# The Promotion Effect of Computer-Aided Technology Combined with RBF Neural Network Algorithm on Art Design

Authors: Chunliang Pan; Yuning Pan
Journal: Advances in Multimedia
 (2022)

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## Abstract

In order to improve the effect of fractal art pattern design, this study combines RBF neural network algorithm to design and evaluate CAD fractal art pattern to improve the effect of fractal art pattern design. Moreover, this study proposes a dynamic weight-based NSC method. Because noise is ubiquitous in reality, the evolution of system dynamics often cannot be carried out strictly according to the iterative rules. In addition, this study shows the changing rules of the competition relationship and the distribution of competitors under the interference of noise. Finally, this study controls the style of the art pattern design in the spatial domain and uses the mask image pattern to disguise the art style. The experimental study shows that the CAD fractal art pattern design system based on the RBF neural network algorithm proposed in this study has a good effect on fractal art pattern design.


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## Body

## 1. Introduction

Digital art graphics can be created by continuous iteration of numbers and computer programs. This iteration is not only a nonlinear transformation, but can also be expressed as a process in a broad sense. The theory of fractals and their meanings have extensive and far-reaching influence, and it can be said that the idea of fractals is a worldview. It is one of the basic theories that constitute nonlinear science, and its appearance also clarifies digital art completely. For example, the famous Mandelbrot set and Julia set can generate breathtaking fine structures with the aid of computers, which have infinite possible shapes. If the geometric method of perspective projection is the first encounter between science and art, then fractal geometry can be said to be the second combination of science and art. Fractal graphics can be generated by recursive algorithm, grammar composition algorithm, iterative function system algorithm, escape time algorithm, and cell evolution algorithm [1]. These computer-generated graphics are designed in a virtual electronic space, using some abstract digital rules or databases, and are immaterial in nature. Since the appearance of this graphic generation method, it has been widely used by designers in the creation of two-dimensional and three-dimensional images and finally formed the concept of “digital art.” Digital art is not only used to generate fantasy-like effects but also needs to study the characteristics of changes in the design process. This interactivity is the main difference between digital art and traditional art, and it changes the way people perceive art [2]. Moreover, digital art graphics generation technology can be used in textile printing and dyeing, industrial design or clothing design, and computer art teaching, and its economic and social benefits are obvious, and it has a very broad application prospect [3].The computer can generate infinitely varied, mysterious, and beautiful digital art graphics through certain numerical calculation methods. These graphics have partial or overall self-similarity, dynamic balance, symmetry, and fine structure, with harmonious rhythm and unique rhythm, reflecting artistic and artistic harmony. The perfect combination of science: graphics in general concept refer to a specific type of image, its main visual feature is shape, the outer contour lines that constitute the main body are obvious, there is no complex texture inside, and the color is simple, so general graphics can be classified or retrieved by shape features [4]. Digital art graphics [5] are generated by the computer’s continuous iteration of digital rules, so it also has a certain degree of mechanical and randomness. Digital art graphics naturally have complex characteristics such as color, shape, and texture. When artists are looking for materials, they hope to find graphics in the graphics database that match their ideas and creative inspiration, but most graphics cannot automatically conform to the user’s subjective thoughts and feelings [6]. Using digital image processing technology to carry out objective aesthetic evaluation and emotional retrieval of digital art graphics by computer can not only facilitate designers to select suitable graphics but also play a guiding role in graphics design and stimulate creative inspiration [7].Fractal geometry can provide realistic descriptions for various scenes that exist in nature. The shapes of these scenes are complex and irregular. It is very rough, which makes it extremely difficult to describe with traditional geometric tools, while fractal models can describe natural scenes very well; because many actual scenes in nature are generally fractal, or vice versa, the shape constructed according to the fractal geometry method is very similar to many natural scenes [8]. Several experiments on a computer to draw self-similar sets have been able to produce surprisingly good patterns of objects that exist in nature. In addition to the simulation of natural scenes, fractal geometry can also be applied to the extraction and recognition of various image information such as biology, medical images, satellite images, and image processing and transmission in television and communications, thus forming fractal geometry and computer. A new field of research combined with graphics has moved from theoretical research to applied research [9]. Fractal solves the problem of the relationship between the whole and the part with its unique means, using the symmetry and self-similarity of the spatial structure and using various simulated real graphic leopard models, so that the entire generated scene presents the nature of infinite regression of details, richly colorful, with wonderful artistic charm. With the help of computer generation of fractals, we can generate complex natural scene graphics from a small amount of data, which brings us a big step forward in simulation [10]. It is certain that the fractal pattern has a wide range of application value in the simulation of real objects in nature, the generation of simulated shapes, computer animation, art decoration texture, pattern design, and creative production, which has attracted the attention of many scientists at home and abroad, especially graphics experts [11].Fractal art is a form of artistic design with its own unique charm. It lacks its own perfect and powerful theoretical system for a long time. It does not establish concepts and ideas that adapt to the characteristics and development of fractal art design and does not introduce these concepts and ideas into fractal in art design. In creation and appreciation, fractal art design cannot exert its advantages and charm [12]. Furthermore, due to the troubles of technology for designers, the application of fractals in art design has been greatly hindered, which is unfavorable to the development of fractal art design itself and the whole art design [13]. Art design needs to interact with science needs to promote the rapid development of fractal art design. On the one hand, the development of science needs the support of the market, and art is an important way for scientific and technological achievements to be brought to the market and transformed into social benefits. On the other hand, the progress of design art depends on the innovation of design art means, and science just provides an unprecedented artistic means for design art [14].Based on the special structure of fractal graphics, it will inevitably lead to the following artistic features in vision: first, the dynamic balance of fractal graphics. Generally speaking, traditional graphics pursue a static balance, and each element in fractal graphics does not exist in isolation and needs an internal dynamic to maintain it. If any element changes, it will affect the overall structure of the graph itself, thus constructing an almost completely different graph [15]. Facing the fractal graphics, viewers can feel a dynamic change that is endless and infinitely multiplying. The second is brilliant colors. Since fractal graphics are generated based on computer technology, the color of each point is generated by computer through iterative function calculation. At the same time, fractal graphics have randomness and variability, and their colors are also richer in variability and diversity. Unexpected color artistic effects are created [16].Fractal graphics need to be modeled according to mathematical formulas and implemented using computer programming languages. Compared with traditional graphics, fractal graphics have unique modeling and decorative features [17]. However, the random form and mechanized characteristics of fractal graphics also require certain artistic treatment when they are used in pattern design. Otherwise, it will run counter to our design creation, making the pattern itself too monotonous and affecting the artistic quality of the work itself. Therefore, a good fractal art work is the product of the fusion of logical thinking and artistic feeling, that is, relying on computer operations to iterate gorgeous and regular graphics, and then supplemented by the designer’s creative thinking to meet the needs of human art and aesthetic spirit [18].


