# Rendering Strategy to Counter Mutual Masking Effect in Multiple Tactile Feedback
## 3 The Proposed Rendering Algorithm
### Generalized Algorithm  
Table 3 lists the variables to be used in this section. The proposed algorithm can be summarized as searching for the values of \(m_{i}\) and \(e_{i}\) (i.e., the required input intensities) based on given values of the variables \(m_{p}\) and \(e_{p}\) (i.e., the desired/designed perceived intensities), as shown in Figure 5.  
Figure 4: Input and perceived intensity when both geometry and texture information are provided: (**a**) ideal case, (**b**) without the algorithm, and (**c**) with the algorithm. To generalize our approach, it should be possible to compute the required input intensity of both the MV and EV for any arbitrary perceived intensity desired. First, we need to derive a generalized form of the masking functions. The masking function investigated in Section 2 above tended to function linearly in a certain region but had a value of zero in other regions, i.e., the form of a ramp function. These results are consistent with the results of earlier studies [23; 24]. Thus, the generalized masking function can be expressed in the form of conditional equations as follows,  
\[f(e_{i})=\left\{ \begin{array}{cl}0,&\quad\text{if }e_{i}<-a/b\\ a+be_{i},&\quad\text{otherwise}\end{array} \right. \tag{1}\]  
\[g(m_{i})=\left\{ \begin{array}{cl}0,&\quad\text{if }m_{i}<-c/d\\ c+dm_{i},&\quad\text{otherwise}\end{array} \right. \tag{2}\]  
where \(a\) and \(b\) denote the intercept and slope of the masking function of the MV, respectively, and \(c\) and \(d\) denote the same for the EV. The relationship between the input and perceived intensities of the MV and EV, using the generalized masking functions in (1) and (2), can be defined as follows. \[m_{p}=m_{i}-f(e_{i}) \tag{3}\]  
\[e_{p}=e_{i}-g(m_{i}) \tag{4}\]  
Equations (3) and (4) show that the values of \(f(e_{i})\) and \(g(m_{i})\) change in two ways according to the interval in which the solution, \(e_{i}\) and \(m_{i}\), is located. Therefore, the proposed algorithm is defined as solving one of the four linear systems, depending on the values of \(m_{i}\) and \(e_{i}\) to be found. If the four conditions are named 1-4, then the exact solution for each of the four conditions can be derived as listed in Table 4. As shown in the table, an exact solution exists for each of the four conditions. As, in our problem, the given variables are a perceived intensity pair, the third column in the table  
Figure 5: Schematic illustration of the proposed rendering algorithm. \begin{table}
\begin{tabular}{l l} \hline
**Term** & **Description** \\ \hline \(m_{p}\) & perceived intensity of MV \\ \(m_{i}\) & input intensity of MV required to produce \(m_{p}\) \\ \(e_{p}\) & perceived intensity of EV \\ \(e_{i}\) & input intensity of EV required to produce \(e_{p}\) \\ \(f(e_{i})\) & masking function of MV \\ \(g(m_{i})\) & masking function of EV \\ \hline \end{tabular}
\end{table} 
Table 3: Variables used in the proposed algorithm. presents rearranged conditions as a function of \(m_{p}\) and \(e_{p}\). It is important to note that condition #4 is subdivided into two cases based on the sign of \(bd\), the multiplication of the two masking functions slope values. To determine the required input intensity to satisfy the desired perceived intensity, there should exist a unique solution pair, \(m_{i}\) and \(e_{i}\) for arbitrary \(m_{p}\) and \(e_{p}\). To clarify this, we examined the region where a solution exists for all possible cases. Figure 6 shows the graphical representation of the region of \(m_{p}\) and \(e_{p}\) that satisfies each of the four conditions. The conditions (third column in Table 4) are represented by four lines. The intersection between them is always \((-a/b,-c/d)\). This intersection can be in different quadrants, depending on the sign of \(a\) and \(c\) (the respective intercepts of the two masking functions). As shown in Figure 6, when \(bd<1\), the regions satisfying each condition are completely separated, and cover all real numbers of \(m_{p}\) and \(e_{p}\).