Research on the Application of Computational Geometry Technology in the Design of Complex Garment Structures

NURBS surfaces are non-uniform rational B-spline surfaces. In current CAD/CAM systems, B-spline curves and surfaces have become a core part of geometric modeling, and among the B-spline methods, NURBS is the most general one, which covers both polynomial and non-uniform and rational B-spline methods. The NURBS method has the following characteristics: 1)

A uniform expression can be used to accurately represent both standard analytic forms (e.g., conic lines, surfaces of rotation, etc.) and free curves and surfaces.

In the system, once the surface data points are given, NURBS surfaces can be constructed to form human body surfaces and 3D clothing piece surfaces.

NURBS surfaces are defined as follows: 1P(u,w)=∑i=0m∑i=0nBi,k(u)Bi,j(w)Wi,jVi,j∑i=0m∑j=0nBi,k(u)Bi,j(w)Wi,j

where V_t,I is the control vertex, W_I,j is the weighting factor, and B_i,k(u) and B_j,J(w) are the k-times along the u-way and l-times along the w-way B-spline basis functions, respectively. The node vectors in the u- and w-way directions are, respectively: 2U=[ 0=u0=u1=⋯=uk,uk+1,⋯,ur−k−1,ur−k=ur−k+1=⋯=ur=l] 3W=[ 0=w0=w1=⋯=wl,wl+1,⋯,ws−l−1,ws−l=ws−l+1=⋯=ws=l]

In the above equation, the number of nodes of the node vectors along the u-way and w-way directions are (r + 1) and (s + 1), respectively. Where r = n + k + 1, s = m + l + 1.

The chi-square coordinates of this surface model are denoted as: 4P(u,w)=∑i=0m∑j=0nBi,m(u)Bj,n(w)Vi,j

Here P(u, w) is a surface in four dimensions, which can be mapped to produce a rational surface. The only difference between a non-rational B-spline and its positive NURBS is the number of coordinates. The following account is based on chi-square coordinates.

The definition of the B-spline basis functions and their recursive formulas are given in terms of B_i,k(u).

B spline can be expressed as: 5Bi,0(u)={ 1, t,≤u≤ti+1 0, other 6Bi,k(u)=x−titi+k−tiBi,k−1(u)+ti+k+1−xti+k+1−ti+1Bi+1,k−1(u),k>0

Convention: 0/0 = 0. Where k denotes the power of the B-spline, t is the node, and subscript i is the serial number of the B-spline.

OpenGL is the best badlands for the development of concise and interactive 2D and 3D graphics applications.OpenGL makes its applications more innovative and dramatically accelerates the development of graphics applications by integrating a large number of rendering, texture mapping, special effects and other powerful visualization functions. In this paper, we call the OpenGL graphics library through VC++ to dust into the desired surface. Programming OpenGL in VC++ using MFC necessitates some setup, which is mainly done in the View class. The basic steps for constructing a NURBS surface are as follows: 1)

If you do light processing, first generate the normal vector of the surface.

A NURBS surface is applied to construct a three-dimensional clothes piece, and the necessary parameters of the surface are: the order of the surface, the control points, and the node vectors. For clothing pieces, type-valued points are necessary to obtain a 3D surface. Once the mannequin has been generated, the information about the points on the mannequin is known. At this point, the corresponding points of the clothing piece and the mannequin are connected. Based on the information provided by the piece of clothing, the corresponding data of the mannequin is added to a certain amount of wealth, and the type value points of the surface of the piece of clothing are obtained.

How to find the control points from the type value points? The theory of NURBS shows that the type value points of curves and surfaces have the following relationships with the control points.

Take NURBS curve as an example, the mathematical form of a general NURBS curve is as follows: 7Pi(u)=∑j=0kBi,k(u)Vi+i

where P_i(u) denotes the ind NURBS curve segment, u is a parameter variable that takes a value between 0 and 1. B_i,k(u) is the k-times B-spline basis function, and the subscript j is the serial number of the B-spline. V_i+j is the control point vector.

Take three times curve as an example, that is, k = 3, substituting into formula (7), it can be seen that at this time a section of curve segment is affected by 4 control points. Make u = 0, then it can be obtained: 8Pi(0)=aVi+bVi+1+cVi+2+dVi+3

where a, b, c, d is the value obtained by substituting u = 0 into the corresponding B-spline basis function. P_i(0) = P_i can be set. P_i indicates that the point is the starting point of this NURBS curve segment.

