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Magnetic Resonance in Medicine 59:571–580 (2008)
Joint Estimation of Water/Fat Images and Field
Inhomogeneity Map
D. Hernando,1,2∗J. P. Haldar,1,2B. P. Sutton,2,3J. Ma,4P. Kellman,5and Z.P. Liang1,2
Water/fat separation in the presence of B0field inhomogeneity
is a problem of considerable practical importance in MRI. This
articledescribestwocomplementarymethodsforestimatingthe
water/fat images and the field inhomogeneity map from Dixon
typeacquisitions.Oneisbasedonvariableprojection(VARPRO)
and the other on linear prediction (LP). The VARPRO method is
very robust and can be used in low signaltonoise ratio condi
tions because of its ability to achieve the maximumlikelihood
solution. The LP method is computationally more efficient, and
isshowntoperformwellundermoderatelevelsofnoiseandfield
inhomogeneity. These methods have been extended to handle
multicoil acquisitions by jointly solving the estimation problem
for all the coils. Both methods are analyzed and compared and
results from several experiments are included to demonstrate
their performance.Magn Reson Med 59:571–580, 2008. © 2008
WileyLiss, Inc.
Key words:dixonimaging;linearprediction;variableprojection;
field map estimation; IDEAL
INTRODUCTION
Methods
nents in MR imaging based on their different reso
nance frequencies have been under intensive investiga
tion for a number of years. Dixon’s original method for
water/fat separation (1) acquires two images with differ
ent echo time shifts, and models the signal in a given
voxel as:
s(tn) = ρW+ ei2πfFtnρF
where tnis the echo time shift. The water component has
intensity ρWand is assumed to be exactly on resonance,
while the fat component has intensity ρF with a known
frequency shift fF(in Hz). If the echo time shifts are cho
sen such that 2πfFtn = {0,π}, then the two images thus
for separating thewaterandfatcompo
[1]
1Department of Electrical and Computer Engineering, University of Illinois at
UrbanaChampaign, Urbana, Illinois
2Beckman Institute for Advanced Science and Technology, University of Illinois
at UrbanaChampaign, Urbana, Illinois
3Department of Bioengineering, University of Illinois at UrbanaChampaign,
Urbana, Illinois
4Department of Imaging Physics, University of Texas M. D. Anderson Cancer
Center, Houston, Texas
5Laboratory of Cardiac Energetics, National Heart, Lung and Blood Institute,
National Institutes of Health, Department of Health and Human Services,
Bethesda, Maryland
Contract grant sponsor: NIH; Grant numbers: P41EB0363116, R01
CA098717
*Correspondence to: Diego Hernando, Beckman Institute for Advanced Sci
ence and Technology, 405 N. Mathews Ave, Urbana, IL 61801. Email:
dhernan2@uiuc.edu
Received 22 June 2007; revised 27 November 2007; accepted 28 November
2007.
DOI 10.1002/mrm.21522
Published online in Wiley InterScience (www.interscience.wiley.com).
obtained will have the water and fat signals in phase and
in opposed phase, respectively, and can be combined to
obtain the individual water and fat components. The main
limitationofthissimplemethodarisesfromthepresenceof
B0field inhomogeneity, which introduces additional phase
shifts that prevent Dixon’s method from correctly separat
ing the water and fat signals. A more realistic signal model
is
s(tn) = ρWei2πfBtn+ ρFei2π(fB+fF)tn
[2]
where fBis the local frequency offset due to field inhomo
geneity. In Ref. (2), a threepoint method is introduced to
allow estimation of fBat each voxel. In this method, the
relative phase shifts (2πfFtn) of the water and fat signals
are typically set to {−π,0,π} or {0,π,2π}. These choices
simplify the estimation of the nonlinear parameter fB(2).
In recent years, there has been considerable interest in
alternative choices of echo shifts for threepoint Dixon
imaging.Forarbitraryechoshifts,determinationofthefield
inhomogeneity at each voxel cannot be decoupled from the
estimation of the water/fat contributions, and the nonlin
earityofthemaximumlikelihood(ML)estimationproblem
cannot be avoided. In Refs. (3,4), a method to solve the non
linear fitting problem, termed iterative decomposition with
echo asymmetry and leastsquares (IDEAL), is proposed.
