Isoplane paving digital art pattern generation method and device based on partial differential equation

Abstract

The invention discloses an isoplane paving digital art pattern generation method and device based on a partial differential equation. The method comprises the following steps: taking an isoplane body shape after interactive curve editing as a paving unit basic domain; a Laplacian equation is introduced into a basic domain, geometric constraints of paving units are converted into Dirichlet boundary conditions of a Laplacian operator, then smooth invariant mapping meeting isohedron symmetric constraints is constructed, and seamless transition of textures at joints of the paving units is guaranteed; and finally, finishing texture coloring in combination with the power system model and the quasi-regular pattern model. The method has the core advantages that the method can adapt to any isohedron type, the symmetry, the texture details and the visual effect of the pattern can be flexibly regulated and controlled by adjusting the constraint vertex, the power system function and the quasi-regular pattern function, a brand new solution is provided for dynamic real-time generation and personalized design of the digital art pattern, and the method can be applied to the field of digital art design. And abundant material sources are provided for the fields of costume design, textile design and the like.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of computer graphics and digital art design, and particularly relates to a method for generating continuous digital art patterns by using partial differential equations and isometric tiling. BACKGROUND

[0002] With the rapid iteration of digital technology, the technological revolution is deeply penetrating into various fields of human society, and the field of artistic creation is also undergoing a revolutionary change. Under the current background of dual driving of efficiency and innovation, the field of textile fabric pattern design presents significant technological transformation: digital art patterns, with their rich visual expression, efficient generation efficiency and flexible editing characteristics, have gradually replaced traditional handcrafting mode and become the core technical means of modern textile pattern design. This technology not only can effectively improve the aesthetic level and personalized customization level of textile products, but also relies on the parameterized generation ability of computers to inject new vitality into the design innovation of clothing and accessories, home wallpaper, game scene map, building decoration fabric and other multi-application fields, and promote the collaborative development of cross-field pattern design. However, the existing traditional textile fabric digital art pattern generation method still has obvious technical shortcomings. Although such methods take nonlinear theory as the core basis, integrate fractal geometry, dynamic system, weak chaos (such as quasi-regular pattern) and other key principles, and can present certain complex properties in the shape, color and texture details of the pattern, the lack of underlying space layout control ability becomes the core bottleneck restricting the development of the technology. Specifically, although some methods attempt to introduce the symmetry group theory and geometric tiling structure to enhance the editability of the pattern, they are limited by the inherent constraints of "straight edge polygon primitives", which not only severely limits the breakthrough of the pattern in the diversity of topological structure and the complexity of morphology, but also requires the design of exclusive invariant mapping algorithms for each tiling mode, resulting in a tedious, time-consuming and labor-intensive design process, which greatly increases the time cost and labor cost of the technology application. The more prominent defect is that the tiling templates used in the existing methods are mostly fixed structures, which are difficult to meet the flexible adjustment requirements. Even a few methods support tiling template editing, but their editing means are limited to simple parameter fine-tuning, which cannot adapt to complex boundary scenarios (such as irregular contours, curved boundaries, etc.), and can only be used with quasi-regular pattern models, with a narrow range of application, and cannot be compatible with other mainstream pattern generation models, further limiting the universality of the technology. In view of the core pain points of the above-mentioned traditional technology, the present application proposes an innovative technical solution: the partial differential equation technology is deeply integrated with the tiling theory to construct a new textile fabric digital art pattern generation system. The scheme breaks through the existing bottleneck through the following key technical innovations: first, the discrete spline editing method is used to realize the flexible adjustment of the shape of the tiling template, breaking through the constraints of traditional straight-edge polygon primitives; second, the geometric constraint conditions of the tiling unit are converted into the Dirichlet boundary conditions of the Laplace operator, and a unified Laplace equation suitable for any shape tiling is constructed, solving the problem of separately designing the mapping algorithm in the traditional method; third, combined with the dynamic system model and the quasi-regular pattern coloring strategy, the shape and color of the tiling unit are optimized, especially for accurately adapting to any complex boundary tiling scene, effectively covering the application requirements such as irregular contours and curved boundaries that cannot be met by traditional methods. The technical solution provides a new idea for breaking through the bottleneck of traditional textile fabric digital art pattern generation technology, and is expected to promote the development of digital pattern design in a more flexible, efficient and universal direction. SUMMARY

[0003] The purpose of the present application is to overcome the shortcomings of the prior art and provide an equal surface tiling digital art pattern generation method and device based on partial differential equations. The core innovation is to decouple the diversity of equal surface tiling structure and the flexibility of shape control, and to construct a unified invariant mapping based on partial differential equations, effectively solving the seamless splicing problem of digital art patterns for textile fabrics.

