Field Maps

John Pauly

October 5, 2005

The acquisition and reconstruction of frequency, orfield, maps is important for both the acquisition ofMRI data, and for its reconstruction. Many of theimaging methods that we will consider later, suchas spiral and EPI, are sensitive to the local reso-nant frequency. Knowledge of the field map canbe necessary for the reconstruction of high-fidelityimages. Measuring field maps is also important forsystem tuning. For example, many lipid suppres-sion methods are sensitive to resonant frequency.By measuring the field map, and then adjustingthe acquisition parameters, lipid suppression canbe made much more robust.

In this section we describe how field maps are mea-sured and reconstructed. Of particular importanceare the estimation of the constant and gradient(linear) components of the field map. This is dueto the fact that these can easily be changed duringthe acquisition of the MRI data, and that simpleand robust methods exist for correcting for theseterms during image reconstruction.

1 Field Map Acquisition

There are many different ways that field maps canbe acquired. A simple method is to collect imagesat two different echo times, as shown in Fig. 1.Assume image m1(x, y) is acquired at time TE,1

and image m2(x, y) is acquired at echo time TE,2.The phase accrued between the two is due to thedifferent precession frequencies at different points

R F

G z

G y

G x

s1(t)

G y

G x

s2(t)

T E 1

T E 2

Figure 1: Simple field map pulse sequence. Images areacquired with two different echo times, TE,1 and TE,2.

in space

∆φ(x, y) = 6 {m∗1(x, y)m2(x, y)}. (1)

If the change in echo times is ∆TE = TE,2 − TE,1,the estimate of local resonant frequency is thechange in phase divided by the difference in echotimes,

ω(x, y) =∆φ(x, y)

∆TE. (2)

where ω(x, y) is in radians/second.

There are two main concerns with this approach.One is that the phase difference can exceed ±πif the frequency ω(x, y) is beyond ± 1

2∆TE. An-

other concern is the presence of multiple chemicalspecies with different resonant frequencies, partic-ularly water and lipids. In this case, chemical shift

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can be confused with field shift, resulting in errorsin the field map.

If the water/lipid issue were not a problem, thesimple approach is to choose a ∆TE such that theentire range of expected frequencies can be unam-biguously resolved. For example, at 1.5T, a ∆TE

of 1 ms allows frequencies from -500 Hz to 500 Hzto be unambiguously resolved. This correspondsto ± 7.8 ppm, which covers the entire range ofspectral shifts, plus the degree of B0 inhomogene-ity that could be expected from a superconductingmagnet.

There are several cases where lipids are not a prob-lem. One is when lipids are being suppressed any-way, such as spiral imaging using a spectral-spatialpulse that only excites water. Another is func-tional imaging, where long echo times are used tocreate the T ∗

2 contrast that is used to infer oxy-gen saturation. The relatively short T2’s of lipidsresults in little lipid signal in this case.

If lipids are present and are producing significantsignal levels, one effective solution is using an echotime that is a multiple of the fat-water differencefrequency. Both fat and water come into phaseat these times. At 0.5 T, the fat water differencefrequency is 72 Hz, so this means that the twofield map images would be at 13.9 ms and 27.8ms. At higher field strengths, these times becomeproportionately shorter, and later rephasing timesmay be used.

An example of a field map acquisition at 0.5T isshown in Fig. 2. These are actually two of the threeimages from a gradient recalled three-point Dixonacquisition that we will be discussing next time.Each of the individual images has an additionallinear phase component that is suppressed in thephase difference. The phase difference wraps, as isseen in the lower left. The range of frequencies inthis case fits within the±36 Hz unambiguous rangeif we assume a center frequency shift of −24 Hz.In general, the range of frequencies can exceed the

Phase at 13.9 ms Phase at 27.8 ms

Wrapped Field Map Shifted Field Map36

-36

0

Hz

Figure 2: Field map acquired at 0.5T using echo timesof 13.9 and 27.8 ms, which correspond to the first twotimes that fat and water are in phase. Both of the indi-vidual images (top) show a linear phase term, which iseliminated in the phase difference (bottom left). How-ever, the relatively limited range of frequencies that canbe unambiguously resolved results in aliasing. In thiscase, simulating a constant frequency shift of -24 Hzallows almost the entire frequency map to be resolvedunambiguously. In general, phase unwrapping will berequired, which will be discussed next time in the con-text of Dixon image reconstruction.

unambiguous range and require phase unwrappingto produce the field map. Phase unwrapping willbe described in the next class.

