The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations

With the continuous change and development of modern information technology, in the field of art technology research, eye movement experimental research methods study the stylised art image transfer art research which has a leading role. At present, good results have been obtained in the study of the image effect. Research using eye movement experimental research methods can be more objectively recorded and studied, but little is involved. The eye-tracking device was used to record the eye-tracking track of the subjects, which was quantified, and the experimental data was used to scientifically demonstrate the reading effect of the learners on the teaching art images. China attaches great importance to the development of information technology, so the study of artistic image effect can not only promote the application of information technology in the process of artistic development but also conform to the development of the times [1]. At present, there are many methods to study the cognitive processing of artistic images in reading, and among which the most mainstream and effective method is the eye movement method [2]. Eye movement experimental method plays an important role in various fields (for example, modern psychology) and has gradually become a mainstream research method. The brain is connected to the eye, and what the eye sees responds to the brain [3]. Therefore, the learner's mental process can be analysed by observing what people pay attention to with the help of an eye tracker. According to the way the visual system works, what you see means what you think, and getting your gaze equals attention. The more the subjects looked, the more they thought, but not that the subjects understood what they saw. The reason is that sometimes there is a lot of casual gazes or gaze that do not understand its meaning.

Eye movement technology refers to the use of eye movement equipment to collect the eye movements of the subjects, analyse and obtain such things as fixation time, fixation point, pupil size, eye jump distance, etc. so as to study the psychological process of the subjects. The line-of-sight tracking system can be divided into three parts: the centre coordinate extraction system of the pupil, the optical system, the superposition system of view and pupil coordinates, and the image and data recording and analysis system [4]. The basic states of eye movement include following the movement, fixation and saccades. It has become a scientific and rigorous quantitative research method for eye movement experiments to analyse the psychological activities of the subjects scientifically by means of fuzzy differential equations. The combination of eye movement experiments and artistic image transfer can effectively improve the readers’ artistic skills. Most of the previous image studies were qualitative and lacked quantitative analysis. The use of eye movement research technology combined with it is just to make up for the lack of quantitative analysis of research. The study of image migration uses new technical routes and research ideas to promote the application of eye movement experimental research in the field of educational technology. In this paper, eye movement experimental research is used as the experimental research method from the perspective of learners, and with the help of fuzzy differential equation theory, the eye movement data of image readers is deeply analysed and the application strategy of artistic image transfer research is obtained. The application of eye movement experiments to artistic image migration and the use of fuzzy differential equations in the field of artistic technology make the analysis of artistic image migration more scientific and intuitive [5].

Fractional order fuzzy differential equation is a fractional order differential equation defined on a specific fuzzy set, also known as a fractional order differential equation with uncertainty. Fractional fuzzy differential equations have the advantages of nonlocal and memory of fractional derivatives and are more widely used. In the field of art, the application of fuzzy differential equations can reflect the subjectivity or uncertainty in the law of determination, and it is more appropriate to describe the problem. The initial value problem of a fractional fuzzy differential equation is one of the basic research objects of qualitative theory of fractional fuzzy differential equation. In this paper, we discuss the existence, non-existence, uniqueness or multiplicity, boundedness, periodicity, vibration, stability and branching of the initial value problem of fractional equations by means of nonlinear analysis [6].

The initial value problem of fractional fuzzy differential equations: (1) {RLDqu=f(t,u(t)),q∈(0,1)limt→0+t1−qu(t)=u0∈E1 \left\{{\matrix{{^{RL}{D^q}u = f(t,u(t)),q \in (0,1)} \cr {{{\lim}_{t \to {0^ +}}}{t^{1 - q}}u(t) = {u_0} \in {E^1}} \cr}} \right.

The RLqD is based on the Riemann-Liouville H- derivative, and the first value problem is solved by studying the E¹ corresponding fuzzy integral equation in one-dimensional fuzzy number space. (2) {GHRLDa+qu(t)=f(t,u(t)),q∈(0,1)GHRLDa+q−1u(t0)=u0∈E1 \left\{{\matrix{{_{GH}^{RL}D_{{a^ +}}^qu(t) = f(t,u(t)),q \in (0,1)} \cr {_{GH}^{RL}D_{{a^ +}}^{q - 1}u({t_0}) = {u_0} \in {E^1}} \cr}} \right.

