Calculate the output produced if a 3 x 3 midpoint filter were applied to a window, w, centred at coordinate (2, 2). Round any answer ∈ R/N to be ∈ N, and round .5 upwards.

Steps to Solve

1. Extract the window values We need to identify the $3 \times 3$ window centered at coordinate $(2, 2)$ in the matrix. The corresponding values are:

   $$ \begin{bmatrix} 88 & 111 & 191 \ 102 & 152 & 191 \ 30 & 166 & 53 \end{bmatrix} $$

2. Determine the maximum value From the extracted window, we find the maximum value:

   • The maximum is $191$.

3. Determine the minimum value Next, we find the minimum value from the same window:

   • The minimum is $30$.

4. Apply the midpoint formula Using the midpoint formula:

   $$ \text{midpoint}(w) = \frac{\text{max}(w) + \text{min}(w)}{2} $$

   We substitute the values obtained:

   $$ \text{midpoint}(w) = \frac{191 + 30}{2} $$

5. Calculate the midpoint Calculating gives:

   $$ \text{midpoint}(w) = \frac{221}{2} = 110.5 $$

6. Round the result Finally, we round $110.5$ to the nearest integer upwards, yielding:

   • Final rounded value is $111$.