When analyzing an image signal as a series of Dirac impulses in the continuous spatial domain, I applied continuous low pass filters (to reconstruct or smooth it) with area equal to 1 to preserve the low frequency amplitudes in the signal spectrum. When using a kernel (which, correct me if I'm wrong, is just the "sampled" discrete version of the original continuous filter in the spatial domain) I was wondering what is the meaning of the sum of the kernel components, and when I should care?
I often see examples with kernels with components which sum to one, stating they preserves the original image brightness. But if the original continuous signal integrates to one, it does not mean the discrete version will sum to one, so this left me a bit confused.