## 2. Fractal Dynamic Model and Algorithm

### 2.1. Two-Dimensional Competition Model under Noise Disturbance

The dual competitor model is proposed as follows:(1)f:xn+1=xn2−yn2+c1,yn+1=2xnyn+c2..Among them,xn,yn,c1,c2∈ℝ, which aims to describe the evolution law of x and y variables with logistic regression competition relationship. It is considered to be the most classic bivariate discrete competition model. It describes the coupling relationship between two competitors x and y under the condition of logistic regression growth law.Considering that noise is ubiquitous in reality, the evolution of system dynamics often cannot be carried out strictly according to the iterative rules. Therefore, it is necessary to study the changing law of competitor’s Bo-Yi relationship and competitor’s distribution under noise interference. It is worth noting that the dynamics of continuous two-dimensional competitive systems under noise interference are reported in the top journal physical reviewE:(2)dxtdt=μxt1−xt−τ−βtyt−τ+xtξxt,dytdt=μyv1−yt−τ−βtxt−τ+ytξyt.Among them,x and y are the population density, μ is proportional to the growth rate, and βt represents the interaction between species. Here ξit represents the effect of noise. In this study, Nie and Mei simulated the effect of noise and time delay on the real ecosystem at the same time and proved the following three points.The time delay causes the two group densities to synchronously oscillate periodically; with the increase in time delay, two competitors show a transition from polystable to monostable; it is proposed that the combination of noise and time delay can provide an effective theoretical basis for understanding the real competition model.Mandelbrot set is the key to ensure the sustainable development of competitive relationship.In view of the importance of the parameters of theM set to the development of the competitive relationship, it is necessary to analyze its internal topological laws. However, noise is ubiquitous in reality, so that the competition relationship is often disturbed by the outside world, and the competition model is often sensitive to the initial value. Therefore, even a small external disturbance can completely change the game relationship of competitors in the future. Stochastic fractal systems have received extensive attention because they can more effectively describe the actual changing laws of competitive processes.According to the definition, the Julia set of system (1) is the initial distribution set that guarantees the stable coexistence of two competitorsx and y in the evolution process; that is, only by selecting the initial value from the Julia set the system can guarantee bounded coexistence. At the same time, it can be seen that the Mandelbrot set is a parameter set to ensure the stability of the connectivity of the attractive domain, and the good connectivity can prevent the attractive domain from collapsing when the system has a small deviation during the evolution process. Therefore, it is necessary to effectively analyze the dynamic characteristics of the parameter set of Mandelbrot set.TheP2 point in the lattice map L is obtained as follows:(3)θ0=c1=−2+i−14P−1,c2=−2+j−14P−1.Mandelbrot setM(f) for noise-free mapping is shown in Figure 1.Figure 1
Mandelbrot setM(f) for noise-free mapping (1).The additive noise represented byfa in formula (1) is defined as follows:(4)fa=xn+1=xn2−yn2+c1+a1wn,yn+1=2xnyn+c2+a2wn.Among them,wn represents the dynamic noise, and the parameter a1=a2=a∈ℝ represents the intensity of the additive noise.The multiplicative noise represented byfm in formula (1) is defined as follows:(5)fm=xn+1=1+m1wnxn2−1+m1wnyn2+c1,yn+1=2+m1wnxnyn+c2.The parameterm1=m2=m∈ℝ represents the strength of the additive noise.Based on the escape time algorithm,Mfua,Mfum and Mfna,Mfnm with different noise intensities a and m are shown in Figure 2. With the increase in the four kinds of noise intensity, the symmetry damage of the Mandelbrot set structure is more obvious. Additive noise destroys the M-set structure from the inside, while multiplicative noise engulfs M(f) more by destroying it from the edges inward. In breaking the symmetry structure, the effect of additive noise seems to be greater than that of multiplicative noise when a = m.Figure 2
Mandelbrot set of noise perturbations.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


