Chapter 10.4, Problem 49E

Digital Images A four-level gray scale is shown below.

1. (a) Use the gray scale to find a 6 × 6 matrix that digitally represents the image in the figure.

1. (b) Find a matrix that represents a darker version of the image in the figure.
2. (c) The negative of an image is obtained by reversing light and dark, as in the negative of a photograph. Find the matrix that represents the negative of the image in the figure. How do you change the elements of the matrix to create the negative?
3. (d) Increase the contrast of the image by changing each 1 to a 0 and each 2 to a 3 in the matrix you found in part (a). Draw the image represented by the resulting matrix. Does this clarify the image?
4. (e) Draw the image represented by the matrix I. Can you recognize what this is? If you don’t, try increasing the contrast.

$I=\left[\begin{array}{rrrrrr}\hfill 1& \hfill 2& \hfill 3& \hfill 3& \hfill 2& \hfill 0\\ \hfill 0& \hfill 3& \hfill 0& \hfill 1& \hfill 0& \hfill 1\\ \hfill 1& \hfill 3& \hfill 2& \hfill 3& \hfill 0& \hfill 0\\ \hfill 0& \hfill 3& \hfill 0& \hfill 1& \hfill 0& \hfill 1\\ \hfill 1& \hfill 3& \hfill 3& \hfill 2& \hfill 3& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 1& \hfill 0& \hfill 1\end{array}\right]$

To determine

To find: The matrix that digitally represent the given figure by using gray scale.

Answer to Problem 49E

The matrix that digitally represents the given figure by using gray scale is

$\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$

Explanation of Solution

Given:

The given digital image is,

Figure (1)

And gray scale is.

Figure (2)

According to given gray scale 0 represents brightest 1 represents light gray, 2 represents dark gray and 3 represents black.

Tabulate the matrix of dimension $6\times 6$ and put the values in the matrix according to the assigned colour in the gray scale,

$\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$

Hence, the matrix that digitally represent the given image is $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$.

To determine

To find: A matrix that represents a darken version of the given digital image.

Answer to Problem 49E

A matrix that represents a darken version of the given digital image is

$\left[\begin{array}{cccccc}2& 1& 2& 1& 2& 2\\ 1& 3& 1& 2& 3& 2\\ 2& 3& 1& 1& 3& 1\\ 2& 3& 3& 3& 3& 1\\ 1& 4& 1& 1& 3& 2\\ 2& 3& 1& 2& 3& 2\end{array}\right]$

Explanation of Solution

To darken the images add 1 to each of the elements of matrix formed in part (a) except the darkest one that is 3.

From part (a) matrix of given image is $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$.

$\left[\begin{array}{cccccc}1+1& 0+1& 1+1& 0+1& 1+1& 1+1\\ 0+1& 3& 0+1& 1+1& 2+1& 1+1\\ 1+1& 2+1& 0+1& 0+1& 3& 0+1\\ 1+1& 3& 2+1& 3& 2+1& 0+1\\ 0+1& 3& 0+1& 0+1& 2+1& 1+1\\ 1+1& 2+1& 0+1& 1+1& 3& 1+1\end{array}\right]=\left[\begin{array}{cccccc}2& 1& 2& 1& 2& 2\\ 1& 3& 1& 2& 3& 2\\ 2& 3& 1& 1& 3& 1\\ 2& 3& 3& 3& 3& 1\\ 1& 3& 1& 1& 3& 2\\ 2& 3& 1& 2& 3& 2\end{array}\right]$

Hence, matrix that represents a darken version of the given digital image is

$\left[\begin{array}{cccccc}2& 1& 2& 1& 2& 2\\ 1& 3& 1& 2& 3& 2\\ 2& 3& 1& 1& 3& 1\\ 2& 3& 3& 3& 3& 1\\ 1& 3& 1& 1& 3& 2\\ 2& 3& 1& 2& 3& 2\end{array}\right]$.

To determine

To find: The matrix that represents the negative of the image.

Answer to Problem 49E

Hence, matrix that represents the negative of the image in the figure is,

$\left[\begin{array}{cccccc}2& 3& 2& 3& 2& 2\\ 3& 0& 3& 2& 1& 2\\ 2& 1& 3& 3& 0& 3\\ 2& 0& 1& 0& 1& 3\\ 3& 0& 3& 3& 1& 2\\ 2& 1& 3& 2& 0& 2\end{array}\right]$

Explanation of Solution

From part (a) matrix of given image is $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$

Reverse the light and the dark to get the negative of the given image, put 0 for 3 and 1 for 2,2 for 1 and 3 for 0. in the matrix formed in part (a).

$\left[\begin{array}{cccccc}2& 3& 2& 3& 2& 2\\ 3& 0& 3& 2& 1& 2\\ 2& 1& 3& 3& 0& 3\\ 2& 0& 1& 0& 1& 3\\ 3& 0& 3& 3& 1& 2\\ 2& 1& 3& 2& 0& 2\end{array}\right]$

Hence, matrix that represents the negative of the image in the figure is

$\left[\begin{array}{cccccc}2& 3& 2& 3& 2& 2\\ 3& 0& 3& 2& 1& 2\\ 2& 1& 3& 3& 0& 3\\ 2& 0& 1& 0& 1& 3\\ 3& 0& 3& 3& 1& 2\\ 2& 1& 3& 2& 0& 2\end{array}\right]$

To determine

To sketch: The contrast image of the given figure.

Explanation of Solution

From part (a) matrix of given image is $\left[\begin{array}{cccccc}1& 0& 1& 0& 1& 1\\ 0& 3& 0& 1& 2& 1\\ 1& 2& 0& 0& 3& 0\\ 1& 3& 2& 3& 2& 0\\ 0& 3& 0& 0& 2& 1\\ 1& 2& 0& 1& 3& 1\end{array}\right]$

Change each 1 to 0 and 2 to 3 to form a matrix of a given image with increased contrast and element 0 and 3 in the matrix is remain same.

$\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 3& 0& 0& 3& 0\\ 0& 3& 0& 0& 3& 0\\ 0& 3& 3& 3& 3& 0\\ 0& 3& 0& 0& 3& 0\\ 0& 3& 0& 0& 3& 0\end{array}\right]$

Use this matrix to draw the image of increased contrast with the help of gray scale.

Hence, the image of the given figure with increased contrast is,

Figure (3)

To determine

To sketch: The digital image from the matrix $I=\left[\begin{array}{cccccc}1& 2& 3& 3& 2& 0\\ 0& 3& 0& 1& 0& 1\\ 1& 3& 2& 3& 0& 0\\ 0& 3& 0& 1& 0& 1\\ 1& 3& 3& 2& 3& 0\\ 0& 1& 0& 1& 0& 1\end{array}\right]$

Explanation of Solution

Draw the image corresponding to the assigned value in the matrix I with the help of gray scale, put white for 0, light gray for 1, dark gray for 2, and black for 3,

Figure (4)

Figure (4) shows the image for given matrix I.

As, image is not recognizable so, change the contrast of the image.

Change each 1 to 0 and 2 to 3 to form a matrix of a given image with increased contrast and element 0 and 3 in the matrix remain same.

$\left[\begin{array}{cccccc}0& 3& 3& 3& 3& 0\\ 0& 3& 0& 0& 0& 0\\ 0& 3& 3& 3& 0& 0\\ 0& 3& 0& 0& 0& 0\\ 0& 3& 3& 3& 3& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$

Again draw the image according to the given matrix using gray scale.

Figure (5)

The Figure (5) shows that the image is of English alphabet E.