Two kinds of nonlinear fractional-order fuzzy integral equations are obtained by using the up-down solution method: (3) u(t)=g(t)⊕1Γ(q)∫0t(t−s)q−1f(t,u(s),Tu(s))dsu(t)=g(t)−1Γ(q)∫0t(t−s)q−1f(t,u(s),Tu(s))ds \matrix{{u(t) = g(t) \oplus {1 \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}f(t,u(s),Tu(s))ds} \hfill \cr {u(t) = g(t) - {1 \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}f(t,u(s),Tu(s))ds} \hfill \cr}

The results of the existence and uniqueness of the solution of the fuzzy integral differential equation is (4) {GHRLDa+qu(t)=f(t,u(t),Tu(t)), q∈(0,1) limt→0+t1−qu(t)=u0∈E1 \left\{{\matrix{{_{GH}^{RL}D_{{a^ +}}^qu\left(t \right) = f\left({t,u\left(t \right),Tu\left(t \right)} \right),\;\;\;q \in \left({0,1} \right)} \hfill \cr \hfill \cr {\mathop {\lim}\limits_{t \to {0^ +}} {t^{1 - q}}u\left(t \right) = {u_0} \in {E^1}\;\;} \hfill \cr}} \right.

Among them, (5) Tu(t)=∫0tT(t,s)u(s)ds Tu(t) = \int_0^t T(t,s)u(s)ds

Based on the application of the eye movement experiment in this paper, the definition of the solution of two set marks and initial value problems is given, namely (6) Ω={u:u|(−∞,0]∈CE(−∞,0],u|[0,a]∈CE[0,a]} \Omega = \left\{{u:u\left| {_{\left({- \infty,\left. 0 \right]} \right.} \in {C^E}\left({- \infty,\left. 0 \right]} \right.,u\left| {_{\left[ {0,a} \right]}} \right. \in {C^E}\left[ {0,a} \right]} \right.} \right\}

Along with, (7) E={u∈CE(−∞,T]|:u(t)=ϕ(t)(−∞,0],(u(t),ϕ(t))≤ρ[0,T]} E = \left\{{u \in {C^E}\left({- \infty,\left. T \right]} \right.\left| {:u\left(t \right) = \phi \left(t \right)\left({- \infty,\left. 0 \right]} \right.,\left({u\left(t \right),\phi \left(t \right)} \right) \le \rho \left[ {0,T} \right]} \right.} \right\}

Suppose u(t)[0, T] as a differential.

Then the upper formula of the initial value problem is equivalent to the fuzzy differential equation (8) u(t)={ϕ(t), t∈(−∞,0]ϕ(0)⊕1Γ(q)∫0t(t−s)q−1r(s)f(t,us)ds, t∈[0,T] u\left(t \right) = \left\{{\matrix{{\phi \left(t \right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t \in \left({- \infty,\left. 0 \right]} \right.} \hfill \cr {\phi \left(0 \right) \oplus {1 \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)f\left({t,{u_s}} \right)ds,\;\;t \in \left[ {0,T} \right]} \hfill \cr}} \right.

In fact, because it is the solution of the initial value problem, it can be u(t) expressed as: (9) u(t)=u(0)⊕1Γ(q)∫0t(t−s)q−1r(s)f(t,us)ds u(t) = u(0) \oplus {1 \over {\Gamma (q)}}\int_0^t {(t - s)^{q - 1}}r(s)f(t,{u_s})ds

Combined with initial conditions: (10) u(t)=ϕ(0)⊕1Γ(q)∫0t(t−s)q−1r(s)f(t,us)ds u(t) = \phi (0) \oplus {1 \over {\Gamma (q)}}\int_0^t {(t - s)^{q - 1}}r(s)f(t,{u_s})ds

In addition, for arbitrary cases, t ∈ [0, T] according to: (11) d(u(t),ϕ(0))=d(1Γ(q)∫0t(t−s)q−1r(s)f(t,us)ds,0^)≤1Γ(q)∫0t(t−s)q−1r(s)d(f(s,us),0^)ds≤KrΓ(q)∫0t(t−s)q−1ds=KrTqΓ(q+1)≤ρ \matrix{{d(u(t),\phi (0)) = d\left({{1 \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)f(t,{u_s})ds,\hat 0} \right) \le {1 \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)d(f(s,{u_s}),\hat 0)ds} \hfill \cr {\le {{Kr} \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}ds = {{Kr{T^q}} \over {\Gamma (q + 1)}} \le \rho} \hfill \cr}

u belongs to E.

Construct the iterative sequence u. Let the initial function be: (12) u(0)(t)={ϕ(t),t∈(−∞,0]ϕ(0),t∈[0,T] {u^{(0)}}(t) = \left\{{\matrix{{\phi (t),t \in (- \infty,0]} \cr {\phi (0),t \in [0,T]} \cr}} \right.

