I am taking a Computer Vision class and the following question about Signal Processing and Filter Kernels is asked on the provided slides:
You are given a generic normalized low pass Filter kernel. Which Filter would be obtained by subtracting 1 from its central element?
However, I do not know the answer to this. Later I was provided with the following master solution:
Let $f$ be a generic low pass filter and $a $ be a generic image, the result of convolving the filter $(f-\delta ) $ with $a$ is: $(f-\delta ) *a$ = $f*a-\delta*a = -(a-f*a) $. To derive this expression, we need to use the linearity properties of the convolution and the fact that the convolution with a delta does not alter the original signal. In the end, it results in a high pass filter which inverts the sign of the image (i.e., its phase is delayed by 180 degrees).
But studying the solution I do not understand it either. What exactly is asked and what is meant by subtracting 1 from the central element of a generic low pass filter? How does one come to this solution? Is there maybe a better way to show this?