## Questions

1. Characterize Algorithm 1 according to category and interactivity level. Assuming that the unit of length is such that the distance between the nearest distinct points in the Vs of Fig. 12.20 is 1, we can define a fuzzy

The following figure and definitions are pertinent to Questions 2 to 6.

Figure 12.20: Three examples of a set of spels V; in each case the spels are dots in the plane and V = T U L U R U B U{o}, where T contains the top three dots, L contains the five (a), four (b), or three (c) horizontally centered dots on the left, R contains the three horizontally centered dots on the right, B contains the three vertically centered dots on the bottom, and o is the dot on the bottom-right.

spel affinity on V as any of the following:

I 1/|| c — d|| otherwise, where ||c — d|| is the Euclidean distance between the dots c and d, f (c, d) = ( f (c, d) if ||c — d||<3' (12.22)

0 otherwise, and

0 otherwise.

2. Are the sets V of Fig. 12.20 ^-connected? If not, why?

3. Are the sets V of Fig. 12.20 ^-connected or ^-connected? If not, why?

4. Consider the seeded 2-fuzzy graph (V, V) where V is the set (a) of Fig. 12.20, ^ = ^), Vi contains the leftmost spel ofV and V2 contains the rightmost spel ofV. Compute the 2-segmentation a using Theorem 1.1.

5. Does the 2-segmentation a change if we use ^ = ^)?

6. Is (V, (ty, ^), V), where V1 contains the leftmost spel ofV and V2 contains the rightmost spel of V, a connectable 2-fuzzy graph for any of the sets of Fig. 12.20?

7. What does the concept of blocking of chains mean?

8. Why should one use thefcc grid instead of the traditional sc (cubic) grid?

9. Suppose that the fuzzy spel affinities defined for a specific application can only assume values from a small set (around 1000 elements). Discuss an alternative data structure for implementing the algorithm more efficiently.

The following definitions are pertinent to Questions 10 and 11.

Using the notation of this chapter, the Relative Fuzzy Connectedness (RFC)9 of [27] defines a 2-segmentation as follows. For 1 < m < 2 and for any c e V, let iicm denote the ^-strength of the strongest chain from (the unique element of) Vm to c. Then, let f-^1 if 1 >> ' 0 otherwise, if < 0 otherwise,

The Iterative Relative Fuzzy Connectedness (IRFC) of [27] produces a sequence ,... of spel-adjacencies and a sequence of 0o,1o,... of

2-segmentations defined as follows: = ^ and 0o is the 2-segmentation defined by RFC. Now assume that, for some i > 0, we have already obtained and i-1o. For all c, d e V, we define it2(c, d) =

3 The definitions of RFC and IRFC of [27] are restricted to 2-fuzzy graphs where fa = fa with a single seed spel per object.

Then ia is defined just as a is defined in RFC using (12.24) and (12.25), but with ixcm replaced by iim everywhere. Whenever ia = i-1a, then that 2-segmentation is considered to be the final output oflRFC.

10. Consider the seeded 2-fuzzy graph (V, V) where V is the set (c) of Fig. 12.20, ^ = Vi contains the leftmost spel ofV and V2 contains the rightmost spel ofV. Compute the 2-segmentations a using Theorem 1.1 and RFC and compare them.

11. Consider the seeded 2-fuzzy graph (V, V) where V is the set (a) of Fig. 12.20, ^ = V1 contains the leftmost spel of V and V2 contains the bottommost spel of B. Compute the 2-segmentations a using Theorem 1.1 and IRFC and compare them.

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