Texture mapping of warp knitted shoe upper based on ARAP parameterization method

1 Introduction

The upper of a sneaker appears as a complex 3D curved surface in space, but in the manufacturing process, it is actually made from a flat piece of fabric. When the 3D surface model of sneakers is expanded into a 2D upper graph, there are inevitable issues such as stretching and deformation, which affect the precise jacquard pattern location of the upper. To solve this problem, researchers employ 3D texture mapping technology. 3D texture mapping is widely used in face recognition, film, television advertising, model inpainting, virtual reality, and other fields. Moreover, in recent years, many constrained texture mapping methods have been proposed.

Manning [1] proposed a surface expansion method based on the isometric tree algorithm and applied it to the shoe industry. Lipman [2] used a geometry-based optimization approach. The method maintains the relative minimum deformation of angle, area, and side length during the parametric expansion of the upper.

Lévy [3] proposed a relatively mature constrained texture mapping method, Least square conformal parametric maps (LSCM). First, the feature points are specified by the user, and then these correspondences are added as constraints to the parameterized objective function, and the corresponding coordinate values are calculated. LSCM can ensure the correspondence of feature points in texture mapping. However, when the number of constraint points specified by the user is large, the validity of its mapping cannot be completely guaranteed. Wang et al. [4] proposed a bijective and low distortion method, which uses an iterative block coordinate descent algorithm based on the spherical as-rigid-as-possible (ARAP) method to further reduce the distortion and ensure the bijective parameterization. However, it still results in different degrees of flip and overlap when maintaining equidistance and area preservation. Wang et al. [5] proposed a new local/global mesh parameterization method ARAP++. ARAP++ adopts the idea of ARAP, approximates the Jacobian matrix with the fitting matrix, and introduces a stretching operator to eliminate the influence of overlap and flip, obtaining the global planarization result. However, the ARAP++ algorithm has high computational complexity and slow convergence speed; the rotation matrix and energy function need to be calculated many times in the iteration process. Sawhney and Crane [6] proposed an interactive editable bounding-first fattening algorithm (BFF). Compared with the traditional conformal parameterization method, BFF can adapt to arbitrary boundary conditions, and can directly and completely control the boundary length and angle through sparse matrix decomposition, which means that the boundary can be topologically circular. In this study, the mesh is manually edited. However, BFF can only deal with surfaces topographed to disks.

In this study, the initial parameterization of the mesh is carried out using the discrete harmonic method, and the initial parameterization data are used as input parameters for the ARAP method. High-quality parameterization results are finally obtained by utilizing the characteristics of maintaining area and shape. Based on the parametric results, the jacquard notation of the warp knitted upper is drawn to generate a planar simulation effect, and ultimately, the warp knitted shoe upper is mapped to the three-dimensional shoe upper, and the fabric structure of the shoe upper is designed and simulated. This has significant practical implications for the development of upper products.

2 Acquisition of upper model

Solid modeling is a modeling technique that defines basic voxels and generates a complex solid through set or deformation operations of these basic voxels [7]. A key characteristic is that the surface of the 3D solid is generated at the same time as the solid. Because solid modeling can define the internal structure and shape of 3D objects, it can completely describe all the geometric and topological information of the object, including the information of the body, face, edge, and vertex of the object.

First, the collection of sampling points is carried out on the drawing plane. The composition of the 2D sketch is shown in Figure 1, and the distribution of sampling points is shown in Figure 1a, and some of the sampling points are marked in red circles. Next through Blender’s geometric constraint function, the sample points and sketches can be further finely connected and constrained. A two-dimensional sketch of the bottom and sides of the shoe body is shown in Figure 1b.

Figure 1

Flowchart of solid modeling for a 3D sneaker model. (a) Sampling point collection diagram, (b) 2D sketches, (c) 3D sketches, and (d) Shoe body model.

After the 2D sketch is drawn, the 2D sketch is gradually transformed into a 3D shoe model with volume and structure by adjusting the position of the collected points, constructing new edges and surfaces between points, stretching and rotating these points, lines, and surfaces, as shown in Figure 1c. After obtaining the preliminary 3D model, it is necessary to further refine the model by editing the geometric elements. At the same time, a triangular mesh is used to express the surface morphology of the upper, so the quadrilateral mesh needs to be triangulated, and finally the shoe body model shown in Figure 1d is obtained.

