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Digital Image Processing

D. Sundararajan

Chapter 9

Edge Detection - all with Video Answers

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Chapter Questions

Problem 1

Find the $6 \times 6$ filtered magnitude output and the edge map of $x(m, n)$ obtained using the Sobel filters. The threshold is 1.2 times the average of the magnitude of the values of the filtered output.
*(i)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
227 & 179 & 45 & 10 & 14 & 15 & 14 & 12 \\
227 & 227 & 177 & 18 & 15 & 17 & 15 & 14 \\
227 & 227 & 225 & 56 & 8 & 15 & 18 & 15 \\
227 & 227 & 227 & 98 & 9 & 17 & 18 & 16 \\
227 & 227 & 227 & 141 & 7 & 17 & 18 & 17 \\
227 & 227 & 227 & 194 & 26 & 9 & 14 & 14 \\
227 & 227 & 227 & 214 & 92 & 0 & 11 & 11 \\
227 & 227 & 227 & 212 & 184 & 64 & 5 & 14
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
168 & 153 & 111 & 58 & 0 & 0 & 0 & 42 \\
159 & 161 & 114 & 79 & 9 & 2 & 5 & 42 \\
134 & 181 & 124 & 67 & 86 & 20 & 15 & 26 \\
117 & 180 & 122 & 79 & 74 & 47 & 46 & 47 \\
152 & 181 & 132 & 70 & 36 & 38 & 40 & 49 \\
156 & 171 & 117 & 59 & 73 & 77 & 61 & 67 \\
174 & 166 & 124 & 84 & 69 & 57 & 55 & 55 \\
172 & 159 & 129 & 93 & 84 & 31 & 22 & 23
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
13 & 10 & 13 & 9 & 7 & 5 & 4 & 2 \\
6 & 10 & 12 & 10 & 6 & 6 & 2 & 0 \\
19 & 19 & 12 & 11 & 6 & 3 & 2 & 2 \\
19 & 19 & 18 & 18 & 10 & 5 & 2 & 3 \\
19 & 18 & 16 & 17 & 14 & 10 & 7 & 6 \\
17 & 14 & 15 & 15 & 10 & 9 & 8 & 6 \\
13 & 10 & 11 & 13 & 12 & 9 & 8 & 3 \\
14 & 9 & 9 & 9 & 10 & 10 & 8 & 5
\end{array}\right]
$$

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Problem 2

Find the filtered output and the edge map of $x(m, n)$ obtained using a $5 \times 5$ LoG filter with $\sigma=1$. The threshold is 0.75 times the average of the magnitude of the values of the filtered output. Assume replication at the borders.
(i)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
25 & 17 & 22 & 8 & 118 & 186 & 136 & 133 \\
15 & 38 & 18 & 13 & 152 & 147 & 131 & 150 \\
33 & 32 & 14 & 9 & 115 & 165 & 143 & 173 \\
34 & 14 & 14 & 12 & 21 & 64 & 41 & 179 \\
15 & 14 & 7 & 18 & 106 & 137 & 92 & 195 \\
15 & 24 & 39 & 156 & 188 & 194 & 191 & 197 \\
12 & 24 & 129 & 204 & 204 & 206 & 208 & 195 \\
0 & 7 & 145 & 206 & 206 & 209 & 211 & 200
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
147 & 163 & 179 & 186 & 191 & 194 & 197 & 157 \\
160 & 175 & 182 & 184 & 184 & 186 & 162 & 50 \\
141 & 163 & 170 & 175 & 174 & 133 & 38 & 3 \\
91 & 127 & 135 & 124 & 85 & 16 & 0 & 7 \\
113 & 126 & 121 & 117 & 18 & 0 & 1 & 10 \\
136 & 135 & 125 & 151 & 99 & 54 & 8 & 9 \\
148 & 150 & 159 & 161 & 149 & 106 & 89 & 20 \\
142 & 164 & 178 & 181 & 168 & 113 & 120 & 91
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{rrrrrrrr}
72 & 77 & 66 & 62 & 50 & 43 & 66 & 66 \\
75 & 65 & 61 & 65 & 50 & 36 & 64 & 64 \\
67 & 57 & 62 & 67 & 48 & 31 & 63 & 63 \\
55 & 62 & 64 & 62 & 16 & 34 & 61 & 61 \\
41 & 46 & 47 & 46 & 11 & 33 & 59 & 62 \\
39 & 28 & 58 & 54 & 25 & 7 & 50 & 61 \\
41 & 0 & 49 & 48 & 28 & 9 & 6 & 29 \\
63 & 20 & 6 & 23 & 16 & 10 & 7 & 15
\end{array}\right]
$$

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