[0004] The purpose of the present application is achieved by the following technical solution: an equal surface tiling digital art pattern generation method and device based on partial differential equations, comprising the following steps: Step one, selecting an equal surface tiling template as a basic domain: selecting one from 93 equal surface tiling unit templates (polygons) as a basic domain for tiling, and obtaining the symmetry information of the tiling unit boundary edge; Step two, editing the shape of the basic domain under symmetry constraints: discretizing the edges of the basic domain, and using a curve modeling tool to edit the shape of the basic domain according to the symmetry constraints of the equal surface tiling template; Step three, discretizing the basic domain: keeping the boundary shape of the basic domain unchanged, and performing triangular subdivision on the interior of the basic domain to realize the discretization of the basic domain; Step four, constructing a smooth invariant mapping in the basic domain: under the constraints of boundary symmetry and position, constructing a Laplace equation with respect to two groups of coordinate directions, and solving the equation numerically to obtain the solution as an invariant mapping; Step five, coloring in the basic domain based on a dynamic system and a quasi-regular pattern model: according to the mapping coordinates generated by the invariant mapping, visualizing the coloring in the basic domain using a dynamic system model or a quasi-regular pattern model to generate the texture of the digital art pattern; Step six, periodically tile the basic domain to form the final digital art pattern: according to the symmetry constraints of the isometric tiling, translate, rotate or reflect the colored basic domain to obtain a translation unit composed of multiple tiling units, and periodically translate the translation unit through two linearly independent translation vectors to tile the entire drawing space to form a digital art pattern. Further, in step one, the isometric tiling unit is modeled as a N polygon, and each edge of the polygon is subdivided according to a step size to make the polygon have m vertices described by coordinate vectors: , wherein is the coordinate of the i point, and the step size can be determined according to the length of the intended curve shape (uniform sampling) or set to a fixed value.

[0005] Further, in step two, based on the symmetry constraints of the isometric tiling unit, the tiling edges can be divided into I , J , U , S four categories, which are defined as follows: S edge, characterized by 180-degree symmetry about its midpoint; U edge, characterized by symmetry about an axis passing through its midpoint and perpendicular to it; I edge: defined as a straight edge, i.e., the edge is limited to a straight shape; J edge, which can take any shape, but must correspond to an edge of the same length and shape. For any point on the edge of the tiling unit, its symmetric point is determined by the prescribed symmetry transformation T : . Thus, for all vertices on the polygon, the corresponding symmetry constraints can be constructed and expressed as the following linear equations: where the constant matrix encodes the condition for maintaining symmetry through its null space, d denotes the number of symmetry constraints.

[0006] Further, in step two, to make the shape of the isometric tiling unit more flexible and varied, the discretized boundary is interactively deformed. Specifically, a certain vertex on the edge of the tiling unit can be moved and the following Laplace energy is minimized to achieve curve editing and form the final basic domain : where, u ( t ) is the parametric curve of the tessellation element, is a positive weight similar to the area of the Voronoi region associated with vertex i . Equations for all linear constraints, including the fixed point position constraints and the symmetry constraints expressed in equation (1), are combined to allow the user to manipulate the shape as desired.

[0007] Further, in step three, an invariant mapping is defined as a function that satisfies the invariance of a specific symmetry group, i.e., for a group , the invariant mapping maps both the symmetric points p and on the plane to the same image, i.e.: The present application uses tessellation elements as the basic domain, and constructs an invariant mapping on the basic domain, i.e., maps the basic domain to another two-dimensional region. According to the definition, this mapping preserves the equivalence between any point and its symmetric point under the group , i.e.: and the tessellation edges with different symmetry types must satisfy the following condition: the invariant mapping of the boundary points and their symmetric points must be equal, i.e.: At the same time, the tessellation element is triangulated to form a triangular mesh , and the vertex set of the mesh is defined as , where is the number of mesh vertices.