A complete NURBS curve can be obtained from the above equation with the following relation: 9P=AV

where A is the corresponding relationship matrix, V is the control point vector, and P is the type-valued point vector. By solving this system of equations, the group of control points V can be found. A The matrix is related to the set of node vectors. The algorithm requires that the two endpoints of the desired curve pass through the first and last points of the known type-value points.

Clothesplicing, the problem of splicing surfaces, can be solved by taking the same type points for both surfaces. Since the interpolation algorithm ensures that the boundary of the resulting surfaces is strictly over the known type points, the two surfaces with the same boundary type points must be spliced together at the boundary. For example, if the shoulder lines of the front and back pieces are at the same location, and the corresponding type points are the same, then the two surfaces will be joined at the shoulder lines. The surface interpolation algorithm is further improved to achieve second-order continuity so that the smoothness at the intersection of the surfaces is improved.

A regular curve with a deployment function is called a free curve [23], then every point on the free curve is a linear combination of type-valued points, and the combination coefficient is also the function value of the deployment function, so if the linear combination of type-valued points is changed, the free curve can be deformed.

Let P_i(i = 0, 1,⋯, m) be a type-valued point, t be a covariate, and the deployment function corresponding to each type-valued point P_i be f_i(t), which gives a free curve expression of the following form: 10P(t)=∑i=0mPifi

In the above equation, each type-valued point P_i is assigned a random weight coefficient w_i(t), which randomly changes the linear combination relationship of type-valued points P_i, and randomly discretizes the free curve. As a result, the expression for the random discretization of the free curve is given: 11R(t)=∑i=0mPifi(t)wi(t)

In w_i(t), the covariate t is not the independent variable of the random weight coefficients, but only at a certain value of the covariate t, the direction of the random discretization of the free curve can be determined as needed to transform the free curve into a random curve, e.g., combine Eq. (10) and Eq. (11) to generate the random curve RC(P_x(t), R_y(t)).

A single-layer Adaline network [24] was applied to control the values of the randomized weight coefficients w_x(t). In setting up the Adaline network, the function value of the provisioning function f_i(t) is used as the Adaline neuron input times number vector, the random weight coefficients w_i(t) are used as the network connection weight vector W(t), Y_k(t) is the actual output simulation, Y_r(t) is the ideal response value, and the gradient descent method is used to adjust the value of the neural network connection weight coefficients w_i(t) between the values of the network, and after adaptive learning by the Adaline network, a set of random weight coefficients is output w_i(t) In the network learning, the broad value is taken as zero. Then the neural network unit input and output relationship equation is: 12Yi(t)=∑i=0∞fi(t)wi(t)

The steps of the algorithm of its body are as follows: 1)

Set the initial value of network connection right w_i(t) according to certain rules;

Let k be the number of selected generations, then the vector expression for the adjustment of weights is: 13wk+1(t)=wk(t)+adk(t)fk(t)|fk(t)|2,0<a<1 14dk(t)=Yr(t)−Yk(t)

Where, W_k + 1(t) is the value of the next weight vector, W_k(t) is the current weight vector, f_a(t) is the current input vector, d_k(t) is the current error value, a is the learning step, |f_ik(t)| is the mode of the neuron input vector.

The stochastic weight coefficient w_(t) is pu closely related to the waveform ups and downs and the magnitude of the fluctuations of the stochastic curve, and it also determines the nature of the stochastic curve.

The initial value W₀(t) of the Adaline network connection weight vector W(t) and the ideal response value Y_s(t) are set according to the following equation: 15W0(t)=A+BX(t) 16Ye(t)=c+dX(t)

where X(t) is a random quantity, A(a_i), B(b_i) is a vector of coefficients, c, d is a constant, and the values of b_i and d determine the randomness of W₀(t) and Y_e(t), respectively.

1)

Light modeling

When light strikes the surface of an object, the light may be absorbed, reflected and transmitted. The part that is absorbed by the object is converted into heat. The reflected and transmitted light enters the human visual system, enabling us to see the object. To simulate this phenomenon, we build some mathematical models to replace complex physical models. These models are called light and dark effect models or lighting models. The realization of high realism of 3D objects is largely determined by the light and dark treatment. When modeling an object, it is important to take into account reflections of light onto the surface of the object, among other things. The most commonly used lighting model is the Phong model: 17I=KαIα+KdIlcosθ+KdIl(cosα)n

1 is the value of lightness and darkness, K_α, K_d is the diffuse reflection constant, which is related to the nature of the object surface, I_α is the incident flood light intensity, which is related to the lightness and darkness of the environment, I_l is the light intensity of the light source, θ is the angle of incidence, K_d is the specular reflection coefficient of the surface of the object, α is the angle of the line of sight to the direction of reflection, and n is the specular highlight index. 2)

