Research on Visual Communication Effect of Advertising Art Design Based on Fractal Geometry

In terms of the structure of the graph, fractal graphs are characterized by the following three main aspects.

The first is self-similarity, which means that the fractal graph will have some similarity with the whole when it is localized and enlarged.

The second is the infinite fineness of the graphic structure, that is, any part of the fractal graphic will be enlarged, it will be found that the localization of the graphic contains a more minute similar structure, and the division of this structure is theoretically infinite.

Third, irregularity, or fractal graphics, do not have the European style geometric figures with a regular, smooth form, but with an irregular structure to achieve a lifelike description of natural objects.

Based on the special structure of fractal graphics, it inevitably leads to the following artistic characteristics in visualization.

First, the dynamic balance of fractal graphics. Generally speaking, traditional graphics aim for a static balance. But each element in the fractal graphic is not an isolated existence, and needs an inner momentum to maintain. If any one of the elements changes, it will affect the overall structure of the graphic itself, thus creating an almost completely different graphic.Therefore, in the face of fractal graphics, viewers can feel a kind of endless, infinite multiplicity of dynamic changes.

Second, the brilliant colors. As fractal graphics is based on computer technology to generate, which each point of the color is generated by the computer through the iterative function calculation. At the same time, fractal graphics with randomness and variability, its color is also more rich in variability and diversity, can create an unexpected color art effect.

Since fractal graphs are generated by computers through iterative function calculations, they have obvious numerical characteristics. Specifically, the digital nature is manifested in the following aspects.

First, the randomness of the generation process of fractal graphics is the most obvious difference between other art forms.

The second advantage of fractal graphics is that they can be infinitely subdivided and developed, provided that the computer has enough computing power and storage space.

In the process of advertising art design, abstract and figurative art forms can be comprehensively combined.In the process of advertising design can be the integration of dots, lines and surfaces, to promote the audience to produce strong perceptual feelings.This factor is a key factor in the success or failure of creativity. The impact of relatively effective visual intensity can stimulate the audience’s senses and attract attention and interest.Texture is due to the different materials and surface organization and structure of different situations. In general, texture refers to roughness, smoothness, and softness.This is the ability to touch different textures, which can be shown through the depiction of the situation.

Advertising visual design mainly plays a specific role in publicity through visual communication. Understanding the good visual effect can attract people’s attention in certain situations.Vision is a positive organ, with visual senses that can form perceptual interaction at the perception level to form a positive role and choice.

Text in advertising needs to reflect the scale. Text is one of the tools for information exchange and promotes psychological and emotional aspects of the formation of a comprehensive communication role.

In the process of graphic design in advertising, the main thing is to accurately convey information, that is to say, to convey the real attributes of the goods, as well as all the information about the service. The most important thing in modern advertising is to emphasize the innovation of graphics. In the process of visual communication of advertisements, it is necessary to highlight the color factor and combine functional risks, applicability, and aesthetics in a comprehensive manner.

Iterated function systems (IFS) arise from the combination of computer graphics, affine geometry, and fractal theory, which is an important branch of fractal theory. It is also one of the most vital and promising areas of application in fractal images.

Affine transformation is an essential component of geometric transformation in computer graphics, and IFS theory is founded on it, which involves rotating the figure around the origin, scaling up and down, and translating. Similarity transformation and affine transformation are two important concepts of IFS. The geometric properties of affine transformations are:

I: The inverse of an affine transformation is still an affine transformation.

The key to generate fractal graphs using the IFS method is to find out the corresponding IFS codes, for those iterations of regular graphs, their obvious geometric relations can be utilized to determine the corresponding IFS codes. However, for those graphs with irregular boundaries, the collage method can be used to approximate the target graph.

There are two methods for determining the general IFS codes, the interactive determination method and the computational determination method. The following collage of its target graph with four subgraphs results in an affine transformation as shown in the following equation. I.e: (2) ω1:{ x′=0.25x+b1y+e1y′=c1x+0.25y+f1${{\omega }_{1}}:\left\{ \begin{array}{*{35}{l}} x\prime =0.25x+{{b}_{1}}y+{{e}_{1}} \\ y\prime ={{c}_{1}}x+0.25y+{{f}_{1}} \\ \end{array} \right.$ (3) ω2:{ x′=0.2x+b2y+e2y′=c2x+0.2y+f2${{\omega }_{2}}:\left\{ \begin{array}{*{35}{l}} x\prime =0.2x+{{b}_{2}}y+{{e}_{2}} \\ y\prime ={{c}_{2}}x+0.2y+{{f}_{2}} \\ \end{array} \right.$ (4) ω3:{ x′=0.25x+b3y+e3y′=c3x+0.25y+f3${{\omega }_{3}}:\left\{ \begin{array}{*{35}{l}} x\prime =0.25x+{{b}_{3}}y+{{e}_{3}} \\ y\prime ={{c}_{3}}x+0.25y+{{f}_{3}} \\ \end{array} \right.$ (5) ω4:{ x′=0.50x+b4y+e4y′=c4x+0.50y+f4${{\omega }_{4}}:\left\{ \begin{array}{*{35}{l}} x\prime =0.50x+{{b}_{4}}y+{{e}_{4}} \\ y\prime ={{c}_{4}}x+0.50y+{{f}_{4}} \\ \end{array} \right.$

