Frequency Domain Filtering

What is Filtering?

Filtering is used to decrease statistical noise and/or enhance edges

à enhance edges = increased “Resolutions”

We can “filter” in 2 different realms:

1. Spatial domain

2. Frequency domain

Spatial Domain Filtering:

à Nonuniformity corrections.

à Background subtraction techniques.

à “smoothing” techniques. (i.e. 5-point or 9-point smoothing kernels)

à Edge Enhancement techniques.

Frequency Domain Filtering:

à In frequency domain filtering, we must convert our spatial

image (in counts and location parameter) into a frequency

domain image (in frequency parameters)

à We do this conversion with the Fourier Transform.

Fourier Transform

The basis of Fourier Transformation is that any curve or function can be represented as a series of sines and cosines with different amplitudes and frequencies.

Example: Square wave function

Note: This square wave function is similar to the count profile of a flood source (i.e. counts vs. distance across crystals)

Note: High frequency components

à Responsible for simulating the rapid changes in intensity

à Edges and noise.

Low frequency components

à Responsible for simulating amplitude of waveform

à Contrast and intensity

Remember what an activity profiles along a row of an image looks like:

This activity profile can then be thought of as a smooth curve or function. can then be thought

This smooth curve can then be transformed or converted by a Fourier transform into a frequency domain signal.

(i.e. frequency and amplitude info)

This can be done for every row in the image.

What we end up with is a data set (or matrix of data) that is now in cycles/pixel

=> cycles/pixel is a unit of “spatial frequency”

(relating the rate of change in intensity (or counts) with distance)

How do we get air spatial frequency units (cycles/pixel)?

For this example, let’s say this distance is the number of pixels in our image (i.e., 4 pixels).

The Nyquist Frequency

According to discrete sampling theory, the maximum usable frequency is 0.5 cycles per pixel because at least two pixels are required to define the one cycle of the wave.

This frequency is called the "Nyquist Frequency" (0.5 cycles/pixel) and is the highest frequency (of intensity variation) that can be accurately reproduced in our data.

à If our source has more variations than this, then the information will be lost and the image will not be faithfully reproduced.

à This is called Aliasing.

If the Nyquest frequency is given in cycles/cm then it will vary for different pixel sizes.

If we know the size of the pixels, then we can convert the

cycles/pixel à cycles/cm units.

Example:

For a 400 mm FOV, 64 x 64 matrix:

0.5 cycles/pixel = 0.5 cycles/pixel * 1 pixel/6.25 mm

= 0.08 cycles/mm

= 0.8 cycles/cm

So, spatial-frequency units can be in

Cycles/pixel

-OR-

Cycles/cm

SUMMARY

So, to recap:

1. We take a spatial domain image set.

2. Make an activity profile for each row of data in the image matrix.

3. FT that activity profile to give us our image data in the frequency domain. (i.e. the spatial-frequency domain).

Now we have an image in the frequency domain.

We can now apply various “filters” to remove or alter the magnitude of selected frequencies in our frequency domain data set.

So, what does a spatial-frequency data set look like?