DOI:10.2214/AJR.07.2874
AJR 2008; 190:1396-1405
© American Roentgen Ray Society
An Introduction to the Fourier Transform: Relationship to MRI
Thomas A. Gallagher1,
Alexander J. Nemeth1,2 and
Lotfi Hacein-Bey1
1 Department of Radiology, Loyola University Medical Center, 2160 S First Ave.,
Maywood, IL 60153.
2 Present address: Department of Radiology, Northwestern University, Chicago,
IL.
Received July 16, 2007;
accepted after revision November 20, 2007.
CME
The article is available for CME credit.
See
www.arrs.org
for more information.
Address correspondence to T. A. Gallagher
(tgalla1{at}lumc.edu).
Abstract
OBJECTIVE. The Fourier transform, a fundamental mathematic tool
widely used in signal analysis, is ubiquitous in radiology and integral to
modern MR image formation. Understanding MRI techniques requires a basic
understanding of what the Fourier transform accomplishes. MR image encoding,
filling of k-space, and a wide spectrum of artifacts are all rooted in the
Fourier transform.
CONCLUSION. This article illustrates these basic Fourier principles
and their relationship to MRI.
Keywords: Fourier transform • Gibbs artifact • k-space
Introduction
Joseph Fourier (1768–1830) is credited with observing that a complex
signal can be rewritten as the infinite sum of simple sinusoidal waves
[1]. Fourier himself is most
famous for applying this principle to solving an array of differential
equations governing heat dissipation. However, applications of this concept
are invaluable to anyone who wishes to study the composition of a complex
signal, whether it is in the form of music, voices, images, or digital medical
imaging, including MRI.
Any complicated signal or wave can be rewritten as the sum of a series of
simple waves. An approximation of a complicated wave can be achieved by adding
together very simple sine and cosine waves
(Fig. 1) with varying
combinations of frequencies and amplitudes
(Fig. 2). The Fourier series
(Fig. 3) provides a means to
describe a complicated wave in terms of simple sines and cosines. The more of
them we add together, each with successively higher frequency, the better the
approximation (Fig. 4A,
4B,
4C).
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Fig. 3 —Fourier series for g(t). A complicated wave
g(t) can be rewritten as an infinite sum of simple cosine and sine
waves by progressively increasing their fundamental frequency f by
integers n, and by varying their amplitudes, an
and bn. If we substitute g(t) = 1 (a square wave)
into the equations shown here, we obtain expressions for a0,
an, and bn that can be inserted into
the Fourier series. After simplifying, we are left with a Fourier
approximation for a square wave.
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Fig. 4A —Fourier series for a square wave. A square wave, given by y
(equal to 1 for 0 < x <  and equal to 0 everywhere else),
can be approximated with increasing accuracy by addition of simple sine and
cosine waves of progressively increasing frequency.
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The Fourier transform decomposes a complicated signal into the frequencies
and relative amplitudes of its simple component waves.
The Fourier transform (Fig.
5A) allows us to study the frequency content of a variety of
complicated signals [1]. We can
view and even manipulate such information in a Fourier or frequency space
(Fig. 5B). A familiar clinical
example of a Fourier space is the frequency and amplitude spectrum obtained
during MR spectroscopy (Fig.
5C). The multiple peaks (amplitudes) in the MR spectroscopy
spectrum represent the relative contributions of particular biologic
metabolites in a given region of interest (ROI), each differing slightly in
resonance frequency as a result of its unique chemical structure
[2].
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Fig. 5A —Fourier transform (FT). Fourier transform of a complicated
signal g(t), which exists in time (t) or spatial domain,
gives an expression for frequency domain G(f). When plotted,
frequency domain displays individual frequencies and relative amplitudes of
simpler waves constituting g(t). Inverse Fourier transform (iFT) of
G(f) restores the time domain. No information is gained or lost in
mathematic transforms; they merely change the way we see the same
information.
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Fig. 5B —Fourier transform (FT). Fourier transform (FT) extracts the
frequencies and relative amplitudes of the simpler waves hidden in a
complicated wave g(t). Inverse Fourier transform (iFT) restores the
time domain. In this example, Fourier transform of three cosine waves of
different frequencies results in three delta functions.