### 2.2. SC and NSC

The “symmetry index”symc1μ0,v0 is defined as follows:(6)symc1μ0,v0=2,IffNx0,y0,μ0<RAndfNx0,y0,v0<R,0,IffNx0,y0,μ0<RAndfNx0,y0,v0≥R,IffNx0,y0,μ0≥RAndfNx0,y0,v0<R,IffNx0,y0,μ0≥RAndfNx0,y0,v0≥R.Mfua is denoted by SCc1Mfua0.The “Symmetry Criterion” about thec1 axis is calculated as follows:(7)symc1fua0=1numMfua0∑Lsymc1μ0,v0.Among them,numMfua0 is the total number of points in Mfua0numMfua0. Obviously, ∑Lsymc1μ0,v0 is the number of symmetry points in Mfua0. Therefore, SCc1fua0 varies in the range [0, 1]. The closer SCc1fua0 is, the more symmetrical the structure Mfua0 is.Finally, the model calculates all the values ofSCc1fua0 when a0∈0,1 in increments of 0.01 and obtains the change curve of SCc1fua0.Forwn∼U0,1 and wn∼N0,1, it can be seen that SCc1fa is larger than SCc1fm and has the same noise intensity α. This confirms the conclusion in Figure 3.Figure 3
(a)SCc1f curve and wn∼U0,1 increase with the intensity of the reel noise: the additive noise is represented by a cool color, and the multiplicative noise is represented by a warm color. (b) The SCc1f curve and wn∼N0,1 increase with the noise intensity.
(a)(b)BothSCc1fu and SCc1fn remain at values close to 1. That is, the noise-perturbed Mandelbrot set with more peripheral point asymmetry may have a high quantization symmetry index.The above simulation results reveal that the peripheral points are more susceptible to noise interference because they are far from the attraction point. We show some zoomed-in details ofMfa=0.8 in Figure 4.Figure 4
Three partial enlarged details.The weight ofSCc1 of edge area points should be greater than the weight of SCc1 of core area points. In Figure 4, most of the noise-perturbed Mandelbrot sets remain at [−1.5, 1.5] on the c2 axis. Therefore, the appropriate weight of SCc1, denoted by w, can be calculated as w=imgμ0/1.5img⋅; we propose an improved “new symmetry criterion” (NSC) method as follows:(8)countα0=w,iffNx0,y0,α0<R0,iffNx0,y0,α0≥R.For any two initial pointsμ0∈L+ and v0∈L− symmetrical about the c1 axis, the “new symmetry index” symc1Visμ0,v0 is defined as follows:(9)symc1Visμ0,v0=2w,iffNx0,y0,μ0<RandfNx0,y0,v0<R0,Other casessame as SCsymmetry criterion method.Then, we set the “new symmetry criterion” ofMfua expressed by symc1Visfua0 about the c1 axis to be calculated as follows:(10)symc1Visfua0=1∑Lcountα0∑Lsymc1Visμ0,v0.


### 2.3. Fractal Analysis and Optimal Control of Symmetrically Coupled Three-Competitor Model

We consider system (1–1),N = 3 is taken, and assume that the three-competitor competition relationship presents a symmetrical coupling structure, and the following three-participant model can be obtained:(11)F:xn+1=xn+Rxn1−xn−αyn−βzn,yn+1=yn+Ryn1−βxn−yn−αzn,zn+1=zn+Rzn1−αxn−βyn−zn.Whenxi,yi,zi≠0, we can get the fixed point 1/1+α+β,1/1+α+β,1/1+α+β, and the three competitors can coexist.Taking parameters asR=1,α=0.4745,andβ=3, the original Julia set of system (1) is shown in Figure 5. In Figure 5(b), the padding K(F) with z0=0 is shown. In fact, when one of the three competitors is eliminated, it can be regarded as a simplified case of system (1). Different initial states A and B are selected, and their competitive relationship trends are shown in Figures 6 and 7. The two figures show trajectories for A∈KF and B∉KF, respectively, where we can see that the trajectories remain closed until 90 iterations. However, when B’s trajectory starts to deviate, the iteration continues. It can be seen from the figure that starting from the fractal attraction domain, the stable coexistence of the three competitors is guaranteed. However, the fractal attraction domain cannot guarantee the stability of the three-competitor relationship.Figure 5
(a) Spatial Julia setJF with R=1,α=0.4745,andβ=3; (b) 2D-filled slice KF with z0=0; (c) 2D-filled slice K(F) with x0=0.01; (d) local zoom of c, where we choose point A=0.22,0.19∈KF and point B=0.23,0.195∉KF.Figure 6
Competitive relationship trajectory of A ((a) 20 iterations; (b) 40 iterations; (c) 100 iterations).
(a)(b)(c)Figure 7
Competitive relationship trajectory of B ((a) 20 iterations; (b) 40 iterations; (c) 91 iterations).
(a)(b)(c)


