Apply rotate -90° to the following 8-bit grayscale image matrix: L = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
Understand the Problem
The question is asking to apply a -90 degree rotation to a given 8-bit grayscale image matrix. The matrix provided is 3x3 and requires transformation according to the specified rotation.
Answer
$$ \begin{bmatrix} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{bmatrix} $$
Answer for screen readers
The rotated matrix is
$$ \begin{bmatrix} 3 & 6 & 9 \ 2 & 5 & 8 \ 1 & 4 & 7 \end{bmatrix} $$
Steps to Solve
- Understand the rotation direction
To rotate a matrix -90 degrees (counterclockwise), we need to rearrange its elements.
- Identify the new positions
For a 3x3 matrix, the elements will move as follows:
- The first column will become the last row
- The second column will become the middle row
- The third column will become the first row
- Apply the rotation to the matrix
Starting with the matrix:
$$ L = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} $$
After a -90 degree rotation, the new arrangement will be:
- First row will take values from the third column: [3, 6, 9]
- Second row will take values from the second column: [2, 5, 8]
- Third row will take values from the first column: [1, 4, 7]
Thus, the new matrix will look like:
$$ L' = \begin{bmatrix} 3 & 6 & 9 \ 2 & 5 & 8 \ 1 & 4 & 7 \end{bmatrix} $$
The rotated matrix is
$$ \begin{bmatrix} 3 & 6 & 9 \ 2 & 5 & 8 \ 1 & 4 & 7 \end{bmatrix} $$
More Information
This transformation is common in image processing and is often used to correct image orientation. The rotation changes the spatial arrangement of the pixels in a way that can emphasize different features or aspects of the image.
Tips
- Confusing the rotation direction; ensure you always rotate in the specified direction (-90 degrees means counterclockwise).
- Failing to maintain the original element placement accurately; carefully track where each element moves.
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