[0008] Further, in step four, in order to ensure the global smoothness of the mapping and guarantee the uniqueness and non-degeneracy of the solution, additional smoothness constraints need to be imposed while fixing the positions of two non-coincident polygon vertices. Thus, the following constraint optimization problem is obtained: where , are the two fixed polygon vertices. By applying the Euler-Lagrange equation, the Laplace equation with Dirichlet boundary conditions can be further derived: ​Optimizing equation (6) can be expressed as a quadratic programming (QP) problem with linear equality constraints. By using the Lagrange multiplier method, these QP equations can be converted into a Karush-Kuhn-Tucker (KKT) system, by solving the minimum value of the Lagrange function, finally get the definition on the grid with boundary symmetry properties of invariant mapping , and the mapping coordinates of each grid vertex, then according to the barycentric coordinates interpolation triangular mesh Each pixel point in the interior.

[0009] Further, in step five, the basic domain can be colored by using a dynamical system model. The dynamical system contains rules describing how a particular quantity evolves over time. The core is to establish the mapping relationship between the state variable and time, expressed in the form of ordinary differential equations or discrete mapping; among them, the discrete mapping is represented in the following iterative form: Where, is the initial point coordinate (corresponding to equation 8), is a function used in the two iteration processes, n is the iteration number. For each pixel point in the basic domain, its mapping coordinates are converted to the normalized space through an affine transformation: Where, and are the width and height of the canvas, is the offset, is the normalized ratio. The normalized coordinates are updated through an iterative process to update the coordinates of the starting point. When certain conditions are met, the iteration process is terminated and the color is finally assigned to the point according to the number of iterations.

[0010] Further, in step five, the coloring can also be performed by using a quasi-regular pattern model. Quasi-regular pattern is a digital art pattern between regular pattern and random pattern, which is obtained by visualizing the function based on ZZSUC mapping under the condition of resonance. The following formula is the basic model of quasi-regular pattern: Where, represents the resonance number, is the coordinate value. As a common generation method of quasi-regular patterns, in practical applications, various transformations can be performed on formula (9) to make it have the ability to generate more diverse textures. Similarly, the mapping coordinates are normalized using formula (8), and the normalized coordinates are substituted into formula (9) to obtain the and according to the interval in which the value of is located, a color is assigned. Similarly, various variants of the quasi-regular pattern model can be used for visualization to generate textures.

[0011] Further, in step six, the colored paving unit is copied and composed into a translation unit according to the symmetry transformation of the isohedral type, and then the translation unit is periodically moved on the plane through two linearly independent translation vectors, and finally a complete digital art pattern is formed.

[0012] In a second aspect, the present application also provides an isohedral paving digital art pattern batch generation device based on partial differential equations, which comprises a memory and one or more processors; wherein the memory stores executable code, and when the processor executes the executable code, the above-mentioned isohedral paving digital art pattern generation method based on partial differential equations can be realized, and the batch generation operation of the digital art pattern can be completed.

[0013] The beneficial effects of the present application are as follows: (1) The technical innovation is outstanding: the present application proposes an isohedral paving digital art pattern generation method and device based on partial differential equations, which realizes real-time editing of paving shapes by moving the control points on the paving edge and combining with the curve generation tool, successfully decouples the diversity of isohedral paving and the deformation control; at the same time, based on partial differential equations, an invariant mapping generation algorithm for any isohedral paving is constructed - the algorithm first deforms the basic domain to another space, and then converts the geometric constraints of the paving unit into the Dirichlet boundary conditions of the Laplace operator, on the basis of fixing two non-overlapping polygon vertices, additional smoothness constraints are applied, which not only ensures the global smoothness of the mapping, but also guarantees the uniqueness and non-degeneracy of the solution, filling the technical gap of traditional methods in complex paving mapping design. (2) Breakthrough of traditional technology limitations: this method retains the inherent advantages of isohedral paving technology in terms of textile fabric texture continuity and computational efficiency, breaks through the limitation of traditional methods that can only use simple straight-sided polygons, significantly improves the control freedom of pattern space composition and local details, and can flexibly meet the diversified pattern design needs. (3) The generated textile fabric pattern has excellent pattern generation effect: the generated textile fabric pattern has outstanding performance in terms of boundary texture continuity, symmetry and texture diversity; and by adjusting the dynamic system function or quasi-regular pattern function and parameters thereof, diversified artistic effects with different forms can be generated, thereby providing rich visual expression possibilities for textile fabric pattern design. (4) The method has obvious contrast advantage: compared with the existing dynamic system method based on symmetric group, the method breaks through the limitation of fixed simple straight edge polygon template, gives the pattern stronger visual tension and artistic expression potential under the premise of maintaining mathematical rigor, and further expands the design boundary of digital artistic pattern. (5) The method has high universality and flexibility: the generated textile fabric pattern can adapt to any curved edge tiling structure under the symmetry constraint while maintaining mathematical rigor; and by adjusting the coloring model function and model parameters, the local details and global symmetry of the pattern can be flexibly controlled, thereby providing a new idea for dynamic generation and personalized design of textile fabric digital artistic pattern, and the method can be widely applied to textile fabric design scenes in many fields such as clothing, home and decoration. BRIEF DESCRIPTION OF DRAWINGS