### 2.4. Optimal Control of Julia Sets for Three-Competitor Systems

By summarizing the above 8 situations, obviously, we hope that these three competitors can coexist continuously in the actual competition process. Therefore, the control process ofJ(F) focuses on the stability of the “coexistence point” 1/1+α+β,1/1+α+β,1/1+α+β; to keep the fixed point constant, we apply the following gradient optimization controls:(12)un1=−k1+kfxn,yn,zn−x∗,un2=−k1+kgxn,yn,zn−y∗,un3=−k1+kqxn,yn,zn−z∗.Sincex∗=y∗=z∗=1/1+α+β, by adding control term (2) to system (1), the following optimized system can be obtained:(13)P:xn+1=11+kfxn,yn,zn+k1+kx∗yn+1=11+kgxn,yn,zn+k1+kx∗zn+1=11+kqxn,yn,zn+k1+kx∗.Based on the above analysis,J(F) is the domain of attraction that attracts fixed points. Therefore, by adding a control term, it is possible to control the Julia set in which the distribution density of the three competitors remains stable by affecting the iterative trajectory of the initial point in system (3), while achieving the stability of the fixed point.The Jacobi matrix of system (3) at the fixed point x∗,y∗,z∗=1/1+α+β,1/1+α+β,1/1+α+β is considered as follows:(14)J=1+R1+k−2R1+kx∗−αR1+ky∗−βR1+kz∗−αR1+kx∗α∗−βR1+ky∗1+R1+k−2R1+ky∗−αR1+kz∗−βR1+kx∗−βR1+kx∗−αR1+kz∗−αR1+kz∗1+R1+k−2R1+kz∗−αR1+kx∗−βR1+ky∗.The characteristic formula of the Jacobi matrixJ is expressed as follows:(15)Δλ=λ3+P˜λ2+Q˜λ+S˜=0.At this point, there are(1)
T=2+α+βRx∗−1+R(2)
P˜=3T/1+k(3)
Q˜=3T2−3αβR2x∗2/1+k2(4)
S˜=T3−α3R3x∗3−β3R3x∗3−3TαβR2x∗2/1+k3To determine that the controlled system (1) is stable at fixed points 1/1+α+β,1/1+α+β,1/1+α+β, the following three conditions need to be satisfied:(1)
Δ1>0.Δ1=1+P˜+Q˜+S˜>0(2)
1−P˜+Q˜−S˜>0(3)
S˜<1 and S˜2−1>S˜P˜−Q˜For two-competitor and multiple-competitor systems, the complex coupling behavior between competitors increases the difficulty of analyzing the topological properties of Julia sets.It simplifies to a competing behavior involving chain coupling, as follows:(16)Fx1x2x3x1n+1=x1n+Rx1n1−x1n−αx2n−βx3n,x2n+1=x2n+Rx2n1−αx3n−x2n,x3n+1=x3n+Rx3n1−x3n.Among them,R,α,β have the same meaning as system (3), x1,x2,x3 represent three competitors, respectively, which means that x1 to x3 are a progressive competition relationship. Among them, x3 is the highest level competitor, that is, the single-headed competitor. Its competitiveness is stronger than that of x1 and x2, and the competitiveness of x2 is stronger than that of R.The upper bound of theJFx1x2x3 system is given by the following theorem.Theorem 1.
We have(17)JFx1x2x3⊂x10,x20,x30i0<Bi,i=x1,x2,x3.Among them,Bx3=R+1/R,Bx2=R+1+RαBx3/R,Bx1=R+1+RαBx2+RβBx3/R.The proof is as follows.Forx3 without a stronger competitor, if x30>R+1/R, we can get the following:(18)x31≥x302−R−x301+R.At this time,∃ϵ makes limn⟶∞1+ϵnx30⟶∞, so x30<R+1/R is a necessary condition to make JFx1x2x3 bounded. If x20≥Bx2, we can get the following:(19)x21=1+Rx20−Rx202−Rαx30x20≥Rx202−1+Rx20+Rαx30x20≥Rx202−1+R+RαBx3x20.At this time,∃ϵ makes limn⟶∞x2n+1=limn⟶∞1+εnx20⟶∞, so obviously x20<Bx2 is a necessary condition to make JFx1x2x3 bounded. Likewise, if x10>Bx1, we can get the following:(20)x11=1+Rx10−Rx102−Rαx20x10≥Rx102−x101+R+RαBx2+RβBx3.At this time,∃ϵ makes limn⟶∞x1n+1=limn⟶∞1+εnx10⟶∞, so obviously x10<Bx1 is a necessary condition to make JFx1x2x3 bounded.


