Apply upsample by nearest neighbour replication to the following 8-bit grayscale image matrix that has order 2 x 2 to make it 4 x 4.
Understand the Problem
The question is asking to apply the nearest neighbour replication method to upsample a 2x2 grayscale image matrix to a 4x4 matrix. This involves duplicating the pixel values in such a way that the expanded matrix maintains the original pixel representations at a larger scale.
Answer
The upsampled 4x4 matrix is: $$ \begin{bmatrix} 34 & 34 & 11 & 11 \\ 34 & 34 & 11 & 11 \\ 97 & 97 & 128 & 128 \\ 97 & 97 & 128 & 128 \end{bmatrix} $$
Answer for screen readers
The upsampled 4x4 matrix is: $$ \begin{bmatrix} 34 & 34 & 11 & 11 \ 34 & 34 & 11 & 11 \ 97 & 97 & 128 & 128 \ 97 & 97 & 128 & 128 \end{bmatrix} $$
Steps to Solve
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Identify the Original Matrix
The original 2x2 grayscale image matrix is: $$ L = \begin{bmatrix} 34 & 11 \ 97 & 128 \end{bmatrix} $$
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Set Up the New Matrix Size
We want to upsample this matrix to a 4x4 matrix. Each pixel in the original matrix will be represented in a 2x2 block in the new matrix.
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Replicate the Pixels
Each value from the original matrix will be copied into a 2x2 block in the new matrix:
- For element 34: $$ \begin{bmatrix} 34 & 34 \ 34 & 34 \end{bmatrix} $$
- For element 11: $$ \begin{bmatrix} 11 & 11 \ 11 & 11 \end{bmatrix} $$
- For element 97: $$ \begin{bmatrix} 97 & 97 \ 97 & 97 \end{bmatrix} $$
- For element 128: $$ \begin{bmatrix} 128 & 128 \ 128 & 128 \end{bmatrix} $$
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Construct the 4x4 Matrix
Now we arrange these blocks into the new 4x4 matrix: $$ \begin{bmatrix} 34 & 34 & 11 & 11 \ 34 & 34 & 11 & 11 \ 97 & 97 & 128 & 128 \ 97 & 97 & 128 & 128 \end{bmatrix} $$
The upsampled 4x4 matrix is: $$ \begin{bmatrix} 34 & 34 & 11 & 11 \ 34 & 34 & 11 & 11 \ 97 & 97 & 128 & 128 \ 97 & 97 & 128 & 128 \end{bmatrix} $$
More Information
The nearest neighbor replication method is a simple and effective technique for upsampling images. Each pixel from the smaller matrix is duplicated according to the desired size, maintaining the visual appearance of the original matrix.
Tips
- Overlapping Values: It's easy to mistakenly overlap the values when constructing the new matrix. Make sure each original pixel is turned into a distinct 2x2 block.
- Incorrect Block Size: Make sure to correctly replicate each pixel into a 2x2 block, rather than other sizes (e.g., 1x1 or 3x3).
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