Clearly, the d(u(0)(t),0∧)≤ρ d\left({{u^{\left(0 \right)}}\left(t \right),\mathop 0\limits^ \wedge} \right) \le \rho , t ∈ [0, T] definition: (13) um+1(t)={ϕ(t), t∈(−∞,0]ϕ(0)⊕1Γ(q)∫0t(t−s)q−1r(s)f(s,usm)ds, t∈[0,T] {u^{m + 1}}\left(t \right) = \left\{{\matrix{{\phi \left(t \right),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;t \in \left({- \infty,\left. 0 \right]} \right.} \hfill \cr {\phi \left(0 \right) \oplus {1 \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)f\left({s,u_s^m} \right)ds,\;\;t \in \left[ {0,T} \right]} \hfill \cr}} \right.

Among them m = 0, 1,.., here t ∈ [0, T]: (14) d(u(1)(t),u(t)(t))=d(1Γ(q)∫0t(t−s)q−1r(s)f(t,us(0))ds,0∧)≤1Γ(q)∫0t(t−s)q−1r(s)d(f(s,us(0)),0∧)ds≤KrΓ(q)∫0t(t−s)q−1ds=KrtqΓ(q+1) \matrix{{d\left({{u^{\left(1 \right)}}\left(t \right),{u^{\left(t \right)}}\left(t \right)} \right) = d\left({{1 \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)f\left({t,u_s^{\left(0 \right)}} \right)ds,\mathop 0\limits^ \wedge} \right)} \hfill \cr {\le {1 \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)d\left({f\left({s,u_s^{\left(0 \right)}} \right),\mathop 0\limits^ \wedge} \right)ds \le {{Kr} \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}ds = {{Kr{t^q}} \over {\Gamma \left({q + 1} \right)}}} \hfill \cr}

Because the fuzzy numerical function f satisfies the local conditions, there are t ∈ [0, T]: (15) d(u(2)(t),u(1)(t))≤1Γ(q)∫0t(t−s)q−1r(s)d(f(s,us(1)),f(s,us(0)))ds≤LΓ(q)∫0t(t−s)q−1r(s)D(us(1),us(0))ds≤LΓ(q)∫0t(t−s)q−1r(s)supτ∈(−∞,0]d(us(1)(τ),us(0)(τ))ds≤LΓ(q)∫0t(t−s)q−1r(s)supτ∈(−∞,0]d(us(1)(s+τ),us(0)(s+τ))ds≤LΓ(q)∫0t(t−s)q−1r(s)supθ∈(−∞,s]d(us(1)(θ),us(0)(θ))ds≤Kr2Γ(q+1)Γ(q)LΓ(q)∫0t(t−s)q−1sqds=KLr2L2t2qΓ(2q+1) \matrix{{d({u^{(2)}}(t),{u^{(1)}}(t))} \hfill & {\le {1 \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)d(f(s,u_s^{(1)}),f(s,u_s^{(0)}))ds} \hfill \cr {} \hfill & {\le {L \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)D(u_s^{(1)},u_s^{(0)})ds} \hfill \cr {} \hfill & {\le {L \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)\mathop {\sup}\limits_{\tau \in (- \infty,0]} d(u_s^{(1)}(\tau),u_s^{(0)}(\tau))ds} \hfill \cr {} \hfill & {\le {L \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)\mathop {\sup}\limits_{\tau \in (- \infty,0]} d(u_s^{(1)}(s + \tau),u_s^{(0)}(s + \tau))ds} \hfill \cr {} \hfill & {\le {L \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}r(s)\mathop {\sup}\limits_{\theta \in (- \infty,s]} d(u_s^{(1)}(\theta),u_s^{(0)}(\theta))ds} \hfill \cr {} \hfill & {\le {{K{r^2}} \over {\Gamma (q + 1)\Gamma (q)}}{L \over {\Gamma (q)}}\int_0^t {{(t - s)}^{q - 1}}{s^q}ds = {K \over L}{{{r^2}{L^2}{t^{2q}}} \over {\Gamma (2q + 1)}}} \hfill \cr}

Assumption: (16) d(u(m)(t),u(m−1)(t))≤=KLrmLmtmqΓ(mq+1) d\left({{u^{\left(m \right)}}\left(t \right),{u^{\left({m - 1} \right)}}\left(t \right)} \right) \le = {K \over L}{{{r^m}{L^m}{t^{mq}}} \over {\Gamma \left({mq + 1} \right)}}

