Magnetic Resonance Imaging and the Fourier Transform

Patrick Foley - University of California Santa Barbara

March 13, 2016

Abstract

Magnetic resonance imaging is the process in which images of tissue are produced by taking ad-vantage of the magnetic properties atoms, hydrogen in particular. Since people are mostly madeof hydrogen, magnetic resonance imaging works well to form images of human tissue.

Hydrogen is the most simple atom, consisting of just a proton and an electron. It is the behaviorof the proton in a magnetic field that makes magnetic resonance imaging possible. When in thepresence of a strong magnetic field, the proton absorbs and emits electromagnetic radiation atspecific frequencies. These frequencies can be detected and stored as data.

The magnetic resonance phase and frequency signal comes in as a complicated sum of manysimple signals and is stored that way as data. The way that this complicated data is organizedand turned into an image is by way of Fourier transform. Fourier transforming the data takes thecomplicated signal and breaks it up into specific frequencies so it can be organized as pixel data.This pixel data can then be displayed as an image.

The technology for magnetic resonance imaging has been being developed over the past centurystarting with early research in nuclear magnetic resonance. The ability to produce images frommagnetic resonance signals would not be possible without the Fourier transform.

1 Introduction

1.1 Nuclear Magnetic Resonance

Magnetic resonance imaging came about fromresearch in nuclear magnetic resonance. Nu-clear magnetic resonance is the the study of howatomic nuclei absorb and emit electromagneticradiation in the presence of a magnetic field.This branch of study was pioneered by WalterGerlach and Otto Stern in the 1920’s. In 1924,they published the results of an experiment thatdemonstrated the quantum nature of the mag-netic moment of silver atoms by molecular beamdeflection in an inhomogeneous magnetic field.[4]

This research paved the the way for the study ofnuclear magnetic resonance.

In the mid-1900s two physicists, Edward M.Purcell and Felix Bloch, found that when certainnuclei were placed in a magnetic field they would

absorb electromagnetic energy and re-emit theenergy when they would relax back to their origi-nal state in the magnetic field.[4] This absorptionand emission of electromagnetic energy turnedout to be the key to magnetic resonance imag-ing.

As nuclear magnetic resonance developed,many wondered whether or not you could createan image out of magnetic resonance signals. In1975, Richard Ernst proposed the use of Fouriertransform of the phase and frequency data inorder to organize spatial data to reconstruct a2 dimensional image. This research led to thefirst images being reconstructed from magneticresonance signals. Richard Ernst was awardedthe 1991 nobel prize in chemistry for his FourierTransform method of image reconstruction.[4]

Without the Fourier transform, magnetic reso-nance imaging would not be possible.

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1.2 Magnetic Resonance Imaging

It is the behavior of the proton in a magnetic fieldthat makes magnetic resonance imaging possible.Protons have a magnetic moment which will lineup with the magnetic field of a strong magnetand begin to precess with their angular veloc-ity in the same direction of the field, and witha frequency dependent on the strength of themagnetic field at that point. If you then dis-turb the protons with a radio frequency pulse,they begin to precess at a higher energy as aresult of absorbing the electromagnetic energy.When the pulse is turned off, protons in a higherenergy state relax back down to a lower energystate. When the proton relaxes to a lower energystate, it gives off electromagnetic radiation at aspecific frequency dictated by its position in themagnetic field gradient. A detector called a ra-diofrequency coil is used to turn the frequenciesgiven off by the relaxing protons into an elec-tronic signal. A radiofrequency coil consists oftwo electromagnetic coils, a transmitter and re-ceiver that generate and receive electromagneticfields. [3,4]

This signal comes in as a very complicatedwave form and is stored as data. The way inwhich the data is organized is by the use or theFourier transform, f̃(ξ) =

∫∞−∞ f(x)e−2πixξdx,

which allows software to break up the signal intoindividual frequencies so that they can be orga-nized into pixel data that can be inverse trans-formed back into an image.