After obtaining the shoe body model by using the solid modeling method on blender, the model is exported in OBJ file format, and the OBJ file mainly includes various object information of the 3D model. The key contents are as follows: object information “o,” vertex 3D coordinate information “v,” vertex mapping coordinate “vt,” vertex normal “vn,” and polygon information “f” that make up the 3D model. Read the information of the “o upper” part to get the vertex, vertex normal, patch, and other information of the upper part. This is shown in Figure 2.

Figure 2

Upper model part. (a) Upper model, (b) Upper mesh.

3 Selection of upper unfolding seam

In order to complete the design of the jacquard notation of the upper, it is necessary to unfold the 3D upper model into a 2D image and then stitch the 2D notation to form a textured 3D upper. In the conversion between 3D and 2D, the position of unfolding and stitching is the same, forming a line called the seam. A seam is a line formed by the edges of all the individual triangles along the unfolding and stitching path.

The seam should be chosen to minimize the deformation of the upper when it is unfolded, ensuring that the basic shape of the upper is not changed after cutting. Additionally, the cutting needs to meet the needs of the shoe-making process. Specifically, the two-dimensional upper should be approximately symmetrical to maximize the utilization of the cloth and avoid obvious protrusion or depression. The selection of the final cutting area is shown in Figure 3.

Figure 3

Illustration of the cutting area.

A triangulated mesh upper is a discontinuous surface model with two closed continuous boundaries, where the longer continuous boundary is defined as the sole edge and the shorter continuous boundary is defined as the collar edge. In the initialization phase before unrolling, the vertices on the sole edge and the collar edge of the shoe need to be properly connected to form a cutting path to facilitate parametric unrolling. While forming a cutting path, new boundary points and boundary edges will be generated. However, the choice of suture line can affect the structure of the upper, so when determining the cutting area, it is important to connect the sole edge and the collar edge boundary points to obtain symmetrical results. The inner and outer boundary points of the cutting area, as shown in Figure 4, are obtained, and then the sole edge and the short toe edge, identified by the search method of the middle boundary edge, are connected with the inner and outer boundary points close to the approximate symmetry axis to form the cutting line of the 3D upper model to be developed.

Figure 4

Schematic diagram of shoe vector calculation.

First, rotation and position adjustment operations are performed on the obtained 3D upper model so that the sole is parallel to the XOY plane. This requires first determining the current three-dimensional coordinate axis of the upper.

The center of gravity of the sole edge serves as a point on the plane, and the average normal of all the faces on the bottom edge is the normal of the plane, and the unit normal is the local Z-axis direction of the three-dimensional upper. The unit vector from root to toe box serves as the local Y-axis direction of the three-dimensional upper.

Map the entire upper vertically to the two-dimensional plane, obtain the coordinates of all vertices on the edge of the sole in the two-dimensional plane, and record the first vertex at any beginning as P ₁, the adjacent second vertex as P ₂, and so on. The last vertex is recorded as P _n, and P ₁ and P _n are adjacent vertices, as shown in Figure 4a. The distance of each vertex from the other vertices is then calculated in turn, and the two vertices with the largest distance are found, denoted as P _max and P _min. Using P _max and P _min, we get a vector along the local Y-axis of the three-dimensional upper, denoted Y _shoe. Take the three equal points of the sole edge P _i , P i + n 3 , P i + 2 n 3 , the three points form a plane α _i , as shown in Figure 4b, to obtain several planes. Each plane normal vector can be calculated by the cross-product formula (1) of the vector, and then its average normal vector is the normal vector of the upper.

where N and L are two vectors determined by the three bisection vertices of the plane α _i , N = P i P i + n 3 and L = P i + n 3 P i + 2 n 3 . Z _shoe is a plane normal vector.

Finally, we bring the above unitized vectors Y _shoe and Z _shoe into formula (1) to obtain the unit vector X _shoe. The direction of the unit vectors X _shoe, Y _shoe, and Z _shoe is the direction of the local space coordinates of the upper X, Y, and Z axes, and the local space coordinates of the upper are defined as X _local = X _shoe, Y _local = Y _shoe, and Z _local = Z _shoe. Let X _space, Y _space, and Z _space be the axis vectors of the space coordinate system. There is a transformation matrix R, so that the local space coordinates and space coordinates of the shoe surface can be converted to each other, i.e., using the following formula:

Through equation (2), the transformation matrix R is obtained, where R _ij represents the rotation coefficient and M _i represents the movement coefficient.