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Fig. 5C —Fourier transform (FT). MR spectroscopy. In contrast to MRI,
which uses resonance frequencies and phase to encode an image, MR spectroscopy
addresses a smaller region of interest (ROI) with a specific radiofrequency
pulse bandwidth. Multiple neuronal metabolites (mI, myoinositol; Cho, choline;
Cr, creatine; Glx, glutamate and glutamine; NAA, N-acetyl aspartate;
Lac, lactate; Lip, lipid) resonate at characteristic frequencies on the basis
of their unique chemical structure. The returning MR spectroscopy echo is a
composite signal of many different echoes from metabolites in the ROI, which
is resolved into individual resonance frequencies and their relative
amplitudes (abundance) by the Fourier transform. The term
"relative" is an important qualifier because the Fourier transform
cannot measure the absolute nature of any frequency. The height of a peak in
the MR spectroscopy Fourier spectrum makes sense only relative to another
peak.
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MR Image Encoding and Filling k-Space
With regard to MRI, the complicated signal we wish to decompose is the MR
echo containing the frequency- and phase-encoded spatial information necessary
to construct an image. Following the slice selection gradient for a typical
spin-echo sequence, which isolates a particular imaging plane, all spin
systems precess at the same frequency and phase as dictated by the main
magnetic field (Bo). At this point, all protons in the desired
imaging plane look the same to the Fourier transform. The application of
superimposed, dynamically changing gradient fields introduces spatially
dependent variations in frequency and phase across the ROI, effectively
interrogating the anatomy for all different spatial frequencies. The steeper
the applied gradient, the greater degree of achievable separation of spin
systems is possible. Strong gradients are necessary to seek out high spatial
frequencies (detail), whereas less steep gradients bring out lower spatial
frequencies (contrast). In addition, these smaller gradients generate much
more collective signal than steep gradients because precessional frequencies
and phase are overall more similar across the ROI (this will be evident when
we compare the central portions of k-space with the periphery, discussed
later). The amplitude of the returning echoes will vary with tissue
composition, TR, and TE [3,
4].
This otherwise hopelessly complicated signal is digitized, dismantled by
the Fourier transform, and entered into k-space, a 2D Fourier space that
organizes spatial frequency and amplitude information (Fig.
6A,
6B,
6C,
6D,
6E). One pixel in k-space,
when inverse-transformed, contributes a single, specific spatial frequency
(alternating light and dark lines) to the entire image. A 2D inverse Fourier
transform of the entirety of k-space combines all spatial frequencies, and
results in the image we see. Depending on where a pixel resides in k-space,
the lines will be of varying frequency and orientation. By convention, high
spatial frequencies are mapped to the periphery of k-space and low spatial
frequencies are mapped near the origin. The relative intensity of a pixel
reflects its overall contribution to the image, with brighter pixels
contributing more of a particular spatial frequency.
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Fig. 6B —Fourier spaces and k-space
[7,
8]. Fourier transform of a
single spatial frequency in the image domain is simple. Three bright dots are
seen in the Fourier space as a consequence of symmetry properties inherent to
the Fourier transform.
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Fig. 6C —Fourier spaces and k-space
[7,
8]. Fourier transform (FT) of
an image is represented by a 2D gray-scale magnitude image in which each pixel
represents a particular spatial frequency. By convention, high frequencies are
mapped to the periphery and low frequencies to the origin. Pixel intensity
corresponds to the relative contribution of that frequency to the entire
image. Any image (which can be thought of as a complicated wave of varying
pixel intensity) can be constructed by the combination of different spatial
frequencies (simple waves). Fourier transform of a simple white square on a
black background, for instance, shows a cruciate pattern of increased
intensity along the traditional x- and y-axes. This reflects
the contribution of spatial frequencies (given by the inverse FT = iFT) most
necessary to recreate the image, which happen to be orthogonal to the edges of
the square. Because essentially no diagonals or curves are present in the
image, these spatial frequencies are not as highly represented in the Fourier
space. (Fourier transform and inverse Fourier transform images (iFT) generated
with ImageJ, National Institutes of Health, Bethesda, MD)
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Fig. 6D —Fourier spaces and k-space
[7,
8]. Fourier transform (FT) of
photograph of Lincoln. All spatial frequency information necessary to create
this image of Lincoln is stored in his Fourier space (right). As
discussed previously, a single pixel in the image does not have a single pixel
correlate in the Fourier space. Rather, each pixel in Fourier space
contributes a spatial frequency to the overall image of Lincoln.