IDEAL consists of repeated linearizations of the original
nonlinear problem, alternatively estimating the water/fat
signals and the field map. Initially, this algorithm was pro
posed for use with fast acquisition schemes (e.g., SSFP
or FSE) which constrain echo time increments to rela
tively small values (3,5). An analysis of the performance
of water/fat decomposition under Gaussian noise is pro
vided in Ref. (6), with the conclusion that, for threepoint
acquisitions, the optimal phase shifts are {−π/6+πk,π/2+
πk,7π/6 + πk}, where k is an integer.
However, the use of arbitrary phase shifts yields a non
linear, nonconvex optimization problem for estimating the
water/fat contributions and the field map. Specifically, the
ML cost function will generally contain multiple local
optima, and therefore iterative, descentbased algorithms
cannotguaranteeconvergencetotheglobalminimum.This
articleaddressesthewater/fatestimationproblemwithtwo
complementary methods, which are extended to impose
field map smoothness constraints and to handle multicoil
acquisitions. These methods are described and analyzed in
the subsequent sections.
METHODS
Under the assumption of white additive Gaussian noise,
the ML estimate estimate for {ρW,ρF,fB} is obtained by
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© 2008 WileyLiss, Inc.
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572Hernando et al.
minimizing the following cost function:
R0(ρW,ρF,fB) =
N
?
n=1
s(tn) − ei2πfBtn(ρW+ ρFei2πfFtn)2
[3]
where N is the number of acquisitions with different echo
times (typically N = 3). Note that the water–fat frequency
shift fFis assumed known a priori.
This formulation corresponds to a nonlinear least
squares (NLLS) optimization problem, which does not gen
erally have a closedform solution. In this section, we
introduce two complementary algorithms to solve this
problem: (a) a variable projection (VARPRO) algorithm that
finds the globally optimal solution with moderate compu
tational requirements, and (b) a computationally efficient
linear prediction (LP) algorithm.
Global Optimization Using Variable Projection
Estimation of {ρW,ρF,fB} by minimizing the cost function
in Eq. [3] is a separable NLLS problem. Specifically, rewrite
Eq. [3] as
R0(ρ,fB) = ?s − ?(fB)ρ?2
where ρ = [ρWρF]T, s = [s(t1)···s(tN)]T, and
ei2πfBtN
2
[4]
?(fB) =
ei2πfBt1
ei2πfBt2
...
ei2π(fF+fB)t1
ei2π(fF+fB)t2
...
ei2π(fF+fB)tN
. [5]
For a given value of fB, the least squares (LS) solution for
the linear parameters ρ is given by ?†(fB)s, where†denotes
pseudoinverse. Therefore, we can remove ρ from Eq. [4]:
R(fB) = ?s − ?(fB)?†(fB)s?2
where I is the N × N identity matrix. This is the socalled
VARPRO formulation of the original NLLS problem in Eq.
[4]. It has been shown that R0(ρ,fB) and R(fB) have the
same global minimum (7,8). Using Eq. [6], the optimal lin
ear and nonlinear parameters in Eq. [4] can be determined
separately as follows
2= ?[I − ?(fB)?†(fB)]s?2
2
[6]
fo
B= argmin
ρo= ?†?fo
fB
R(fB)[7]
B
?s. [8]
This formulation has several desirable features:
a. Minimization of R(fB) in Eq. [7] is a nonconvex prob
lem (with multiple local optima), but it is only a
onedimensional problem, which can be effectively
solved using a search method. Specifically, assuming
suitable bounds fB,MIN and fB,MAX on the field inho
mogeneity, we can evaluate R(fB) on a set of L closely
spaced points {fB,l}L
and select the minimum. Thus, the presence of multi
ple local minima in R(fB) does not hinder attaining the
global minimizer fo
b. Oncefo
to a small matrix multiplication.
l=1on the interval [fB,MIN,fB,MAX]
B.
Bisknown,estimationofρofromEq.[8]reduces
c. Computation of ?†(fB) for the values {fB,l}L
performed efficiently by rewriting ?(fB) = ?(fB)?,
where
1ei2πfFtN
0
Therefore,thedesired
?†(fB) = ?†?(−fB),notingthat?(fB)isaunitarymatrix
and also ?−1(fB) = ?(−fB). Furthermore, only one
pseudoinverse needs to be computed for the entire
decomposition, even if ρ is estimated for many differ
ent values of fBat each voxel, since ? does not depend
onfBors.If?hasfullcolumnrank,thispseudoinverse
computation reduces to ?†= [?H?]−1?H.
