Chapter 4, Image Enhancement in the Frequency Domain Video Solutions, Digital Image Processing

Problem 1

Compute the linear convolution of $x(n), n=0,1, \ldots$, and $h(n), n=0,1, \ldots$, using the DFT and IDFT and verify your answer by directly computing the convolution. Assume zero-padding at the ends.
(i) $x(n)=\{2,1,3\}$ and $h(n)=\{1,-2\}$.
(ii) $x(n)=\{-1,3\}$ and $h(n)=\{1,3,-2\}$.
*(iii) $x(n)=\{4,-1\}$ and $h(n)=\{-3,1,-2\}$.
(iv) $x(n)=\{-1,2,3\}$ and $h(n)=\{-2,3\}$.
(v) $x(n)=\{2,4\}$ and $h(n)=\{4,3,-2\}$.

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Problem 2

Compute the linear convolution of $x(m, n), m, n=0,1,2$, and $h(m, n), m$, $n=0,1$ using the DFT and IDFT and verify your answer by directly computing the convolution. Assume zero-padding at the borders.
*(i)
$$
x(m, n)=\left[\begin{array}{rrr}
1 & 2 & 3 \\
2 & -3 & 1 \\
1 & 2 & 1
\end{array}\right] \text { and } h(m, n)=\left[\begin{array}{rr}
2 & -1 \\
1 & 2
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{rrr}
2 & 2 & 3 \\
2 & -3 & -1 \\
1 & -2 & 1
\end{array}\right] \text { and } h(m, n)=\left[\begin{array}{rr}
-2 & -1 \\
1 & 3
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{rrr}
1 & -4 & 3 \\
-2 & 3 & 1 \\
1 & -3 & 1
\end{array}\right] \text { and } h(m, n)=\left[\begin{array}{rr}
4 & 1 \\
-1 & 2
\end{array}\right]
$$
(iv)
$$
x(m, n)=\left[\begin{array}{rrr}
2 & 1 & 4 \\
1 & -3 & 2 \\
3 & 2 & 1
\end{array}\right] \text { and } h(m, n)=\left[\begin{array}{rr}
1 & -3 \\
2 & 2
\end{array}\right]
$$
(v)
$$
x(m, n)=\left[\begin{array}{rrr}
1 & 1 & 1 \\
2 & -3 & 4 \\
2 & 2 & 2
\end{array}\right] \text { and } h(m, n)=\left[\begin{array}{rr}
3 & 1 \\
1 & -3
\end{array}\right]
$$

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Problem 3

Using the DFT and IDFT, convolve $x(m, n)$ and a $3 \times 3$ Gaussian lowpass filter with $\sigma=1$. Assume periodicity at the borders.
(i)
$$
x(m, n)=\left[\begin{array}{rrrr}
2 & 1 & 3 & 4 \\
1 & 1 & 4 & 2 \\
1 & -1 & 2 & -2 \\
3 & 2 & -2 & 1
\end{array}\right]
$$
*(ii)
$$
x(m, n)=\left[\begin{array}{rrrr}
1 & 2 & 3 & 4 \\
2 & 1 & 1 & 4 \\
1 & -1 & 0 & -2 \\
0 & 2 & -2 & 1
\end{array}\right]
$$

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Problem 4

Using the DFT and IDFT, convolve $x(m, n)$ and a $3 \times 3$ averaging lowpass filter. Assume periodicity at the borders.
(i)
$$
x(m, n)=\left[\begin{array}{rrrr}
1 & -1 & 3 & 4 \\
1 & 2 & -4 & 2 \\
1 & -1 & 3 & -2 \\
1 & -2 & -2 & 1
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{rrrr}
1 & 3 & -3 & 4 \\
2 & 0 & 1 & 4 \\
1 & -1 & 0 & -2 \\
0 & 2 & -2 & 3
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{rrrr}
2 & 0 & 3 & -2 \\
2 & -1 & 2 & 2 \\
4 & -1 & 3 & -2 \\
1 & 0 & -2 & 3
\end{array}\right]
$$

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Problem 5

Using the DFT and IDFT, convolve $x(m, n)$ and a $3 \times 3$ Laplacian enhancement filter. Assume periodicity at the borders.
$$
\left[\begin{array}{rrr}
0 & -1 & 0 \\
-1 & 5 & -1 \\
0 & -1 & 0
\end{array}\right]
$$
*(i)
$$
\left[\begin{array}{llll}
7 & 7 & 5 & 1 \\
4 & 4 & 4 & 1 \\
0 & 8 & 4 & 3 \\
5 & 0 & 2 & 1
\end{array}\right]
$$
(ii)
$$
\left[\begin{array}{llll}
1 & 2 & 5 & 1 \\
4 & 1 & 0 & 1 \\
0 & 0 & 3 & 3 \\
5 & 0 & 2 & 4
\end{array}\right]
$$
(iii)
$$
\left[\begin{array}{rrrr}
2 & 3 & 1 & 1 \\
4 & 0 & -4 & 1 \\
0 & 1 & 4 & 3 \\
4 & 0 & -2 & 1
\end{array}\right]
$$

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Problem 6

Let an ideal 2-D lowpass filter $H(k, l)$ is defined between the limits $\{-4,-3$, $-2,-1,0,1,2,3\}$ in both the directions. Find the filter coefficients for a given cutoff radius $r$ such that $H(k, l)=1$ inside the circle defined by $r$ and zero elsewhere.
(i) $r=1$
(ii) $r=2$
(iii) $r=3$

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Problem 7

Let an ideal 2-D highpass filter $H(k, l)$ is defined between the limits $\{-4,-3$, $-2,-1,0,1,2,3\}$ in both the directions. Find the filter coefficients for a given cutoff radius $r$ such that $H(k, l)=1$ outside the circle defined by $r$ and zero elsewhere.
(i) $r=1$
(ii) $r=2$
(iii) $r=3$

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