1. INTRODUCTION

National patterns design has an element of fuzziness and uncertainty to them that is related to subjective psychology and image perception [1]. Traditional pattern design is subject to the functional and aesthetic demands of manual design, resulting in low design efficiency and strong subjectivity. However, in recent decades, with the development of computer graphics, machine learning, and the combination of digital and graphic design, computer-aided design (CAD) technology has been able to help designers meet their design requirements [2,3]. Shape grammar, with its grammar rule base, recursive operation mechanism, and intuitive visualization, can explain design well through formal description. In addition, it can creatively support designers to better understand and perceive design and to promote the development of CAD tools, making design more scientific, reliable, and universal [4–6].

In the literature, research on combining shape grammar with other optimization algorithms focuses on combining shape grammar with parallel particle swarm optimization (PSO), which was proposed by Keles [7], in order to solve the problem of embedding and replacing the emergent part after the design is generated with shape grammar. Chau et al. [8] proposed a method that combines shape grammar with lattice theory and uses lattice-based rational quadratic Bézier curves to represent three-dimensional circular arc shape grammar, which solves the complexity of geometric operations related to specific syntactic shapes in shape computation. Jia and Ming [9] proposed the dynamic shape representation based on shape grammar and extent the theory of shape grammar and further enriches the form of grammar rules. To improve traditional shape grammar, Abdelbar and Rachid [10] proposed a parametric shape grammar method to generate a template motif. Jia and Ming [11] proposed hierarchical shape grammar and used descriptive language concepts to apply shape grammar to the creative design of Zhuang embroidery patterns. Zahra et al. [12] proposed the parametric shape grammar design. Although traditional shape grammar has great advantages in design evolution, for the art of graphic design, the degree of abstraction of traditional shape grammar is low. Therefore, for the innovative design of national patterns, the traditional and single use of shape grammar cannot adequately meet the needs of pattern reuse and rapid design.

Therefore, based on the above analysis and considering the problem of low abstraction and reuse of national pattern design, the main contributions of this study are as follows:

• An improved shape grammar of comprehensive predicate grammar is developed to represent the generation process of national pattern redesign.

• Taking the Miao batik pattern as the application carrier, the number of pattern elements in the pattern is taken as the optimization basis for the first time, the mathematical model is established, and the pattern arrangement parameters are optimized by using the PSO algorithm, so as to obtain the optimized predicate shape grammar parameters and realize the reuse and design generation of pattern elements in the Miao batik pattern. Improved shape grammar is introduced into the design system to realize the rapid design of different patterns.

• Pattern elements are obtained that captured high user attention, as demonstrated by an eye movement experiment.

2. RESEARCH BACKGROUND

2.2. Euclidean Transformation

Since this study only focuses on the variation of the state model in the two-dimensional space, we only consider Euclidean transformation in the two-dimensional space. That is, Euclidean plane is regarded as a set of points satisfying special relations expressed by distance and angle. A new shape is created by varying the displacement, angle, scale or a limited combination thereof of the initial shapes [14]. The Euclidean distance formula is given in Equation (1):

d=x2−x12+y2−y12(1)

Figure 1 illustrates the definition of shape grammar and the application of Euclidean transforms in shape grammar.

Figure 1

Examples of traditional shape grammar applications. (a) Shape rules; (b) Initial shape; (c) Pattern derivation using shape grammar rules.

The graph generated in Figure 1(c) was derived from the rules of Figure 1(a) and (b). That is, the final graph of Figure 1(c) is obtained by repeatedly applying the rule shown in Figure 1(a), starting from the initial shape shown in Figure 1(b). Each time, the rule uses a different Euclidean transformation to apply a combination of translation and rotation to the left-hand side of the shape rule, with the nested small diamond corresponding to the marker point l.

3. IMPROVED SHAPE GRAMMAR APPLICATION RULES

Traditional shape grammar uses marker points to control the application of shape rules in the design process. Although the sequential programming style can be well used to describe the design process by using marker points, it relies on specific shapes, which makes the design process too rigid [1]. Therefore, this study separate the shape knowledge and transformation process knowledge of patterns, focus on the application of transformation process knowledge; that is, the transformation process τ is regarded as a transformation verb (i.e., “predicate”), which concretizes the Euclidean transformation parameters and constitutes an improved shape grammar with comprehensive predicate grammar coding. The code of improved shape grammar rules is shown in Table 1.