[0014] Figure 1 It is an equal surface tiling digital artistic pattern generation flowchart based on partial differential equation.

[0015] Figure 2 It is a heat map of an invariant mapping result.

[0016] Figure 3 It is a coloring result schematic diagram of the invariant mapping of the tiling structure.

[0017] Figure 4 It is a part of beauty pattern schematic diagram generated by the equal surface tiling digital artistic pattern generation method based on partial differential equation.

[0018] Figure 5 It is a computer program user end visual result schematic diagram provided by the present application.

[0019] Figure 6 It is an effect diagram of the equal surface tiling digital artistic pattern generation method and device based on partial differential equation provided by the present application in clothing material application.

[0020] Figure 7 It is a structure diagram of the equal surface tiling digital artistic pattern generation device based on partial differential equation provided by the present application. DETAILED DESCRIPTION

[0021] In view of the deficiencies in the background art, the present application aims to make the "equal surface tiling" planar tiling unit become an efficient and editable primitive for digital pattern design, while taking into account artistic diversity and industrial generation efficiency.

[0022] The application discloses a method and device for generating an isometric tiling digital art pattern based on a partial differential equation. Firstly, a freely editable control point is arranged on the tiling edge of an inputted arbitrary isometric tiling unit, and the geometric constraint of isometric tiling is converted into a Dirichlet boundary condition of a Laplace operator to construct a seamless and invariant unified invariant mapping, and the basic domain is deformed to another two-dimensional region. Then, an editable dynamic system iteration model and a quasi-regular pattern model are embedded in the region, and real-time response coloring is realized by means of GPU parallel iteration. Finally, a high-resolution pattern with continuous boundary, symmetry and controllable local details is obtained through one-time generation. The method eliminates the cumbersome process of rederiving the mapping for each isometric shape, supports interactive design and parameter adjustment, and can batch output standard image textures, thereby rapidly constructing a high-quality pattern material library for the textile, building decoration and digital media industries.

[0023] The advantages of the application include solving the problems of difficulty in ensuring boundary continuity and complex mapping analysis for arbitrary isometric tiling, establishing a unified invariant mapping by converting the geometric constraint of isometric tiling into a Dirichlet boundary condition of a Laplace operator, not needing to repeatedly derive the mapping for different shapes, opening the coloring model function and iteration parameters, supporting interactive adjustment of texture complexity and color rhythm, reducing manual joint repair and repeated trial and error, and finally batch outputting standard images to rapidly construct a high-quality digital art pattern material library for the textile printing, building decoration and other industries, thereby significantly saving labor and time costs.

[0024] The application will be further described in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the application and not to limit the scope of the application.

