2.1.1. Linear Filters

Linear Filters

A linear filter modifies an image by applying a weighted sum of pixel intensities in a local neighborhood. The same operation is repeated across all pixels, making it shift-invariant and linear. Mathematically, the operation we are performing here is called a convolution:
$I’(x,y)={\sum }_{u,v}K(u,v)I(x-u,y-v),$
where K is the kernel (filter mask) and I’ is the filtered image.

Linear filters have various uses:

• Edge detection, using finite difference operators or the Sobel filter.
• Blurring using a box filter or a Gaussian blur.
• Sharpening using a Laplacian filter.