In Table 1, τ represents all two-dimensional Euclidean transformation modes, 1/0 represents whether the initial shape is retained (1 is retained, 0 is not retained), +θ,−θ represents clockwise or counterclockwise rotation about the center of the shape itself, x0,y0 is the center of the circumferential array, N represents the number of arrays, dX,NX;dY,NY represent the spacing and number in the X and Y directions, respectively, and up, down, left, right represent which boundary of the shape is mirrored as the axis of symmetry.

Use V=IS,τ to represent the application rule of improved shape grammar (IS represents the initial shape set) and call it the predicate grammar expression. The basic expression is shown in Equation (4), and the illustration is shown in Figure 2:

Figure 2

A graphical representation of the basic predicate shape grammar. (a) Initial shape is1; (b) Transformed graph sτ.

The basic two-dimensional Euclidean transform matrix is expressed as

The transformation process of Figure 3 is shown in Equation (5):

If there is a circumferential array of shape transformation, that representation is defined as V=IS,Circx0,y0,N. If the transformation form is a linear array, the definition representation is V=IS,LnardX,NX;dY,NY. If the shape transformation is in the form of a mirror, the definition representation is V=IS,Mirrup/down/left/right,1/0.

Since most graphic transformations use transformation τ multiple times for the initial graphic IS, the connection between multiple transformations is represented by “⊕,” and expressed as V=IS,τ1⊕IS,τ2.

If the initial shape IS overlapped with the new shape Sτ generated after V, the overlapped shape is called emergent shape SESE=se1,se2,…,sen, that is, SE=IS∩Sτ. If SE is also transformed, the expression is V=is1,Trandx,dy,1Rota±θ,1Scalsx,sy,1⊕se1,Trandex,dey,1Rota±θe,1Scalsex,sey,1. The illustration is shown in Figure 3.

Figure 3

SE transform graphic demonstration.

V of the initial shape hexagon is represented by Equation (6):

To eliminate overlap between shapes, define IF IS, Sτ∈Shape, then ST=Shape−SE, where ST represents the terminal shape, expressed as V={(is1,Tran(dx,dy,1/0)Rota(θ,1/0)Scal(sx,sy,1/0),Eli)}, where; Eli=eli1,eli2,…,elin. N represents eliminating the overlap of several transforms. An example illustration is shown in Figure 4.

Figure 4

Eliminate overlapping graphic presentations demonstration.

The transformation process of Figure 4 is shown in Equation (7).

If the shape rule is applied to the graph transformed by τ transformation, the expression is defined as V=IS,τ1⇒ISnm,τ2.

Where; ISnm=isn1,isn2,…,isnm, n represents the initial shape and m represents the graph after several transformations.

4. APPLICATION EXAMPLE AND OPTIMIZATION DESIGN

Miao batik is mainly distributed in Sichuan, Guizhou, and Yunnan Provinces of China and has a long history. They have numerous design themes and rich patterns, mainly including natural forms such as flowers, leaves, birds, and fish. Batik products include tablecloths, clothing, aprons, cloth bags, and other daily necessities [17].

Taking Miao batik pattern elements as an example, the predicate shape grammar is used to reuse the patterns, and PSO is used to optimize the number of patterns to determine the Euclidean transformation parameters, so as to determine the Euclidean transformation parameters, so that the patterns can be transformed and reused according to the optimized parameters.

4.1. Batik Pattern Extraction

This paper selects the pattern elements of batik from the website of “Intangible Cultural Heritage of Guizhou Province” as the basis of reuse design. Website system as shown in Figure 5.

Figure 5

System home.

Batik patterns are selected from the batik productions example library, as shown in Figure 6.

Figure 6

Batik products. (a) Batik product I; (b) Batik product II; (c) Batik product III.

