PERSISTENTE
B. LA COMPLEJIDAD
3. LA EVENTUAL INCOMPATIBILIDAD DE LOS INTERESES PROTEGIDOS
Suppose that the real space function represents some sort of signal in time, that is, a represen-tation of the signal intensity as it varies with time, like an MR signal. The Fourier space rep-resentation does not have the same units. The units on the horizontal axis in Fourier space are inverse of the units in real space. In this case, the horizontal real space unit is time (e.g., sec-onds). Therefore the unit in Fourier space is 1/time (e.g., 1/seconds).
The quantity 1/time (e.g., cycles/second, hertz) occurs often and is frequency. A plot of intensity versus frequency is a spectrum. The FT can take a picture of intensity versus time (i.e., the time domain) and create a picture of the same signal represented as intensity versus frequency (i.e., the frequency domain).
Once again, real space and Fourier space views are two different representations of the same real-world object. For example, the MR sig-nal is a real space representation of how that signal varies with time. The FT shows how the same signal varies with frequency, that is, what frequencies are present in the signal.
The concept of frequency domain or spatial frequency domain is easy to recognize. For example, ears do a frequency transformation of the signals (i.e., sounds) that they receive. The time domain representation of the sound gen-erated by a complex source such as a sym-phony orchestra can be represented by a rapidly varying oscillating signal. This signal alone conveys little meaning. However, human
hearing takes this time-varying signal and transforms it into frequencies.
In the complex sound from a symphony, the ear can distinguish between the high treble pitch of a violin and the deeper bass pitch of a tuba. Indeed, a small percentage of people have absolute pitch, the ability to tell the pre-cise pitches (i.e., frequencies) of the sounds they hear. In this sense, ears “view” the world in the frequency domain.
Too Small to See
On the left side of Figure 8-5 are three smooth, bell-shaped (i.e., Gaussian) functions in real space. On the right side are their corresponding Fourier transforms. If these real-space functions represent a signal (i.e., the representation of intensity with time), then the Fourier space rep-resentation is a frequency spectrum (i.e., repre-sentation of the frequencies present).
The time domain representation in the top pair of curves shows a wide curve that changes slowly over time. The corresponding Fourier space representation shows a narrow function in the frequency domain. This means that the signal contains only a narrow range of fre-quencies. As the time domain signal narrows (i.e., the signal is made to change faster in time), the curves in the frequency domain become broader, indicating that a wider range of frequencies is required.
In extreme cases where a sharp signal spike exists in the time domain, the range of fre-quencies contained in that spike approaches infinity. Thus the more localized a signal is in time, the wider the range of frequencies that must be handled. In other words, the sharp edges of objects contain more extremely wide ranges of frequencies and higher frequencies than smooth objects.
A bone-soft tissue interface is a high spa-tial frequency object.
This simple fact has profound implications for viewing the world. Consider again the square wave of Figure 8-1. This sharp-edged object might be an MR signal of a fluid-filled cyst. In problem
solution
real space Fourier space
“problem”
“solution”
FT
FT-1
Figure 8-4 A magnetic resonance image is obtained by transforming a signal into Fourier space, reassembling the data, and computing the inverse transform.
Figure 8-1, the Fourier space representation shows what frequencies are present in the object as given by the wavy function to the right. Note that this representation only shows part of the Fourier space function; the ripples diminish in height to the right and left but never totally disappear until reaching plus and minus infinity. For this object to be truly rep-resented, an infinite range of frequencies must be handled.
This is unfortunate because no system can handle an infinite range of frequencies. When an object is detected with the radio receiver of an MRI system, the frequencies inherent in the object must pass through the imaging system.
If the electronics of the system do not pass all the frequencies in the object, then part of the structure of that object is lost.
The system could be designed more care-fully, but there is always a finite limit to the range of frequencies that it allows to pass.
Therefore some part of the information from the object (i.e., the part of the object contained in the high frequencies) is always lost. Various curves show the effect of this loss (Figure 8-6).
From a curve in Fourier space, the view of that curve in real space can be obtained by
applying the inverse FT. In the top set of curves, the high frequencies have been abruptly chopped off. This is truncation, and the signal is said to be truncated. Truncation results in large oscillations at the sharp edges of the real space square wave.
If an MRI system has a sharp cutoff of frequencies, a false, sharp ringing will be detected every time the signal changes abruptly. Because this ringing is so objection-able, systems are designed so that they do not cut off sharply at the edge of their frequency range; rather, they fade away gradually. A sharp square wave received by such a system would come out with rounded edges. The information that forms the sharp edges of the object is lost, thereby causing the object to become blurred. Spatial resolution is reduced.
Consequently, there is a limit to the ability to image sharp edges and produce fine detail in that image. This situation can be improved by increasing the ability of the MRI system to han-dle high frequencies. Because there is always a limit to the frequencies that any MRI system can handle, there is always a limit to the object size that can be imaged.