This application of VARPRO is similar to a recently pro
posed method for estimation of the T1relaxation constant
(9). Note that this formulation is flexible on the choice of
the echo time shifts, although obviously at least N = 3
acquisitions are required for the estimation problem to be
wellposed in voxels containing both water and fat.
l=1can be
? =
1
1
...
ei2πfFt1
ei2πfFt2
...
···
···
, and
?(fB) =
ei2πfBt1
0
...
00
0
...
ei2πfBt2
...
0
···
ei2πfBtN
.[9]
pseudoinverseissimply
Imposing Spatial Constraints
VoxelbyvoxelMLestimatesmayleadtoerroneousdecom
positions at some voxels due to the presence of noise and
the inherent ambiguity of the model (Eq. [2]) when a sin
gle component is present (10). To improve the water/fat
separation, spatial smoothness is commonly imposed as
prior knowledge on the estimate of the field map (3,11–13).
Clearly,fieldmapsmoothingcanbeperformedinaseparate
step, similarly to the original IDEAL algorithm (3). How
ever, as noted in (10), this approach can be problematic if
thefieldinhomogeneityislarge(e.g.,fB≥ fF)becausephase
shifts due to field inhomogeneity become indistinguish
able from those due to the presence of different chemical
species. A regiongrowing method is proposed in (10) to
address this problem.
Within the VARPRO formulation, decoupling the estima
tion of fBand ρ allows us to effectively impose smoothness
constraints on the field map. Here, we propose to use a
Markov random field (MRF) prior on the field map (14).
The MRF prior can be imposed efficiently using the well
known iterated conditional modes (ICM) algorithm (15).
Let us denote fB= {fq
all Q voxels. ICM iterates several times through all the
voxels, updating one at a time. In this paper, we have con
sidered simple Gaussian smoothness priors, resulting in
the following update for voxel q (with neighborhood δq):
B}Q
q=1as the complete field map for
fq,new
B
= argmin
fq
B
R?fq
B
?+ µ
?
j∈δq
wq,j
??fq
B− fj,cur
B
??2
[10]
where fj,cur
at neighboring voxel j, wq,j are weights that control the
B
is the current field inhomogeneity estimate
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Joint Estimation of Water/Fat Images and Field Map573
field difference between voxels q and j, and µ controls the
amountofsmoothnessimposedbytheMRF.Notethat,from
a regularization standpoint, squared differences in the field
map are penalized to enforce smoothness. Also, the results
will be generally less sensitive to the choice of a few seed
voxels than a regiongrowing method, since the complete
field map estimate will be recomputed iteratively.
VARPRO requires setting several parameters. The field
inhomogeneity bounds were set to almost the chemical
shift between water and fat, i.e. ±200 Hz for 1.5 T acqui
sitions and ±400 Hz for 3 T acquisitions. These bounds
were sufficient for all the cases presented in this article.
The number of discretized field values used in the opti
mization was L = 300, which provides a good tradeoff
between estimation accuracy and computational efficiency
(see Appendix for mathematical details on the selection
of the discretization step size). The MRF neighborhood
employed was the square of size 5 × 5 voxels centered at
each voxel (excluding the center). The weights wq,j were
set to the inverse of the distance between voxels q and j.
Finally, the regularization parameter µ was set to σ2/30,
where σ2is the estimated noise variance. The value of σ2
can be estimated from the data itself, or alternatively from
prescan noise only data (16). According to the MRF model,
the value of µ is the ratio between the noise variance in the
acquired images and a measure of the variability of the field
map. Even though these values can be further optimized
for each particular acquisition, we chose to maintain them
constant throughout the results presented in this paper to
highlight the robustness of the proposed method.
The VARPRObased method with MRF prior is summa
rized below:
1. Initialize the field map estimate fB(e.g., all zeros).
2. Precompute the cost function {R(fB,l)}L
for a set of field inhomogeneity values fB,l
[fB,MIN,fB,MAX], for all voxels.
3. For each voxel, update the field map estimate using
Eq. [10].
4. Repeat step (3) until the overall field map change
falls below some small threshold ε > 0:
l=1(Eq. [6])
∈
Q
?
q=1
??fq,new
B
− fq,cur
B
??< ε. [11]
5. For each voxel, estimate ρWand ρFgiven the esti
mated field map using Eq. [8].
