Dixon Reconstruction

John Pauly

October 11, 2005

One of the important applications of phase sen-sitive reconstruction is Dixon imaging. Multipleimages are acquired at different echo times. Lin-ear combinations of these images produce waterselective and fat selective images.

This approach is particularly important at lowerfield strengths, where it is the preferred methodof lipid suppression. At 0.5T and below, chemi-cally selective excitation or saturation is impracti-cal since the small fat/water spectral shift wouldrequire a very long RF pulse. For example, theminimum time for a spectral-spatial pulse at 0.5Tis about 24 ms. Inversion recovery methods be-come less attractive as the T1’s decrease with de-creasing field strength.

1 Basic Two Point DixonMethod

The simplest approach uses only two images [1].The basic assumption is that there are only twocomponents in the image, water and fat. If mw isthe image of the water component, and mf is theimage of the fat component, the combined imageat an echo time TE,i is

mi = mw + e−iωf TE,imf (1)

where we have assumed, for convenience, that weare exactly on the water resonance, so only thelipid component precesses at ωf . We choose TE,i

so thatωfTE,i = 0, π mod 2π. (2)

The two images that result are

m1 = mw + mf (3)m2 = mw −mf . (4)

We can then combine these to make images thatcontain only water, or only fat

mw =12

[m1 + m2] (5)

mf =12

[m1 −m2] . (6)

The problem with this approach is that there areother factors that confound the decomposition.The major factor is the off-resonance frequencyshift ω that produces phase errors. Another factoris the T2 decay of signal at increasing echo times.For spin echoes, the decay may be kept constant bychanging the echo time by shifting the refocusingpulse. In gradient-recalled acquisitions, T2 decaywill be a concern. The actual signal for the ith TE

is then

mi = mwe−TE,i/T ∗2,we−iωTE,i

+mfe−TE,i/T ∗2,f e−i(ω+ωf )TE,i .

Of these two factors, the problem of the off-resonance frequency ω is by far the most im-portant. The estimation and correction for T ∗

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degrades the separation between fat and waterslightly, but is of secondary concern. We will ig-nore this effect here, but see [3] for a detailed anal-ysis of the magnitude of the errors introduced, andthe effect on the SNR of the resulting images. Ourprimary concern here is correcting for the effect ofoff-resonance frequency.

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Water Lipid

Figure 1: A simple two-point Dixon reconstructionof gradient echo data acquired on a 0.5T system. Al-though the predominant component in the water imageis water, and the fat image is fat, there is significantcontamination of each by the other. This is due to theoff-resonance frequency ω across the head.

The two images are then

m1 = (mw + mf )eiφ0 (7)

m2 = (mw −mf )ei(φ0+φ) (8)

where

φ0 = −ωTE,1, (9)

and the phase due to the precession between twoechoes is

φ = −ω(TE,2 − TE,1). (10)

The initial phase φ0 can be computed from m1 andeliminated from both terms, so we neglect it. Theestimate m̂w is then

m̂w =12(m1 + m2) (11)

=12(mw(1 + eiφ) (12)

+mf (1− eiφ)). (13)

Hence, even for small off-resonance precession an-gles, the water image is contaminated by a signifi-cant component of the fat image. The lipid imageis similarly contaminated by the water image. Thisis illustrated in Fig. 1.

2 Three Point Dixon Methods

One alternative is to add an additional measure-ment to allow the off-resonance frequency ω tobe estimated. The original proposal [2] was forspin-echo acquisitions, and used measurements at−π, 0, and π. The echoes were symmetrically dis-tributed about the spin echo.

A better choice that also works for gradient echoesis to pick TE,i’s such that

ωfTE,i = 0, π, 2π. (14)

Neglecting φ0 as before, the three images can bewritten

m1 = (mw + mf ) (15)m2 = (mw −mf )eiφ (16)m3 = (mw + mf )ei2φ. (17)

From images m1 and m3 we can estimate

2φ̂ = 6 m∗1m3. (18)

We can then phase correct m2, and combine withm1 to compute m̂w,

m̂w =12(m1 + m2e

−iφ̂) (19)

=12((mw + mf ) (20)

+(mw −mf )eiφe−iφ̂) (21)

=12((mw(1 + ei(φ−φ̂)) (22)

+mf (1− ei(φ−φ̂))). (23)

A similar expression can be written for m̂f . Ifφ = φ̂ the reconstruction is correct. However thereis a problem if the phase of φ exceeds 2π.

