Let’s calculate the output of the sharpen kernel for the matrix provided. The sharpen kernel is applied centered at coordinate (2,3).

Steps to Solve

1. Define the Sharpen Kernel

The sharpen kernel is a 3x3 matrix that enhances the edges in an image. A typical sharpen kernel is defined as:

$$ K = \begin{bmatrix} 0 & -1 & 0 \ -1 & 5 & -1 \ 0 & -1 & 0 \end{bmatrix} $$

2. Extract the 3x3 Region

Next, we need to extract the 3x3 region from the original matrix centered at the coordinate (2,3). Assuming the matrix is:

$$ M = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \ 1 & 2 & 2 & 2 & 1 \ 1 & 2 & 3 & 2 & 1 \ 1 & 2 & 2 & 2 & 1 \ 1 & 1 & 1 & 1 & 1 \end{bmatrix} $$

The 3x3 region centered at (2,3) (using 0-based indexing) is:

$$ R = \begin{bmatrix} 2 & 2 & 1 \ 2 & 3 & 2 \ 2 & 2 & 1 \end{bmatrix} $$

3. Perform the Convolution Operation

The convolution operation involves multiplying corresponding elements in the kernel and the region, and then summing the results.

Let’s calculate this step-by-step:

• For the center of the sharpen kernel and extracted region:

$$ (0 * 2) + (-1 * 2) + (0 * 1) + (-1 * 2) + (5 * 3) + (-1 * 2) + (0 * 2) + (-1 * 2) + (0 * 1) $$

4. Sum the Products

Calculating the sum:

• Contributing products:
  • $0*2 = 0$
  • $-1*2 = -2$
  • $0*1 = 0$
  • $-1*2 = -2$
  • $5*3 = 15$
  • $-1*2 = -2$
  • $0*2 = 0$
  • $-1*2 = -2$
  • $0*1 = 0$

Now sum these results:

$$ 0 + (-2) + 0 + (-2) + 15 + (-2) + 0 + (-2) + 0 = 7 $$

5. Output the Final Result

The final result after applying the sharpen kernel at the coordinate (2,3) is 7.