Box Filter

Notes from Cyrill Stachniss.

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

For 1D input $g(i)=\frac{1}{K}{\sum }_{k}f(i-k)$

For 2D input (image) $g(i)=\frac{1}{KL}{\sum }_{k,l}f(i-k,j-l)$

We can formulate the box filter by using a weighting function $w$ $g(i,j)={\sum }_{k,l}w(k,l)f(i-k,j-l)$ 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 … $L({\alpha }_1{f}_1)$

Filters of the form $g(i,j)={\sum }_{k,l}w(k,l)f(i-k,j-l)$ are convolutions of the function $f$ and a kernel function $w$ $g=w\ast f$