Fourier transform is used to decompose thecomplicated signal received from magnetic res-onance echoes. These echoes contain encodedfrequency and phase spatial information used toconstruct an image. The Fourier Transform sep-arates all encoded frequencies so that the datacan be stored and organized as pixels in what isknown as k-space. When inverse-transformed,the pixel contains one spatial frequency con-tributing to the image. A spatial frequency cor-responds to the number of alternating light anddark bands per meter in the pixel.[3,5]

2 Materials and Methods

2.1 The Fourier Transform

The Fourier transform is a mathematical toolthat is used to decompose a function or signalinto its constituent sinusoids. As it turns out, allperiodic functions can be represented as a sum ofsimple sinusoids of different frequencies (fig. 1).This is a great tool for magnetic resonance imag-ing since the incoming signal is a sum of simplesinusoids of different frequencies.[2]

Figure 1: Composition of a complicated waveform from

simple sinusoids.[2]

The Fourier transform is typically used to an-alyze functions of time or space by looking atthem in their frequency domain then transform-ing them back to their time domain. Here is the

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Fourier transform of a function in the time do-main and it’s inverse:

F{g(t)} =

∫ ∞−∞

g(t)e−2πiωtdt = G(ω)

F−1{G(ω)} =

∫ ∞−∞

G(ω)e2πiωtdω = g(t)

You can see here that the Fourier transformtakes a function of time or space and transformsit into a function of frequency or spacial fre-quency. What might not be obvious is that thefrequency domain function will be represented ina basis of sines and cosines. This will allow forthe separation of discrete frequencies that makeup a function of time or space (Fig. 2).

Figure 2: 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 trans-

form of three cosine waves of different frequencies results

in three delta functions.[3]

2.2 K-Space

When the signal of the magnetic resonance echocomes in, it is incredibly complicated. As wehave seen from §2.1, the Fourier transform canhelp us with this by breaking up the signalinto its discrete frequencies. Once dismantled,the data is entered into k-space which is a 2D

Fourier space. K-space is used to organize spa-tial frequency and amplitude information intopixels. A single pixel in k-space can be inverse-transformed to contribute one specific spatial fre-quency to the image in the form of alternatinglight and dark bands.[1] The inverse transformof all of the information in k-space combines allspatial frequencies resulting in the entire image.The frequency and orientation of the alternatinglight and dark bands that make up the the im-age depend on where the pixel resides in k-space.Pixels with low spatial frequency are mappednear the origin and pixels with high spatial fre-quency are mapped on the outskirts (fig. 3).[3]

Figure 3: MRI. This coronal slice of a brain is interro-

gated 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 gradi-

ent combinations are necessary to fill k-space with enough

phase and spatial information to create an image. In-

verse Fourier transform (iFT) of k-space essentially adds

the relative contributions of all spatial frequencies to give

the final image.[3]

2.3 Image Refinement

An important part of constructing an image isencoding phase spatial information. Withoutthis phase spatial information it would be im-possible to reconstruct an image. One way thatthe phase information is encoded in the signalis by using a phase encoding magnetic gradient.This is achieved by quickly activating and deac-tivating the gradient. When the gradient is only

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on for a short time, some of the protons spinsprecess faster or slower than others dependingon their position in the field gradient. When al-lowed to relax in state, a change of phase is im-printed on the protons due to the timing of turn-ing the gradient on and off and that informationis carried over in their given signal.[3] The morephase encoding steps that are taken, the greaterthe image resolution will be due to the gatheringof more phase information.