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Fig. 6E —Fourier spaces and k-space
[7,
8]. MRI. This coronal slice of
a brain is interrogated for all its different spatial frequencies by
successively altering magnetic field gradients (open arrows in top three
images) during frequency- and phase-encoding. Although only three
examples are shown here, many different gradient combinations are necessary to
fill k-space. Inverse Fourier transform (iFT) of k-space essentially adds the
relative contributions of all spatial frequencies to give the final image.
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Radiofrequency spike is an artifact that exemplifies this concept quite
nicely. A spark or other source of radiofrequency noise in the MR scanner room
can contaminate the MR echo. When Fourier-transformed, the frequency of that
spark may be erroneously incorporated into k-space as an abnormal, bright
pixel. The rogue frequency is then inverse-Fourier-transformed into sinusoidal
noise in the image space (Figs.
7A and
7B). Zipper artifact is
another manifestation of radiofrequency leak. It is caused by a constant,
narrow range of radiofrequency emission that is occasionally emanating from
patient monitoring devices. When this unwanted signal is
inverse-Fourier-transformed into the image space, it manifests as persistent,
thin, hyperintense lines in the frequency-encoding direction that are thought
to resemble a zipper [3]
(Fig. 7C).
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Fig. 7A —Radiofrequency spike artifact
[7,
8]. One abnormal, bright pixel
in a Fourier space is transformed into sinusoidal noise in the image space. If
moved slightly farther away from the origin, spatial frequency is higher. A
spark in the MR scanner room, erroneously integrated into k-space, may result
in radiofrequency spike artifact. FT = Fourier transform.
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Fig. 7B —Radiofrequency spike artifact
[7,
8]. Door to MR scanner was left
open just a crack during this acquisition. Notice regular pattern of
striations (arrows) present in image, a result of radiofrequency
leak.
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Fig. 7C —Radiofrequency spike artifact
[7,
8]. "Zipper"
artifact from radiofrequency leak. During this sagittal FLAIR acquisition,
radiofrequency noise from a patient monitor is transformed into intense thin
bright lines through the image. Artifact reflects a narrow range of
contaminant frequencies manifest in frequency-encoding direction. Every image
during this acquisition was degraded by the same intense lines in the same
location.
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Depending on how and when we choose to activate a particular combination of
phase- and frequency-encoding gradients, we have the option of filling k-space
in several creative ways. If we inspect the Fourier space of a photograph of
Lincoln (Fig. 8A), the most
intense portions (brightest pixels) are located centrally where low
frequencies reside (contrast). This is where the most essential components of
the image are stored. By corollary, when filling k-space, we may choose to
fill the central, high-signal-to-noise portions and ignore the less important,
low-signal-to-noise regions to reduce acquisition time. This can be
accomplished by activating gradients corresponding to the center of k-space,
perhaps in a spiral fashion [3]
(Fig. 8B). Furthermore, the
symmetric organization of k-space, a direct consequence of complex conjugate
symmetry properties inherent to the Fourier transform, has been used to
decrease acquisition time by acquiring only half of k-space
[3]
(Fig. 8C). A mirror image of
the remaining half can then be generated, saving time at the expense of the
signal-to-noise ratio.
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Fig. 8A —Differential filling of k-space. When low frequencies are
removed from Fourier space of Lincoln (upper left), sharp edges are
preserved in image at the expense of contrast resolution. When high
frequencies are removed, image contrast is preserved; however, it is blurry
and demonstrates Gibbs artifacts (see Fig.
10A,
10B,
10C). Observe how few spatial
frequencies are actually necessary to recreate a recognizable image of
Lincoln. FT = Fourier transform, iFT = inverse Fourier transform.