Multicoil Acquisitions
Following (3), let us consider a multicoil acquisition with
P distinctcoils,whichproduceP imageswithindependent
amplitude weightings and phase offsets. Thus, the signal at
a given voxel corresponding to coil p with echo time tncan
be modelled as
sp(tn) = ei2πfBtn(ρW,p+ ρF,pei2πfFtn)
where ρW,pand ρW,pare the the water and fat signal inten
sities, weighted by the complexvalued sensitivity of coil p
at the location of the voxel under consideration.
The proposed VARPRO formulation can be extended to
optimally (in the ML sense) estimate the field map as well
[12]
as the P sensitivityweighted water/fat images. According
to the signal model in Eq. [12], the new cost function is
RMC,0(ρ1,...,ρP,fB) = R1,0(ρ1,fB) + ··· + RP,0(ρP,fB) [13]
where ρp= [ρW,pρF,p]Tand Rp,0(ρp,fB) is the singlecoil
cost function for the signal sp, as defined in Eq. [4]. Clearly,
fB is the only nonlinear parameter under consideration
and thus the VARPRO approach discussed earlier can be
naturally extended by simply minimizing the sum of the
individual cost functions. Since for each value of fBall the
linear parameters {ρ1,...,ρP} are obtained immediately by
solution of P linear LS problems, we can express the com
bined cost function in the VARPRO formulation RMC(fB)
(similarly to Eq. [6]), and again a global onedimensional
search is possible to find the optimal fBestimate.
As in the singlecoil case, the water/fat amplitudes can
be determined efficiently once fB is estimated, by solv
ing the corresponding linear problem (Eq. [8]) for each
coil. After the P sensitivityweighted water/fat images are
obtained, they can be combined using standard multicoil
combination techniques (3,17–19).
Here, the coil sensitivities are assumed unknown. If they
areknown,theVARPROformulationcanstillbeused,with
thedifferencethatonlytwocomponentamplitudes,ρWand
ρF, and the field inhomogeneity, fB, need to be estimated at
each voxel.
Efficient Solution Using Linear Prediction
If the images are acquired at uniformly spaced echo times,
a computationally faster solution is possible. Assuming the
echo times are tn= t0+n?t,n = 1,...,N, the signal model
in Eq. [2] can be rewritten as follows:
s(tn) =
2
?
m=1
amzn
m,n = 1,...,N [14]
where a1= ρWei2πfBt0, a2= ρFei2π(fF+fB)t0, z1= ei2πfB?t, and
z2= ei2π(fF+fB)?t.
This signal, in the absence of noise, is linearly pre
dictable with coefficients {g1,g2}, i.e.
s(tn) = g1s(tn−1) + g2s(tn−2),
and, since z1 = z2 = 1 it is also backwardpredictable
with the same prediction coefficients:
n = 3,...,N [15]
s∗(tn) = g1s∗(tn+1) + g2s∗(tn+2),
Furthermore, it can be shown that the polynomial
n = 1,...,N − 2.[16]
G(z) = 1 − g1z−1− g2z−2
[17]
has its roots at z1and z2(see, e.g., (20) for details).
This formulation enables an efficient determination of
the parameters in the signal model (Eq. [2]). Similarly
to the VARPRO method described earlier, the problem is
solvedbyestimatingthelinearandnonlinearparametersin
two separate steps. First, the prediction coefficients {g1,g2}
are estimated using the socalled forwardbackward LP
by simultaneously solving Eqs. [15] and [16] (21). Next,
the estimates for zm are computed as the roots of G(z),
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574Hernando et al.
and the linear parameters amare obtained by solving the
corresponding linear problem (Eq. [14]).
In the absence of noise, the parameters {ρW,ρF,fB} are
obtained directly from the LP estimates {am,zm}, as long as
a1?= 0 and a2?= 0. Denoting φm= ∠zm/(2π?t), then either
φ2 = φ1+ fFor φ2 = φ1− fF. Without loss of generality,
we can sort {φm} so that the former is satisfied. Thus, the
signal parameters are obtained as follows: fB = φ1, ρW =
a1e−i2πfBt0, and ρF= a2e−i2π(fB+fF)t0.
If one of the components is absent, the signal model
becomes ambiguous (10) and assigning the observed com
ponent to water or fat requires some prior knowledge.
