How does Correlation for 1D Signals work?

On page 2 of the slide, I don't understand the example. Like $$j$$ and $$s$$ are both on the x-axis, what does j+s mean? The filter is $$(f(−1), f(0), f(1)) = (−0.5, 0, 0.5)$$. How can we use it? Could anyone explain how we get $$r(j)$$ with numbers?

For the simple $$3$$-tap filter given in the example, each output value $$r(j)$$ is computed as
\begin{align}r(j)&=f(-1)I(j-1)+f(0)I(j)+f(1)I(j+1)\\&=-\frac12 I(j-1)+\frac12 I(j+1)\tag{1}\end{align}