How many of the 64 possible assignments of grey levels to two adjacent pixels satisfy this rule?

by

Each of the pixels can take on one of eight grey levels ranging from 0 (white) to7 (black). In order to prevent sharp discontinuities of shade, the softwaresystem that causes pictures to be displayed enforces the rule that the greylevels of two adjacent pixels cannot differ by more than two. How many of the 64 possible assignments of grey levels to two adjacent pixels satisfy this rule?
(A) 24
(B) 32
(C) 34
(D) 40
(E) 64

Answer:

To find out how many of the 64 possible assignments of grey levels to two adjacent pixels satisfy the rule that the grey levels of two adjacent pixels cannot differ by more than two, we need to count the number of valid assignments.The grey levels for each pixel can range from 0 (white) to 7 (black). For any given grey level (0 through 7), there are three possible grey levels for the adjacent pixel that satisfy the rule:

1. The adjacent pixel has the same grey level (difference of 0).
2. The adjacent pixel has a grey level that differs by 1.
3. The adjacent pixel has a grey level that differs by 2.

However, there are two cases where this doesn’t hold true: for grey levels 0 and 1, and grey levels 6 and 7. In these cases, the adjacent pixel can differ by up to two grey levels. Therefore, we have:

  • For grey levels 0 and 1, there are four possible valid assignments: (0,0), (0,1), (1,0), and (1,1).
  • For grey levels 2 through 5, there are six possible valid assignments: (i, i), (i,i±1), and (i,i±2), where i is the grey level.
  • For grey levels 6 and 7, there are four possible valid assignments: (6,6), (6,7), (7,6), and (7,7).

Thus, the total number of valid assignments is:
2(4) + 4(6) + 2(4) = 8 + 24 + 8 = 40.
Therefore, the correct answer is (D) 40.

More Questions And Answers:

Leave a Comment