The pattern element identified by the red box in Figure 6 was selected as the extraction object. When extracting, each pattern element was regarded as a separate closed contour, and the shape context method is used to describe and match the pattern elements [18,19]. The Canny operator [20] is also used to obtain the contour of the shape of the pattern by sampling to obtain the contour sampling points, and then plane software was used to modify the extracted patterns. The results are shown in Figure 7.

Figure 7

Pattern matching and extraction results. (a) Patterns extracted from batik product I; (b) Patterns extracted from batik product II; (c) Patterns extracted from batik product III.

4.2. Application of Improved Shape Grammar

Batik patterns include the main, corner, and border pattern elements [21], with special attention to the pattern element arrangement symmetry, single pattern irregularity, and the overall spatial arrangement of strong features. Based on these characteristics, it is feasible to characterize the reuse process of pattern elements with improved shape grammar.

Taking the pattern elements are shown in Figure 7(a) and (b) as examples for reuse design. The Figure 7(a) pattern element is set as pattern1, and the Figure 7(b) pattern element is set as pattern2. First, the predicate shape grammar vocabulary Vpattern1 and Vpattern2 are defined, as shown in Equations (8) and (9), and then the pattern elements of Figure 7(a) and (b) are transformed according to the rules of Vpattern1 and Vpattern2. The result is shown in Figure 8.

Vpattern1=V1=pattern1,Scal1/2,1/2,0⇒pattern11,Circ1,−0.5,4V2=pattern12,Tran−4,0,1⇒pattern13,Circ1,0,4V3=pattern13,Tran14,0,1⇒pattern14,Mirrdown,1⇒pattern15,Rota+45°,0.(8)

Vpattern2=pattern2,Scal3/5,3/5,0⇒pattern21,Mirrdown,1Tran(3,0,1).(9)

Similarly, the reused graphics can be added or deleted as required by the design, as shown in Figure 8(c). If the final patterns shown in Figure 8(c) and (d) are subjected to linear array rule processing and then to combination processing, the two patterns can be reused to design articles such as simple tablecloths or wall decorations, as shown in Figure 9.

Figure 8

Predicate shape grammar instance application. (a) Pattern formed according to V1 rules; (b) Pattern formed according to V2,V3 rules; (c) Deletion processing; (d) Pattern formed according to Vpattern2 rules.

Figure 9

Reuse design result.

4.3. Application of Optimization Algorithm

According to the type of batik pattern and the characteristics of the pattern elements in Figure 7(a)–(c), the pattern of Figure 7(a) is used as the main pattern, and Figure 7(b) and (c) are used as the border element pattern. As shown in Figure 10, the circular configuration mode of the batik pattern is selected. The corresponding position is selected according to the different functions of each pattern, and corner elements are not arranged for circular configuration.

Figure 10

Configuration and regionalization.

When Figure 7(a) (pattern1) is used as the main pattern element, two transformations are defined and as shown in Equations (10) and (11):

V 1pattern1=pattern1,Circ0,0,N.(10)

V 2pattern1=pattern1,Scal1/2,1/2,0⇒pattern11,Circx0,y0,8⇒pattern12,Circ0,0,N.(11)

When Figure 6(b) and (c) are used as border pattern elements, two transformation modes were defined, as shown in Equations (12–14), respectively:

V 1pattern2=pattern2,Lnardx,nx.(12)

V 2pattern2=pattern2,Lnardx,nx⇒pattern21,Scal1/2,1/2,0⇒pattern22,Mirrup/down,1.(13)

V 1pattern3=pattern3,Lnardx,nx.(14)

4.3.1. Mathematical model construction

In this study, the PSO algorithm is used to optimize the number of layouts of each pattern at a specified position. First, according to the size parameters shown in Figure 11, the overall mathematical model is established, as shown in Equation (15):

FXn=fx1,fx2,…,fxnK⋅x1=XFXn≥0∑KuFxn=FX(15)

Figure 11

Dimensional parameters. (a) Each frame size parameters; (b) Initial size parameters of each patterns.

Where; n=1,2,…,5, m=1,2,…,5, K represents the original number, and u represents the number of variables.

