# Box Filter

Notes from Cyrill Stachniss.

“Replace an intensity value by the mean intensity value of the neighbourhood”.

For 1D input $g(i)=K1 ∑_{k}f(i−k)$

For 2D input (image) $g(i)=KL1 ∑_{k,l}f(i−k,j−l)$

We can formulate the box filter by using a weighting function $w$
$g(i,j)=∑_{k,l}w(k,l)f(i−k,j−l)$
This weighting function is called *kernel* (or kernel function)

Often, these filtering operators involve weighted combinations of intensity values in a neighborhood.

A filter $L$ that transforms $g(i,j)=l(f(i,j))$ is called linear and shift invariant if … $L(α_{1}f_{1})$

Filters of the form $g(i,j)=∑_{k,l}w(k,l)f(i−k,j−l)$ are convolutions of the function $f$ and a kernel function $w$ $g=w∗f$