Perform convolution and correlation on the given image matrix with the provided kernel, noting the differences in the outputs.

Steps to Solve

1. Define the Image Matrix and Kernel

The given image matrix is:

$$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 \ \end{bmatrix} $$

The kernel to be used is:

$$ \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \ \end{bmatrix} $$

2. Perform Correlation

In correlation, the kernel is applied directly to the image without flipping. The process involves sliding the kernel over the image and computing the sum of products for each overlapping position.

For example, for the position where 1 (from the kernel) overlaps with the value at (1, 2) in the image, you calculate:

$$ 1 \cdot 1 + 2 \cdot 0 + 3 \cdot 0 + 4 \cdot 0 + 5 \cdot 0 + 6 \cdot 0 + 7 \cdot 0 + 8 \cdot 0 + 9 \cdot 0 = 1 $$

Repeat the process for all overlapping positions.

3. Perform Convolution

For convolution, first flip the kernel horizontally and vertically, resulting in the flipped kernel:

$$ \begin{bmatrix} 9 & 8 & 7 \ 6 & 5 & 4 \ 3 & 2 & 1 \ \end{bmatrix} $$

The convolution process is similar to correlation but uses this flipped kernel. Calculate the overlapping sums similarly:

$$ 9 \cdot 1 + 8 \cdot 0 + 7 \cdot 0 + 6 \cdot 0 + 5 \cdot 0 + 4 \cdot 0 + 3 \cdot 0 + 2 \cdot 0 + 1 \cdot 0 = 9 $$

And repeat for all positions.

4. Fill in Outputs

Using both methods (correlation and convolution), fill in the resulting matrices based on calculated values for all positions.