Physics, Techniques and Procedures

Fourier transformation (FT)

extremely useful mathematical tool used in the quantitative analysis of many physical processes. In radiology the Fourier transformation is most prominently used in MR imaging.

Fourier transformation departs from Fourier analysis, which states that any function f(x) defined within an interval or period 0 £ x £ r can be represented as a series of sine and cosine functions (trigonometric functions), the so called Fourier series with a series index n. These sine and cosine functions are functions either in space or time. The spatial period is r, with n/r being the spatial frequency, and 2pn/r is called kn. When dealing with functions of time, the period is a time interval T. The frequency n is n/T and the angular frequency w = 2pn/T with w = 2 pn. The Fourier series can then be written as

f(x) = A0 + S (An cos nkx + Bn sin nkx), or

f(t) = A0 + S (An cos nwt + Bn sin nwt), (1)

where the sum S extends from n = 1 to ¥ (infinity). More elegantly, the Fourier series can be expressed as a series of imaginary exponential functions (imaginary number)

f(x) = S an exp(inkx), or f(t) = S an exp(inwt) (2)

where the sum extends from n = - ¥ to n = + ¥. An, Bn and an are related as follows:

a0 = A0, an = (An - iBn)/2, a -n= (An + iBn)/2

Think of any image line in an MR image with linear field of view FOV dimensions of r, where the function f(x) is the signal intensity variation along this image line in the horizontal direction. The coefficients an are called Fourier coefficients or amplitudes, and are computed as follows:

an = k/2pò f(x) exp(-inkx)dx,

or an = w/2pò f(t) exp(-inwt)dt (3)

where the integral extends over the sampled interval (i. e. from 0 to r or T). The mathematical operation in Eq. 3 is called the Fourier transformation. When the sampling points in space are discrete, as is the case with a digital imaging modality, x and t are discrete functions, which are sampled at intervals d between 0 and r = md (d: pixel resolution). The function f(ld), l=0,...,m is given as

f(ld) = S an exp (inlkd) (4)

and the Fourier coefficients are then given by

an = k/2pS f(ld) exp (-ilnkd) (5)

with the sum extending from l = 0 to n. This operation is the discrete Fourier transformation.

In MRI data are obtained in k space because of the nature of data acquisition, that is, an is measured rather than f(ld). f(ld) is obtained by reverse Fourier transformation. an can also be looked at as a discrete function a(nk) defined for all values of n between 0 and m. a(nk) is a spatial frequ face="symbol">ò

A(w) exp (iwt) dw (6)

with the integral extending from - ¥ to +¥ and

A(w) = (1/Ö (2p)) ò f(t) exp (-iwt) dt (7)

where 1/Ö(2p) is a normalization factor.

As an example, we consider the radiofrequency excitation pulse which is to be radiated into a patient in MR imaging. The selection of a section occurs by a spatially varying magnetic gradient field. To excite a section with a rectangular excitation profile, the envelope f(t) of the radiofrequency pulse has to be such, that spins in a limited frequency range w£w£w2 corresponding to the desired section thickness, are excited uniformly (excitation pulse (I), Fig. 1) and with the same amplitude A(w) so as to obtain a uniform flip angle in the section. The function A thus takes a finite value for frequencies between w£w£w2 and has to be zero for w < w and for w > w2. The temporal envelope f(t) which excites such a profile A is found by means of a Fourier transformation and turns out to be a so called sinc function (see excitation pulse).

Other frequently encountered functions have the following corresponding functions when subject to Fourier transformation and are called Fourier pairs.

Delta functiond(t) (infinitesimally narrow, area of 1)«f(w) = 1 constant at level 1
exponential f(t) = exp(ct)cos(w0 t) with /c/ £ 0, for t ³ 0 and /c/ ³ for t < 0 (envelope of an echo in spin-echo imaging)«Lorentzian function f(w) = c/(c2 + (w + w0)2)
Gaussian distribution function f(t) = exp(-t2)«Gaussian distribution function f(w) = exp(-w2)
f(t) = sin(t)/t«f(w) =1 for w 1)

Using the Fourier transform technique we can therefore calculate any radiofrequency pulse shape given the frequencies which the pulse is to excite. All the information present after MR data acquisition is transformed into a standard image by a Fourier transformation of the k-space data, hence the expression Fourier Transform imaging.

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