Similarly, in the presence of noise, the frequency sepa
ration of the observed components will not be exactly
fF. Assuming a limited range on the field inhomogeneity,
[fB,MIN,fB,MAX], we propose to assign to water and fat the
component with estimated frequency φmclosest to 0 and
fF, respectively, i.e., the component minimizing
φW= arg min
φ∈{φ1,φ2}φ,s.t.φ ∈ [fB,MIN,fB,MAX]
[18]
is assigned to water (with corresponding amplitude ρW=
aWe−i2πfBt0), and the component minimizing
φF= arg min
φ∈{φ1,φ2}φ − fF,s.t.φ ∈ [fF+ fB,MIN,fF+ fB,MAX]
[19]
is assigned to fat (with amplitude ρF= aFe−i2π(fB+fF)t0).
If no estimated frequency φm lies within the specified
bounds for a given component, this component is assumed
not present at the current voxel. Subsequently, the field
map value can be estimated at each voxel by weighted aver
aging of the individual estimated field inhomogeneities:
fB,v=φWρW + (φF− fF)ρF
ρW + ρF
.[20]
Imposing Spatial Constraints
The voxelbyvoxel LPbased method typically produces
rougher field map estimates than VARPRO, due to the sub
optimal nature of LP for solving the NLLS fitting problem
(minimizing Eq. [3]) in the presence of noise. Thus, the LP
field map estimate can benefit from spatial regularization.
However, the LP formulation does not provide the same
flexibility as VARPRO for incorporating spatial constraints.
Here, we propose to impose smoothness on the field map
in a separate step, by penalizing deviations from the voxel
byvoxel estimates as well as field map roughness. This can
be formulated effectively as a regularized LS problem:
ˆfB= argmin
fB
[?W(fB− fB,v)?2+ λ?DfB?2]
[21]
where fBis the complete field map (a lengthQ vector cor
responding to the Q voxels in the image), fB,vis the rough
field map estimated independently at each voxel (Eq. [20]),
W isadiagonalweightingmatrixusedtoplacemoreweight
on field map estimates from voxels where the signal level
is higher, D computes spatial finite differences in the field
map, and λ is a regularization parameter controlling the
tradeoff between field map smoothness and data fidelity.
This minimization reduces to a linear problem:
(WHW + λDHD)ˆfB= WHWfB,v
and can be solved efficiently using, e.g., a conjugate
gradient method (22). This method is similar to the one
proposed in (3), where the water/fat images and field map
are estimated point by point and field map smoothing
is performed separately. Decoupling both steps simplifies
the algorithm and reduces the computational burden. It
must be noted that this “twostep” method is suboptimal,
and is expected to perform well only in cases of moder
ate field inhomogeneity. Specifically, the smoothing step
will generally not be able to correct large errors in the
field map estimate, e.g., in voxels where water and fat are
swappedduringvoxelbyvoxelprocessing(10,13).Finally,
thewaterandfatcomponentsshouldbereestimatedateach
voxel using the regularized field mapˆfBby solving the
corresponding linear problem (Eq. [8]).
In this article, we have employed the following parame
tersforLP:theweightingmatrixW wassettothesumofthe
signal amplitudes at each voxel, normalized to have a max
imum value of 1 (thus assigning more weight to the field
map estimates from voxels containing higher signal ampli
tude).Theregularizationparameterλwassetto1.Similarly
to VARPRO, the parameters for LP were fixed throughout
the results.
Since the proposed regularization of the field map will
place very little weight on estimates from voxels that
contain only noise (e.g., voxels where the signal ampli
tude is below a noise threshold), these voxels may be
skipped (and their field inhomogeneity set to zero) during
the voxelbyvoxel estimation, for increased computational
efficiency.
The LPbased algorithm for regularized estimation of
water/fat images and field map can be summarized as
follows:
[22]
1. At each voxel with signal amplitude above a noise
threshold,performforwardbackwardLPtoobtainzk
and akfollowing Eqs. [15], [16], and [14]. Assign the
estimated components to water/fat using Eqs. [18]
and [19].
2. Obtain the regularized field mapˆfBby imposing
spatial smoothness (Eq. [21]).
3. Reestimate the water/fat components ρWand ρFat
each voxel using the regularized field map (Eq. [8]).
Multicoil Acquisitions
The single coil LP algorithm can easily be extended to
handle multicoil acquisitions. According to the multicoil
signal model in Eq. [12], the field inhomogeneity effect is
the same for all coils, and furthermore the signals detected
at a particular voxel by the different coils are different lin
ear combinations of the same complex exponentials. Thus,
the prediction coefficient vector is the same for all coils,
which can simply be enforced by solving Eqs. [15] and
[16] simultaneously for all coils. Once the prediction coef
ficients(andthusthezk)areobtained,theamplitudesofthe
different chemical species can be estimated independently
for each coil using Eq. [14].