[0025] Embodiment: The application provides a method for generating an isometric tiling digital art pattern of a textile fabric based on a partial differential equation. The specific technical process is as follows. Firstly, a unified tiling shape parameterized general model is constructed according to the deformation characteristics of isometric bodies to provide a standardized basic framework for subsequent tiling design. Secondly, the geometric constraint condition of the tiling unit is converted into a Dirichlet boundary condition of a Laplace operator, and a seamless mapping is obtained by solving the boundary condition to ensure the consistency of the connection between the tiling units. Then, an editable dynamic system model and a quasi-regular pattern model are applied to determine the textile fabric pattern generation parameters with aesthetic value and design adaptability by parameter adjustment. Finally, a textile fabric digital pattern with continuous boundary texture and rich detail performance is output in one step, which can efficiently support the construction of a large-scale textile pattern design material library and meet the batch design requirements. Figure 1A flowchart for generating a digital art pattern of an isometric tiling textile fabric based on partial differential equations in the embodiment is shown in FIG. 1. The generation process of the digital art pattern of the textile fabric includes the following six steps: Step one: select a target isometric tiling template and determine it as the basic domain for pattern generation to provide an initial carrier for subsequent shape editing and parameterized design; Step two: edit and adjust the shape of the basic domain under the preset symmetry constraint condition. The shape of the basic domain can be flexibly optimized by moving the control points on the tiling edge and combining the curve generation tool; Step three: discretize the edited basic domain to convert the continuous basic domain into discrete calculation units to provide a data basis for the construction of smooth invariant mapping in the subsequent step; Step four: construct a smooth invariant mapping in the discretized basic domain. By converting geometric constraints into Dirichlet boundary conditions of the Laplacian operator, the global smoothness of the mapping and the uniqueness of the solution are ensured; Step five: complete the pattern coloring operation in the basic domain based on the dynamic system model and the quasi-regular pattern model. The individualized design of color and texture is realized by adjusting the model parameters; Step six: perform periodic tiling processing on the colored basic domain to repeatedly arrange the single basic domain according to the preset rule and form the final digital art pattern of the textile fabric. Specifically, in one embodiment, the method for generating a digital art pattern of an isometric tiling textile fabric based on partial differential equations includes the following steps: Step one, select an isometric tiling template as the basic domain of the textile fabric pattern, specifically the 21st of the 93 isometric templates (IH21): model the isometric tiling unit as a N edge polygon, and divide each edge of the polygon according to a step size step= 0.01, evenly sample according to the length of the edge, and the polygon after subdivision has m= 230 vertices, which are described by coordinate vectors: wherein is the coordinate of the i th point.

[0026] Step two, edit the shape of the basic domain under the symmetry constraint: Based on the symmetry constraint feature of the isometric tiling unit, the tiling edge can be divided into I , J , U , S four categories, which are defined as follows: S edge, which is characterized by 180-degree symmetry about its midpoint; UAn edge is characterized by being axially symmetric about a line passing through its midpoint and perpendicular to it; I Edge: Defined as a straight edge, meaning that the edge is restricted to a straight line shape; J An edge can be of any shape, but it must correspond to an edge of equal length and the same shape. For any point on the edge of a paving unit... Its symmetrical point Through the prescribed symmetry transformation T Sure: Therefore, for all vertices of the polygon, corresponding symmetry constraints can be constructed, which can be expressed as the following system of linear equations: Wherein, constant matrix Its null space encodes the conditions for maintaining symmetry. d This indicates the number of symmetric constraints.

[0027] To make the shapes of the isoplanar tiling units of textile fabric patterns more flexible and varied, the discretized boundaries are interactively deformed. Specifically, a vertex on the edge of the tiling unit can be moved, and the curve is edited by minimizing the following Laplace energy, thus forming the final basic domain. : in, u ( t ) is the parameter curve of the uniform paving element. It is a positive weight, similar to that of a vertex. i The area of ​​the relevant Voronoi region. The equations formed by all linear constraints include fixed point position constraints and symmetry constraints represented by formula (1), which allows users to manipulate the shape of the paving units such as textile fabric patterns as needed.

[0028] Step 3: Discretize the basic domain: An invariant mapping is defined as a function that satisfies the invariance of a specific symmetric group, i.e., for the group... Invariant mapping It can connect points p and symmetric points on a plane. All are mapped to the same image, that is: This invention uses the equiplanar tiling units of textile fabric patterns as the basic domain, and constructs an invariant mapping on the basic domain. This involves mapping the basic domain to another two-dimensional region. By definition, this mapping preserves... any point in the middle with its group The symmetrical point below The equivalence between them is as follows: and the paving edges with different symmetry types must satisfy the following condition: the invariant mapping of the boundary point and its symmetric point must be equal, i.e.: Meanwhile, the paving unit of the textile pattern is triangulated to form a triangular mesh and the vertex set of the mesh is defined as where is the number of mesh vertices, denotes the n mesh vertex.