According to the placement position of each pattern in Figure 11 and the size parameters of each part in Figure 11, a mathematical model was established according to the transformation mode of Equations (10) and (12–14) as the basis of reuse transformation, as shown in Equation (16):

15≤x1≤374≤x2≤711≤x3≤245≤x4≤122≤x5≤5FXn=2x1+6x2+6x3+6x4+6x5.(16)

Where; x1,…,x5 is a design variable representing the number of rows of fill patterns in each of the five portions in Figure 10(a), FXn=2x1+6x2+6x3+6x4+6x5 is the objective function, and 15≤x1≤37,…,2≤x5≤5 are the constraints.

4.3.2. Result analysis

MATLAB R2017 software is used as the optimization tool, the local and global acceleration constants of the PSO were set as c1=c2=0.149445, and the population size was S=200. The pattern formed by the transformation rule of Equation (10) is used as the main element, and the optimization data are shown in Table 2.

Variable	x1	x2	x3	x4	x5
Number of Groups
Group 1	20.431	5.181	22.288	7.360	2.060
Group 2	33.271	6.269	18.184	7.216	4.425
Group 3	26.166	4.235	13.233	11.985	3.063
Group 4	15.326	5.236	16.459	9.425	4.383
Group 5	18.415	6.806	23.775	6.347	4.977

Table 2

Selecting five sets of PSO data optimized according to Equation (16).

Since x1,…,x5 represent the number of layouts, rounding is performed. After the five sets of algorithm data generated in Table 2 are rounded, the transformations are arranged in accordance with Figure 10 and Equations (10) and (12–14). If there is a scaling factor for any patterns, the number of additions or subtractions of the pattern is inversely proportional to the scaling factor.

The first set of optimized data was combined with the predicate shape grammar vocabulary V, and the undefined transformation parameters in V are defined according to the optimized number of patterns and the defined scaling ratio. The expression VG is defined according to the five sets of data in Table 2, thereby generating a pattern, as shown in Figure 12(a)–(e), respectively. The first set of data was used and expression VG1 is defined, as shown in Equation (17). The rest of the expressions are similar.

VG1=VIpattern3=pattern3,Lnar3.65,20VIIpattern2=pattern2,Lnar12,5⇒pattern21,Scal1/2,1/2,0⇒pattern22,Mirrdown,1Tran−3,0,1VIIIpattern1=pattern1,Circ0,0,22VIVpattern1=pattern1,Scal1/2,1/2,0⇒pattern11,Circ0,0,14VVpattern1=pattern1,Scal1/3,1/3,0⇒pattern11,Circ0,0,6.(17)

Figure 12

Pattern element reuse design result I. (a) Group 1 data pattern; (b) Group 2 data pattern; (c) Group 3 data pattern; (d) Group 4 data pattern; (e) Group 5 data pattern.

If the pattern formed by the transformation rule of Equation (11) is used as the main element, the constraints x3,x4,x5 were changed, as shown in Equation (18). The final optimization data is shown in Table 3, and the design pattern reuse results are shown in Figure 13.

15≤x1≤374≤x2≤711≤x3≤245≤x4≤122≤x5≤5FXn=2x1+6x2+6x3+6x4+6x5.(18)

Figure 13

Pattern element reuse design result II. (a) Group 1 data pattern; (b) Group 2 data pattern; (c) Group 3 data pattern; (d) Group 4 data pattern; (e) Group 5 data pattern.

Variable	x1	x2	x3	x4	x5
Number of Groups
Group 1	19.785	4.881	9.713	4.231	1.985
Group 2	32.871	6.417	6.845	5.437	1.667
Group 3	26.166	4.235	8.143	3.415	2.015
Group 4	15.326	5.236	11.306	5.843	1.788
Group 5	17.935	6.746	9.451	3.472	2.855

Table 3

Selecting five sets of PSO data optimized according to Equation (18).

As seen from that above reuse results, the same transformation rules were applied to the body patterns in Figures 12 and 13, respectively. However, for the border pattern pattern2, different transformation rules are adopted to observe the influence of different border configurations on the pattern and to explore the transformation rules of border styles.

4.4. Construction of Pattern Reuse Design System

In order to apply the predefined improved shape grammar to the reconfiguration design of different pattern frames quickly, the predicate shape grammar expression is applied to the process implementation, and the batik pattern reuse design system is constructed to achieve the rapid pattern frame reuse design. The system implementation is shown in Figure 14.

