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The Fourier Transform
Fourier transforms (FTs) play an important role in tomographic reconstruction (see Chapter 16 ) and in computer implementation of convolutions (see Appendix G ). As well, they are the basis for the modulation transfer function (MTF), one of the methods for evaluating spatial resolution of imaging systems (see Chapter 15 ) and many other methods for image analysis and image processing. This appendix is intended to provide an intuitive description of what the FT is, how it is calculated, and some of its properties. For simplicity, the analysis is presented for one-dimensional examples. The extension to higher dimensions is relatively straightforward. A detailed treatment is beyond the scope of this text. The interested reader is referred to the excellent texts by Bracewell for further information.
A
The FOURIER TRANSFORM: What It Represents
The FT is an alternative manner for representing a mathematical function or mathematical data. For example, suppose that the function f ( x ) represents an image intensity profile. It can be shown that, so long as f ( x ) has “reasonable properties,” the profile can be represented as a sum of sine and cosine functions of different frequencies extending along the x-axis. The FT of f ( x ), denoted as F ( k ), represents the amplitudes of the sine and cosine functions for different spatial frequencies, k. Spatial frequency reflects how rapidly a sine or cosine function oscillates along the x-axis and has units of “cycles per distance,” such as cycles per cm, * or cm −1 .
The concept of spatial frequency is illustrated in Figure F-1 . Slow oscillations represent low spatial frequencies and rapid oscillations represent high frequencies. If f ( x ) represents an image profile, the former would represent primarily the coarse structures, whereas the latter would represent fine details.
* As noted in Chapter 15 , the notation k is used in physics to denote “spatial radians per distance” and the notation 
, or “k-bar,” is used to denote “cycles per distance.” Mathematically, 
= k /2π, because there are 2π radians per cycle. However, for notational simplicity, we use k for “cycles per distance” in this text.
Thus F ( k ) is a representation of the image profile in k-space, or spatial-frequency space, whereas f ( x ) is a representation of the profile in object space (sometimes also called distance space ). Either f ( x ) or F ( k ) is a valid representation of the image intensity profile and, as shown subsequently, either one can be derived from the other. Another notation for the FT is
where the symbol 
denotes the operation of computing an FT.
FTs can be computed for functions in other coordinate spaces as well. For example, in audio technology, signal intensity varies as a function of time, t, and its FT describes the function in terms of temporal frequencies, v, expressed as cycles per second, or Hz (sec −1 ). Coordinate pairs that are represented by a function and its FT, such as x and k, or t and v, are referred to as conjugate variables.
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