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Joint Estimation of Water/Fat Images and Field Map575
FIG.1.
ical MSE for amplitude estimation. The solid line indicates the
CRLB, while the circles and asterisks represent the empirical results
obtained from VARPRO and LP, respectively.
Comparisonbetweentheoreticalbounds(CRLB)andempir
RESULTS AND DISCUSSION
A simulation study was done to test the performance
of the proposed methods for singlevoxel decomposition
in the presence of noise. Figure 1 shows a comparison
of the CramérRao lower bound (CRLB, a lower bound on
the variance of any unbiased estimator (6,23)) and mean
squared error (MSE) simulation results for amplitude esti
mation using three samples (N = 3) with phase shifts
{−π/6,π/2,7π/6}. This choice of phase shifts is optimal for
all water/fat ratios, as shown in (6). The simulated signal
contains two components with amplitudes ρW = ρF = 1.
Complex Gaussian noise with a range of different vari
ances was added to the signal. The VARPRO estimate
appears to be efficient (i.e., unbiased and with MSE match
ing the CRLB) for all SNR values. The LP estimate becomes
more robust as the SNR increases. Figure 1 shows that, at
lower SNR, the VARPRO approach is preferable. On the
other hand, at higher SNR, LP provides a competitive and
computationally efficient solution to the water/fat imaging
problem.
A quantitative comparison of the accuracy of the pro
posed methods and IDEAL (including spatial regulariza
tion of the field map) was performed using synthetic
data. Three synthetic datasets were generated, based on
brain, abdominal, and cardiac acquisitions, respectively.
The water and fat images were obtained by wavelet denois
ing the estimated water/fat components (obtained using
VARPRO with no spatial regularization on the field map)
from each in vivo dataset. The synthetic field maps were
obtained by smoothing the corresponding voxelbyvoxel
estimated field maps. This was done by applying a Ham
ming window in the Fourier domain. Note that the field
map smoothing step in our implementation of the IDEAL
algorithm was performed with the same Hamming window
used to generate the synthetic field maps (3). Several field
maps, simulating increasing severity of field inhomogene
ity, were obtained in each case by scaling each synthetic
field map. The water/fat images were then combined with
each field map according to the signal model in Eq. [2] to
obtain datasets with increasing levels of field inhomogene
ity. The water/fat chemical shift was 215 Hz and the the
TEs produced water/fat phases {−π/6,π/2,7π/6}. Finally,
complex Gaussian noise was added to each of the datasets
(SNR = 20). The noisy datasets were then processed using
IDEAL, VARPRO and LP, and the resulting decompositions
compared to the true images. Figure 2 shows the relative
norm of the error (averaged for the brain, abdominal, and
cardiac simulated datasets) in the resulting water image
produced by IDEAL, VARPRO, and LP. Note how VARPRO
performs almost uniformly well for all levels of field inho
mogeneity, whereas the errors of LP and IDEAL increase
sharply as the maximum field inhomogeneity becomes
larger than fF/2. This is due to the misclassifications that
occur in the voxelbyvoxel decomposition, which are only
partially removed in the field map smoothing step. Also,
for low field inhomogeneities, LP performs nearly as well
as VARPRO and IDEAL.
To test the proposed methods on in vivo data, several
brain images were obtained using optimal echo spacings
(4), with TE values {3.38,4.17,4.97} ms, corresponding
to water/fat phases {7π/6,π/2,−π/6}. Data were acquired
on a 3 T Siemens Allegra head scanner in accordance
with the local institutional review board. Figure 3 shows
the water/fat decomposition and field map obtained with
IDEAL (3) and the proposed VARPRO and LP methods,
respectively. The water/fat decompositions using all three
methods are very similar. The differences observed in the
estimated field map are due to the different strategies
for imposing field map smoothness: (a) IDEAL filters the
raw field map with a smoothing kernel (3), and thus the
estimates from voxels where the signal is mostly noise
are preserved; (b) VARPRO imposes a smoothing MRF
prior on the field map (Eq. [10]), automatically assigning
more weight to field inhomogeneity estimates from vox
els with higher signal intensity; (c) LP applies a weighted
LS smoothing (Eq. [21]) which has a similar effect to the
MRFbasedapproach,sincetheweightsappliedonthefield
FIG. 2.
water/fat decomposition including spatial smoothness constraints on
the field map. Relative errors are shown for water image reconstruc
tion using the three methods, for different levels of field inhomogene
ity, and averaged for three different synthetic datasets.
