Step-by-step solution for the sharpen kernel calculation given an 8-bit greyscale matrix and a target pixel position centered at (2,3).

Steps to Solve

1. Extract the 3x3 Submatrix First, we need to extract the 3x3 submatrix from the greyscale matrix centered at the pixel position (2,3). The submatrix can be obtained as follows:

Submatrix from pixel (2,3):

[ \begin{bmatrix} M[1][2] & M[1][3] & M[1][4] \ M[2][2] & M[2][3] & M[2][4] \ M[3][2] & M[3][3] & M[3][4] \end{bmatrix} ]

Assuming our greyscale matrix ( M ) is:

[ M = \begin{bmatrix} 50 & 60 & 70 & 80 & 90 \ 60 & 70 & 80 & 90 & 100 \ 70 & 80 & 90 & 100 & 110 \ 80 & 90 & 100 & 110 & 120 \ 90 & 100 & 110 & 120 & 130 \end{bmatrix} ]

The submatrix will be:

[ \begin{bmatrix} 70 & 80 & 90 \ 80 & 90 & 100 \ 90 & 100 & 110 \end{bmatrix} ]

2. Define the Sharpen Kernel Next, define the sharpen kernel used for convolution. The typical 3x3 sharpen kernel is:

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

3. Apply Convolution Now, we apply the convolution of the submatrix with the sharpen kernel. Each element of the result is computed by taking the dot product of the kernel with the corresponding 3x3 area:

[ \text{Result} = \sum_{i=0}^{2}\sum_{j=0}^{2} M[i][j] \times K[i][j] ]

Calculating the convolution for each pixel:

[ \text{Result} = (70 \times 0) + (80 \times -1) + (90 \times 0) + (80 \times -1) + (90 \times 5) + (100 \times -1) + (90 \times 0) + (100 \times -1) + (110 \times 0) ]

4. Calculate the Final Result Now we perform the operations step-by-step:

[ = 0 - 80 + 0 - 80 + 450 - 100 + 0 - 100 + 0 ] [ = 0 - 80 - 80 + 450 - 100 - 100 ] [ = 0 - 160 + 450 - 200 ] [ = 290 ]

5. Clip the Result Since we are dealing with an 8-bit image, we clip the result to ensure it falls within the allowable range [0, 255].

The clipped value of 290 is equal to 255, which is the maximum value.