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An Introduction to the Fourier Transform: Relationship to MRI

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.

View larger version (19K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 1 —Adding simple waves. Example sine wave shown here, 1 + 2sin(2ft), has frequency f = 1 /(2), period T = 2, amplitude = 2, and is centered at g(t) = 1.

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 2 —Complicated waves. A complicated wave g(t) can be obtained by adding together simpler waves.

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (8K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (21K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 4B —Fourier series for a square wave. In waves shown in B, n = 1, 3, 9; in C, n = 51.

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 4C —Fourier series for a square wave. In waves shown in B, n = 1, 3, 9; in C, n = 51.

View larger version (5K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (26K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (10K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 6A —Fourier spaces and k-space [7, 8]. Fourier transform of a blank canvas (left) is one bright dot at the origin in the Fourier space (right).

View larger version (12K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (49K): [in this window] [in a new window] [as a PowerPoint slide]   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)

View larger version (52K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (48K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (66K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (114K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (169K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (61K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (53K): [in this window] [in a new window] [as a PowerPoint slide]   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).

View larger version (89K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (28K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (25K): [in this window] [in a new window] [as a PowerPoint slide]   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].

View larger version (39K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (33K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 10B —Gibbs artifact (also called truncation or ringing artifact). Gibbs phenomenon is evident mathematically and in manipulated image of Lincoln (arrows).

View larger version (123K): [in this window] [in a new window] [as a PowerPoint slide]   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).

View larger version (22K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (121K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (42K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 12A —Wraparound. Wraparound (aliasing). Only phase shifts between 2 radians or 360° are available to encode an image. The phase shift in this image (depicted by waves A–H) covers the field of view (black rectangular outline). Just outside the field of view, wave "I" has assumed a phase shift of 2 radians (360°) and is mathematically identical to wave "A" on the opposite side of the field of view. Fourier transform assigns structures encoded by "I" to positions encoded by "A," giving the wraparound phenomenon.

View larger version (107K): [in this window] [in a new window] [as a PowerPoint slide]   Fig. 12B —Wraparound. Axial T2-weighted image shows back of head (excluded from field of view) wrapping around to the front.

View larger version (16K): [in this window] [in a new window] [as a PowerPoint slide]   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.

View larger version (42K): [in this window] [in a new window] [as a PowerPoint slide]   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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