### 2.5. Synchronization of Julia Sets for Three-Competitor Systems

Synchronization is considered as another optimal control method, whose purpose is to make the dynamics of the controlled system behave the same as the ideal system. For the purposes of this section, synchronization aims to steer a competitive relationship with a specified steady state. Synchronization of Julia sets can be achieved by adding some coupling terms. System (4) with the same structure as system (3) but with different parameters is considered as follows:(21)F˜:un+1=un+R˜un1−un−α˜vn−β˜wn,vn+1=vn+R˜vn1−β˜un−vn−α˜wn,wn+1=wn+R˜wn1−α˜un−β˜vn−wn.To relate system (5) to the ideal system (3), we introduce two coupling terms in it to get the following:(22)F˜0:un+1=2un−un2−α˜unvn−β˜unwn−L1un−xn−L˜1βxnzn+αxnyn+x2−β˜unvn−α˜unvn−un2vn+1=2vn−vn2−α˜unwn−β˜unvn−L2vn−yn−L˜2βxnyn+αynzn+y2−β˜unvn−α˜vnwn−vn2wn+1=2wn−wn2−α˜unwn−β˜vnwn−L3wn−zn−L˜3βynzn+αxnzn+z2−β˜vnwn−α˜unwn−wn2.The Julia sets of systems (6) and (3) are denoted as JF˜0 and J(F), respectively, and the unified expression L of the coupling parameters Lj and L˜jj=1,2,3 is given, and the following lemmas and theorems are obtained.Lemma 1.
If there is someL0 satisfying the following formula, it is said to achieve synchronization between JF˜0 and J(F).(23)limL⟶L0JF˜O∪JF−JF˜O∩JF=∅.Theorem 2.
JF˜ and J(F) are synchronized when Li⟶1,L˜l⟶2.The proof is as follows.Obviously, the Julia set is obtained by iterating over points within a bounded region, which is denoted byΩ. Since ψ0∉JF, the presence of n0 will cause Fn0ψ0 to escape from the Ω region. Therefore, in the synchronization process, only the points of the trajectory within Ω need to be considered. Therefore, ∃N1 makes βxnzn+β˜unwn<N1,αxnyn+αunvn<N1,xn2+un2<N1.Then, we have the following:(24)un+1−xn+1≤Z−L1un−xn+1−L˜1αxnyn−α˜xnyn+1−L˜1βxnyn−β˜unwn+1−L˜1xn2−un2≤2−L1un−xn+1−L˜1βxnzn+β˜unwn+1−L˜1αxnyn+α˜unvn+1−L˜1xn2+un2≤2−L1un−xn+31−L˜1N1≤2−L12−L−1u−n−1−x−n−1+31−L˜1N1+31−L˜1N1≤2−L12un−1−xn−1+31−L˜1N11+2−L1≤⋯…≤2−L1nu1−x1+31−L˜1N11+2−L1+2−L12+⋯+2−L1n=2−L1nu1−x1+31−L˜1N11−2−L1n1−2−L1.That is,un+1−xn+1≤2−L1nu1−x1+31−L˜1N11−2−L1n/1−2−L1, the right side constraint satisfies L1⟶2, and the trajectories of L˜1⟶1,un, and xn are synchronized. It can also be concluded that when L2,3⟶2,L˜2,3⟶1, the trajectories of vn,yn,wn,zn are synchronized.The algorithm camouflages the artistic style using a masked image pattern by controlling the style of the artistic pattern design on the spatial domain. Figure8 is a comprehensive flow chart of artistic pattern design, and mask image and color controls are optional. If the mask image is invalid, the style transfer can be performed without spatial region control.Figure 8
CAD fractal art pattern design system fused with RBF neural network algorithm.


## 2.1. Two-Dimensional Competition Model under Noise Disturbance

The dual competitor model is proposed as follows:(1)f:xn+1=xn2−yn2+c1,yn+1=2xnyn+c2..Among them,xn,yn,c1,c2∈ℝ, which aims to describe the evolution law of x and y variables with logistic regression competition relationship. It is considered to be the most classic bivariate discrete competition model. It describes the coupling relationship between two competitors x and y under the condition of logistic regression growth law.Considering that noise is ubiquitous in reality, the evolution of system dynamics often cannot be carried out strictly according to the iterative rules. Therefore, it is necessary to study the changing law of competitor’s Bo-Yi relationship and competitor’s distribution under noise interference. It is worth noting that the dynamics of continuous two-dimensional competitive systems under noise interference are reported in the top journal physical reviewE:(2)dxtdt=μxt1−xt−τ−βtyt−τ+xtξxt,dytdt=μyv1−yt−τ−βtxt−τ+ytξyt.Among them,x and y are the population density, μ is proportional to the growth rate, and βt represents the interaction between species. Here ξit represents the effect of noise. In this study, Nie and Mei simulated the effect of noise and time delay on the real ecosystem at the same time and proved the following three points.The time delay causes the two group densities to synchronously oscillate periodically; with the increase in time delay, two competitors show a transition from polystable to monostable; it is proposed that the combination of noise and time delay can provide an effective theoretical basis for understanding the real competition model.Mandelbrot set is the key to ensure the sustainable development of competitive relationship.In view of the importance of the parameters of theM set to the development of the competitive relationship, it is necessary to analyze its internal topological laws. However, noise is ubiquitous in reality, so that the competition relationship is often disturbed by the outside world, and the competition model is often sensitive to the initial value. Therefore, even a small external disturbance can completely change the game relationship of competitors in the future. Stochastic fractal systems have received extensive attention because they can more effectively describe the actual changing laws of competitive processes.According to the definition, the Julia set of system (1) is the initial distribution set that guarantees the stable coexistence of two competitorsx and y in the evolution process; that is, only by selecting the initial value from the Julia set the system can guarantee bounded coexistence. At the same time, it can be seen that the Mandelbrot set is a parameter set to ensure the stability of the connectivity of the attractive domain, and the good connectivity can prevent the attractive domain from collapsing when the system has a small deviation during the evolution process. Therefore, it is necessary to effectively analyze the dynamic characteristics of the parameter set of Mandelbrot set.TheP2 point in the lattice map L is obtained as follows:(3)θ0=c1=−2+i−14P−1,c2=−2+j−14P−1.Mandelbrot setM(f) for noise-free mapping is shown in Figure 1.Figure 1
Mandelbrot setM(f) for noise-free mapping (1).The additive noise represented byfa in formula (1) is defined as follows:(4)fa=xn+1=xn2−yn2+c1+a1wn,yn+1=2xnyn+c2+a2wn.Among them,wn represents the dynamic noise, and the parameter a1=a2=a∈ℝ represents the intensity of the additive noise.The multiplicative noise represented byfm in formula (1) is defined as follows:(5)fm=xn+1=1+m1wnxn2−1+m1wnyn2+c1,yn+1=2+m1wnxnyn+c2.The parameterm1=m2=m∈ℝ represents the strength of the additive noise.Based on the escape time algorithm,Mfua,Mfum and Mfna,Mfnm with different noise intensities a and m are shown in Figure 2. With the increase in the four kinds of noise intensity, the symmetry damage of the Mandelbrot set structure is more obvious. Additive noise destroys the M-set structure from the inside, while multiplicative noise engulfs M(f) more by destroying it from the edges inward. In breaking the symmetry structure, the effect of additive noise seems to be greater than that of multiplicative noise when a = m.Figure 2
Mandelbrot set of noise perturbations.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


