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Digital Image Processing

D. Sundararajan

Chapter 6

Geometric Transformations and Image Registration - all with Video Answers

Educators


Chapter Questions

Problem 1

Using nearest-neighbor interpolation, find the $7 \times 7$ interpolated version of the image $x(m, n)$.
(i)
$$
\left[\begin{array}{llll}
43 & 50 & 50 & 52 \\
45 & 49 & 51 & 50 \\
46 & 46 & 49 & 48 \\
43 & 44 & 47 & 42
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
68 & 80 & 83 & 78 \\
39 & 54 & 61 & 66 \\
41 & 44 & 44 & 67 \\
55 & 46 & 34 & 66
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
63 & 64 & 64 & 64 \\
62 & 62 & 62 & 61 \\
61 & 61 & 61 & 58 \\
62 & 61 & 60 & 58
\end{array}\right]
$$

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Problem 2

Using bilinear interpolation, find the $7 \times 7$ interpolated version of the image $x(m, n)$.
(i)
$$
\left[\begin{array}{llll}
34 & 51 & 56 & 53 \\
38 & 53 & 57 & 54 \\
40 & 52 & 56 & 52 \\
39 & 48 & 52 & 49
\end{array}\right]
$$
*(ii)
$$
\left[\begin{array}{llll}
53 & 42 & 39 & 58 \\
51 & 46 & 44 & 49 \\
54 & 52 & 58 & 46 \\
63 & 57 & 63 & 52
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{rrrr}
82 & 84 & 86 & 97 \\
80 & 80 & 85 & 103 \\
79 & 77 & 90 & 114 \\
80 & 84 & 102 & 118
\end{array}\right]
$$

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Problem 3

Using nearest-neighbor interpolation, find the scaled version of the image $x(m, n)$.
(i) $a=0.5, e=0.75$
$$
\left[\begin{array}{llll}
52 & 61 & 57 & 66 \\
58 & 64 & 69 & 64 \\
45 & 60 & 74 & 61 \\
56 & 63 & 74 & 63
\end{array}\right]
$$
*(ii) $a=-0.5, e=-0.5$
$$
\left[\begin{array}{rrrr}
71 & 56 & 47 & 92 \\
66 & 51 & 47 & 108 \\
64 & 55 & 70 & 122 \\
73 & 57 & 81 & 127
\end{array}\right]
$$
(iii) $a=0.75, e=0.25$
$$
\left[\begin{array}{llll}
48 & 53 & 46 & 62 \\
54 & 54 & 54 & 80 \\
64 & 53 & 59 & 78 \\
57 & 46 & 56 & 55
\end{array}\right]
$$

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Problem 4

Using nearest-neighbor interpolation, find the sheared version of the image $x(m, n)$.
(i) $b=0, d=1$
$$
\left[\begin{array}{llll}
17 & 20 & 26 & 25 \\
18 & 23 & 30 & 24 \\
17 & 24 & 32 & 27 \\
20 & 28 & 30 & 32
\end{array}\right]
$$
(ii) $b=0, d=0.5$
$$
\left[\begin{array}{llll}
63 & 49 & 51 & 54 \\
66 & 60 & 52 & 56 \\
57 & 62 & 62 & 57 \\
57 & 56 & 64 & 61
\end{array}\right]
$$
(iii) $b=0, d=0.3$
$$
\left[\begin{array}{llll}
179 & 178 & 179 & 184 \\
177 & 178 & 179 & 189 \\
176 & 177 & 180 & 193 \\
174 & 175 & 184 & 190
\end{array}\right]
$$

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Problem 5

Using nearest-neighbor interpolation, find the sheared version of the image $x(m, n)$.
(i) $b=1, d=0$
$$
\left[\begin{array}{rrrr}
41 & 36 & 123 & 151 \\
27 & 10 & 79 & 136 \\
17 & 17 & 33 & 91 \\
17 & 30 & 17 & 70
\end{array}\right]
$$
(ii) $b=0.7, d=0$
$$
\left[\begin{array}{lllll}
172 & 157 & 115 & 62 \\
163 & 165 & 118 & 83 \\
138 & 185 & 128 & 71 \\
121 & 184 & 126 & 83
\end{array}\right]
$$
*(iii) $b=0.3, d=0$
$$
\left[\begin{array}{rrrr}
95 & 111 & 48 & 32 \\
96 & 115 & 59 & 26 \\
89 & 90 & 37 & 24 \\
86 & 73 & 15 & 21
\end{array}\right]
$$

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04:06

Problem 6

Using nearest-neighbor interpolation, find the rotated version of the image $x(m, n)$ in the counterclockwise direction. $\theta=90^{\circ}, \theta=180^{\circ}$ and $\theta=45^{\circ}$.
(i)
$$
\left[\begin{array}{rrrr}
95 & 47 & 65 & 55 \\
74 & 60 & 60 & 47 \\
105 & 103 & 67 & 46 \\
103 & 78 & 67 & 58
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
74 & 81 & 67 & 75 \\
77 & 77 & 77 & 83 \\
58 & 69 & 69 & 80 \\
46 & 61 & 69 & 82
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
66 & 77 & 79 & 64 \\
56 & 68 & 61 & 69 \\
43 & 51 & 49 & 66 \\
39 & 45 & 43 & 55
\end{array}\right]
$$

Emily Min
Emily Min
Numerade Educator
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Problem 7

Find the cross-correlation and the correlation coefficients of $x(m, n)$ and $h(m, n)$. Assume zero-padding at the borders.
$$
h(m, n)=\left[\begin{array}{lll}
1 & 1 & 1 \\
3 & 1 & 0 \\
1 & 2 & 1
\end{array}\right]
$$
*(i)
$$
x(m, n)=\left[\begin{array}{llll}
2 & 1 & 3 & 2 \\
3 & 2 & 1 & 0 \\
1 & 2 & 1 & 2 \\
1 & 3 & 0 & 2
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
3 & 1 & 1 & 2 \\
0 & 2 & 1 & 2 \\
1 & 1 & 1 & 2 \\
1 & 2 & 0 & 2
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 2 & 3 & 0 \\
0 & 2 & 1 & 0 \\
2 & 2 & 1 & 2 \\
1 & 1 & 0 & 2
\end{array}\right]
$$

Victor Salazar
Victor Salazar
Numerade Educator
03:29

Problem 8

Find the autocorrelation of $x(m, n)$. Assume zero-padding at the borders.
(i)
$$
x(m, n)=\left[\begin{array}{llll}
2 & 1 & 3 & 2 \\
2 & 1 & 1 & 0 \\
1 & 0 & 1 & 2 \\
1 & 1 & 0 & 2
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
3 & 1 & 1 & 2 \\
0 & 3 & 1 & 2 \\
1 & 2 & 1 & 0 \\
1 & 2 & 0 & 2
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
3 & 2 & 1 & 0 \\
0 & 2 & 1 & 0 \\
2 & 0 & 1 & 2 \\
3 & 1 & 0 & 2
\end{array}\right]
$$

Arpit Gupta
Arpit Gupta
Numerade Educator