texture information is, the lower the simplification priority is, otherwise, the higher the simplification priority is. 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, the quadratic error matrix Q in the prior art is redefined after introducing texture error adjustment factors as
Qv＝P(α)Q ⑧
Because the adjustment effect of the adjustment factor may not be obvious when the value of α is small, the texture adjustment factor is not directly multiplied by Q to maintain good discrimination; where P (alpha) is TαT may be a positive integer greater than 1; changing Q in the formula (II) to QvObtaining a new geometric error calculation formula as
As shown in FIG. 5, a texture error Er of the three-dimensional mesh model is obtaineduvIn the process of (2), the texture error caused by folding the edge is defined as the texture color difference between two vertexes of the folded edge. First, an RGB color space is defined, and the RGB three values are respectively used as coordinate axes of the color space, so that each color corresponds to a point in the color space, as shown in fig. 3. The greater the difference between the colors of two points, the greater the distance between their corresponding points in color space, and vice versa. And the distance between the points corresponding to black (0, 0, 0) and white (255, 255, 255) with the largest color difference is also the farthest. For one e in the model, the embodiment uses D (e) to represent the vertex P1、P2Distance in three-dimensional geometric space, DRGB(e) Represents a vertex P1、P2The distance of the color of (b) in the RGB space, i.e., the color difference of two points. In addition, because human eyes experience and acquire three primary colors of red, yellow and blue to different degrees, in order to enable the simplified texture picture to better conform to human vision, the spatial distance of two points of color is weighted, as shown in formula ninx:
w in formula ninthlyr、wg、wbAs weighting coefficients, here (w) is takenr,wg,wb) Has a value of (3,4, 2). Then the folding edge P is defined1、P2The resulting texture error is
Wherein the content of the first and second substances,
the two factors d (e) and phi are used to balance the geometric error and the texture error, so that the measurement values of the two errors are in the same level.
And step five, calculating the edge collapse error of each edge by adopting a formula I:
Cost(e)＝ω(Errorv+λ*Eruv) Ⅰ
cost represents edge collapse error; ω is a threshold value when processing the boundary edge, and λ is an artificially set parameter, which is a scale factor representing the importance of the texture relative to the geometrical importance according to the specific situation of the geometry and the texture of a certain network model, and the specific flow is shown in fig. 6.
When the grid is simplified, the boundary points and boundary edges of the grid model should be preferentially ensured not to be deleted, because once the boundary of the model is changed, the edge characteristics of the model are likely to be influenced; the traditional quadratic error measure method can generate larger influence on the boundary characteristics of the open boundary grid, so that for the boundary vertex of the three-dimensional grid, the embodiment adopts a method of directly taking the boundary point as a new vertex position after the edge is collapsed, and the integral quality of the simplified model can be well ensured; for the boundary edge, if a non-folding method is adopted, for some grids with more boundaries and obvious boundaries, the overall simplification degree of the grid model can be influenced, for the processing of the boundary edge, a threshold value omega is added on the basis of the collapse error of the boundary edge, and when a certain triangular edge is in the grid, the value of omega is set to be 1; when the triangular edge is a boundary edge, let ω be a real number greater than 1, so that the collapse cost of the boundary edge becomes large and the collapse priority is lowered. By the processing mode, the detail characteristics at the boundary of the model can be maintained conveniently and simply.
Claims (4)
1. A three-dimensional grid simplification method applied to a VR virtual bank is characterized by comprising the following steps of:
reading related information of a three-dimensional grid model, and calculating the number of simplified theoretical triangular surfaces according to a set model simplification precision epsilon;
wherein, the epsilon is more than 0 and less than 100 percent, and the related information of the three-dimensional mesh model comprises vertex information and texture mapping information;
secondly, calculating an error matrix M of each original triangle contained in each triangular mesh through the read-in related information of the three-dimensional mesh modelp；
Thirdly, obtaining the geometric Error of the three-dimensional grid modelvAnd texture error Eruv；
Fourthly, calculating the edge collapse error of each edge;
fifthly, constructing a minimum stack by using the edge collapse error;
sixthly, selecting the edge with the minimum collapse cost to perform collapse simplification;
seventhly, performing side collapse operation, selecting a new vertex, and recording a corresponding relation after collapse;
eighthly, deleting the edge associated with the collapsed edge from the stack, recalculating the collapse error of the edge where the new vertex is located, and inserting the minimum stack to obtain the number of faces of the simplified triangular plate;
and ninthly, if the number of the simplified triangular faces is larger than the number of the simplified theoretical triangular faces, continuing to simplify the process by turning to the sixth step, and otherwise, ending the simplified process.
2. The method of claim 1, wherein the step three comprises obtaining a three-dimensional gridGeometric Error of modelvThe process of (2) is carried out according to the following steps:
the degree of regularity of (a) the original triangle is expressed as
re＝2(cosα+cosβ+cosγ-1) ④
Wherein alpha, beta and gamma are degrees of the inner angle of the triangle respectively;
secondly, setting the error adjustment factor as the average value of the regularization degrees of all the generated triangles in the neighborhood of the new vertex selected after the edges are collapsed, namely
(III) determining a new error calculation method based on the quadratic error measure and the triangle regularization judgment, namely
(IV) use
Representing mesh reduction weights
Wherein f is a single triangular mesh in the three-dimensional mesh model, sfIs the area, s, of the triangular mesh ftexfThe area of the texture triangle of the triangular mesh f on the projection plane is shown;
(V) defining the texture error adjustment factor as
Sixthly, texture error adjustment factors are introduced, and a quadratic error matrix Q in the prior art is redefined into
Qv＝P(α)Q ⑧
(VII) changing Q in the formula (III) to QvTo obtain a new geometric error meterThe formula is
3. The three-dimensional mesh simplification method applied to VR virtual banks of claim 2, wherein the texture error Er of the three-dimensional mesh model is obtained in the third stepuvThe process of (2) is carried out according to the following steps:
s1, weighting the distance of two vertexes of one e in the three-dimensional mesh model in the color space to obtain
Wherein D isRGB(e) Represents a vertex P1、P2Distance of the color of (1) in RGB space, wr、wg、wbIs a weighting coefficient;
s2, defining folding edge P1、P2The resulting texture error is
Wherein D (e) represents a vertex P1、P2Distance in three-dimensional geometric space.
4. The three-dimensional grid reduction method applied to VR virtual banks of claim 3, wherein in the fourth step, the edge collapse error is calculated by the formula
Cost(e)＝ω(Errorv+λ*Eruv) Ⅰ
Wherein Cost represents the edge collapse error; omega is a threshold value when processing boundary edges, lambda is a scale factor which expresses the importance of the texture relative to the geometrical importance according to the specific situation of the geometry and the texture of a certain network model, and when a certain triangular edge is in the interior of a mesh, the value of omega is set to 1; when the triangle edge is a boundary edge, let ω be a real number greater than 1.
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