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Fig. 8B —Differential filling of k-space. By steering frequency- and
phase-encoding gradients appropriately during MR image acquisition, k-space
can be filled not only sequentially line by line, but also in a spiral fashion
about the origin. Filling the essential, high-signal-to-noise, central
portions of k-space can save considerable time and result in a recognizable
image. This comes at the expense of fine detail, which is stored in the
periphery of k-space (as depicted in A).
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Fig. 8C —Differential filling of k-space. Fourier transform formula
makes use of exponentials of imaginary numbers (ei) to represent
simple waves, and as a result the Fourier transform yields both real and
imaginary information displaying complex conjugate symmetry. Half-Fourier
techniques exploit this symmetry by acquiring only half of k-space and
generating a mirror image of the remaining half. Such a time-saving mechanism
comes at the expense of signal-to-noise, however, because only half of the
potential signal is actually acquired.
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Phase-encoding involves quickly activating, then deactivating a gradient.
While the gradient is transiently on, some spin systems will precess faster
than others, depending on their location along the gradient. When turned off,
the rate of precession across the ROI will equilibrate; however, a spatially
dependent, gradual change in phase will have been "imprinted" on
the protons [3,
4]. Successively increasing the
phase-encoding gradient amplitude will create a varying rate-of-change of
phase across the ROI. This rate-of-change of phase translates into a kind of
frequency that the Fourier transform resolves into different spatial
frequencies [5]
(Fig. 9A). The greater number
of phase-encoding steps performed, the greater the resulting spatial
resolution (Fig. 9B).
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Fig. 9A —Phase-encoding. Spin systems are purposely dephased across a
region of interest to create spatial variation in phase-encoding direction.
The four cosine waves in this figure are shifted slightly out of phase. If a
line intersects the middle of the waves, and the changing amplitude along this
line is plotted, it corresponds to its own wave. This rate-of-change of phase
corresponds to a frequency that Fourier transform can resolve. Each
phase-encoding step is performed at different gradient amplitudes, resulting
in differing degrees of phase change.
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Fig. 9B —Phase-encoding. Columns in 4 x 4 matrix image each
correspond to a specific frequency, depending on location (frequency-encoded).
Without phase-encoding (left), Fourier transform (FT) cannot resolve
any differences in brightness in vertical direction because all frequencies
are identical and all amplitudes (brightness) are blurred together.
The addition of a phase shift (middle, implied by a shift in the
boxes to the right) imparts uniqueness to the boxes in the vertical
(phase-encoding) direction so brightness is partially resolved. The greater
the number of phase-encoding steps, the better the resolution (right)
[8].
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Artifacts
Gibbs artifact is an imperfect approximation of sharp edges by a Fourier
series lacking an adequate number of high-frequency terms. This effect is
easily shown by removing high spatial frequencies from the Fourier space of an
image of Lincoln and inverse-transforming the result (Figs.
10A and
10B). In MRI, this is commonly
referred to as truncation or ringing artifact, and it becomes noticeable when
too few phase-encoding steps are performed. It is often seen near the inner
table of the calvarium (Fig.
10C) or in the upper cervical spinal cord, where it can be
mistaken for a syrinx. Increasing the number of phase-encoding steps (e.g.,
from 128 to 256) will ameliorate this artifact.
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Fig. 10A —Gibbs artifact (also called truncation or ringing artifact).
High spatial frequencies were removed from this image of Lincoln. When
inverse-transformed, not enough frequencies are available to approximate sharp
edges, resulting in Gibbs artifact and blurring.
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Fig. 10C —Gibbs artifact (also called truncation or ringing artifact).
Axial gradient-echo image of brain obtained at 256 x 160 matrix. Gibbs
artifact near inner table of calvarium manifests as subtle hypointense lines
overlying cortex (arrows).
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Because it consumes the most time in signal acquisition, motion artifact or
ghosting most noticeably occurs in the phase-encoding direction. In the time
it takes to apply a new phase-encoding amplitude step (approximately seconds),
a moving structure may have assumed a new position and thus a new resonance
frequency. Phase-encoding an abrupt change in position is essentially
approximating a sharp edge (frequency shift) with sinusoidal waves
(phase-encoding), lead ing to ringing artifact in its Fourier series
[6] (Fig.