## 2.2. SC and NSC

The “symmetry index”symc1μ0,v0 is defined as follows:(6)symc1μ0,v0=2,IffNx0,y0,μ0<RAndfNx0,y0,v0<R,0,IffNx0,y0,μ0<RAndfNx0,y0,v0≥R,IffNx0,y0,μ0≥RAndfNx0,y0,v0<R,IffNx0,y0,μ0≥RAndfNx0,y0,v0≥R.Mfua is denoted by SCc1Mfua0.The “Symmetry Criterion” about thec1 axis is calculated as follows:(7)symc1fua0=1numMfua0∑Lsymc1μ0,v0.Among them,numMfua0 is the total number of points in Mfua0numMfua0. Obviously, ∑Lsymc1μ0,v0 is the number of symmetry points in Mfua0. Therefore, SCc1fua0 varies in the range [0, 1]. The closer SCc1fua0 is, the more symmetrical the structure Mfua0 is.Finally, the model calculates all the values ofSCc1fua0 when a0∈0,1 in increments of 0.01 and obtains the change curve of SCc1fua0.Forwn∼U0,1 and wn∼N0,1, it can be seen that SCc1fa is larger than SCc1fm and has the same noise intensity α. This confirms the conclusion in Figure 3.Figure 3
(a)SCc1f curve and wn∼U0,1 increase with the intensity of the reel noise: the additive noise is represented by a cool color, and the multiplicative noise is represented by a warm color. (b) The SCc1f curve and wn∼N0,1 increase with the noise intensity.
(a)(b)BothSCc1fu and SCc1fn remain at values close to 1. That is, the noise-perturbed Mandelbrot set with more peripheral point asymmetry may have a high quantization symmetry index.The above simulation results reveal that the peripheral points are more susceptible to noise interference because they are far from the attraction point. We show some zoomed-in details ofMfa=0.8 in Figure 4.Figure 4
Three partial enlarged details.The weight ofSCc1 of edge area points should be greater than the weight of SCc1 of core area points. In Figure 4, most of the noise-perturbed Mandelbrot sets remain at [−1.5, 1.5] on the c2 axis. Therefore, the appropriate weight of SCc1, denoted by w, can be calculated as w=imgμ0/1.5img⋅; we propose an improved “new symmetry criterion” (NSC) method as follows:(8)countα0=w,iffNx0,y0,α0<R0,iffNx0,y0,α0≥R.For any two initial pointsμ0∈L+ and v0∈L− symmetrical about the c1 axis, the “new symmetry index” symc1Visμ0,v0 is defined as follows:(9)symc1Visμ0,v0=2w,iffNx0,y0,μ0<RandfNx0,y0,v0<R0,Other casessame as SCsymmetry criterion method.Then, we set the “new symmetry criterion” ofMfua expressed by symc1Visfua0 about the c1 axis to be calculated as follows:(10)symc1Visfua0=1∑Lcountα0∑Lsymc1Visμ0,v0.


## 2.3. Fractal Analysis and Optimal Control of Symmetrically Coupled Three-Competitor Model

We consider system (1–1),N = 3 is taken, and assume that the three-competitor competition relationship presents a symmetrical coupling structure, and the following three-participant model can be obtained:(11)F:xn+1=xn+Rxn1−xn−αyn−βzn,yn+1=yn+Ryn1−βxn−yn−αzn,zn+1=zn+Rzn1−αxn−βyn−zn.Whenxi,yi,zi≠0, we can get the fixed point 1/1+α+β,1/1+α+β,1/1+α+β, and the three competitors can coexist.Taking parameters asR=1,α=0.4745,andβ=3, the original Julia set of system (1) is shown in Figure 5. In Figure 5(b), the padding K(F) with z0=0 is shown. In fact, when one of the three competitors is eliminated, it can be regarded as a simplified case of system (1). Different initial states A and B are selected, and their competitive relationship trends are shown in Figures 6 and 7. The two figures show trajectories for A∈KF and B∉KF, respectively, where we can see that the trajectories remain closed until 90 iterations. However, when B’s trajectory starts to deviate, the iteration continues. It can be seen from the figure that starting from the fractal attraction domain, the stable coexistence of the three competitors is guaranteed. However, the fractal attraction domain cannot guarantee the stability of the three-competitor relationship.Figure 5
(a) Spatial Julia setJF with R=1,α=0.4745,andβ=3; (b) 2D-filled slice KF with z0=0; (c) 2D-filled slice K(F) with x0=0.01; (d) local zoom of c, where we choose point A=0.22,0.19∈KF and point B=0.23,0.195∉KF.Figure 6
Competitive relationship trajectory of A ((a) 20 iterations; (b) 40 iterations; (c) 100 iterations).
(a)(b)(c)Figure 7
Competitive relationship trajectory of B ((a) 20 iterations; (b) 40 iterations; (c) 91 iterations).
(a)(b)(c)


