An introduction to smoothing--forcked from matthew-brett

Smoothing is a process by which data points are averaged with their neighbors in a series, such as a time series, or image. This (usually) has the effect of blurring the sharp edges in the smoothed data. Smoothing is sometimes referred to as filtering, because smoothing has the effect of suppressing high frequency signal and enhancing low frequency signal. There are many different methods of smoothing, but here we discuss smoothing with a Gaussian kernel. 

Why smooth?

The primary reason for smoothing is to increase signal to noise. Smoothing increases signal to noise by the matched filter theorem. This theorem states that the filter that will give optimum resolution of signal from noise is a filter that is matched to the signal. In the case of smoothing, the filter is the Gaussian kernel. Therefore, if we are expecting signal in our images that is of Gaussian shape, and of FWHM of say 10mm, then this signal will best be detected after we have smoothed our images with a 10mm FWHM Gaussian filter.

Sometimes you do not know the size or the shape of the signal change that you are expecting. In these cases, it is difficult to choose a smoothing level, because the smoothing may reduce signal that is not of the same size and shape as the smoothing kernel. There are ways of detecting signal at different smoothing level, that allow appropriate corrections for multiple corrections, and levels of smoothing. This Worsley 1996 paper describes such an approach: Worsley KJ, Marret S, Neelin P, Evans AC (1996) Searching scale space for activation in PET images. Human Brain Mapping 4:74-90

source: https://matthew-brett.github.io/teaching/smoothing_intro.html