# Matched a phase only correlation filters

## Matched Filter Results

The easiest way to analyses the correlator is with a computer
simulation. If our input signal is the the letters ‘A’ and ‘B’

and the target filter is the letter ‘A’

The intensity of the cross correlation (matched filter) is then

Correlation plane image

Correlation output plot mesh

Note that the left hand peak is the correlation of the two A’s
and the right is the cross correlation of ‘A’ and the ‘B’. There is some
similarity between the ‘A’ and ‘B’ so the peak is not zero in height. This example
shows how the matched filter is not very good at discriminating between
objects.

## Phase Only Filter

One of the short comings of most realistic optical filters is
that they can only modulate one parameter. Typically this is either phase or
amplitude. For a practical application the filter is usually dynamically updatable
so we use an SLM (Spatial Light Modulator). Liquid crystal SLMs can modulate
the phase of light and these are commonly used as filters for correlators.
The just after the filter the signal is then

$$F(u,v)exp(-i*\theta(-u,-v))$$

where $$\theta$$ is the phase of $$G$$, the filter function. Now when the two signals match the phase components cancel to zero and the result is the Fourier transform of the amplitude of $latex F$. This usually a much smaller width peak than the matched filter result.

Here is an example result with the same inputs as before.

Phase only correlation

This filter is much more discriminating than the matched
filter. The correlation of the ‘A’ and ‘B’ is now smaller and the peak for the ‘A’ correlation is sharp making it much easier to locate. However, the phase only filter is perhaps too sensitive. If we replace the input with the following:

Rotated A

The second ‘A’ has been rotated by 5 degrees and the resultant correlation is

Rotated phase only correlation

The small change in rotation has resulted in the peak collapsing.

The reason for this is missing amplitude modulation in the filter. The amplitude component acts to attenuate the higher frequencies, without it they are over emphasised resulting in the filter been extremely sensitive to any small change in the signal. If the input is also represented by a phase only SLM, the result is a phase-phase correlator which is even more sensitive.

If we add some Gaussian noise to one of the ‘A’s and remove the rotation so the input is now

Noise in the input

The phase only filter result is

Noisy phase correlation

So even with the small amount of noise, the peak drop is noticeable.  On the other hand if we use a matched filter on the same input we get

Noisy matched correlation

The peak with the noise is slightly higher! This is because the matched filter is very good at ignoring noise and adding the noise has increased the overall power contained within the input signal. The reason the phase filter is so sensitive can be understood by looking at the missing amplitude. Have another look at the intensity pattern of the letter A above. In the full complex filter, this intensity pattern (actually its square root) acts to attenuate the high frequency components in the signal. Without it, the high frequency components are given equal weights to the low frequency components. The high frequency componets represent the fine detail of the image and so the phase only filter is over sensitive to these. Any small change in the fine detail causes the correlation to collapse.

### Binary Phase Only Filters

Some types of SLM (FLC) can modulate phase but only have two stable states 0 and π. They are however extremely fast so are still useful. If we take the our previous phase only filter and set it so θ can only be 0 or π our correlator still works (but not quite as well the maximum peak height drops from 100% to 40.5%). Here is the binary filter working on the original ‘A’ and ‘B’ input image we used earlier.

Binary phase only filter

Clearly these basic filters serve a purpose, they are easy to implement but they are a bit limited. Something bit better than telling the difference between ‘A’s and ‘B’s in needed.

Next I will look real world examples.