Calculate the output produced if a 3 x 3 median filter were applied to a window, w, centred at coordinate (2, 2).
Understand the Problem
The question is asking to calculate the output of a 3 x 3 median filter applied to a specific window of a grayscale image matrix. The focus is on the pixel value at the coordinate (2, 2) using the provided median filtering formula to derive the result.
Answer
The output produced by the 3 x 3 median filter at coordinate (2, 2) is $127$.
Answer for screen readers
The output produced by the 3 x 3 median filter at coordinate (2, 2) is $127$.
Steps to Solve
- Identify the 3x3 window centered at (2, 2)
Select the 3x3 matrix portion of the grayscale image centered around the pixel at coordinate (2,2). The window includes the pixels: $$ \begin{bmatrix} 127 & 107 & 194 \ 186 & 3 & 70 \ 164 & 162 & 6 \end{bmatrix} $$
- List the pixel values
Extract the values from the selected window: $$ {127, 107, 194, 186, 3, 70, 164, 162, 6} $$
- Sort the pixel values
Sort the extracted values in ascending order: $$ {3, 6, 70, 107, 127, 162, 164, 186, 194} $$
- Determine the median value
Since there are 9 values (odd count), the median value is the middle element, which is the 5th element in the sorted list: $$ \text{Median} = 127 $$
The output produced by the 3 x 3 median filter at coordinate (2, 2) is $127$.
More Information
The median filter is often used in image processing to reduce noise while preserving edges. It works by replacing each pixel with the median value of neighboring pixels, which is effective in eliminating outliers.
Tips
- Forgetting to include the correct neighboring pixels when constructing the 3x3 window, which can lead to incorrect median calculations.
- Misidentifying the median value in the sorted list; it’s crucial to correctly find the middle element when the number of values is odd.
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