In this example an additional complete image ac-quisition was used to determine the field map. Inmany applications, such as real-time and high-resolution spiral imaging, a lower resolution fieldmap acquisition is used in order to minimize thescan time penalty. In practice, the field map is ac-quired using two single shot acquisitions that arecollected during the magnetization transient at thestart of the scan, or when the scan plane changes,and hence the field map is likely to change.

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2 Linear Fit

One of the most important uses of a field map is toestimate the constant and linear components of thefield inhomogeniety [1]. There are two reasons forthis. One is that these terms are easily correctedin the acquisition. The constant term is a centerfrequency shift, and the linear terms are constantbiases to the gradient waveforms. Each of thesecan be changed on a TR by TR basis by the pulsesequence. The other reason is that these terms areeasy to correct in reconstruction. If these termsare known, it is easy to make corrections for theseerrors during reconstruction with almost no costin computation time.

We start by assuming that an unwrapped phasedifference map ∆φ(x, y) has already been com-puted. We want to approximate this as a constantplus linear terms in x and y, and perhaps higherorder terms. As an example, we use a field map ac-quired during an fMRI study on a 3T system. Themagnitude, phase, and frequency maps are shownin Fig. 3

The basic idea is that we want to approximatethe field map ω(x, y) by a linear combination ofa constant term, the linear terms, and higher or-der terms if desired. We can write this as

∆φ(x, y) ≈ a0P0 + a1P1 + a2P2 + · · · (3)

where P0 is a matrix that is a constant 2π over theFOV, P1 is a ramp from −π to π in the x direc-tion, and constant in the y direction, and P2 is aconstant in x, and a ramp in y. These are illus-trated in Fig. 4. The next three terms would bethe quadratic terms x2, y2, and xy. The coefficienta0 gives the offset frequency, in cycles over ∆TE .The coefficients a1 and a2 give the gradient termsin cycles per FOV over ∆TE .

We wish to determine the coefficients ai suchthat the approximation is optimum in a leastsquares sense. The basis set {Pi} consists of two-

Constant x Gradient y Gradient

Figure 4: Constant and linear basis functions for thefield map fit.

dimensional matrices. For convenience, we definepi to be a one dimensional vector that containsall the columns of Pi sequentially in one long col-umn vector. In addition, f is a column vector cor-responding to ∆φ(x, y). In this case we want toapproximate

f ≈ a0p0 + a1p1 + a2p2 + · · · (4)≈ Ha (5)

where H is the model matrix with columns

H = (p0,p1,p2, · · ·) (6)

and a is a column vector of coefficients of the leastsquares fit. We wish to minimize the error

e = f −Ha (7)

in the least squares sense. This is done by mini-mizing

Q = e∗e = (f −Ha)∗(f −Ha). (8)

If we expand this expression, compute the gradientwith respect to the vector a, and then set the resultto zero

∂Q

∂a=

∂

∂a(f∗f − a∗H∗f − f∗Ha + a∗H∗Ha)

= −2H∗f + 2H∗Ha

= 0.

Then, after moving the negative quantity to theother side of the equation,

H∗Ha = H∗f (9)

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Magnitude Phase, TE1 Phase, TE2 Frequency Map62.5

-62.5

0

Figure 3: Field map for an fMRI study performed at 3T.

we geta = (H∗H)−1H∗f , (10)

which is the familiar form for a least-squares fit.