At this point the 4 sub-maps were 1/4, 1/5, 1/4, and 1/2 sized replicas of the target maple map. The affine transformation parameters of each sub-map are then adjusted interactively on the screen in sequence and essentially cover the target map. There can be overlap between subgraphs, but the overlap should be kept as small as possible in order to minimize the amount of repetitive computation so that the IFS code for the graph can be obtained.

The method of solving the system of linear equations by three-point correspondence is used to find the affine transform coefficients and obtain the system of equations: (6) { ax1+by1+e=r1ax2+by2+e=r2ax3+by3+e=r3\[\left\{ \begin{array}{*{35}{l}} a{{x}_{1}}+b{{y}_{1}}+e={{r}_{1}} \\ a{{x}_{2}}+b{{y}_{2}}+e={{r}_{2}} \\ a{{x}_{3}}+b{{y}_{3}}+e={{r}_{3}} \\ \end{array} \right.\] (7) { cx1+dy1+f=s1cx2+dy2+f=s3cx3+dy3+f=s3\[\left\{ \begin{array}{*{35}{l}} c{{x}_{1}}+d{{y}_{1}}+f={{s}_{1}} \\ c{{x}_{2}}+d{{y}_{2}}+f={{s}_{3}} \\ c{{x}_{3}}+d{{y}_{3}}+f={{s}_{3}} \\ \end{array} \right.\]

The values of a, b and e can be obtained by solving the above system of linear equations (6). Solving the system of linear equations (7) yields the values of c, d, and f, and the corresponding IFS codes can be obtained.

The attraction of the IFS method is that it is the “inverse problem” of fractal iterative generation, according to the splicing theorem, for a given image, to find its IFS code, you can greatly compress the information.

The flow of Markov chain based fractal graph generation algorithm is shown in Fig. 1. First the coefficient matrix is created (the number of rows of matrix M is n and the number of columns is 6). The number of rows n depends on the graph and the graph is divided into n parts. Then the number of rows of M is n and each row of M represents a data processing method. The larger the n, the richer the graph that can be represented. Each column of M is denoted as m_i1, m_i2, m_i3, m_i4, m_i5, m_i6(i = 1,2,3,⋯,n), which represents the coefficients a, b, α, β, E, F, respectively.This coefficient matrix can establish several state transfer functions, which control the main information of the image such as the position and angle of the image, and the results of the algorithms in this paper are mainly determined by this coefficient matrix.

Figure 1.

The fractal graphics generation algorithm process of the markov chain

The fractal code matrix is M, and each row vector of the two-dimensional coefficient matrix M is adjusted to a 2×3 matrix, and the resulting matrix M is adjusted to a three-dimensional matrix A^[i] of 2×3×i(i = 1,2,3,⋯,n). For example, the elements of the first row of M are M₁ converted to A^[i]: (8) M1=[ m11m12m13m14m15m16 ]\[{{M}_{1}}=\left[ \begin{array}{*{35}{l}} {{m}_{11}} & {{m}_{12}} & {{m}_{13}} & {{m}_{14}} & {{m}_{15}} & {{m}_{16}} \\ \end{array} \right]\] (9) A[i]=[ A11A13A15A12A14A16 ]=[ m11cos(m13)−m12sin(m14)m15m11sin(m13)m11cos(m14)m16 ]\[{{A}^{[i]}}=\left[ \begin{array}{*{35}{l}} {{A}_{11}} & {{A}_{13}} & {{A}_{15}} \\ {{A}_{12}} & {{A}_{14}} & {{A}_{16}} \\ \end{array} \right]=\left[ \begin{matrix} {{m}_{11}}\cos \left( {{m}_{13}} \right) & -{{m}_{12}}\sin \left( {{m}_{14}} \right) & {{m}_{15}} \\ {{m}_{11}}\sin \left( {{m}_{13}} \right) & {{m}_{11}}\cos \left( {{m}_{14}} \right) & {{m}_{16}} \\ \end{matrix} \right]\]

Where m₁₁ and m₁₂ are the scaling factors for x and y axes respectively. m₁₃, m₁₄ are the rotation angles along the counterclockwise direction for the x and y axes, respectively. m₁₅ and m₁₆ are the translation coordinates along the x and y axes, respectively.