Fourier space
Figure 8-5 The Fourier transformation of a broad gaussian distribution results in a nar-row frequency spectrum and vice versa.
No imaging system can pass an infinite range of frequencies.
Chemistry’s Signature
The MR signal is profoundly affected by the chemical bonding of the atoms generating the signal. If a complex molecule emits the MR sig-nal, the signal would have a correspondingly complex structure.
A nucleus that is bound inside a complex molecule would resonate at a slightly different frequency than a nucleus in a simple water molecule. This is due to the magnetic fields of electrons in the atom shielding the nucleus from the B0field (see Chapter 9).
These changes in resonance are changes in frequency, but the signal received is changing in time. The FT is the bridge connecting the time domain signal to the frequency domain representation.
The sample FID is the plot of signal inten-sity versus time for a complex molecule (Figure 8-7). If an FT is applied to this signal, a plot of signal intensity versus frequency is obtained. The clear arrangement of sharp peaks of various heights should be noted
because this particular arrangement of peaks is the unique chemical signature for that molecule.
A trained NMR chemist learns to recognize the standard arrangements of these peaks. The relationships of these peaks to one another and their widths and heights indicate the nature of the bonding between atoms. All this same information is contained in the original FID, although in an obscured form. The FT pro-duces a new and useful view of the data.
SPATIAL LOCALIZATION AND THE FOURIER TRANSFORM
For an MR image to be made, the origin in space for each part of the signal must be known. Unfortunately, only one signal at a time is received from the patient. Therefore spatial information must be encoded into each such signal (see Chapters 15 and 16).
In a uniform external magnetic field, all nuclei resonate at the same frequency, called the resonant frequency. If a gradient magnetic field that varies uniformly in the Z direction is added to this primary magnetic field, the spins
Roll-off Truncated Off
Fourier space Real space
FT-1
FT-1
Figure 8-6 If the high frequencies of a signal in Fourier space are chopped off (i.e., truncated), the inverse Fourier transform (FT−1) results in a ringing appearance at the sharp boundaries in an image.
on the −Z-axis would be in a lower magnetic field than those on the +Z-axis. Therefore they will resonate at a lower frequency. This differ-ence in frequency directly relates to the posi-tion of the spins along the Z-axis and the amplitude of the gradient magnetic field.
Two globs of fat are shown at different posi-tions in the Z direction (Figure 8-8). With no Z gradient magnetic field, the two globs resonate at the same frequency and contribute to a single peak in the frequency domain. When a gradient magnetic field, GZ, is added, this single peak splits into two peaks, one peak for each glob, each of which is now at a different frequency.
The frequency difference between the peaks directly relates to the distance between the globs of fat in the Z direction. The stronger the gradient magnetic field, the further apart the peaks in the frequency domain; thus it is easier to separate objects in space.
Gradient magnetic fields provide spatial localization of the MR signal.
This process relies on the FT. Because an MR signal is a variation of intensity as a function of time, an FT must be applied to the signal to view the frequency distribution that is related to the spatial distribution.
FT Signal
intensity
Signal intensity
Time Frequency
Figure 8-7 The Fourier transform (FT) of a free induction decay results in a nuclear magnetic resonance spectrum, which is in the frequency domain.
FT
FT
glob A + glob B
frequency domain without gradient
frequency domain with Z gradient
A B
A B
BZ
glob A glob B
Figure 8-8 When a Z gradient magnetic field is applied, the same tissue results in dif-ferent peaks in the frequency domain.
In the example found in Figure 8-8, each fat glob generated a signal. A gradient magnetic field was used to determine where along the Z direction the signal from each originated.
Another approach can also be taken. Such a method would energize only one part of the object so that any signal received would only come from that part, rather than from the entire object.
This method uses the same gradient mag-netic field system as seen in the previous meth-ods. For a signal to be received from the spins in only one object, the initial RF pulse must only excite those spins. For example, if the spins in object B are energized and the spins in object A are left undisturbed, any signal received would come from object B alone.
For the precessing nuclei to absorb energy, the RF signal must exactly match the fre-quency of precession of the spins. Because objects A and B are in a gradient magnetic field, the spins in the objects have slightly dif-ferent resonant frequencies. Therefore an RF pulse with a frequency distribution unique to object B, not that of object A, is required if object B is to be imaged (Figure 8-9).
For the RF signal to be actually generated, however, knowledge of the signal as a function of time is required. The FT allows a change between time and frequency. To go from the frequency domain to the time domain, one must apply the inverse FT. This exercise
pro-vides the shape of the RF pulse that must be used. If this shaped RF pulse were transmitted into a patient, only the spins in a chosen part of the patient would be excited. This isolates the MR signal in a narrow section of the patient.
In these simple examples, spatial informa-tion has been encoded in only one dimension.
Similar methods, which are also heavily based on the use of the FT, are used to obtain spatial information in all three dimensions.