Chapter 7, Image Reconstruction from Projections Video Solutions, Digital Image Processing

Problem 1

Find the equation of the straight line, in the normal form, located at a distance $s$ from the origin and the perpendicular to it makes an angle of $\theta$ degrees with the $x$-axis.
(i) $s=3, \theta=0^{\circ}$.
(ii) $s=1, \theta=45^{\circ}$.
* (iii) $s=2, \theta=-60^{\circ}$.
(iv) $s=5, \theta=315^{\circ}$.
(v) $s=0, \theta=30^{\circ}$.

Carson Merrill

Numerade Educator

03:35

Problem 2

Find the Radon transform of a circular cylinder with radius 6 and height 3 located at the origin.
The cylinder is characterized by
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for } x^2+y^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$
From the transform obtained, and using the Radon transform properties, find the Radon transform of
(i)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for } x^2+y^2 \leq 3^2 \\
0 \text { otherwise }
\end{array}\right.
$$
(ii)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for }(x-1)^2+(y-2)^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$
(iii)
$$
f(x, y)=\left\{\begin{array}{l}
3 \text { for }(3 x)^2+(3 y)^2 \leq 6^2 \\
0 \text { otherwise }
\end{array}\right.
$$

Linda Hand

Numerade Educator

Problem 3

Find the Radon transform of the shifted and scaled impulse.
(i) $\delta(x, y)$
(ii) $\delta(x-4, y-4)$
* (iii) $\delta(x-4, y+4)$
(iv) $\delta(x-1, y)$
(v) $\delta(x, y-1)$

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Problem 4

Find the Radon transform of the line $f(x, y)$ characterized by the given equation. Using the result, find the transforms of $f(a x, a y)$ and $f(x-p, y-q)$ from the properties of the Radon transform. Find the equation of the lines $f(a x, a y)$ and $f(x-p, y-q)$ and determine the Radon transform directly. Verify that the results are the same as those obtained using the properties.
(i)
$$
f(x, y)=3 x+2 y-6=0, x \text { is limited from } 0 \text { to } 2, a=2, p=2, q=3
$$
(ii)
$$
f(x, y)=4 x+2 y-8=0, x \text { is limited from } 0 \text { to } 2, a=3, p=3, q=2
$$
(iii)
$f(x, y)=x+y-1=0, x$ is limited from 0 to $1, a=-3, p=-3, q=2$

Nick Johnson

Numerade Educator

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Problem 5

Find the Radon transform $R(s, \theta)$ of the image $x(m, n)$ and reconstruct the image from its transform by the back-projection method, using the DFT and the IDFT.
(i)
$$
x(m, n)=\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{ll}
3 & 1 \\
2 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]
$$
(iv)
$$
x(m, n)=\left[\begin{array}{ll}
3 & 4 \\
1 & 2
\end{array}\right]
$$
(v)
$$
x(m, n)=\left[\begin{array}{ll}
4 & 3 \\
3 & 2
\end{array}\right]
$$

Nick Johnson

Numerade Educator

Problem 6

Detect the lines in the $4 \times 4$ binary image $x(m, n)$. Choose a suitable threshold. * (i)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0
\end{array}\right]
$$
(ii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0
\end{array}\right]
$$
(iii)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1
\end{array}\right]
$$
(iv)
$$
x(m, n)=\left[\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array}\right]
$$
(v)
$$
x(m, n)=\left[\begin{array}{llll}
0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right]
$$

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