Light Source Creation

OpenGL decomposes natural light into three RGB components when simulating light sources. For each light, it can be represented by the ratio of red, green and blue light in it. For the description of the material, it can be represented by the intensity of light reflection. Ambient light and scattered light pass through a material and produce different colors depending on the material’s reflectivity. In general, ambient and diffuse light are the same for objects of the same material, so white light is observed when specular reflection occurs on the material.OpenGL divides the lighting model into the following four categories: specular and diffuse light, and scattered light and ambient light. The computation of these four types of light is done separately and overlaid. Before creating a light source, we first have to place the objects, i.e. Define the direction of the objects in relation to the light. First, we make normal vectors for the vertices of each object. The direction of the normal vectors determines which side of the object will be facing the light and which side will be backlit. Next, we create the desired light source and set its position. To create a light source use the function: 18void glLightfv(GLenum light, GLenum pname, TYPE param)

The equation can be expanded to ax + by + cz + d = 0, where: 20a=| y1 z1 1 y2 z2 1 y3 z3 1|=0b=| z1 x1 1 z2 x2 1 z3 z3 1|=0c=| x1 y1 1 x2 y2 1 x3 y3 1|=0

(a, b, c) is unitized to the normal vector of the triangle.

A texture map is a 2D map, the color value of no point in the text on the 2D map needs to be mapped to the polygon and then modulated with the color of the polygon itself to get the final color, i.e., texture mapping, which is the process of assigning a 2D texture or data to a 3D object. The polygon mesh model needs to be textured to become the final object. By utilizing texture mapping, the image’s realism can be improved without increasing the model’s complexity.

The OpenGL texture setup process is shown in Figure 2. In OpenGL, there exists a texture memory, which we may set up in the middle of the video memory of the graphics card for the purpose of hardware acceleration. Since some graphics cards support texture compression technology, the texture memory image has an internal specific format, in the application program, when we store the texture pixels into the texture memory, it is necessary to convert the format, so that it can be compatible with that specific format.

Figure 2.

Texture setting process

After extracting the human body feature points and curve fitting, the human body height (length) data and width data are calculated similarly as follows: 1)

Height is the straight-line distance between the point (X_head, Y_head, Z_head) at the top of the head and the point (X_fωιt, Y_fωι, Z_fωι) at the bottom of the feet, then the formula is: 21Height=(Xhead−Xfoot)2+(Yhead−Yfoot)2+(Zhead−Zfoot)2

Then the formula for calculation is: 22Larm=(Xarm−Xwrist)2+(Yarm−Ywrist)2+(Zarm−Zwrist)2

Since human height, arm length, leg length, shoulder width, chest width, waist width and other dimensional data are calculated using the formula for calculating the straight line distance between two points in space. 3)

The human body circumference data is difficult to calculate in three-dimensional space and the error is relatively high, so in this paper, before calculating the circumference, the fitted curves and the corresponding point cloud coordinates of the Z-axis coordinate data are not saved, and the point cloud to be fitted has only two-dimensional coordinate data, which is equivalent to the fitting of the curves under the right-angle coordinate system. At the same time, the fitted curves are split under the two-dimensional right-angled coordinate system with the coordinates of the feature points as the splitting points, and then the length of each segment of the curves is solved by integrating the lengths of each segment, and the upper and lower limits of the integrals are the x-coordinates of the feature points, respectively. Finally, the length of each segment of the curve is totalized to get the total length of the curve, i.e. the circumference data of the human body. Taking the fitting curve of waist circumference as an example, the formula equation of human body circumference is as follows: 23L=∫ab1+(f′(x))2dx

where y = f(x) is the equation of the fitted curve, a, b is the x coordinate data of the curve splitting point, and b > a > 0.

The schematic diagram of the waist circumference fitting curve splitting and summation is shown in Fig. 3. The results show that, with the left waist point and the right waist point as the splitting point, the waist circumference fitting curve is split into the upper waist line and the lower waist line, and the corresponding fitting equations are y = f₁(x) and y = f₂(x), respectively, and Eq. (23) can be used to obtain the lengths of the upper waist line and the lower waist line l₁ and l₂, respectively, and thus the total length of the waist circumference is L = l₁ + l₂. Similarly, the circumference data of the other parts of the human body can be measured.

Figure 3.