11A,
11B). When these
phase-encoding errors are inverse-Fourier-transformed, the structure appears
to be spread out over the image in the phase-encoding direction, regardless of
the original direction of motion.
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Fig. 11A —Motion. Ghosting (motion). An abrupt change in the position
of a structure results in a shift along the frequency-encoding gradient and a
change in precessional frequency. Phase-encoding an abrupt shift in position
is similar to approximating a sharp edge with a Fourier series. Ripples in its
Fourier series propagate in the phase-encoding direction. For a structure with
periodic motion such as aortic pulsation, these errors are incorporated into
k-space in a periodic fashion, resulting in duplicates of the moving structure
propagating in the phase-encoding direction, regardless of the direction of
the original motion.
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Fig. 11B —Motion. In this axial T1-weighted MR image, pulsation
artifact from aorta simulates a hypointense epidural lesion (arrow).
Swapping frequency- and phase-encoding directions can often redirect this
artifact away from target anatomy.
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Aliasing or wraparound artifact (Fig.
12A,
12B) is also related to
phase-encoding and Fourier misregistration. To understand this artifact,
recall that a phase shift of exactly 2 radians or 360° will
superimpose two waves exactly, and thus negate any benefit of imparting
spatial variation based on phase. This leaves – to +
(–180° to + 180°) available to phase-encode a given field of
view. Aliasing artifact occurs when excited spin-systems from outside the
field of view (less than – or more than +) overlap with those of
identical phases inside the field of view. Mathematically indistinguishable,
these structures are assigned by the Fourier transform to the same spatial
position in the image space, causing them to wrap around to the other side
[3].
Chemical shift (india ink artifact) is a spatial misregistration phenomenon
occurring in the frequency-encoding direction. Protons in fat and water
precess at slightly different resonance frequencies (the gap becoming more
prominent with increasing main magnetic field strength), with fat precessing
slower than water by about 3.5 ppm. During frequency-encoding, signal from a
single voxel containing fat and water is assigned two discrete spatial
positions based on these two resonance frequencies
[3]
(Fig. 13A). The result is an
accentuated bright or dark margin corresponding to fat–water inter
faces. If resonance frequencies from fat and water are not resolved as
different, additive fat and water signals from a given voxel will result in
oscillating peaks and troughs depending on the TE, forming the basis for
in-phase and out-of-phase imaging (Fig.
13B).
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Fig. 13A —Chemical shift. Chemical shift artifact occurs when a voxel
in the body contains both fat and water. When signal from such a voxel is
Fourier-transformed, peaks corresponding to both fat and water (each differing
in amplitude, depending on TR and TE) resonate at slightly different
frequencies, separated by 3.5 ppm at 1.5 T). Fourier transform (FT) assigns
two separate spatial locations to a single voxel on the basis of these
different frequencies (chemical shift), despite their common origin. iFT =
inverse Fourier transform.
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Fig. 13B —Chemical shift. Axial gradient-echo out-of-phase
(left, TR/TE, 150/2.236) and in-phase (right, 150/5.516) images
through abdomen. When fat (lower frequency) and water (higher frequency)
signals from a single voxel are added, alternating peaks and troughs occur at
regular time intervals. At TE of 2.236 (left image), a sharp, dark
margin delineating fat–water interfaces (around liver, kidneys, muscles,
and so forth) represents signal trough from voxels sharing water and fat. At
TE of 5.516 (right image), the restored signal replaces the sharp
interface.
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Summary
The Fourier transform is a fundamental tool in the decomposition of a
complicated signal, allowing us to see clearly the frequency and amplitude
components hidden within. In the process of generating an MR image, the
Fourier transform resolves the frequency- and phase-encoded MR signals that
compose k-space. The 2D inverse Fourier transform of k-space is the MR image
we see. A grasp of the Fourier transform is essential to understanding several
MR artifacts and the myriad of methods of signal acquisition in practice
today.
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Abstract
Introduction
