## 2 Models of Emotion in Pictures
### The Discrete Emotion Model  
The discrete or categorical model classifies emotions into specific labels, referred to as basic emotions, discrete emotions, emotion norms, or norms for short [31]. Categorical emotion theories claim that dimensional models do not accurately reflect the neural systems underlying emotional responses [8]. Proponents of these theories propose instead that there is a universal set of emotions [8]. However, the exact number of basic emotions is a point of dispute among experts. Contemporary psychological theories propose discrete emotion models with up to 24 different norms [6]. Nevertheless, most researchers agree that six primary emotions--happiness, surprise, fear, sadness, anger, and disgust--are universal across cultures and have an evolutionary and biological basis [32]. According to the universality hypothesis, these six basic internal human emotions are expressed with the same facial movements in all cultures, which supports a universal recognition of emotions in facial expressions. Moreover, some experimental evidence exists for mixed emotion models, at least during some life stages [33]. However, in all models, the simultaneous presence of more than one discrete emotion is allowed but with different intensities. The emotion with the highest intensity is called the dominant emotion. A clear example of how a discrete model is used in computer systems is the NimStim stimuli set of facial expressions [34] as illustrated in Figure 2. The NimStim picture dataset was created to provide facial expressions that untrained individuals would recognize as typical of  
Figure 1: Picture stimuli from the OASIS dataset as an example of the dimensional emotion model (measured on a 1–7 Likert scale) with valence on the \(x\)-axis and arousal on the \(y\)-axis. The markers denote four different image categories: object, person, animal, and scene. Approximate positions of discrete emotions are indicated. For more information on the dimensional model, see [28; 29; 30]. Reproduced with permission from [PERSON]; [PERSON]; [PERSON], Introducing the open affective standardized image set (OASIS); published by Springer, 2017.  
research participants. This set is comprehensive, multiracial, and available online to the scientific community together with the accompanying pictures' normative ratings [34]. In general, datasets similar to NimStim use the discrete model. They are important because behavior, cognition, and emotion experiments often rely on highly standardized picture datasets to understand how people respond to different faces that convey visually specific, discrete emotions of varying intensity. These datasets have played an essential role in the development of face science [35].  
Individual norms can be mathematically described as mutually orthogonal unit vectors that collaboratively define an \(n\)-dimensional emotion space. Intensity of each vector is a dimensionless value normalized in the range [0.0, 100.0]. Therefore, a discrete emotion \(a^{dis}\) in a 6-component emotion space defined with norms of happiness, sadness, surprise, anger, fear, disgust is a vector: \(a^{dis}=\left[a_{nappiness}^{dis},a_{saddness}^{dis},a_{surprise}^{dis},a_{ anger}^{dis},a_{fear}^{dis},a_{disgust}^{dis}\right]^{T}\). Consequently, given two discrete emotions \(a^{dis}\) and \(b^{dis}\) where each contains \(n\) components, the distance \(dist\left(a^{dis},b^{dis}\right)\) between them can be written as:  
\[\begin{split} dist\left(a^{dis},b^{dis}\right)=1-sim\left(a^{dis},b^{dis}\right)\\ sim\left(a^{dis},b^{dis}\right)=\cos(\Theta)=\frac{d^{dis},b^{ dis}}{\|a^{dis}\|\|^{b^{dis}}\|}=\frac{\sum_{i=1}^{n}a_{i}^{dis}b^{dis}}{\sqrt{\sum_{i=1}^{n}a_{i}^{ dis}\sqrt{\sum_{i=1}^{n}b_{i}^{dis}}}}\end{split} \tag{2}\]  
The distance \(dist\left(a^{dis},b^{dis}\right)\) is a numerical value in the range [0.0, 1.0].