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:

\[ R_{f,g}(u,v) = \sum_{m,n} f(m,n) g(u+m,v+n). \]

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:

\[ \tilde{f}(m,n) = \frac{ f(m,n)-\mu_f }{ \sigma_f } \qquad \tilde{g}(m,n) = \frac{ g(m,n)-\mu_g }{ \sigma_g } \]

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:

\[ \tilde{R}_{f,g}(u,v) = \sum_{m,n} \tilde{f}(m,n) \tilde{g}(u+m,v+n). \]

Fig. 107 gives an example of matched filter.

Fig. 107 Normalized cross-correlation with the pattern shown top-left (the letter G).

As seen in Fig. 108, the major limit of the matched filter is that it is sensitive to variations in orientation, size, etc.

Fig. 108 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!