[0029] Step four, constructing a smooth invariant mapping in the fundamental domain. To ensure the global smoothness of the mapping and guarantee the uniqueness and non-degeneracy of the solution, additional smoothness constraints are imposed while fixing the position of two non-coincident polygon vertices. Thus, the following constrained optimization problem is obtained: where , are the two fixed polygon vertices, U is the aforementioned invariant mapping. By applying the Euler-Lagrange equation, the Laplace equation with Dirichlet boundary conditions can be further derived: The optimization formula (6) can be expressed as a quadratic programming (QP) problem with linear equality constraints. By using the Lagrange multiplier method, these QP formulas can be converted into a Karush-Kuhn-Tucker (KKT) system, by solving the minimum value of the Lagrange function, finally obtaining the invariant mapping defined on the mesh with boundary symmetry properties and the mapping coordinates of each mesh vertex, then the mapping coordinates of each pixel point in the triangular mesh are interpolated according to the barycentric coordinates. Figure 2 is the invariant mapping heat map of the paving unit in two dimensions.

[0030] Step five, coloring in the fundamental domain based on the dynamical system and quasi-regular pattern model: The fundamental domain of the textile pattern can be colored using a dynamical system model. The dynamical system contains rules that describe how a particular quantity evolves over time. Its core lies in establishing the mapping relationship between the state variable and time, which is represented in the form of ordinary differential equations or discrete mappings; where the discrete mapping is represented in the following iterative form: where, These are the initial point coordinates (corresponding to formula (8)). These are functions used in the two iterations. n It is the iteration number. For each pixel within the basic domain of the textile fabric pattern. Map its coordinates Transform to the normalized space through an affine transformation: in, and These are the width and height of the canvas. It's the offset. This is a normalized scale. The normalized coordinates are iteratively updated to continuously update the coordinates of the starting point. When certain conditions are met, the iteration process terminates, and finally, a color is assigned to the point based on the number of iterations.

[0031] Alternatively, a quasi-regular pattern model can be used for coloring. Quasi-regular patterns are a type of digital art pattern that falls between regular and random patterns; they are visualized based on... The function is derived by Hamiltonian transformation and smoothing of the ZZSUC mapping when the secondary resonance condition is met. The following formula is the basic model of the quasi-regular pattern: in, This indicates the resonance number (taken as 5 in this example). For the coordinate values, the mapped coordinates are also normalized using formula (8). Substituting the normalized coordinates into formula (9) yields the coordinates for each point. And according to The range of values ​​is used to assign colors. Similarly, various variants of the quasi-regular pattern model can be used for visualization to generate textile fabric textures, such as... Figure 5 As shown.

[0032] Step six: Periodically tile the basic fields to form the final digital art pattern for the textile fabric: The tiling units of the colored textile fabric are copied and combined into a translation unit based on the symmetry transformation of the isohedral type. This translation unit is then periodically moved on the plane using two linearly independent translation vectors, ultimately forming a complete digital art pattern of the textile fabric. The tiling result obtained in this example is shown below. Figure 3 .

[0033] Figure 4 This showcases some aesthetically pleasing graphics generated by a digital art pattern generation method for isosurface paving textile fabrics based on partial differential equations. These graphics can be used as design materials in the textile fabric field, and their application in clothing materials is illustrated in the following images. Figure 6As shown.

[0034] Corresponding to the foregoing embodiment of the isoperimetric tiling digital art pattern batch generation method based on partial differential equations, the present application also provides an embodiment of an isoperimetric tiling digital art pattern batch generation device based on partial differential equations.

[0035] Referring to Figure 7 , the embodiment of the present application provides an isoperimetric tiling digital art pattern generation device based on partial differential equations, comprising a memory and one or more processors, the memory stores executable code, and the processor executes the executable code to implement the high-aesthetic-appeal regular pattern batch generation method in the above embodiment.

[0036] The embodiment of the isoperimetric tiling digital art pattern generation device based on partial differential equations provided by the present application can be applied to any device with data processing capability, which can be a device or apparatus such as a computer. The device embodiment can be realized by software, or by hardware or a combination of software and hardware. Taking software realization as an example, as a logical device, it is formed by reading the corresponding computer program instructions in the non-volatile memory into the memory for execution by the processor of the device with data processing capability where it is located. From the hardware level, as shown in Figure 7 , it is a hardware structure diagram of the device with data processing capability where the isoperimetric tiling digital art pattern generation device based on partial differential equations provided by the present application is located, in addition to the processor, memory, network interface, and non-volatile memory shown in Figure 7 , the device with data processing capability where the device in the embodiment is located usually includes other hardware according to the actual functions of the device with data processing capability, and details are not described here.

