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Flipping the Kernel Matrix

In 2-D convolution, we flip the kernel matrix two times – once horizontally and once vertically. This process is
called Flipping the kernel matrix. In the vertical flip, we exchange the top row with bottom, second from top,
second from bottom, and so on. In the horizontal flip, the rightmost column is flipped with the leftmost, and
so on for the remaining. This in effect, causes the kernel matrix to be turned through 180 degrees. For ease
of operations, the Kernel matrix is always kept as a square matrix with rows = columns = an odd number. The
number is 3 or 5 in most of the cases.

Flipping of the kernel is important in context of convolutions. If we don’t flip the kernel, the process will be of
correlation, i.e., measuring similarities between two signals, rather than processing the signal to extract features.
The flip ensures that the convolution operation correctly captures the relationship between the input and the
kernel, allowing for proper feature extraction and signal processing.

1 2 3 3 2 1 9 8 7

4 5 6 6 5 4 6 5 4

7 8 9 9 8 7 3 2 1

To produce the output image, follow the given steps:

Step 1: Select a cell from the input image matrix.
Step 2: Take a kernel matrix of suitable size and flip it.

Step 3: Place the center of the kernel matrix at the selected cell in the image matrix.
Step4: Perform element-wise multiplication of the input matrix and the kernel matrix and add the products
to produce a single value as the output.
Step 5: Store the computed value in the output matrix in the corresponding cell as the input matrix.

Solved Example

Let us compute the convolution of an input matrix and kernel matrix.
1 2 1 1 –2 1
1 2 3 11 15 13

–1 0 –2 –2 0 –1
4 5 6 17 18 12

1 –2 1 1 2 1
7 8 9 –11 –23 –23

Input Image Kernel Image Flipped Kernel Convolution Output

175
175