The order of entry into each state transfer function is not deterministic, noting that the probability of entering a A^[i] state transfer function for an operation is p_i, the vector p(p₁,p₂,⋯,p_n) controls the distribution of the overall graph sparsity. Where p the sum of all elements in the vector is 1, representing that each iteration of the operation must correspond to a state transfer function. If p_i is large, the probability of entering the corresponding state transfer function is large, and the more points there are in controlling the graph at the corresponding location, the finer the graph is at that location. Setting the number of iterations N, each iteration generates the coordinate information of a point, and the result is used as the state input for the next time, then the number of iterations N corresponds to the coordinate information of N points in the result of the operation. Therefore, the larger the value of N, the finer the graph produced by the operation.

Create an iterative matrix G (G is a matrix of rows 2 and columns N) and set the initial value of all elements of matrix G to 0 as the initial state of the set of states. A set of column vectors obtained from each iteration operation is used as the position information of a point in the image (a set of horizontal and vertical coordinates) to replace the column vectors in the matrix sequentially, and finally the termination state set is generated.

The method of randomly entering the state transfer function generates the grid matrix q (q is a square matrix with rows and columns all n) All the elements in each column vector have the values of the elements of the corresponding column vectors in p. The probability accumulation is obtained by taking the lower triangular matrix for matrix q and summing the row vectors as shown in equation (10). At this point q is a column vector of length n. i.e: (10) q=[ p1+0+0+0p1+p2+0+0p1+p2+p3+0p1+p2+p3+p4 ]\[q=\left[ \begin{array}{*{35}{l}} {{p}_{1}} & + & 0 & + & 0 & + & 0 \\ {{p}_{1}} & + & {{p}_{2}} & + & 0 & + & 0 \\ {{p}_{1}} & + & {{p}_{2}} & + & {{p}_{3}} & + & 0 \\ {{p}_{1}} & + & {{p}_{2}} & + & {{p}_{3}} & + & {{p}_{4}} \\ \end{array} \right]\]

On the interval [0,1], take any value k and differentiate k from the q vector. From the above, we can see ∑i=1npi=1$\sum\limits_{i=1}^{n}{{{p}_{i}}}=1$, so the result produced by doing the difference must start from a row with a negative value, remember that the number of rows here is d, then if the value of the value of d,k is taken to satisfy the value: (11) qd−1<k<qd\[{{q}_{d-1}}<k<{{q}_{d}}\] (12) qd−1=∑i=1d−1pi\[{{q}_{d-1}}=\sum\limits_{i=1}^{d-1}{{{p}_{i}}}\] (13) qd=∑i=1dpi\[{{q}_{d}}=\sum\limits_{i=1}^{d}{{{p}_{i}}}\]

The random number k corresponding to d can take a sequence of length: (14) qd−qd−1=pd\[{{q}_{d}}-{{q}_{d-1}}={{p}_{d}}\]

It can be seen that the probability of taking row d corresponds to the value in row d of matrix p, i.e., p can be used as the probability distribution column for d and q as the probability distribution function for d. The assignment of values to matrix G is done in the following way: (15) Gj+1=[ xj+1yj+1 ]=A[d]×[ xjyj1 ]\[{{G}_{j+1}}=\left[ \begin{matrix} {{x}_{j+1}} \\ {{y}_{j+1}} \\ \end{matrix} \right]={{A}^{[d]}}\times \left[ \begin{matrix} {{x}_{j}} \\ {{y}_{j}} \\ 1 \\ \end{matrix} \right]\]

To wit: (16) xj+1=A[d](1,1)×xj+A[d](1,2)×yj+A[d](1,3)\[{{x}_{j+1}}={{A}^{[d]}}\left( 1,1 \right)\times {{x}_{j}}+{{A}^{[d]}}\left( 1,2 \right)\times {{y}_{j}}+{{A}^{[d]}}\left( 1,3 \right)\] (17) yj+1=A[d](2,1)×xj+A[d](2,2)×yj+A[d](2,3)\[{{y}_{j+1}}={{A}^{[d]}}\left( 2,1 \right)\times {{x}_{j}}+{{A}^{[d]}}\left( 2,2 \right)\times {{y}_{j}}+{{A}^{[d]}}\left( 2,3 \right)\]

The reason for the limited ability to describe the results produced under a deterministic state system is that with a limited number of state initial values and state transfer functions, they can only be described by strict lines, or straight line segments, or a small number of curved segments with defined paths. A small number of lines can appear to have a rigid outline, or even a graph that is difficult to describe. The fractal geometry under the nondeterministic state system generated using Markov chains can theoretically have an infinite number of initial values, and it continues the iterative operation by using the state values at any one time node as the initial values, which makes up for the disadvantage of the limited amount of data that can be used for the graphs generated under the deterministic state system.