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One‐Dimensional Fourier Imaging and k‐Space
Publication Name:
Current Protocols in Magnetic Resonance Imaging
Unit Number:
Unit B4.2
DOI:
10.1002/0471142719.mib0402s00
Online Posting Date:
May, 2001
Abstract
Magnetic resonance imaging offers the possibility to obtain spatially resolved anatomical information. This is accomplished by taking advantage of the Larmor relationship, which dictates that the frequency of the spins depends on the local magnetic field. This unit discusses the one dimensional Fourier imaging based on this relation. The one-to-one mapping of the signal from a given frequency to a given spatial location is explained. The image reconstruction based on well-known Fourier transform reconstruction method is described in detail. The Fourier transform takes the MR signal as acquired in the time domain (usually referred to as the k-space domain) and converts it to the frequency domain where 1-D spatially resolved information can be obtained.
Figures
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Figure B4.2.1 Example of a 1-D MRI experiment. The positions of two-point spins (depicted as circles on a “spin dumbbell” with no MRI signal from the connecting rod) are to be determined by imaging. The z-component of the laboratory magnetic field is plotted to the left of the dumbbells. The rotating frame orientation of the spin in each dumbbell is displayed at the right of the diagram. The sequence diagrams underneath represent the rotating frame input RF (assumed to be along x¢ and producing a /2 rotation), the applied gradient, and the laboratory received signal (with T 2 decay neglected), before demodulation, as a function of time. Steps (A), (B), and (C) are described in the text. Note that Gz has the constant value G in part (C) between the times t1 and t2 . -
Figure B4.2.2 A consecutive set of time events in a 1-D MRI experiment. This illustration follows the description of Figure 2 decay, and (C) and (D) are described in the text. The magnetic field plotted at the left of the cylinder in (C) refers to the field in the time interval (t1 , t2 ), and, in (D), to the interval (t3 , t4 ). For comparison, the spin isochromats are pictured lying in a single plane, despite their different z coordinates. Note that the amplitude of the gradient is –G in part (C) and has both a negative and a positive lobe of strength G in part (D). -
Figure B4.2.3 The k-space coverage for an FID imaging experiment. The time t² = (t – t 1 ) is defined from the beginning of the gradient. The total sampling time is Ts . Notice that only half of k-space is covered by this experiment. There is no progression through k-space until time t1 in Figure -
Figure B4.2.4 The k-space coverage for the basic gradient echo experiment and the two variants of the spin echo experiment presented in the text. Diagram (A) applies to the basic gradient echo experiment and to the spin echo variant where all frequency encoding occurs after the -pulse. Diagram (B) shows the spin echo variant where both gradient lobes are positive. In all cases, k max =
G(t4 –t3 )/2. The dashed line represents the action of the -pulse which changes kmax to –kmax .
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Literature Cited
| Key References | |
| Haacke, E.M., Brown, R.W., Thompson, M.R., and Venkatesan, R. 1999. Magnetic Resonance Imaging: Physical Principles and Sequence Design. John Wiley & Sons, New York. | |
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This book covers the technical discussion here as well as other advanced materials in detail. | |
| Lauterbur, P.C. 1973. Image formation by induced local interactions. Examples employing magnetic resonance. Nature 243:190. | |
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This paper demonstrates the first imaging experiment using MRI. | |
