An Improved Three-Dimensional Model for Emotion Based on Fuzzy Theory

Emotion Model is the basis of facial expression recognition system. The constructed emotional model should not only match facial expressions with emotions, but also reflect the location relationship between different emotions. In this way, it is easy to understand the current emotion of an individual through the analysis of the acquired facial expression information. This paper constructs an improved three-dimensional model for emotion based on fuzzy theory, which corresponds to the facial features to emotions based on the basic emotions proposed by Ekman. What’s more, the three-dimensional model for motion is able to divide every emotion into three different groups which can show the positional relationship visually and quantitatively and at the same time determine the degree of emotion based on fuzzy theory.


Introduction
The Emotion Model uses mathematical ideas to formalize changes in emotions, carries out quantitative analysis of emotions, and then reflects the changing trends of people's psychology and emotions. At present, the existing emotion models have one-dimensional, two-dimensional, three-dimensional, four-dimensional representations, including OCC models

Six Emotions Proposed by Ekman
Human beings express their emotions through facial expressions mainly by using special movement changes in various parts of the human face. Therefore, the premise of obtaining emotional information is that the emotional types need to correspond to facial expressions. By studying the different expressions of the faces of people from different countries, Ekman defines six types of emotions corresponding to the associated facial expressions: anger, disgust, fear, happiness, sadness, and surprise [8]. Table 1 shows the correspondence between the Ekman's basic emotion type and the facial expression. Through these correspondences, the emotional information to be expressed can be obtained from the facial expression.

Three-Dimensional State-Space Emotion Model
Emotion has multiple dimensions, which can be deemed as a feature of emotion.   coordinates, representing 27 emotions [9].
As one's emotions change, the corresponding emotional position in the three-dimensional state-space emotion model will move to another position. Similar to the above process, the process of emotional change can be referred to as the Markov state-transition matrix at various locations in this confined region [10].
There are m kinds of emotion types, and each type of emotion can be divided into n levels, and the emotion space contains m n state. Let m l n = the Markov probability transfer matrix of the l dimension will be the formula (1).
The above process is a description of the probability that emotions change from one type to another. In this way, each emotion in the emotional model space can be divided into various degrees, and the possibility of emotional change is affected by the position and distance of coordinates corresponding to different emotions in three-dimensional space.
In order to evaluate and measure the expression of emotions in the emotional space, the mathematical tool of entropy can be used. For the sentiment model space containing l points and m dimensions expressed in formula (2), if the emotion is the i category at a certain time, the probability of the transition to other categories constitutes the probability vector of the emotional change: Emotional entropy is defined as: In formula (4), the emotional entropy of i state is i A so the probability that i emotional state changes to j emotional state is , 1) The possibility that an emotional category in an emotional space is transferred to another category (including itself) decreases as the distance between the category and other categories increases.
2) If there is no interference from the outside world, the emotion gradually approaches the vicinity of the origin (calm point), that is, the vicinity of the origin is a stable point, and the emotion gradually becomes stable.
3) If there is a contradiction between emotions, the probability of simultaneous performance is extremely small. For example, the two emotions of happiness and fear have a very low probability of simultaneous performance.

Fuzzy Theory
In actual daily life, lots of things, laws, and phenomena are described as uncertain, transitional, and non-absolute. For example, the temperature, the degree of beauty, the degree of youthfulness, etc., cannot be accurately quantified by traditional mathematics description. In order to solve these uncertainties, non-absoluteness, and transitional problems, fuzzy theory is adopted [11]. Accurate models and methods can be used to describe, process, and study certain unclear boundaries, incomplete information, and inaccurate data.
Set a domain as U, for any u U ∈ define a mapping to the closed interval [0, 1] as shown in equation (5): In formula (5), A is a fuzzy set on the domain U, also called fuzzy subset, as the membership function A of the mapping shown in formula (5) 2) In the case that U is finite set, the fuzzy set is: In the formula (7), ∑ on the right side represents a symbol, and does not represent the addition of numbers. Or it can be expressed as a vector form of Journal of Computer and Communications Equation (8), in which case the order of the elements in each set has been determined: 3) If the domain U is an infinite set, you can represent the fuzzy set A as: The premise of establishing a fuzzy set is to construct the membership degree.
So far, the methods of constructing the membership function include a variety of methods. The fuzzy statistics method and fuzzy distribution are introduced below.
The fuzzy statistical method to obtain the mapping function from the domain U to the closed interval [0, 1] mainly includes four elements: the domain U; one element 0 u in the domain; a variable set * A of boundaries, * A is corresponding to a fuzzy concept a and fuzzy set A; condition s. In the idea of fuzzy statistical method, the appearance of ambiguity is caused by the uncertainty of the division s of corresponding concepts a in statistics.
In the process of realizing the fuzzy statistical method, it is necessary to judge whether 0 u belongs to by multiple tests * A , the number of 0 u A ∈ in the subtest n is calculated, and the membership frequency of 0 u can be obtained.

Improved Three-Dimensional Emotion Model
In the application of human-computer interaction systems, it is very important • According to the actual situation, we can know that each type of emotion has a certain range of variation. Therefore, in order to distinguish the intensity of each type of emotion, combined with the fuzzy theory, each type of emotion Journal of Computer and Communications is defined to have a certain spherical range, and different areas in the range are expressed. The degree of strong, medium, or weak emotions of a certain kind of emotion is defined by fuzzy rules. The emotional model obtained by combining Ekman's basic emotion, state space method emotion model and fuzzy theory is shown in Figure 2. The schematic diagram of each type of emotion degree is shown in Figure 3. In Figure 3, Let D show the Euclidean distance from the measured emotion to the standard emotion [12], that is, the distance between the projection point corresponding to the tested emotion and the basic standard emotional center, as shown in formula  , then the degree of emotion is weak.
The fuzzy rules defined above are organized into the form shown in Table 2.
Among them, each type of emotion is divided into different types, that is, each type of emotional degree is divided into six types of emotions 1  Through the establishment of the above model, the basic process of using the • Obtaining a useful facial area by detecting a face.
• Correspond different types of facial expression characteristics to emotion types, and perform rough classification recognition on facial expressions to obtain the main emotion types of facial expressions.
• Understand the emotion division based on the fuzzy rules so as to get detailed information.
It can be seen from the above process that the model is not only used for the rough classification of emotions and facial expressions in the expression recognition process, but also the fuzzy rules in the model are used to implement the emotional classification of emotions.

Conclusion
This paper introduces a three-dimensional emotion model based on the improvement of fuzzy theory. Firstly, the two basic emotion models of Ekman's six basic emotions and three-dimensional state space emotion model are introduced.
Then the basic knowledge of fuzzy theory is introduced, including the definition and determination of membership function, the definition of fuzzy rules, etc.
Finally, the merits as well as demerits of Ekman's six basic emotion models and three-dimensional state space emotion models in the background of human-computer interaction system are analyzed. Moreover, an improved three dimensional model for emotion based on fuzzy theory is generated, which can map Ekman's six emotions to three dimensional space-state model and realize a win-win situation. Furthermore, by defining the rules of fuzzy theory, we can make the emotion division more detailed and get more consistent with the actual situation. At the same time, the improved model can determine the basic process to get more detailed emotion information by using expression recognition.