Graph Representation of Tactile Codes

We represented the tactile shapes as nodes and the relationship between them as edges in a undirected graph. In order to construct a graph with 512 nodes, we need a incidence matrix of size 511 ∗ 511. Since differentiating shape i from shape j is the same as differentiating shape j from shape i, the edges of the graph are symmetric and undirected. So to construct a graph with 511 shapes, we require 511 ∗ 256 incidence matrix. The data collection study was performed with eight users. From each response provided by a user, we can determine the relationship between one tactile code and four other tactile codes. Each tactile code was presented four times to a user. Thus, for each tactile code in the code space, the data collection study yielded a relationship between that code and 8 ∗ 4 ∗ 4 = 128 other codes in the graph. The study yielded eight response time (RT ) matrices and eight perception error matrices (E ). To combine the partial incidence matrix from eight users to create a final incidence matrix, we have used the approach proposed by Ternes [91].

Ternes approach calculates a final incidence matrix of perceptual difference from partial incidence matrices of six users by finding the average of the edge weights from six incidence matrices. We have used the same approach to calculate the average response time to distinguish two shapes and the average number of perception errors between two shapes.

In a undirected graph, the relationship between tactile shapes was represented by the weight of the edge connecting the nodes. The weight of the edges can be represented in one of the following three methods:

1. Error matrix (E) - The number of errors in distinguishing the shapes(nodes) connected by an edge

2. Response time matrix (RT) - The time taken to distinguish shapes(nodes) connected by an edge

3. Error + Response time matrix (E + RT) - The combination of the number of errors and the time taken to distinguish the shapes(nodes) connected by an edge

Given the graph representation of tactile codes, our goal of finding a good tactile code is defined as finding a cluster of nodes in the graph that minimizes the number of high weight edges and maxmizes the low weight edges between nodes. We use the spring-electrical technique to represent the edge weights as repulsive force between nodes and Hu’s fast force approximation algorithm [32] to perform clustering. The spring-electrical model is proven to work well for sparse incidence matrices and scales better than other models for large graphs [32].

figure 5.1(a). The output clusters the shapes that were distinguished without errors. The clustering algorithm was evaluated based on four parameters: network diameter, modularity, average clustering coefficient, and average path length. Table 5.1 shows the values of four parameters used to evaluate the three graph models. The network diameter (8), average path length (3.811) and Modularity (0.393) of GE were high,

and the average clustering coefficient (0.013) was low, showing that the error data alone did not suffice in creating a good cluster of shapes.

GRT was formed with a response time incidence matrix (RT). Figure 5.1(b)

shows the output of Hu’s algorithm on GRT . The algorithm clustered the shapes

that were easily distinguished. The time taken to distinguish two shapes within a cluster is much smaller than the time taken to distinguish two shapes from two different clusters. The resulting graph had a low network diameter (2), average path length (1.772), and modularity (0.084) were low, and the average clustering coefficient (0.227) was high, showing that GRT formed clusters of shapes. The disadvantage of

using clustering on GRT is that it does not take into account the errors in perceiving

the difference between two shapes. The shapes that are hard to distinguish can also be easily mistaken for each other. Such shapes will have low response time and will fall in the same cluster.

Table 5.1: Network Diameter, Modularity, Average Clustering Coefficient and Average Path Length of the Graph Models.

Cluster Type Network

Diameter Modularity Average Clustering Coefficient Average Path Length Error 8 0.393 0.013 3.811 Time Taken 2 0.084 0.227 1.772

(a) Graph GE formed with errors (b) Graph GRT

(c) Graph GC

Figure 5.1: Figures show the visualization of three graph GE, GRT and GC formed

with the error matrix (E), the response time matrix (RT), and the combination of the error matrix and the response time matrix (E + RT) respectively. The figures also illustrates the visualization of the graphs after the application of Hu’s fast force algorithm.

We used a linear combination of the average response time by users to distinguish the tactile codes and the number of errors committed by users while distinguishing the codes to form graph GC. For any pair of tactile codes i and j in the code space,

the weight W(i,j) or W(j,i) of the edge between the nodes is given by the following equation:

RT(i,j) is the average response time taken by users to distinguish i and j during the user study, E(i,j) is the number of errors committed by users while distinguishing the codes, and C is a constant value equal to 100s. The weight of an edge is the time taken to distinguish codes i and j. Each error committed adds a 200ms response time to the edge weight. This equation is chosen to guarantee the following two properties:

1. If two tactile codes i and j are easy to distinguish, the value of W(i,j) is small due to one or both of the following reasons. The users take less time to distinguish i from j and/or commit no errors in distinguishing i and j.

2. If two tactile codes i and j are hard to distinguish, the value of W(i,j) is large because users take more time to distinguish i from j and/or commit more than one error in distinguishing i and j. The penalty (C ) of 100 seconds guarantees W(i,j) is a large value when there is more than one error to distinguish i and j.

Equation 1 defines the relationship between the weight of each node in the graph, the average response time, and the average perception error. The output of the clus- tering algorithm is presented in the figure 5.1(c). The output combined the positives from graphs GE and GRT . It had a low network diameter(2), low modularity(0.201),

low average path length(0.23), high average clustering coefficient(1.769), and also included the error information in clustering the shapes.

Based on the graph network measures—network diameter, modularity, average path length, and average clustering coefficient—we choose graph GC over graphs

GE and GRT . As discussed earlier in this chapter, GC combines the information

represented by both GE and GRT . In the next chapter, we discuss the validation of

rate and accuracy. Figure 5.2 shows a method of selecting tactile codes from GC

clusters. Each cluster represents shapes that are dissimilar and distinguishable from each other. The tactile codes selected from one cluster of the model represent easy to distinguish codes. We have selected ten shapes from one cluster to validate the output of the model in validation user studies. The ten shapes are highlighted as green nodes in figure 5.2.

Figure 5.2: The graph shows the output of the clustering algorithm. Each node in the graph is one of the 512 shapes that can be presented with three-by-three tactile display. An edge between two shapes is weighted by the dissimilarity between the shapes. The weight (i,j) for an edge between node i and node j is calculated using equation 1. The red box on the right is zoomed in view of two clusters inside the red box on the left. The thickness of the edges correspond to the edge weight. The green nodes are marked to show the tactile codes selected for evaluation. The nodes that are closer to each other represent shapes that are easy to distinguish from each other.