Fourier Transform Examples
Example 1 - Gaussian Function
The Fourier transform of the Gaussian function is another Gaussian:
Note that the width sigma is oppositely positioned in the arguments of the exponentials. This means the narrower a Gaussian is in one domain, the broader it is in the other domain.
The Gaussian function can approximate the behavior of an imaging system. In particular, if we think of a very narrow slit of x-rays as being a line of delta functions, an x-ray screen will blur this delta line into a broader "ridge". It should be obvious that we want this ridge to be as narrow as possible. The imaging system's response to a delta function line input is called the LINE SPREAD FUNCTION, or LSF in the spatial domain. The magnitude of the complex function which is the Fourier transform of the LSF is the frequency-dependent function known as the MODULATION TRANSFER FUNCTION, or MTF.
Using what we have just learned about Gaussian functions, we conclude that the narrower the LSF, the broader the MTF in frequency space. Since we want narrow LSF's to produce sharper images, we want MTF's to stay high until a high spatial frequency is reached before it falls to zero. High frequencies are associated with sharp features in the image, and the MTF is the system's ability to record information as a function of frequency.
HEISENBERG UNCERTAINTY PRINCIPLE
In Quantum Mechanics, the Heisenberg uncertainty principle states that we cannot simultaneously know a particle's position and momemtum (or direction of motion). This is because the position wave function and the momentum wave function are Fourier transform pairs. The narrower one function becomes, the wider the pair becomes. The better we know position, the worse we know momentum.
Example 2 - Square Wave Function
The Fourier transform of an ideal-edged square wave function is the sinc function:
Keep in mind that the Fourier transform edges will be the sinc function and associated "ringing" will be introduced in the opposite domain.
RINGING IN AN IMAGE IS MOST LIKELY CAUSED BY AN EDGE BEING APPLIED IN THE OPPOSITE DOMAIN.
In Chapter 16 of Parker's book it is shown that the complicated integral of the convolution (shift, scale, add) is simply the product of the Fourier transforms in the opposite domain. That is, convolution of two functions in one domain is the multiplication of the Fourier transform of the two functions in the other domain:
This is very useful in analyzing systems conprised of a sequence of components. For example, an imaging system for x-rays might look something like this:
and we can analyze the system imaging performance using either the LSF or the MTF:
The ability to perform the mutliplication in the frequency domain is a considerable simplification.
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Douglas J. Wagenaar, Ph.D., email@example.com