I'm having a little trouble understanding why we even want to use convolution in the first place. I undrstand that we need to perform linear shift-invariant image filtering, which involves applying the same filter on some kernal size smaller than the image, and calculating new pixels as linear combinations of the adjacent pixels in that pixel's adjacent kernal.. I do not directly see from this equation how convolution allows us to accomplish this. Could you elaborate on why we can use convolution?

@mariecurie I may not understand your question completely here, but let me give it a shot!

This equation is the definition of a convolution operation. And according to this equation, a convolution performs linear shift-invariant image filtering. Why is the convolution operation linear? Well, the equation itself is linear (it involves a linear combination of pixel values). Why is the convolution operation shift-invariant? Well, if you shift the input, the result will simply shift the output.

What do the variables x and y here represent? Does x represent a pixel location and y the offset similar to the box filtering example, or something else?

@zyl yes, you are correct.