Matched filter#
When a pattern (French: motif), perfectly known as a sub-image \(g\), is searched for in an image \(f\), then the cross-correlation (French: corrélation croisée) between \(f\) and \(g\) is a very efficient technique. This technique is often known as matched filter (French: filtre adapté). The cross-correlation between \(f\) and \(g\) gives a new image \(R_{f,g}\) defined as:
The cross-correlation can be calculated as a convolution, hence the term “filter” in the name of this technique.
Usually, \(f\) and \(g\) are normalized into:
where \(\mu_f\) and \(\sigma_f\) are respectively the mean and the standard deviation of the image \(f\). This results in the normalized cross-correlation (French: corrélation croisée normalisée) which is insensitive to changes in amplitude:
Fig. 87 gives an example of matched filter.
Fig. 87 Normalized cross-correlation with the pattern shown top-left (the letter G).#
As seen in Fig. 88, the major limit of the matched filter is that it is sensitive to variations in orientation, size, etc.
Fig. 88 Normalized cross-correlation with the pattern shown top-left (the digit 0).#
To overcome this limit, one can apply several matched filters, each representative of all the variations of the patterns. However, this idea is very time-consuming!