All vertices are left multiplied by the transformation matrix R to obtain the transformed upper. Figure 5 shows the 3D upper diagram before and after position adjustment, and the viewing angle is the Z-axis direction. Figure 5a is the 3D upper before adjustment and Figure 5b is the 3D upper after adjustment.

Figure 5

Schematic diagram of the three-dimensional upper before and after adjustment. (a) Before adjustment. (b) After adjustment.

The search method for the boundary edge is as follows:

1. Iterate over all edges of the triangle mesh upper, starting from the first edge. If the edge belongs to only one triangle, it can be determined that the edge is the boundary edge of the surface, and the vertex information of the edge is stored in the edge boundary container. Continue traversing the next edge.

2. Iterate over all edges in the edge boundary container, starting from the first edge, find the edges adjacent to this edge, i.e., the edges with the same vertex, and so on. The last edge is adjacent to or identical to the first edge, and these edges form a boundary edge, and the other edges are the other boundary edges.

3. Calculate and compare the length of the two boundary edges. The longer continuous boundary is the edge of the sole, which is called the sole edge. And the shorter continuous boundary is the edge of the shoe, which is called the collar edge.

After cutting, as shown in Figure 6, Figure 6a is the schematic of the upper model after cutting, and Figure 6b is the schematic of the upper mesh after cutting.

Figure 6

Cutting area. (a) Schematic diagram of the cutting model area. (b) Schematic diagram of the cutting grid area.

4 Application of ARAP parameterization method

The ARAP parameterization method uses the local/global idea to perform a subset-orthogonal transformation of two-dimensional spatial affine on each triangle, and the shape change in the global solution parameterization is affected by local constraints, and the conformal parameterization results are obtained [8].

4.2 Discrete harmonic parameterization

ARAP is designed to keep 3D models as rigid as possible, which means that it is suitable for applications where the rigidity of objects needs to be maintained, such as angle and length, such as bone animation of models and object deformation. Because ARAP is able to maintain the natural appearance of objects, it is suitable for applications that need to maintain the authenticity of objects, such as human deformations or facial animations, to make models look more realistic. Compared with other deformation techniques, ARAP is easy to implement, and it usually uses local coordinate system for deformation, which reduces the complexity of the algorithm. However, ARAP is a local deformation method, so there may be limitations when dealing with global deformation. There are some limitations in its performance in the case of large-scale deformation or global deformation.

Therefore, before using ARAP expansion, an initial expansion is performed using discrete harmonic parameterization, and the result of discrete harmonic parameterization is taken as the input to ARAP.

The core principle of discrete harmonic parameterization is based on discrete harmonic functions and Laplace operators. Specifically, when it parameterizes a three-dimensional grid to a two-dimensional plane, the goal is to minimize deformation such that the two-dimensional coordinates of each vertex are the harmonic average of the coordinates of its neighbors [14]. This can be done by solving a set of linear equations. The details are as follows.

4.2.1 Discrete harmonic function

In discrete geometry, a harmonic function is a function that satisfies the discrete Laplace equation in a grid. For a given three-dimensional mesh M, the goal of discrete harmonic parameterization is to find the two-dimensional coordinates (u _i , v _i ) of each vertex such that these coordinates satisfy the discrete harmonic condition.

4.2.2 Discrete Laplacian operator

The discrete Laplacian operator is used to define the relationship between a vertex and its neighbors. In the discrete case, the Laplacian operator can be represented by the weights of the vertices and edges of the mesh. For a vertex i and its set of neighbor vertices N(i), the discrete Laplacian is defined as

(5) ∇ f ( i ) = ∑ j ∈ N ( i ) w i j ( f ( j ) − f ( i ) ) ,

where f is a function defined on the vertices of the mesh (the 2D coordinates of the vertices u or v), w _ij is the weight, and different weights can be selected according to different methods, here the weight of the edge (i, j) is taken, and w i j = 1 2 ( cot α i j + cot β i j ) , where α _ij and β _ij are the two opposite angles of the edge (i, j). From the vector inner product formula, we can calculate cot.

(6a) cot α i j = v k v i ∥ v k × v i ∥ ,

(6b) cot β i j = v i v j ∥ v i × v j ∥ ,

where v _i , v _{j ,} and v _k are the position vectors of vertices i, j, and k, respectively.