## 2.4. Optimal Control of Julia Sets for Three-Competitor Systems

By summarizing the above 8 situations, obviously, we hope that these three competitors can coexist continuously in the actual competition process. Therefore, the control process ofJ(F) focuses on the stability of the “coexistence point” 1/1+α+β,1/1+α+β,1/1+α+β; to keep the fixed point constant, we apply the following gradient optimization controls:(12)un1=−k1+kfxn,yn,zn−x∗,un2=−k1+kgxn,yn,zn−y∗,un3=−k1+kqxn,yn,zn−z∗.Sincex∗=y∗=z∗=1/1+α+β, by adding control term (2) to system (1), the following optimized system can be obtained:(13)P:xn+1=11+kfxn,yn,zn+k1+kx∗yn+1=11+kgxn,yn,zn+k1+kx∗zn+1=11+kqxn,yn,zn+k1+kx∗.Based on the above analysis,J(F) is the domain of attraction that attracts fixed points. Therefore, by adding a control term, it is possible to control the Julia set in which the distribution density of the three competitors remains stable by affecting the iterative trajectory of the initial point in system (3), while achieving the stability of the fixed point.The Jacobi matrix of system (3) at the fixed point x∗,y∗,z∗=1/1+α+β,1/1+α+β,1/1+α+β is considered as follows:(14)J=1+R1+k−2R1+kx∗−αR1+ky∗−βR1+kz∗−αR1+kx∗α∗−βR1+ky∗1+R1+k−2R1+ky∗−αR1+kz∗−βR1+kx∗−βR1+kx∗−αR1+kz∗−αR1+kz∗1+R1+k−2R1+kz∗−αR1+kx∗−βR1+ky∗.The characteristic formula of the Jacobi matrixJ is expressed as follows:(15)Δλ=λ3+P˜λ2+Q˜λ+S˜=0.At this point, there are(1)
T=2+α+βRx∗−1+R(2)
P˜=3T/1+k(3)
Q˜=3T2−3αβR2x∗2/1+k2(4)
S˜=T3−α3R3x∗3−β3R3x∗3−3TαβR2x∗2/1+k3To determine that the controlled system (1) is stable at fixed points 1/1+α+β,1/1+α+β,1/1+α+β, the following three conditions need to be satisfied:(1)
Δ1>0.Δ1=1+P˜+Q˜+S˜>0(2)
1−P˜+Q˜−S˜>0(3)
S˜<1 and S˜2−1>S˜P˜−Q˜For two-competitor and multiple-competitor systems, the complex coupling behavior between competitors increases the difficulty of analyzing the topological properties of Julia sets.It simplifies to a competing behavior involving chain coupling, as follows:(16)Fx1x2x3x1n+1=x1n+Rx1n1−x1n−αx2n−βx3n,x2n+1=x2n+Rx2n1−αx3n−x2n,x3n+1=x3n+Rx3n1−x3n.Among them,R,α,β have the same meaning as system (3), x1,x2,x3 represent three competitors, respectively, which means that x1 to x3 are a progressive competition relationship. Among them, x3 is the highest level competitor, that is, the single-headed competitor. Its competitiveness is stronger than that of x1 and x2, and the competitiveness of x2 is stronger than that of R.The upper bound of theJFx1x2x3 system is given by the following theorem.Theorem 1.
We have(17)JFx1x2x3⊂x10,x20,x30i0<Bi,i=x1,x2,x3.Among them,Bx3=R+1/R,Bx2=R+1+RαBx3/R,Bx1=R+1+RαBx2+RβBx3/R.The proof is as follows.Forx3 without a stronger competitor, if x30>R+1/R, we can get the following:(18)x31≥x302−R−x301+R.At this time,∃ϵ makes limn⟶∞1+ϵnx30⟶∞, so x30<R+1/R is a necessary condition to make JFx1x2x3 bounded. If x20≥Bx2, we can get the following:(19)x21=1+Rx20−Rx202−Rαx30x20≥Rx202−1+Rx20+Rαx30x20≥Rx202−1+R+RαBx3x20.At this time,∃ϵ makes limn⟶∞x2n+1=limn⟶∞1+εnx20⟶∞, so obviously x20<Bx2 is a necessary condition to make JFx1x2x3 bounded. Likewise, if x10>Bx1, we can get the following:(20)x11=1+Rx10−Rx102−Rαx20x10≥Rx102−x101+R+RαBx2+RβBx3.At this time,∃ϵ makes limn⟶∞x1n+1=limn⟶∞1+εnx10⟶∞, so obviously x10<Bx1 is a necessary condition to make JFx1x2x3 bounded.


