Chapter 2, Image Enhancement in the Spatial Domain Video Solutions, Digital Image Processing

Problem 1

Find the complement of the $4 \times 48$-bit gray level image and verify that the image can be restored by complementing the complemented image.
(i)
$$
\left[\begin{array}{rrrr}
112 & 148 & 72 & 153 \\
120 & 125 & 30 & 99 \\
95 & 120 & 89 & 33 \\
170 & 99 & 109 & 40
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{rrrr}
164 & 127 & 117 & 59 \\
154 & 122 & 104 & 83 \\
129 & 136 & 100 & 60 \\
117 & 128 & 80 & 48
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
46 & 48 & 46 & 45 \\
42 & 49 & 46 & 45 \\
64 & 73 & 60 & 43 \\
94 & 69 & 63 & 37
\end{array}\right]
$$

Katelyn Chen

Numerade Educator

03:35

Problem 2

Find the complement of the $4 \times 4$ binary image and verify that the image can be restored by complementing the complemented image.
(i)
$$
\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
1 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 \\
1 & 1 & 0 & 0
\end{array}\right]
$$

Bryan Lynn

Numerade Educator

Problem 3

For the list of gray levels, apply gamma correction and find the corresponding new gray levels. Apply the inverse transformation to the new gray levels and verify that the given gray levels are obtained.
(i) $\gamma=0.8$.
(ii) $\gamma=1.1$.
(iii) $\gamma=1.8$.

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00:52

Problem 4

Given a $4 \times 4$ 4-bit image, find the histogram equalized version of it.
*(i)
$$
\left[\begin{array}{llll}
4 & 4 & 3 & 3 \\
4 & 4 & 4 & 3 \\
5 & 4 & 4 & 4 \\
4 & 4 & 4 & 4
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
1 & 1 & 0 & 1 \\
1 & 1 & 0 & 3 \\
1 & 0 & 0 & 2 \\
1 & 0 & 0 & 2
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{llll}
3 & 5 & 5 & 3 \\
4 & 4 & 4 & 3 \\
4 & 2 & 3 & 4 \\
4 & 2 & 2 & 3
\end{array}\right]
$$

Trinity Steen

Numerade Educator

Problem 5

Given $4 \times 4$ 4-bit reference and input images, use histogram matching to restore the input image.
*(i) The reference and input images, respectively, are
$$
\left[\begin{array}{llll}
4 & 4 & 3 & 3 \\
4 & 4 & 4 & 3 \\
5 & 4 & 4 & 4 \\
4 & 4 & 4 & 4
\end{array}\right]\left[\begin{array}{rrrr}
15 & 15 & 0 & 0 \\
15 & 15 & 15 & 0 \\
15 & 15 & 15 & 15 \\
15 & 15 & 15 & 15
\end{array}\right]
$$
(ii) The reference and input images, respectively, are
$$
\left[\begin{array}{llll}
3 & 3 & 3 & 3 \\
3 & 3 & 3 & 3 \\
3 & 2 & 2 & 3 \\
2 & 2 & 2 & 2
\end{array}\right]\left[\begin{array}{rrrr}
15 & 15 & 15 & 15 \\
15 & 15 & 15 & 15 \\
15 & 0 & 0 & 15 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
(iii) The reference and input images, respectively, are
$$
\left[\begin{array}{llll}
3 & 5 & 5 & 3 \\
4 & 4 & 4 & 3 \\
4 & 2 & 3 & 4 \\
4 & 2 & 2 & 3
\end{array}\right]\left[\begin{array}{rrrr}
0 & 15 & 15 & 0 \\
15 & 15 & 15 & 0 \\
15 & 0 & 0 & 15 \\
15 & 0 & 0 & 0
\end{array}\right]
$$

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04:55

Problem 6

Given a $8 \times 8$ 8-bit image, find the binary, hard and soft thresholded versions with the threshold $T=160$.
$$
\left[\begin{array}{rrrrrrrr}
255 & 255 & 255 & 117 & 50 & 39 & 50 & 56 \\
255 & 255 & 255 & 194 & 45 & 26 & 48 & 54 \\
255 & 255 & 255 & 241 & 61 & 25 & 53 & 57 \\
255 & 255 & 255 & 255 & 104 & 32 & 64 & 64 \\
255 & 255 & 255 & 255 & 154 & 37 & 59 & 61 \\
255 & 255 & 255 & 255 & 199 & 54 & 55 & 61 \\
255 & 255 & 255 & 255 & 230 & 71 & 59 & 64 \\
255 & 255 & 255 & 255 & 250 & 95 & 60 & 68
\end{array}\right]
$$

Stanley Enemuo

Numerade Educator

Problem 7

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ averaging filter with the borders zero-padded.
$$
x(m, n)=\left[\begin{array}{llll}
70 & 62 & 51 & 45 \\
71 & 62 & 57 & 55 \\
73 & 65 & 56 & 60 \\
68 & 69 & 63 & 66
\end{array}\right]
$$

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Problem 8

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ averaging filter with the borders replicated.
$$
x(m, n)=\left[\begin{array}{llll}
41 & 43 & 45 & 43 \\
40 & 41 & 42 & 41 \\
42 & 38 & 39 & 42 \\
39 & 33 & 37 & 36
\end{array}\right]
$$