The core idea of harmonic mapping is to keep the mapping as smooth as possible by minimizing some kind of energy function. Specifically, for each internal vertex i, we want its two-dimensional coordinates (u _i , v _i ) to satisfy the discrete harmonic equations:

(7a) ▽ u i = 0 ,

(7b) ▽ v i = 0 .

Expanding the Laplacian operator, these equations can be written as

(8a) ∑ j ∈ N ( i ) w i j ( u j − u i ) ,

(8b) ∑ j ∈ N ( i ) w i j ( v j − v i ) .

This means that for each internal vertex i, its u and v coordinates are the weighted average of the coordinates of its neighboring vertices.

This is equivalent to constructing a system of linear equations: Lu = 0 and Lv = 0, where L is a Laplace matrix constructed to solve the above harmonic equations, for a mesh with n vertices, the Laplace matrix is a sparse matrix n × n with elements L _ij being − ∑ k ∈ N ( i ) w i k if i = j w i j if i and  j 0 otherwise are adjacent u and v are the u and v coordinate vectors for all vertices, respectively.

4.2.3 Boundary conditions

To ensure the uniqueness and stability of parameterization, it is necessary to apply constraints on the boundary vertices. Typically, boundary vertices are pinned to a predetermined shape, such as a unit circle or a square, on a two-dimensional plane. These boundary conditions define the u and v coordinates of the fixed boundary vertices, ensuring the uniqueness and stability of the parameterization. Assuming there are m boundary vertices, the boundary condition can be written as u i = cos 2 π i m and v i = sin 2 π i m , where i is the index of the boundary vertices.

4.2.4 Solving linear equations

By combining the discrete harmonic equations of the inner points and the boundary conditions, a system of linear equations can be obtained.

(9a) L u = b u ,

(9b) L v = b v ,

where the matrix L is the Laplace matrix, which contains the weight information of all vertices, and u and v are the coordinate vectors of u and v of all vertices. b _u and b _v are constraint vectors that contain boundary conditions [15].

This system of equations can be solved by conventional linear algebraic methods. LU factorization is used here, and LU factorization is commonly used to solve linear systems, find inverse matrices, or compute determinants [16]. For example, when calculating determinants, for an Angle matrix, the value of the determinant is the product of the elements on the diagonal. So, if you triangulate the matrix, then it becomes very easy to find the determinant. The resulting discretely unfolded upper mesh is shown in Figure 7.

Figure 7

Discrete harmonic expansion.

4.3 Parametric results and simulations

Figure 8 shows the parameterization results under different conditions, Figure 8a and c shows the parameterization results of 200 and 300 ARAP iterations, and Figure 8b and d shows the parameterization results of 200 iterations and 300 iterations after discrete optimization. In contrast, it can be seen that the expansion effect after discrete modulation parameterization is better than that before discrete modulation parameterization under the same number of iterations, and the discrete modulation parameterization can effectively reduce the number of iterations required for 3D mesh expansion. During the experiment, it was found that when the number of iterations reached 600, the difference in the number of iterations was 100 times, and the change in energy difference was less than 0.00003. In view of this, after comprehensive consideration, the result after 600 parametric iterations was selected as the final expansion result. Figure 9 shows the final parameterization results.

Figure 8

Parameterization effect diagram under different condition. (a) 200 iterations, (b) discrete iterations 200 times, (c) 300 iterations, and (d) discrete iterations 300 times.

Figure 9

The final parameterization result.

Tables 1–3 show a comparison of the triangle area, side length, and inner angle of the triangle of the triangle mesh of the original upper model after parametric expansion, and it can be seen from the table that the change difference of the inner angle, area, and side length of the triangle is less than 1%, accounting for 83.95, 70.97, and 84.03% of the total, respectively. The percentages of less than 5% of the total number reached 99.15, 97.23, and 98.68%, respectively. Moreover, the maximum deformation difference is less than 15%, which shows that the deformation and stretch of the triangular mesh of the upper after unfolding are small, and the expansion effect is better, which is conducive to the subsequent research on the design of the upper.