## 2.5. Synchronization of Julia Sets for Three-Competitor Systems

Synchronization is considered as another optimal control method, whose purpose is to make the dynamics of the controlled system behave the same as the ideal system. For the purposes of this section, synchronization aims to steer a competitive relationship with a specified steady state. Synchronization of Julia sets can be achieved by adding some coupling terms. System (4) with the same structure as system (3) but with different parameters is considered as follows:(21)F˜:un+1=un+R˜un1−un−α˜vn−β˜wn,vn+1=vn+R˜vn1−β˜un−vn−α˜wn,wn+1=wn+R˜wn1−α˜un−β˜vn−wn.To relate system (5) to the ideal system (3), we introduce two coupling terms in it to get the following:(22)F˜0:un+1=2un−un2−α˜unvn−β˜unwn−L1un−xn−L˜1βxnzn+αxnyn+x2−β˜unvn−α˜unvn−un2vn+1=2vn−vn2−α˜unwn−β˜unvn−L2vn−yn−L˜2βxnyn+αynzn+y2−β˜unvn−α˜vnwn−vn2wn+1=2wn−wn2−α˜unwn−β˜vnwn−L3wn−zn−L˜3βynzn+αxnzn+z2−β˜vnwn−α˜unwn−wn2.The Julia sets of systems (6) and (3) are denoted as JF˜0 and J(F), respectively, and the unified expression L of the coupling parameters Lj and L˜jj=1,2,3 is given, and the following lemmas and theorems are obtained.Lemma 1.
If there is someL0 satisfying the following formula, it is said to achieve synchronization between JF˜0 and J(F).(23)limL⟶L0JF˜O∪JF−JF˜O∩JF=∅.Theorem 2.
JF˜ and J(F) are synchronized when Li⟶1,L˜l⟶2.The proof is as follows.Obviously, the Julia set is obtained by iterating over points within a bounded region, which is denoted byΩ. Since ψ0∉JF, the presence of n0 will cause Fn0ψ0 to escape from the Ω region. Therefore, in the synchronization process, only the points of the trajectory within Ω need to be considered. Therefore, ∃N1 makes βxnzn+β˜unwn<N1,αxnyn+αunvn<N1,xn2+un2<N1.Then, we have the following:(24)un+1−xn+1≤Z−L1un−xn+1−L˜1αxnyn−α˜xnyn+1−L˜1βxnyn−β˜unwn+1−L˜1xn2−un2≤2−L1un−xn+1−L˜1βxnzn+β˜unwn+1−L˜1αxnyn+α˜unvn+1−L˜1xn2+un2≤2−L1un−xn+31−L˜1N1≤2−L12−L−1u−n−1−x−n−1+31−L˜1N1+31−L˜1N1≤2−L12un−1−xn−1+31−L˜1N11+2−L1≤⋯…≤2−L1nu1−x1+31−L˜1N11+2−L1+2−L12+⋯+2−L1n=2−L1nu1−x1+31−L˜1N11−2−L1n1−2−L1.That is,un+1−xn+1≤2−L1nu1−x1+31−L˜1N11−2−L1n/1−2−L1, the right side constraint satisfies L1⟶2, and the trajectories of L˜1⟶1,un, and xn are synchronized. It can also be concluded that when L2,3⟶2,L˜2,3⟶1, the trajectories of vn,yn,wn,zn are synchronized.The algorithm camouflages the artistic style using a masked image pattern by controlling the style of the artistic pattern design on the spatial domain. Figure8 is a comprehensive flow chart of artistic pattern design, and mask image and color controls are optional. If the mask image is invalid, the style transfer can be performed without spatial region control.Figure 8
CAD fractal art pattern design system fused with RBF neural network algorithm.


## 3. Simulation Test

After extracting the artistic pattern features, when searching, the algorithm will extract the artistic pattern features of the image to be retrieved and then compare them with the artistic pattern features of the image art pattern feature library. There are many factors that affect the retrieval results, such as the amount of image dimensionality reduction and the similarities and differences in similarity matching algorithms. These will more or less have a certain impact on the accuracy of the retrieval results. In view of these factors, this study studies their influence on the retrieval results. Because the classification of images is not involved in this article, the data used in the experiments are image sets with high similarity. The experimental results of nonequal scaling experiments are shown in Table1.Table 1
Experimental results of nonequal scaling.
Compression ratioImage size after compressionAverage value of images related to search resultsAverage retrieval accuracy (%)0.071 × 0.1656.0 × 56.03.774380.0976 × 0.14778.0 × 78.03.978400.1126 × 0.16890.0 × 90.04.488550.142 × 0.213112.0 × 112.07.344730.226 × 0.337180.0 × 180.08.262830.281 × 0.423224.0 × 224.09.282930.376 × 0.566300.0 × 300.09.078910.643 × 0.962512.0 × 512.09.1892From Table1, it can be seen that the model proposed in this study has a good performance in nonequal scaling. On this basis, the effect of the CAD fractal art pattern design system based on the RBF neural network algorithm proposed in this study is verified, and the fractal art pattern design effect is calculated, as shown in Table 2.Table 2
Design effects of fractal art patterns.
NumberDesign effectNumberDesign effectNumberDesign effect186.7121185.3582184.778290.0461284.2572289.849385.3411390.8952384.137490.3631489.6632484.683587.3261585.7382585.313687.9151687.0622690.072789.5221788.9672786.017889.8251888.0822888.070987.6611987.5152986.3691087.5302086.9293085.330It can be seen from the above research that the CAD fractal art pattern design system based on the RBF neural network algorithm proposed in this study has a good effect on fractal art pattern design.


## 4. Conclusion

Fractals are everywhere, and fractal theory is called the geometry of nature, which is essentially a new worldview and methodology. Moreover, it can be used for reference by other disciplines and has penetrated into many disciplines including physics, chemistry, biomedicine, materials science, economic management, and computer graphics. In addition, the fractal theory has broad application prospects. While promoting research in related fields, it also brings forward many new topics that need further research. At present, fractals, especially fractal patterns, are gradually entering people’s lives. People are not only limited to appreciating the wonderful fractal art, but have begun to consciously bring fractals into industrial design. This study combines the RBF neural network algorithm to design and evaluate the CAD fractal art pattern to improve the effect of fractal art pattern design. The experimental results show that the CAD fractal art pattern design system based on the RBF neural network algorithm proposed in this study has a good effect on fractal art pattern design.



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