Smoothing Filters

Smoothing filters are used to reduce noise, remove small details and texture, and produce a blurred version of the image. They act as low-pass filters because they keep slow/smooth changes.

Constant Averaging Filter

An average (mean) filter is the simplest smoothing filter:

$$w=\frac{1}{9}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$$

So each new pixel in the filtered image is the average of the 9 pixels in its neighborhood in the original image.

Gaussian Averaging Filter

Instead of averaging all neighbors equally, we weight neighbors based on distance from the center pixel:

$$G_{\sigma }(x,y)=e^{-\frac{x^2+y^2}{2{\sigma }^2}}$$

Pixels closer to the center get more weight.

For example, for a $3\times 3$ filter with $\sigma =0.85$, we would have

$$G=\frac{1}{16}\left[\begin{array}{ccc}1& 2& 1\\ 2& 4& 2\\ 1& 2& 1\end{array}\right]$$

where all the coefficients are rescaled to sum to 1, which preserves brightness. Furthermore, all numbers are powers of 2, which makes it very efficient to compute.

Increasing the $\sigma$ causes a stronger blur.