The establishment of t ∈ [0, T], there are (17) d(u(m+1)(t),u(m)(t))≤1Γ(q)∫0t(t−s)q−1r(s)d(f(s,us(m)),f(s,us(m−1)))ds ≤LΓ(q)∫0t(t−s)q−1r(s)D∞(us(m),us(m−1))ds =LΓ(q)∫0t(t−s)q−1r(s)supτ∈(−∞,0]d(us(m)(τ),us(m−1)(τ))ds =LΓ(q)∫0t(t−s)q−1r(s)supθ∈(−∞,0]d(us(m)(θ),us(m−1)(θ))ds ≤KLrmLmΓ(mq+1)LΓ(q)∫0t(t−s)q−1smqds=KLrm+1Lm+1tm+1qΓ((m+1)q+1) \matrix{{d\left({{u^{\left({m + 1} \right)}}\left(t \right),{u^{\left(m \right)}}\left(t \right)} \right) \le {1 \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)d\left({f\left({s,u_s^{\left(m \right)}} \right),f\left({s,u_s^{\left({m - 1} \right)}} \right)} \right)ds} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {L \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right){D_\infty}\left({u_s^{\left(m \right)},u_s^{\left({m - 1} \right)}} \right)ds} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {L \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)\mathop {\sup}\limits_{\tau \in \left({- \infty,\left. 0 \right]} \right.} d\left({u_s^{\left(m \right)}\left(\tau \right),u_s^{\left({m - 1} \right)}\left(\tau \right)} \right)ds} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = {L \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}r\left(s \right)\mathop {\sup}\limits_{\theta \in \left({- \infty,\left. 0 \right]} \right.} d\left({u_s^{\left(m \right)}\left(\theta \right),u_s^{\left({m - 1} \right)}\left(\theta \right)} \right)ds} \hfill \cr {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \le {K \over L}{{{r^m}{L^m}} \over {\Gamma \left({mq + 1} \right)}}{L \over {\Gamma \left(q \right)}}\int_0^t {{\left({t - s} \right)}^{q - 1}}{s^{mq}}ds = {K \over L}{{{r^{m + 1}}{L^{m + 1}}{t^{m + 1q}}} \over {\Gamma \left({\left({m + 1} \right)q + 1} \right)}}} \hfill \cr}

According to the mathematical induction method, the inequality holds for all m = 1, and it is proved that the fuzzy differential equation can be applied to this art eye movement experiment [7].

The purpose of this study is to explore how to maximise the effect of artistic images in artistic image migration. To understand the cognitive process of learners in reading artistic images, the following research hypotheses are proposed based on the theoretical review: (1) There is no significant difference between cognitive style and fixation times of artistic images. (2) There was no significant difference between cognitive style and fixation time of artistic images. In this study, an eye movement experiment was used to study the effect of teaching art images. With the help of the eye movement index, the cognitive process of learners is explored, and the eye movement data such as the number of fixation times, fixation time and hot spot chart of learners are recorded by eye movement instrument so as to analyse the eye movement track and the fixation points of learners during the learning process. The research process includes the training process and the experimental process.

During the training, before all the work was started, the experimenter needed to check the eyes and the blue dots showed on the screen. The subjects had to keep their eyes fixed on the blue dots and keep them for a period of time so as to check the eye movement device. The blue dots would appear five times in a row. After verification, the subject maintains his or her posture, and eye movements are recorded by the eye movement device [8]. At this point, the subjects are introduced to the pre-experiment program, and the space bar will control the beginning and end of the experiment. After the subjects fully understood the experiment process, the training process was over. During the experiment, the eyes of the subjects were checked according to the training procedure. After the verification, the subjects were informed to start the experiment officially. When the subject is ready, press the space bar to start. At the end of the experiment, press the space bar again. After the experiment, the observers interviewed the subjects and collected experimental data. In this study, the eye movement analysis software of an eye movement instrument was used to extract the eye movement data of the subjects. Based on the theory of fuzzy differential equations, SPSS statistical software was used to analyse the collected eye movement data. In the eye movement test, the fixation time of the target object is determined by the fixation time of each fixation point [9]. At a single fixation point, the duration of the subject's gaze can be seen in the line of sight as shown in Figure 1. The circle represents each fixation point, and the size of the circle represents the fixation time. The larger the area, the longer the fixation; the smaller the area, the shorter the gaze time. Each circle represents the subject's eye movement fixation point. According to the circular area, the fixation time of each fixation point can be inferred, and the sum of each fixation point time is the total fixation time of the experiment. The first time it reaches the target region is shown in Figure 2. In the experiment, the experimental materials of eye movement were divided into several regions of interest. After the target interest area is specified, the time when the line of sight reaches the specified interest area for the first time represents the time when the target area is first reached. This time is a point in time, not a time period.

Fig. 1

Fig. 2

This study uses the eye movement experiment research method, with the help of fuzzy differential equations, and scientifically analyses the influence of art image application on learners’ learning effect. In art image making, we should follow the learner's learning characteristics, evaluate the quality of art image by ourselves and take eye movement instruments as the assistant. Using line-of-sight tracking technology to accurately record eye movement data can evaluate the level of artistic image production from the perspective of learners and will be a trend of evaluation methods. In this paper, the method of eye movement experiment is related to the effect of stylised art image migration, and the scientific quantitative analysis is carried out so as to explore the deeper meaning and function of art image from a scientific point of view.