In order to further refine images, a threshold-ing technique has been developed to reduce noiseand eliminate false correlation in pixels with verylarge signal changes due to the flow of fluid inthe body. The mechanics of this method canbe best explained using abstract vectors contain-ing time-course data information. The idea is toform a correlation coefficient, which we will callcc, for each pixel, and define a threshold value,TH, where if |cc| > TH then the pixel is thrownout. In order to define the correlation coefficient,we must define the vector ~σf which contains thefunctional time domain magnetic resonance in-formation and the vector ~σr which is the corre-sponding reference vector. ~σr may be derivedfrom the average of several experimentally de-rived ~σf ’s. The correlation coefficient can bewritten:

cc =~σf · ~σr~σf~σr

The angle between the vectors ~σf and ~σf canbe written:

α = cos−1|~σf · ~σr~σf~σr

|

The thresholding value then implies thatdata will be accepted when α < cos−1(TH).The smaller the angle, the more closely shapesresemble each other. The typical angle used forthresholding is π

4 . The goal here is to create soft-ware that will make a digital map of zeros andones with the same dimensionality as the image.Ones are assigned to pixels with similar shapesas defined by the threshold value, and zeros areassigned to the others, essentially removing thedata from those pixels. This method is meant

to remove bad data which in turn refines theimage.[1]

3 Results

Magnetic Resonance Imaging is made possibleby the behavior of a proton in a magnetic field.It is the ability of the proton to absorb and emitelectromagnetic radiation, nuclear magnetic res-onance, that allows us to extract informationfrom inside the body to create an image.

Because the processing of complicated sig-nals is inherent to magnetic resonance imaging,the Fourier transform is clearly necessary in or-der to make sense of the information containedin the signals obtained from magnetic resonanceechoes.

There are many ways to improve the qual-ity of the images obtained in magnetic resonanceimaging, one of those being the encoding of phaseinformation by quickly turning a magnetic fieldgradient on and off. The more phase informa-tion encoded and obtained, the higher the imageresolution will be.

In order to refine magnetic resonance imagesfurther, sophisticated mathematics have beenused to create a theory for image refinement. Acorrelation coefficient is defined using abstractvectors containing time course data. This corre-lation coefficient can be compared to a thresholdvalue in order for software to decide how to keepgood data and throw out bad data. The time-course data in each pixel can be examined us-ing this thresholding technique in both time andfrequency domains by way of Fourier transforma-tion in order to remove bad pixels from images.

4 Discussion

Before the Fourier transform method was cre-ated by Richard Ernst in 1975, a few peopletried their hand at image formation from mag-netic resonance signals. In 1973, Physicist PeterMansfield proposed using magnetic field gradi-ents to acquire spatial information in order toproduce an image. That same year at a con-ference in Poland he showed that he produced

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a magnetic resonance interferogram at a 1mmresolution.[4] This was not an actual image butit was still a pretty good reconstruction of spatialinformation.

The only way in which the magnetic reso-nance signal can be turned into specific frequen-cies and mapped to a pixel is by Fourier trans-formation. There are many methods of refiningmagnetic resonance images and most of them useFourier transformation in one way or another.

5 Acknowledgments

Special thanks to Jared Pagett and Dylan Beardfor their feedback on this paper.

6 References

1. Bandettini, Peter A., Jesmanowicz A.,Wong, Eric C., Hyde, James S. (1993).Processing Strategies for Time-CourseData Sets in Functional MRI of the Human

Brain. Magnetic Resonance in Medicine,30(1), 161-173.

2. Bevel, P. (2010). Fourier Trans-forms. Retrieved February 08, 2016, fromhttp://www.thefouriertransform.com

3. Gallagher, Thomas A., Nemeth, Alexan-der J., Hacein-Bay, Lotfi. (2008).An Introduction to the Fourier Trans-form: Relationship to MRI. AmericanJournal of Roentgenology, 190(5), DOI10.2214/AJR.07.2874

4. Geva, T. (2006). Magnetic resonanceimaging: historical perspective. Journal ofCardiovascular Magnetic Resonance, 8(4),573-580.

5. Oshio, K., and Feinberg, D. A. (1991).GRASE (Gradient and Spin Echo) imag-ing: A novel fast MRI technique. Magneticresonance in medicine, 20(2), 344-349.

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