In practice, we usually want to limit the area of theimage that the fit is performed over. Areas of lowsignal should be excluded because the phase mea-surements are dominated by noise. In addition,certain parts of the image are of greater interestthan others. We can include a weighting term inthe quadratic form

Q = (f −Ha)∗W (f −Ha). (11)

where W is a diagonal matrix. In this case theweighted least squares fit is

a = (H∗WH)−1H∗W f . (12)

There are two things that the weighting factor Ware typically used for. The first is to indicate thereliability of particular phase measurements. Thesecond is to limit the fit to an area of interest.

Using a weighting that reflects the reliability of thedata is reasonable if we are interested in fitting theentire slice, and there is significant variation in sig-nal level in different parts of the image. Since thenoise in MRI is constant across the image, the er-ror in the phase estimate is inversely proportionalto the magnitude of the image at each pixel. If thenoise is normally distributed N(0, σn), the stan-

dard deviation of the phase estimate at a high sig-nal pixel is

σφ(x, y) ≈ σn/m(x, y) (13)

by the small angle approximation, where the angleis measured in radians. Note that only the noisecomponent in quadrature to m(x, y) contributes tothe phase error. This is illustrated in Fig. 5. Tominimize the variance of the estimates the diagonalelements of W are then

1/σ2φ(x, y) =

1σ2

n

m2(x, y). (14)

Since σn is a constant, the optimum weighting issimply the image magnitude squared.

In many respects this does the right thing. Back-ground signal is effectively masked out, and areasof high signal are relied on more heavily in the fit.However, there are many cases where this is unde-sirable [2]. One very important case is when thereceiver coil sensitivity is non-uniform. One exam-ple is cardiac imaging using a surface coil, as shownin Fig. 6. Here the chest wall is by far the brightestarea of the image. Weighting by the signal magni-tude would preferentially fit the chest wall. Sincethis is typically shifted several ppm from the heart,this can be a serious problem.

Another approach is to uniformly weight all of thedata that falls within a region of interest and isabove a noise dependent threshold (a multiple of

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S+n

S

n

nq

Re

Im

fn

fn= nq/|S|

Figure 5: For the high SNR case, the error in angledue to noise is the length of the noise component inquadrature to the signal vector, scaled by the signalvector.

Shim Area

Figure 6: A short axis view of the heart acquired at10 images/s. Because a surface coil is used for receive,the chest wall is very much brighter than the heart.Typically the chest wall can be shifted several ppm fromthe heart. A linear shim using signal intensity weightingwould focus on the chest wall. A better option is toshim over the central part of the image (here shown at60%) of the FOV, and uniformly weight all pixels abovea noise-based threshold.

σn for example). In general, since we are only es-timating a few parameters from a large numberof samples, the variance of the estimated parame-

ters is not a major concern. Far more importantis whether the estimate accurately represents thefeatures of interest in the image.

As an example, we return to the fMRI data setshown previously in Fig. 3. This is the lowestslice in a multislice set. A volume shim was donefirst. This produces a shim that is globally opti-mum over the entire volume. However, additionallinear shims on each slice can still be significant.First we compute the constant and linear termsover the entire slice where the signal level is greaterthan 20% of the peak signal. This excludes back-ground noise, as well as spiral image artifacts. Theinitial shim had an RMS error of 37 Hz, which isreduced to 18.3 Hz by correcting for the constantand linear terms. The initial shim, the fitted lin-ear correction, and the final field map are shownin Fig. 7.

Often in fMRI studies only part of the brain isof interest. In this case the correction can be im-proved by using the weighting function to reducethe region being fit even further. In Fig. 8 onlythe back half of the brain has been used for the fit.The original shim map, the linear correction fit,and the corrected map are shown. Over the backhalf of the brain the RMS frequency error has beenreduced to 12 Hz. The price for this reduction is anincrease in the overall RMS frequency error, whichhas increased to 33 Hz.