Figure 14

System design. (a) Initial interface; (b) Reading batik patterns; (c) Apply rules to generate patterns.

The system is divided into three main areas: read, operation, and display. The reading area is used to display the pattern elements to be arranged, the operation area is used to input relevant parameters to arrange the pattern elements, and the display area is used to display the results of the operation. The system can obtain the coordinates of the corresponding position by clicking the mouse on the canvas of the display area.

5. EXPERIMENTAL COMPARISON AND RESULT ANALYSIS

In this study, the Eyeso eye tracker is used to perform eye movement experiments [22,23] on 10 patterns based on the PSO-optimized data reuse design. Twenty postgraduates are selected and their uncorrected or corrected visual acuity is above 1.0. The subjects were required to sit 0.5 m in front of the screen and adjust the angle of the eye mover to 15°, as shown in Figure 15.

Figure 15

Eye movement experiment scene.

In the experiment, 10 patterns are arranged out of order into three groups to counteract the influence of the different display position of each pattern on the experimental results. The sample code and arrangement of the experimental group are shown in Figure 16. The experiment measured the fixation time of the subjects on different patterns and calculated the average fixation time of each pattern, as well as the hot spot map of the longest fixation pattern. The experimental results are shown in Figure 17.

Figure 16

Sample coding and disordered arrangement of the experimental group. (a) Sample coding; (b) Group 1; (c) Group 2; (d) Group 3.

Figure 17

Eye movement test results. (a) Mean fixation time comparison chart; (b) The hotspot map for the longest gaze pattern.

The results showed that the mean fixation time of different patterns is in the order of  I ＞ B ＞ H ＞ G ＞ C ＞ A ＞ F ＞ J ＞ D. Thus, the subjects pay more attention to the patterns formed from the fourth group of data in Table 3, and the hot spot map shows that the subjects pay more attention to the main pattern elements than the border elements.

The A–E encoded pattern (i.e., the subject pattern formed by the transformation rule of Equation (10)) had a total fixation duration average of 16,357.2 ms. The F–J encoded pattern (i.e., the subject pattern formed by the transformation rule of Equation (11)) had a mean total fixation duration of 20,308.4 ms, indicating that the subjects were more interested in the subject pattern formed by the transformation rule of Equation (11).

6. CONCLUSION AND PERSPECTIVES

1. Aiming at the problem of the low abstraction degree of traditional shape grammar in design pattern reuse, the concept of predicate shape grammar vocabulary is proposed to represent the concrete process of pattern reuse design, which transfers the traditional shape grammar from the initial shape to a transformation process and enhances the abstraction of the shape grammar.

2. The improved shape grammar is applied to the reuse design of Miao batik pattern elements. The results show that the improved shape grammar can satisfy the reuse design of batik pattern elements. It is the first time that improved shape grammar and PSO have been combined to form multiple batik patterns, which proves the good scalability of the method.

3. Through eye movement experiments, it is concluded that the subjects are more interested in the main pattern elements formed by the transformation rules of Equation (11), which shows that complex patterns and compact arrangement can attract people's attention. As for the main pattern formed by the transformation rules of Equations (10) and (11), the pattern formed by the fourth group of data in Table 3 is more noticeable to the subjects according to the second group of Table 2. According to the hot spot map, the subjects pay more attention to the main pattern. Therefore, in the future work, the design emphasis should be placed on the design and coding of the main pattern elements.

4. Future work will focus on deepening the content of Section 4.4, that is, expanding the function of the system for pattern reuse design and improving the flexibility and accuracy of pattern reuse design.

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Supervision, N.D.; conceptualization, N.D. and J.L.; research methodology, N.D. and L.H.; experimental operation, N.D. and L.H.; writing—review and editing, N.D. supervision, J.L.; project administration, J.L.

Funding Statement

This research was supported by the Natural Science Foundation of China (Nos. 51865004, 2014BAH05F01) and the Provincial Project Foundation of Guizhou, China (Nos. [2018]1049, [2016]7467).

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers for their valuable suggestions and important comments which greatly helped them to improve the presentation of this manuscript.