FIGURE 1.1 Schematic representations of the Fourier transform. (a) If a time domain signal contains a tone at a single frequency, its Fourier transform will contain a peak at that frequency, which in this case is 163 Hz. (b) If the signal contains a superposition of two tones, the Fourier transform displays a second peak. In the case shown, a 15-Hz tone with approximately one-quarter the amplitude is modulating the original tone.

TABLE 1.1 Properties of Complex Numbers

1.1.1 THE CONTINUOUS FOURIER TRANSFORM AND ITS INVERSE

Let g(x) be a function of the real variable x. The output of the function g(x) can have complex values. The complex Fourier transform of g(x) is another function, which we call G(k):

     (1.1)

The two real variables x and k are known as Fourier conjugates and represent a pair of FT domains. Examples of domain pairs commonly used in MR are (time, frequency) and (distance, k-space). If the physical units of the pair of variables that represent the two domains are multiplied together, the result is always dimensionless. For example, with the time-frequency pair, the product:

     (1.2)

The two functions g(x) and G(k) in Eq. (1.1) are called Fourier transform pairs. Knowledge about one of the pair is sufficient to reconstruct the other. If G(k) is known, then g(x) can be recovered by performing an inverse Fourier transform (IFT):

     (1.3)

The IFT undoes the effect of the FT, that is:

     (1.4)

and vice versa:

     (1.5)

Note that the right sides of Eqs. (1.4) and (1.5) are simply g(x) and G(k), respectively, and are not multiplied by any scaling factors. This is because the IFT definition in Eq. (1.3) is properly normalized. A further discussion of the normalization is given in subsection 1.1.10.

Note the factor of 2π that appears in the argument of the exponentials in Eqs. (1.1) and (1.3). If instead domain variables such as time and angular frequency (ω, measured in radians/second) are used, then the form of the FT appears somewhat differently. The FT and its inverse become:

     (1.6)

Note the absence of the 2π factor in the exponential in Eq. (1.6) and the extra multiplicative normalization factor in front of the FT. Equation (1.6) could be recast into a more symmetric form by splitting the 2π into equal factors in the denominators of both the FT and IFT definitions. Alternatively, we can recast Eq. (1.6) by making the familiar substitution from angular frequency ω to standard frequency f (measured in cycles/second or hertz):

     (1.7)

Substituting Eq. (1.7) into Eq. (1.6) yields the symmetric FT pairs

     (1.8)

and

     (1.9)

In this book, we mainly use the form of the FT and IFT with the factor of 2π in the exponential, such as Eqs. (1.1) and (1.8).

1.1.2 MULTIDIMENSIONAL FOURIER TRANSFORMS, AND SEPARABILITY

Multidimensional FTs often arise in MRI. For example, the two-dimensional FT (2D-FT) of a function of two variables can be defined as:

     (1.10)

where and are vectors. The inverse 2D-FT is given by:

     (1.11)

Eqs. (1.10) and (1.11) are readily generalized to three or more dimensions.

If the function g is separable in x and y:

     (1.12)

then the FT is also separable:

     (1.13)

An example of a separable two-dimensional function is the Gaussian:

     (1.14)

In contrast,

     (1.15)

is not separable.

1.1.3 PROPERTIES OF THE FOURIER TRANSFORM

An important property of the FT is the shift theorem. A shift or offset of the coordinate in one domain results in a multiplication of the signal by a linear phase ramp in the other domain, and vice versa:

     (1.16)

A second useful property of the FT is that convolution in one domain is equivalent to simple multiplication in the other. If f(x) and g(x) are two functions, then convolution is defined as:

     (1.17)

and

     (1.18)

Parseval’s theorem (named after Marc-Antoine Parseval des Chêsnes, 1755–1836, a French mathematician) is a third commonly used property of the FT. It states that if f and g are two functions with Fourier transforms F and G, respectively, then

     (1.19)

where * denotes complex conjugation. Letting g = f in Eq. (1.19) results in a useful special case, which shows that the FT operation conserves normalization:

     (1.20)

Table 1.2 provides several 1D-FT pairs that are commonly used in MRI. These relationships can be applied to multidimensional FTs if the variables are separable.

1.1.4 THE DISCRETE FOURIER TRANSFORM AND ITS INVERSE

In MRI, the sampling process provides a finite number (e.g., 256) of complex data points, rather than a function of a continuous variable. Consequently, the MR image is normally reconstructed with a DFT. Given a string of N complex data...