3 Matlab Implementation

In implementing the fitting methods describedhere, it is useful to be able to convert betweena vector representation of an image and a matrixrepresentation. Any ordering of the vector repre-sentation will work as long as it is used consis-tently. A particularly convenient ordering is toappend all the columns of an nx by ny image minto one long nx*ny column vector mv. This can

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62.5

-62.5

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Initial Shim Map Linear Correction Corrected Map

Figure 7: A field map acquired using a single-shotspiral sequence at 3T (left). This is the lowest slice ina multislice set. A volume shim had been performed.Since this is a compromise over the entire volume, thelinear shims on individual slices can be improved. Thefit of constant and linear terms is shown in the middle,and the corrected field map on the right. The RMSfrequency error has been reduced from 37 Hz to 18.3Hz over the slice.

62.5

-62.5

0

Initial Shim Map Linear Correction Corrected Map

Figure 8: This is the same data as in Fig. 7, wherethe linear correction has been calculated over the dataincluded in the dashed box. A different correction termresults (center) which reduces the RMS frequency errorto 12 Hz over the box, but increases the overall RMSerror to 33 Hz.

be accomplished by

>> mv = m(:);

The original matrix representation can be restoredby

>> m = reshape(mv,nx,ny);

The basis set for the linear fit consists of a constantand ramps in x and y. These can be generated by

>> P0 = 2*pi*ones(nx,ny);

>> PX = 2*pi*([-(nx/2):(nx/2-1)]’/nx)*ones(1,ny);

>> PY = 2*pi*(ones(nx,1)*[-(ny/2):(ny/2-1)]/ny);

and the model matrix H by

>> H = [P0(:) PX(:) PY(:)];

The phase difference is computed from two imagesm1 and m2 by

>> pdm = angle(conj(m1).*m2)

and converted to a column vector

>> pdv = pdm(:);

The unweighted least squares fit is then computedby

>> a = inv(H’*H)*H’*pdv;

which you probably never want to use, becauseof the phase noise in the background. One possi-ble weighting matrix is simply the image squared.Another is a unit amplitude mask that includes allpixels over a threshold

>> wm = abs(m1)>0.2*max(max(abs(m1)));

which is one for all pixels greater than 20% of themaximum. In computing the weighted estimate,the weighting matrix is a huge diagonal matrix (nx* ny by nx*ny) with wv = wm(:) on the diago-nal. However it never needs to be stored this way.If we look at Eq. 12 we can replace the productW f with an element-by-element product with thecolumn vector wv. Similarly, the product H∗WH

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can be replaced by first computing the element-by-element product of wv with the columns of H.This can be done by creating wv*ones(1,3), a ma-trix that replicates three columns with wv. Theelement-by-element product with H is the same asdiag(wv)*H which would otherwise be required.Then H∗WH can be computed as

>> HpWH = H’*((wv*ones(1,3)).*H);

and the weighted fit is then computed by

>> aw = inv(HpWH)*H’*(wv.*pdv);

Another alternative would be to use the matlableft matrix divide operator, which I can’t figureout how to typeset in LATEX!

>> aw = HpWH \ (H’*(wv.pdv));

The estimated fit is then H*aw, orreshape(H*aw,nx,ny) in image representa-tion. The basis functions have been scaled sothat a is in cycles per ∆TE , which will be dte inmatlab. The center frequency shift is then

>> d omega = a(1)/dte;

where d omega is in kHz if dte is in ms. The linearterms can be converted into gradient amplitudesby realizing that the linear basis images have beenscaled to one cycle over the FOV. The gradientamplitude can be computed by scaling a(2) anda(3) for dte in ms, the field of view fov in cm,and gamma in kHz/G. The result, in G/cm, is

>> Gx = a(2)*(1/dte)*(1/fov)*(1/gamma)

and similarly for Gy.

4 References

1. P. Irarrazabal, C.H. Meyer, D.G. Nishimura,and A. Macovski, Inhomogeneity CorrectionUsing an Estimated Field Map,Magn. Re-son. Med.35(2):278–282, Feb 1996.

2. A.B. Kerr, J.M. Pauly, B.S. Hu, K.C.P. Li,C.J. Hardy, C.H. Meyer, D.G. Nishimura, andA. Macovski. Real-Time Interactive MRI ona Conventional Scanner. Magn. Reson. Med.,38(3):355–67, Sep 1997.

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