The operator arg means that nGTW|PRM(PIDi) is the artwork that gives the maximal value of W(nPIDi, nk)
among nk belonging to the primary cluster cPRM(nPIDi**). There may be multiple gateway artworks. Links are**
drawn between the subject and the gateway artworks in the primary cluster.

The secondary cluster cSCN(nPIDi**) is calculated by Equation (13).**

$$c_{\mathrm{SCN}}(n_{\mathrm{PID}i})=\arg\operatorname*{min}_{c_{k}}\;\;\;\operatorname*{max}_{n_{j}\in c_{k}}W(n_{\mathrm{PID}i},n_{j}).$$
$\left(13\right)$. 
$$(14)$$
W(nPIDi, nj ). **(13)**
It is the cluster whose artworks are preferred by the subject most weakly. Alternatively, the secondary cluster can be the cluster whose artworks are preferred by the subject **most strongly after the primary cluster. It is** calculated by Equation (14) instead of Equation (13).

$$c_{\mathrm{SCN}}(n_{\mathrm{PID}i})=\arg\operatorname*{max}_{c_{k}\neq c_{\mathrm{PBM}}(n_{\mathrm{PID}i})}\operatorname*{max}_{n_{j}\in c_{k}}W(n_{\mathrm{PID}i},n_{j}).$$
W(nPIDi, nj ). **(14)**
The gateway artwork in the secondary cluster nGTW|SCN(nPIDi**) is calculated by Equation (15). All characteristic objects are determined here.**

$$n_{\mathrm{GTW}|\mathrm{SND}}(n_{\mathrm{PID}i})=\arg\operatorname*{max}_{n_{j}\in c_{\mathrm{SND}}(n_{\mathrm{PID}i})}W(n_{\mathrm{PID}i},n_{j}).$$
$$\left(15\right)$$
W(nPIDi, nj ). **(15)**
Finally, links are drawn between the disjoint clusters so that the switch object nSWTi **can connect the subject**
nPIDi and the gateway artwork in the secondary cluster nGTW|SND(nPIDi**), as in Figure 1 [b]. The preference**
diagram uses the spring model [Fruchterman 1991] as a graph-drawing method. The spring model converts the strength of the relationship across the link between two nodes into **Hooke's constant of the spring, which is placed** between the nodes imaginarily, and calculates the equilibrium position of the nodes.

## 2.3 Reflection Process

The reflection process uses the preference diagrams to prompt the subjects' reflective visualization and verbalization. Group discussion is incorporated into the process as a means to understand the frame of the subjects' own perception. The diagram is also meant to help create efficient communication and mutual understanding during group discussion. The sequence in the designed reflection process **is shown by Figure 2. The notations follow**
the UML (unified modeling language) [UML 2.1.2] specifications. It defines a graphical language for visualizing, specifying, constructing, and documenting the artifacts of distributed object systems. The time goes by from top to bottom. The sequence consists of the prior, main, and posterior stages. An organizer coordinates the process. The organizer uses a a questionnaire to ask the subjects eight questions (Q1 to Q8). Questions Q1 to Q4 are essential for the subjects to become aware of their unconscious **preference. The other questions (Q5 to Q8) are** for the purpose of evaluating the designed reflection process in the experiment.

The content of questions Q1 to Q8 is listed in Table 1. The organizer generates the preference diagrams from the answers to question Q1 after the prior stage. Group discussions are carried out in two sub-stages (part 1 and part 2). This is for the purpose of evaluating the experiment. The number of sub-stages can be one or arbitrary. In the group discussions, the organizer asks the subjects to discuss the preferences of themselves or the others freely. Drawing any conclusion is not requested. Questions Q2 to Q4 **are the central drivers in this process,** which promotes reflection. The organizer extracts the topics in which the subjects express interest during the group discussions from the recorded protocols after the main stage. The protocols are the verbal reports from the subjects. The topics are used in questions Q7 and Q8 in the posterior stage.

## 3 Experiment 3.1 Visualization

The experiment was carried out according to the reflection process described in 2.3. Fifty artworks (classical portraits, landscapes, abstract paintings, modern pop art) are **used in Q1 in Figure 2. Thirty-two subjects** participated in the prior stage. The coordinator generated preference diagrams as presented in 2.2. The main stage was carried out three separate times, with four, two, and five subjects. It took sixty to ninety minutes to finish the main stage.

The four diagrams that include the cluster structures were presented in the part 1 group discussion. Finer granularity diagrams (the number of clusters |c|=3, 5) and courser granularity diagrams (|c|**=7, 8) were presented**
at the same time. The subjects could recognize the primary clusters, compare the details of the diagrams, and