Quantitative comparison of IDEAL, VARPRO, and LP for
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576Hernando et al.
FIG. 3.
algorithms. The second column contains the estimated fat component. The third column contains the regularized field map. (a–c) IDEAL
estimates. The smooth field map is obtained by filtering the raw field map (resulting from voxelbyvoxel estimation) with a smoothing kernel (3);
(d–f) VARPRO estimates. The smooth field map is obtained directly by applying an MRF prior; (g–i) LP estimates. The smooth field map is
obtained by weighted LS regularization of the raw field map.
Water/fat decomposition from a brain acquisition. The first column contains the estimated water component using the different
map estimates are proportional to the amplitudes of the
corresponding components.
To test the multicoil version of the proposed methods, a
multicoil acquisition of the abdomen was performed using
six different TE values, {1.5,2.0,3.6,5.1,6.6,8.2} ms, corre
spondingtorelativewater/fatphases{−7π/5,−7π/6,−π/2,
π/6,5π/6,3π/2}. Data were collected on a GE 1.5 Tesla
whole body scanner (GE Healthcare Technologies, Wauke
sha, WI) using a fourchannel torso phasedarray receiver
coil. The pulse sequence used was a 3D fast spoiled gradi
ent echo sequence. Each 3D data set (for a corresponding
echo time) was acquired in a single but separate breath
hold. All data were collected in accordance with the local
institutional review board.
The multicoil results are shown in Fig. 4. Figures 4a–
c show the “gold standard" decomposition obtained from
VARPRO using all six shifts. Figures 4d–f show the results
from VARPRO using only three different echo times (with
water/fat phases {−7π/6,−π/2,π/6}). Figures 4g–i show
the resulting decomposition from the same three echo
times, using the LP method. Note the high quality of the
decompositions obtained with both methods using just
three echo times.
Figure 5 shows a comparison of IDEAL and the proposed
VARPRO method in the presence of high field inhomo
geneity. The images were acquired with water/fat phases
{7π/6,π/2,−π/6}, on a 3 T Siemens Allegra head scanner
in accordance with the local institutional review board.
The field inhomogeneity reached approximately 360 Hz.
TheVARPROmethodincludedspatialregularizationusing
ICM. Note how IDEAL swaps the different components
in part of the image, whereas the proposed method is
able to correctly separate the water and fat signals. This
increased robustness is due to the global optimality of the
VARPROapproach,regardlessofthenonconvexityofR(fB),
and the improved method for imposing spatial smooth
ness on the field map (which is performed jointly with
the water/fat estimation, instead of in a separate step).
A regiongrowing method (10) is also expected to per
form well for this type of dataset. However, the fact that
VARPRO produces good results in this case (using the
same parameters as in the moderate field inhomogeneity
cases) highlights the robustness of the proposed method.
Although ICM only guarantees convergence to a local opti
mum of the a posteriori distribution, ICMbased field map
estimation is less dependent (compared to regiongrowing)
Page 7
Joint Estimation of Water/Fat Images and Field Map577
FIG. 4.
the different algorithms. The second column contains the estimated fat component. Both components are displayed using sum of squares
combination of the multicoil signal. The third column contains the regularized field map. (a–c) VARPRO estimates using all six echo time
shifts; (d–f) VARPRO estimates using three water/fat shifts; (g–i) LP estimates using three water/fat shifts.
Water/fat decomposition from a multicoil abdominal acquisition. The first column contains the estimated water component using
on the initial values assigned to the field map (which is set
to all zeros at the beginning of the ICM iteration for all the
results presented in this paper).
