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Digital Image Processing

D. Sundararajan

Chapter 8

Morphological Image Processing - all with Video Answers

Educators

Chapter Questions

04:06

Problem 1

Find the dilation of $x(m, n)$ and $h(m, n)$.
$$
h(m, n)=\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 1 & 0 \\
1 & 1 & 1
\end{array}\right]
$$
(i)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
1 & 1 & 1 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right]
$$

Amy Monaghan
Amy Monaghan
Numerade Educator

Problem 2

Find the erosion of $x(m, n)$ and $h(m, n)$.
$$
h(m, n)=\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 0 & 0
\end{array}\right]
$$
(i)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 0 \\
0 & 1 & 1 & 0
\end{array}\right]
$$

Check back soon!
03:26

Problem 3

Find the opening of $x(m, n)$ and $h(m, n)$.
$$
h(m, n)=\left[\begin{array}{lll}
0 & 1 & 0 \\
1 & 1 & 1 \\
0 & 1 & 0
\end{array}\right]
$$
(i)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1
\end{array}\right]
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:26

Problem 4

Find the closing of $x(m, n)$ and $h(m, n)$.
$$
h(m, n)=\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 1 & 0 \\
0 & 1 & 1
\end{array}\right]
$$
Verify the output using the equivalent expression in terms of the opening operation.
(i)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 \\
0 & 1 & 1 & 0
\end{array}\right]
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:58

Problem 5

Find the hit-and-miss transformation of $x(m, n)$ and $h_h(m, n)$ and $h_{m s}(m, n)$.
$$
h_h(m, n)=\left[\begin{array}{lll}
1 & 0 & 1 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] \quad h_{m s}(m, n)=\left[\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 0 \\
1 & 0 & 1
\end{array}\right]
$$
* (i)
$$
x(m, n)=\left[\begin{array}{llllllll}
1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 & 1
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 & 1
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\
0 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 1
\end{array}\right]
$$

Arun Bana
Arun Bana
Numerade Educator

Problem 6

Find the thinned version of $x(m, n)$.
(i)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
* (ii)
$$
x(m, n)=\left[\begin{array}{llllllll}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 0
\end{array}\right]
$$

Check back soon!
03:56

Problem 7

Find the skeleton of $x(m, n)$.
(i)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
" (iii)
$$
x(m, n)=\left[\begin{array}{llllllll}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 1 & 1
\end{array}\right]
$$

Bryan Lynn
Bryan Lynn
Numerade Educator
03:56

Problem 8

Extract the boundary of $x(m, n)$.
(i)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llllllll}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llllllll}
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0
\end{array}\right]
$$

Bryan Lynn
Bryan Lynn
Numerade Educator