[0037] The implementation process of the functions and roles of each unit in the above device is specifically described in the implementation process of the corresponding steps in the above method, and is not described here.

[0038] For the device embodiment, since it basically corresponds to the method embodiment, the relevant part can refer to the part of the method embodiment. The device embodiments described above are only schematic, and the units described as separate components can or can not be physically separated, and the components displayed as units can or can not be physical units, that is, they can be located in one place, or distributed on multiple network units. According to actual needs, part or all of the modules can be selected to achieve the purpose of the present application scheme. Those skilled in the art can understand and implement without creative labor.

[0039] The embodiment of the present application further provides a computer readable storage medium, which stores a program, and the program is executed by a processor to realize the method for batch generating isometric tiling digital art patterns based on partial differential equations.

[0040] The computer readable storage medium can be an internal storage unit of any data processing device, such as a hard disk or a memory. The computer readable storage medium can also be an external storage device of any data processing device, such as a plug-in hard disk, a smart media card (SMC), an SD card, a flash card, etc. Further, the computer readable storage medium can include both an internal storage unit and an external storage device of any data processing device. The computer readable storage medium is used to store the computer program and other programs and data required by the data processing device, and can also be used to temporarily store data that has been output or will be output.

[0041] The present application further provides a computer program product, which includes computer programs / instructions, and the computer programs / instructions are executed by a processor to realize the method for batch generating isometric tiling digital art patterns based on partial differential equations.

[0042] Although the present application has been illustrated and described with reference to certain preferred embodiments thereof, it should be understood that various changes in form and detail can be made therein without departing from the spirit and scope of the application.

Claims

1. A method for generating digital art patterns using equal-surface tiling based on partial differential equations, characterized in that, The method includes the following steps: Step 1: Select a tiling template of a digital art pattern for textile fabric as the basic domain: Select one from the polygonal tiling unit template as the basic domain for tiling the textile fabric, and obtain the symmetry information of the boundary edges of the tiling unit. Step 2, Edit the shape of the basic domain under symmetry constraints: Discretize the edges of the basic domain, and edit the shape of the basic domain using curve modeling tools based on the symmetry constraints of the equal-surface tiling template of the digital art pattern of textile fabric. Step 3: Discretize the basic domain: Keep the boundary shape of the basic domain unchanged, and perform triangulation on the interior of the basic domain to achieve the discretization of the basic domain. Step 4: Construct a smooth and invariant mapping within the basic domain: Under boundary symmetry and position constraints, construct the Laplace equation with respect to the X and Y coordinate directions, and solve the equation numerically to obtain the solution as an invariant mapping. Step 5: Coloring within the basic domain based on dynamical system or quasi-regular pattern model: Based on the mapping coordinates generated by the invariant mapping, the dynamical system model or quasi-regular pattern model is used to perform visualization coloring within the basic domain to generate the texture of the digital art pattern of the textile fabric. Step 6: Periodically tile the basic domain to form the final digital art pattern of the textile fabric: According to the symmetry constraint of equal-surface tiling, translate, rotate or reflect the colored basic domain to obtain a translation unit composed of multiple tiling units. Periodically translate the translation unit through two linearly independent translation vectors to fill the entire drawing space and form the digital art pattern of the textile fabric.

2. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step one, the tessellation unit of the digital art pattern on the textile fabric is modeled as an N-sided polygon. Each side of this polygon is subdivided according to the step size, so that the polygon has m vertices, described by coordinate vectors: ； in, It is the coordinate of the i-th point. The step size is determined by uniform sampling based on the expected curve shape and length, or it can be set to a fixed value.

3. The method for generating equal-surface tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step two, based on the symmetric constraint characteristics of the paving unit, its paving edges are divided into four categories: I, J, U, and S, specifically defined as follows: S-edge: symmetric about its midpoint after a 180-degree rotation; U-edge: symmetric about a straight line passing through its midpoint and perpendicular to it; I-edge: defined as a straight line edge, meaning the edge is restricted to a straight line shape; J-edge: exhibits any shape, but must correspond to an edge of equal length and the same shape; for any point on the edge of the paving unit... Its symmetrical point It is determined by the symmetry transformation T: Therefore, for all vertices of the polygon, a corresponding symmetry constraint can be constructed, which can be expressed as the following system of linear equations: ； Wherein, constant matrix The condition for maintaining symmetry through its null space encoding, where d represents the number of symmetry constraints.

4. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step two, the discretized boundary is interactively deformed. Specifically, a vertex on the edge of the paving unit is moved, and the curve is edited by minimizing the following Laplace energy, thus forming the final fundamental domain. : ； Where u(t) is the parameter curve of the uniform paving element, It is a positive weight, similar to the area of ​​the Voronoi region associated with vertex i. The equations, which consist of all linear constraints, include fixed-point position constraints and symmetric constraints, allowing users to manipulate the shape as needed.

5. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step three, the invariant mapping is defined as a function that satisfies the invariance of a specific symmetric group, that is, for the group... Invariant mapping It can connect points p and symmetric points on a plane. All are mapped to the same image, that is: ; Using the equiplanar tiling units of digital art patterns on textile fabrics as the basic domain, an invariant mapping is constructed on the basic domain. This maps the basic domain to another two-dimensional region; by definition, this mapping preserves... any point in the middle with its group The symmetrical point below The equivalence between them is as follows: ； Furthermore, paving edges with different symmetry types must satisfy the following conditions: boundary points and its symmetrical points The invariant mappings must be equal, that is: ； At the same time, the paving units are triangulated to form a triangular grid. and define the vertex set of the mesh. ,in This represents the number of grid vertices.

6. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step four, while fixing the positional conditions of the two non-coincident polygon vertices, an additional smoothness constraint is applied, resulting in the following constrained optimization problem: ； in, , That is, the coordinates of two fixed polygon vertices; by applying the Euler-Lagrange equations, the Laplace equation with Dirichlet boundary conditions is further derived: ； Optimizing the Laplace equations is formulated as a quadratic programming (QP) problem with linear equality constraints. Using the Lagrange multiplier method, these QP formulas are transformed into a Karush-Kuhn-Tucker (KKT) system. By solving for the minimum of the Lagrange function, an invariant mapping with boundary symmetry defined on the mesh is finally obtained. And the mapped coordinates of each grid vertex, and then interpolate the triangular mesh based on the barycenter coordinates. The mapped coordinates of each pixel within the array.

7. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step five, a dynamical system model is used to color the basic domain. The dynamical system contains rules describing how specific quantities evolve with time, establishing a mapping relationship between state variables and time, represented by a system of ordinary differential equations or discrete mappings. The discrete mapping is expressed in the following iterative form: ； in, These are the initial point coordinates. These are functions used in two iterations, where n is the iteration number; for each pixel within the basic domain... Map its coordinates Transform to the normalized space through an affine transformation: ； in, and These are the width and height of the canvas. It's the offset. The normalized ratio is used to continuously update the coordinates of the starting point through an iterative process. When a certain condition is met, the iteration process is terminated, and the point is finally assigned a color based on the number of iterations.

8. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step five, a quasi-regular pattern model is used for coloring; a quasi-regular pattern is a digital art pattern that falls between regular and random patterns, and is visualized based on... The function derived from the ZZSUC mapping under the secondary resonance condition through Hamiltonian transformation and smoothing is obtained; the following formula is the basic model of the quasi-regular pattern: ； in, Indicates the number of resonances. The coordinate values ​​are transformed to enable the generation of more diverse textile fabric textures; similarly, the mapped coordinates are normalized to obtain the height field. And according to The range of values ​​is used to assign colors; similarly, a variant of the quasi-regular pattern model is used for visualization to generate textile fabric textures.

9. The method for generating equiplanar tiling digital art patterns based on partial differential equations as described in claim 1, characterized in that, In step six, the colored paving unit is copied and combined into a translation unit according to the symmetry transformation of the iso-faceted type. Then, the translation unit is periodically moved on the plane by two linearly independent translation vectors to finally form a complete digital art pattern of textile fabric.

10. A device for generating digital art patterns by tiling surfaces based on partial differential equations, comprising a memory and one or more processors, wherein the memory stores executable code, characterized in that, When the processor executes the executable code, it implements a method for generating equiplanar tiling digital art patterns based on partial differential equations as described in any one of claims 1-9.

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