Split and sum diagram of waist fitting curve

In order to verify the feasibility of the 3D human body measurement method proposed in this paper and the accuracy of the measurement data, a total of 10 people were selected as models for the experiments, and Kinect was used to collect the point cloud data, and after the point cloud preprocessing, point cloud alignment and surface reconstruction, the 3D human body model obtained was used as the experimental material, and the height, arm length, leg length, neck circumference, chest circumference, waist circumference, hip circumference, shoulder width, chest width, waist width, hip width and other dimensional data of the human model were measured with the human body dimensional measurement method proposed herein. The measurement data of height, arm length, leg length, neck circumference, chest circumference, waist circumference, hip circumference, shoulder width, chest width, waist width, hip width and other dimensions of the mannequin are measured, and the average of three actual manual measurements is taken as the standard value. In order to further verify the accuracy and error of the measurement data in this paper, the experiment was carried out for a total of 3 measurements of human body dimensions, the average value was taken and compared with the standard value, the average error was calculated and the results were analyzed. The results of the experiment are as follows:

Since the accuracy of the leather ruler used for manual measurement is in millimeters, the data of the standard value can only be retained as one decimal. The height measurement results and errors are shown in Table 1. The results can be obtained, the largest value in the average error is -0.24cm, the reason for the large error in this group of data may be that the hair of the model No. 8 is more fluffy, and the hair was not depressed when the height measurement of the model was performed using Kinect, while in the manual measurement, the hair of the model was depressed, so it caused the error in the measurement results to be larger than that of the other 9 groups of data. Therefore, the experimental results of the body size measurement method used in this paper were as expected when measuring height.

The results of the bust measurements and the errors are shown in Table 2. It can be found that the mean value of the three measurements is greater than the standard value, and the largest error is the data of model No. 7, which is 2.19 cm, and the smallest error is the measurement data of model No. 6, which is 0.08 cm. Therefore, the average error of the 10 sets of data is 0.59 cm, which is in line with the expected range of error for clothing use.

The results of shoulder width measurements and errors are shown in Table 3. The data can be obtained, the average value of the measurement has 8 groups of data is greater than the standard value, there are 2 groups of data is less than the standard value, and the largest average error is 2.02cm, the smallest average error is -0.01cm, so the average error of 10 groups of data is 0.52cm, which is in line with the experimental expectations. By analyzing the above data, the main factors that caused the large error in this group are (1) The clothes worn by the model are thicker and also looser. (2) When using the least squares method to fit the chest curve, it has a large error because of the influence of noise. (3) When using the computer to select the human body feature points, there may be deviation from the feature points selected during manual measurement. (4) When Kinect is calibrated by the camera, the error that occurs in the internal and external parameters is difficult to avoid, thus leading to a certain error in the process of Kinect’s point cloud alignment. From the above experimental results and error analysis, it can be seen that the results of the three-dimensional human body measurement method proposed in this paper in the measurement of human body size data, although there is a certain degree of error with the data measured manually, but the error is within the acceptable range of size measurement error for clothing, in line with the expected results.

The user wears the sample garment and evaluates the design samples from the perception perspective of lightness, elasticity, softness, stiffness, stuffiness, dryness, wetness, and warmth during static wearing experiences. The static comfort evaluation grading results are shown in Table 4.

The static comfort evaluation results of the 10 testers are shown in Figure 4. By evaluating each of the perceptions during static wearing, the users’ overall evaluation means for the optimized design samples were 2.3, 4.3, 2.1, 1.7, 1.9, 4.5, and 4.5 points, respectively, which corresponded to the results of lighter, looser, softer, less stuffy, drier, warmer, and aesthetically pleasing, and therefore a higher overall satisfaction level.

Figure 4.

The static comfort of the 10 testers was evaluated

The exercise comfort evaluation refers to the simulation of certain mountaineering exercises performed by the subjects under the condition of wearing the sample garment in the laboratory. From the perspective of sports comfort of the garment, the A: shoulder, B:elbow, C:chest, D:armpit, E:back width of the design samples, as well as F:sleeve length and G:garment length were tested and subjectively evaluated, and the grading of dynamic comfort evaluation is shown in Table 5.

The results of the dynamic comfort evaluation of the 10 testers are shown in Figure 5. As can be seen from the figure, the overall wearing experience of the design sample sports comfort is good, and the evaluation of sports comfort is high, which basically meets the user’s demand for looseness in the actual sports process. From this, it can be verified that the extended shape derived from the reverse of the actual garment and then put into production is feasible and can meet the needs of the dress.

Figure 5.

The dynamic comfort of 10 testers was evaluated