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Problem 9

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ averaging filter with the borders periodically extended.
$$
x(m, n)=\left[\begin{array}{llll}
45 & 78 & 87 & 51 \\
59 & 56 & 62 & 49 \\
59 & 39 & 44 & 57 \\
56 & 36 & 35 & 51
\end{array}\right]
$$

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Problem 10

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ Gaussian filter ( $\sigma=0.5$ ) with the borders symmetrically extended.
$$
x(m, n)=\left[\begin{array}{llll}
202 & 195 & 192 & 191 \\
216 & 211 & 200 & 209 \\
224 & 212 & 215 & 227 \\
224 & 205 & 227 & 230
\end{array}\right]
$$

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Problem 11

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ Gaussian filter ( $\sigma=0.5$ ) with the borders periodically extended.
$$
x(m, n)=\left[\begin{array}{llll}
202 & 195 & 192 & 191 \\
216 & 211 & 200 & 209 \\
224 & 212 & 215 & 227 \\
224 & 205 & 227 & 230
\end{array}\right]
$$

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Problem 12

Given a $4 \times 4$ image, find the $4 \times 4$ lowpass filtered output using the $3 \times 3$ Gaussian filter ( $\sigma=0.5$ ) with the borders zero-padded.
$$
x(m, n)=\left[\begin{array}{llll}
95 & 82 & 54 & 33 \\
84 & 78 & 56 & 64 \\
73 & 71 & 53 & 60 \\
73 & 73 & 54 & 36
\end{array}\right]
$$

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Problem 13

Given a $4 \times 4$ image, find the $4 \times 4$ highpass filtered output using the $3 \times 3$ Laplacian filter
$$
h(m, n)=\left[\begin{array}{rrr}
0 & 1 & 0 \\
1 & -4 & 1 \\
0 & 1 & 0
\end{array}\right]
$$
with the borders zero-padded.
$$
x(m, n)=\left[\begin{array}{llll}
45 & 52 & 56 & 52 \\
49 & 60 & 55 & 55 \\
47 & 55 & 53 & 46 \\
45 & 48 & 51 & 40
\end{array}\right]
$$

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Problem 14

Given a $4 \times 4$ image, find the $4 \times 4$ highpass filtered output using the $3 \times 3$ Laplacian filter with the borders symmetrically extended.
$$
x(m, n)=\left[\begin{array}{llll}
64 & 62 & 62 & 68 \\
68 & 66 & 58 & 64 \\
75 & 70 & 60 & 58 \\
72 & 69 & 59 & 60
\end{array}\right]
$$

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Problem 15

Given a $4 \times 4$ image, find the $4 \times 4$ highpass filtered output using the $3 \times 3$ Laplacian filter with the borders replicated.
$$
x(m, n)=\left[\begin{array}{llll}
39 & 40 & 35 & 33 \\
31 & 40 & 39 & 37 \\
34 & 38 & 41 & 43 \\
37 & 39 & 42 & 43
\end{array}\right]
$$

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Problem 16

Given a $4 \times 4$ image, find the $4 \times 4$ enhanced output using the $3 \times 3$ Laplacian sharpening filter
$$
h(m, n)=\left[\begin{array}{rrr}
0 & -1 & 0 \\
-1 & 5 & -1 \\
0 & -1 & 0
\end{array}\right]
$$
with the borders replicated.
$$
x(m, n)=\left[\begin{array}{llll}
190 & 206 & 228 & 238 \\
180 & 205 & 227 & 219 \\
182 & 203 & 211 & 159 \\
184 & 212 & 206 & 177
\end{array}\right]
$$

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Problem 17

Given a $4 \times 4$ image, find the $4 \times 4$ enhanced output using the $3 \times 3$ Laplacian sharpening filter with the borders periodically extended.
$$
x(m, n)=\left[\begin{array}{llll}
138 & 163 & 162 & 177 \\
148 & 157 & 167 & 175 \\
153 & 165 & 160 & 178 \\
157 & 162 & 164 & 188
\end{array}\right]
$$

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Problem 18

Given a $4 \times 4$ image, find the $4 \times 4$ enhanced output using the $3 \times 3$ Laplacian sharpening filter with the borders zero-padded.
$$
x(m, n)=\left[\begin{array}{llll}
201 & 195 & 191 & 169 \\
210 & 201 & 181 & 157 \\
213 & 207 & 190 & 166 \\
204 & 204 & 197 & 159
\end{array}\right]
$$
$$

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Problem 19

Given a $4 \times 4$ image, find the $4 \times 4$ median filtered output using the $3 \times 3$ window with the borders zero-padded.
(i)
$$
x(m, n)=\left[\begin{array}{llll}
201 & 195 & 191 & 169 \\
210 & 201 & 181 & 157 \\
213 & 207 & 190 & 166 \\
204 & 204 & 197 & 159
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
138 & 163 & 162 & 177 \\
148 & 157 & 167 & 175 \\
153 & 165 & 160 & 178 \\
157 & 162 & 164 & 188
\end{array}\right]

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