Table 1

Comparison of front and back side lengths before and after unfolding

Degree of change (%)	Quantity	Percentage
<1%	3,000	84.03
1% ≪ 5%	523	14.65
5% ≪ 10%	30	0.84
10% ≪ 15%	17	0.48
>15%	0	0

Table 2

Comparison of the inside angles before and after unfolding

Degree of change (%)	Quantity	Percentage
<1%	5,813	83.95
1% ≪ 5%	1,052	15.19
5% ≪ 10%	50	0.72
10% ≪ 15%	9	0.13
>15%	0	0

Table 3

Comparison of areas before and after unfolding

Degree of change (%)	Quantity	Percentage
<1%	1,638	70.97
1% ≪ 5%	606	26.26
5% ≪ 10%	51	2.21
10% ≪ 15%	13	0.56
>15%	0	0

In order to generate a weave upper jacquard notation based on the parametric results, and to map the 2D Jacquard simulation texture to the 3D model, the parameterized results of the shoe upper need to be normalized.

First, you need to determine the scope of normalization, i.e., to limit the parameterized results to a certain range, which is set to [0, 1]. Then, traverse the UV parameterization coordinates to find the minimum values U _min, V _min and maximum values U _max, V _max in the upper parameterization data. Determine the size of U _max – U _min and V _max – V _min, if U _max – U _min > V _max – V _min, for each parameter P, normalization is performed using the following formula:

(10a) U ′ = U − U min U max − U min ,

(10b) V ′ = V − V min U max − U min ,

where U′ and V′ are normalized parameters.

Figure 10 is a schematic representation of the normalized and texture-mapped coordinate system.

Figure 10

Schematic representation of the normalized and textured mapping coordinate system.

Through normalization, the range of UV parameterized coordinates can be normalized to [0, 1], so that the range of UV coordinates corresponds to operations such as texture mapping, which is convenient for subsequent operations such as texture mapping and coloring. And this will ensure that the UV coordinates do not occur outside the reasonable range.

The final parametric results are imported to CAD for the design of warp knitted jacquard fabrics, the appropriate flower width and height were set, and the model was selected as the RDJP_7/1 of 7 combs and the gauge was 24. The vertical and horizontal densities are 16 and 14, respectively. After setting the basic parameters such as the number and model, it is also necessary to design the process parameters of the upper fabric, and the following are the example process parameters:

(1) Cushion digital:

GB1: Not used

GB2: Not used

GB3: 3-1-1-1/0-2-2-2//

GB4: Not used

JB5.1: 1-0-1-0/1-2-0-1//

JB5.2: 1-0-1-0/1-2-0-1//

GB6: 4-4-4-3/4-4-4-3/4-4-4-3/1-1-1-2/1-1-1-0/2-2-2-3//

GB7: 0-0-0-1/0-0-0-1/0-0-0-1/3-3-3-2/3-3-3-4/2-2-2-1//

(2) Drawing-in: each comb is full of wear, of which JB5.1 wears ordinary polyester and JB5.2 wears cationic polyester. A diagram of the craftsman as shown in Figure 11(a) is obtained.

Figure 11

(a) Jacquard notation and (b) simulation notation.

Specify the number and drawing-in of the carding pad yarn, the raw material information, and calculate the coil hierarchy and key points of the two-needle bed fabric, then describe the deformation of the coil by the spring particle model, then describe the coil path using the spline method, and finally use the sweeping method to generate the coil body, as described by Zhang [17]. From this, a simulation diagram is drawn according to the Yijiang diagram, as shown in Figure 11(b).

The simulation drawing is mapped to the 3D vamp, and the final effect is shown in Figure 12, where Figure 12a is a schematic representation of the side effect of the sneaker model shoe simulation and Figure 12b is a schematic representation of the top-down effect of the sneaker model simulation, and the left side is a local enlarged schematic representation. The details of the model upper are clearly shown in the figure, and after mapping to the three-dimensional upper, the position of the pattern in the simulation drawing is accurate and the deformation is small, which proves the effectiveness of the method.

Figure 12

Simulation effect. (a) Side view and (b) top view.

5 Conclusion

In this study, the ARAP parameterization algorithm is adopted to achieve texture mapping with minimal deformation. Meanwhile, the discrete harmonic initialization results serve as the input for ARAP parameterization, which reduces the number of iterations, improves the accuracy of texture mapping, and holds significant practical value for the manufacture of warp knitted uppers. This approach can enhance production efficiency and product quality, and reduce production costs. Constrained texture mapping involves significant interaction; insufficient iterations can affect the texture mapping quality, making this method less suitable for real-time interactive adjustments. Future research could explore more accurate and efficient methods to minimize iterations and enhance the precision and efficiency of texture mapping. Additionally, optimization algorithms for specific sneaker types could be developed and refined to meet diverse shoe upper design requirements, thereby advancing the field.