For the following 8-bit grayscale image matrix: 107, 82, 57, 87, 244, 48, 159, 225, 195, 139, 73, 232, 7, 166, 110, 9. Calculate the output produced if the 3 x 3 Gaussian kernel de... For the following 8-bit grayscale image matrix: 107, 82, 57, 87, 244, 48, 159, 225, 195, 139, 73, 232, 7, 166, 110, 9. Calculate the output produced if the 3 x 3 Gaussian kernel defined below were applied centered at coordinate (2, 3). Round any answer to be in the natural numbers and round .5 upwards. Kernel: k = [[1/16, 1/8, 1/16], [1/8, 1/4, 1/8], [1/16, 1/8, 1/16]]. Read more

Steps to Solve

1. Identify the relevant pixel values

Extract the pixel values from the 8-bit grayscale image matrix at the coordinate (2, 3) and the surrounding pixels. The relevant 3x3 section of the matrix is:

[ 57,  87, 244 ]
[159, 225, 195 ]
[110,   9, 166 ]

2. Set up the convolution equation

The convolution of the pixel values with the Gaussian kernel is calculated using the formula:

$$ \text{Output} = \sum_{i=0}^{2} \sum_{j=0}^{2} \text{Pixel}(i,j) \times \text{Kernel}(i,j) $$

3. Compute the output using the Gaussian kernel

Substituting the pixel values and the kernel into the convolution equation:

• For kernel $k = [[1/16, 1/8, 1/16], [1/8, 1/4, 1/8], [1/16, 1/8, 1/16]]$, calculate each contribution:

[ \begin{aligned} \text{Output} = & \left( 57 \times \frac{1}{16} \right) + \left( 87 \times \frac{1}{8} \right) + \left( 244 \times \frac{1}{16} \right) \ & + \left( 159 \times \frac{1}{8} \right) + \left( 225 \times \frac{1}{4} \right) + \left( 195 \times \frac{1}{8} \right) \ & + \left( 110 \times \frac{1}{16} \right) + \left( 9 \times \frac{1}{8} \right) + \left( 166 \times \frac{1}{16} \right) \ \end{aligned} ]

Calculating each term:

• $57 \times \frac{1}{16} = 3.5625$
• $87 \times \frac{1}{8} = 10.875$
• $244 \times \frac{1}{16} = 15.25$
• $159 \times \frac{1}{8} = 19.875$
• $225 \times \frac{1}{4} = 56.25$
• $195 \times \frac{1}{8} = 24.375$
• $110 \times \frac{1}{16} = 6.875$
• $9 \times \frac{1}{8} = 1.125$
• $166 \times \frac{1}{16} = 10.375$

Adding these together:

$$ 3.5625 + 10.875 + 15.25 + 19.875 + 56.25 + 24.375 + 6.875 + 1.125 + 10.375 = 149.6875 $$

4. Round to the nearest integer upwards

To round $149.6875$ to the nearest integer upwards, we get $150$.