Matched filter - Basics of Image Processing

Consider the problem where one searches the presence and position of a perfectly known sub-image $g$ in an image $f$ . The sub-image $f$ is called the pattern (French: motif). Then the cross-correlation (French: corrélation croisée) between $f$ and $g$ is a very efficient technique. This technique is often known as matched filter or template matching (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 $\sigma_f$ (resp. $\mu_g$ and $\sigma_g$ ) are respectively the mean and the standard deviation of the image $f$ (resp. $g$ ). 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).

Figure 1 gives an example of matched filter.

Figure 1:Normalized cross-correlation with the pattern shown top-left (the letter G).

As seen in Figure 2, the major limit of the matched filter is that it is sensitive to variations in orientation, size, etc.

Figure 2: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 is very time-consuming!