The source of the problem is that the term 2φ̂ isitself phase wrapped at 2π. This means that thatφ̂ itself wraps at π. After the first wrap of 2φ̂

φ− φ̂ = π. (24)

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If we substitute this in to Eq. 23 we get

m̂w =12(mw(1 + eiπ) (25)

+mf (1− eiπ)) (26)

=12(mw(1 + (−1)) (27)

+mf (1− (−1))) (28)= mf ! (29)

This means that each time 2φ̂ wraps by another 2π,water and fat swap. This is an objectionable arti-fact. Another consequence is that, even if the twodo separate properly, you can’t tell which will bethe water image. This is easily detectable, but un-fortunate, since water is usually the image of inter-est. Figure 2 demonstrates the water/fat switchingdue to a 2π phase wrap in 2φ̂.

3 2D Phase Unwrapping

Two-dimensional phase unwrapping is in generala hard problem. Much of the work in phase un-wrapping has been done in Synthetic ApertureRadar (SAR) imaging, where the phase informa-tion is used to improve range resolution. SAR datasets typically have hundreds to thousands of phasewraps. The phase unwrapping problem in MRI isrelatively easy by comparison, with tens of phasewraps in extreme cases, and typically less than ten.In spite of this, phase unwrapping still fails occa-sionally, so this is an area that could still use im-provement. The algorithms tend to rely on heuris-tics to decide how to proceed with the unwrap-ping. As such, the algorithms can be fairly intri-cate. A description of a large number of differentalgorithms is presented in [5], and “C” code imple-mentations are available on the Wiley ftp server.

The basic idea of phase unwrapping in 2D is verysimilar to phase unwrapping in one dimension.Fig. 3 illustrates the basic idea in one dimension.The points at which the signal wraps are identified

p(t)p

-p

p

-p

p

-p

pw(t)

dp(t)dt

-2p -2p

+2p +2p

Figure 3: A 1D phase unwrapping example. The ini-tial phase waveform has values outside of±π (top). Thewrapped phase 6 eip(t) then has 2π phase jumps wherethe p(t) crosses the ±π boundaries (center). If we differ-entiate p(t) these jumps become impulses of amplitude±2π. If we subtract or add 2π to bring these points intothe ±π range, and then integrate the result, we recoverp(t), but with a 2π shift of the entire waveform.

by differentiating the phase function, and lookingfor samples with values close to ±2π. The sam-ples are wrapped back into the ±π range. Then,integrating restores the original waveform, with apossible offset of a multiple of 2π.

The same idea applies in 2D. The phase differenceis integrated along a path, and the 2π phase jumpsare taken out. The major issue is how to definethe path. There are two basic principles. First,the unwrapping should be path independent. Thisis illustrated in Fig. 4. Integrating the phase dif-ferences from point A to point B should be thesame for any path we take. In addition, integrat-ing around a closed path should be zero.

There are a several areas that cause problems,

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Wrapped Phase "Lipid" Component "Water" Component

Figure 2: A 3-point reconstruction of the data shown in Fig. 1. The phase φ̂ wraps, which causes the waterand fat components of the images to switch.

AA

B

Figure 4: In 2D phase unwrapping, the result inte-grating the phase differences from point A to point Bshould be independent of the path. In addition, inte-gration around a closed path should be zero.

where these conditions don’t hold, and the phaseunwrapping fails. One is low signal areas wherethe phase data is random. Another is areas wherethe spatial frequencies of the phase map are higherthan the resolution of the image. A third problemis unconnected regions.

There are many different algorithms that addressthese problems. Here we will briefly describe two.The first is based on a quality map of the phasedata. The algorithm tries to unwrap using the bestdata it can find, first. Many different functionshave been proposed for quality maps. One is theminimum gradient. Another is the minimum vari-ance of the gradient. The idea is to pick a pixel tostart unwrapping, and then region grow by includ-ing the border pixel with the best quality value.This is a simple approach that works remarkablywell.

Another important class of unwrapping algorithmsis based on branch cuts. The idea here is to firstidentify the points in the phase map that violatethe requirement that the integrated phase differ-ence around a closed path is zero. These points arecalled residues. Since we are working on a discretegrid, these residues are discrete, and are conceptu-ally located at the intersection of four image pix-els. They can be found by integrating around each2×2 element of the phase data. Once the residuesare found, they are grouped into pairs or sets bybranch cuts. If a positive residue and a negative

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Water+Fat Water Fat

Figure 5: A 3-point reconstruction of the data shown in Fig. 1 after the phase has been successfully unwrapped.

residue are connected, and we integrate around thepair, the result is zero. Thus, as long as we don’tintegrate across a branch cut, we have path inde-pendence. Once all the residues have been pairedup, and the unpaired residues connected to theimage edge, we can unwrap with any path thatdoesn’t cross a branch cut.

Once the phase 2φ̂ has been successfully un-wrapped, φ̂ has also been unwrapped, and thephase compensated estimates of m̂w and m̂f canbe unambiguously computed. The result for thedata set that was unsuccessfully decomposed intofat and water in Fig. 2 is shown properly separatedin Fig. 5 after unwrapping 2φ̂ using a branch cutalgorithm. A more challenging problem is shown inFig. 6. The rapid phase changes would cause wa-ter/fat switching. However, after phase unwrap-ping with a quality map algorithm, water and fatare properly separated.

4 Two Point Dixon, Revisited

Once we understand the idea of three-point Dixonwe look back at the two-point approach and effec-tively accomplish the same thing with only twomeasurements [6,7]. The main function of the

real(m1) angle(m1) 2*phi

mw+ml ml mw

Figure 6: Phase artifacts produced by a needle. Theseare successfully unwrapped, and a properly separatedwater and fat images produced.

third measurement m3 in Eq. 17 is to allow theestimation of 2φ̂ by computing

2φ̂ = 6 m∗1m3. (30)

We can also do this with

2φ̂ = 6 (m∗1m2)2. (31)

This effectively synthesizes the third three-pointDixon measurement from the two-point data set.The reason we need to do this using m∗

1m2 is thatwater and fat have opposite polarities, and that

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would appear in the phase estimate. By squaringthe phase difference, water and fat are again inphase, and we can extract the off-resonance com-ponent of the phase. While this is a great idea, inpractice its use is limited due to the performanceat pixels that have a contribution from both waterand fat. Since fat shifts in the readout directionin a 2DFT acquisition, this includes pixels on thefat/water boundary. At these points fat and waterinterfere, resulting in phase anomalies, and sepa-ration failures.

5 Conclusion

Three-point Dixon methods provide an effectivemeans of fat/water imaging. Since the fat/waterdifference frequency is in the order of susceptibilityshifts, the off-resonance phase estimate 2φ̂ mustbe unwrapped to avoid water and fat switching.A number of different phase unwrapping routinesmay be used, and work well. However, there stillare occasional failures.

6 References

1. W.T. Dixon, Simple Proton SpectroscopicImaging, Radiology, 153:189-194, 1984.

2. G.H. Glover and E.Schneider, Three-PointDixon Technique for True Water/Fat De-composition with B0 Inhomogeneity Correc-tion,Magn. Reson. Med, 18(2):371–383,1991.

3. G.H. Glover, Multipoint Dixon Techniquefor Water and Fat Proton and SusceptibilityImaging, J. Magn. Reson. Imag. 1(5):512–530, 1991.

4. Y. Wang, D. Li, E.M. Haacke, and J.J. Brown,A Three Point Dixon Methods for Water andFat Separation Using 2D and 3D Gradient

Echo Techniques, J. Magn. Reson. Imag,8(3):703–710, 1997.

5. D.C. Ghiglia and M.D. Pritt, Two-Dimensional Phase Unwrapping: Theory,Algorithms, and Software, Wiley, 1998.

6. T.E. Skinner and G.H. Glover, An ExtendedTwo-Point Dixon Algorithm for CalculatingSeparate Water, Fat, and B0 Images, Magn.Reson. Med, 37(4):628-630.

7. B.D. Coombs, J. Szumowski, and W. Coshow,Two Point Dixon Technique for Water-FatSignal Decomposition with B0 Inhomogene-ity Correction,Magn. Reson. Med,38(6):884-889.

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