The three methods considered in this article present
different computational requirements. In IDEAL, most of
the computation time is spent solving two small sys
tems of linear equations for updating the estimated field
value and the water/fat amplitudes (3). This is done itera
tively (until a convergence criterion is satisfied) for each
voxel from each coil. While an iterative procedure may
potentially require many iterations to converge, in our
experience convergence typically occurs in a few itera
tions, so the method is quite fast. After the voxelbyvoxel
iterative procedure, lowpass filtering the field map esti
mate and recomputing the water/fat images is done very
rapidly. In VARPRO, first the residual (Eq. [6]) must be
computed for all voxels and field map values, and next
the field map is estimated by repeatedly updating its value
at all voxels according to Eq. [10]. Subsequent estima
tion of the water/fat images given the field map requires
negligible computation. In LP, the main computational
burden consists of solving a small linear system and poly
nomial rooting at each voxel with signal above the noise
threshold, and smoothing the estimated field map by solv
ing a regularized LS problem. We compared our own
Matlab (The Mathworks, Natick, MA) implementations
of IDEAL, VARPRO, and LP. Note that the execution
times reported here are expected to decrease dramatically
when optimized and coded in C on a fast, multiproces
sor architecture. For a singlecoil, threepoint dataset with
images of size 128×128, the voxelbyvoxel IDEAL method
followed by lowpass filtering of the fieldmap and reesti
mation of the water/fat amplitudes throughout the image
(as described in (3)) required 16.5 sec on an Intel Xeon
based desktop PC at 3.6 GHz with 8 GB of RAM. To
solve the same problem, the VARPRO method with MRF
prior, with L = 300 field inhomogeneity values and 30
ICM iterations required 64.3 sec, whereas the LP method
using weighted LS field map regularization required 18.8
sec.
Multicoil acquisitions (number of channels P > 1) are
handled by the proposed methods with a small increase in
computation time. For an eightcoil (P = 8), threepoint
dataset with images of size 128 × 128, the computation
timeforIDEALwas114.4sec,whileVARPROrequired79.8
sec and LP required 39.4 sec. The observed differences are
mainly due to the fact that the proposed methods directly
impose the presence of a unique field map (thus solving the
same nonlinear problem as in the singlecoil case), whereas
IDEAL solves P nonlinear problems and then combines the
resulting field maps. This does not imply any fundamental
limitation of IDEAL: the proposed multicoil formulation
Page 8
578Hernando et al.
FIG. 5.
field map, respectively; (d–f) VARPRO estimates for water, fat, and field map, respectively.
Comparison of IDEAL and VARPRO results in the presence of high field inhomogeneity. (a–c) IDEAL estimates for water, fat, and
used with VARPRO in this paper can be incorporated into
the IDEAL algorithm.
In recent work, Yu et al. proposed an extension of the
IDEAL algorithm to include simultaneous estimation of T∗
(24). Relaxation effects are naturally accounted for within
the LP formulation. However, to take relaxation effects into
account, zn < 1 in Eq. [14], and thus forward and back
ward linear prediction cannot be used simultaneously as in
the zn = 1 case. For example, if two damped components
are to be estimated, at least four uniformly spaced data
points need to be acquired. Clearly, the VARPRO method
can be applied for the estimation of multiple nonlinear
parameters (e.g., relaxation effects). However, the compu
tational demands increase significantly with the number of
nonlinear parameters.
Both VARPRO and LP can be used to estimate multi
ple chemical species. If the chemical shifts are known
exactly, then VARPRO is very efficient and robust (since
the problem still reduces to a onedimensional search). On
the other hand, LP estimates all the chemical shifts and
thus requires a larger number of measurements. Both meth
ods have been studied extensively in the context of MR
spectroscopy quantitation (25).
2
CONCLUSIONS
This article presented two novel methods for joint estima
tion of water/fat images and field inhomogeneity map from
Dixontype acquisitions. The VARPRO method overcomes
several limitations of previously proposed descentbased
algorithms. First, it provides an efficient and globally opti
mal solution to the nonconvex NLLS problem. Second,
spatial smoothness constraints on the field map estimate
are enforced in a statistically meaningful way using an
MRF prior. The LP method produces good results under
moderate noise and field inhomogeneity conditions and is
computationally very efficient. This method can be used
as long as the echo times are uniformly spaced. Both LP
and VARPRO allow natural extensions to handle multicoil
acquisitions, as well as cases in which there are more than
two chemical species, or when relaxation effects cannot be
neglected.
APPENDIX
One key aspect of the proposed VARPRO method is the
discretization of the feasible range of main field values
as a finite set of values {fB,l}L
ment for the minimizing solution to be meaningful is that
the spacing is small enough relative to the variability of
R(fB). In other words, we need to guarantee that R(fB) does
not contain abrupt changes which are not captured by
the discretized version {R(fB,l)}L
consider the derivative of R(fB). From Eq. [6],
l=1. An important require
l=1. For this purpose, let us
dR(fB)
dfB
= −
?
m?=n
m,n
s∗(tn)s(tm)?n,mei2πfB(tn−tm)2π(tn− tm)
where ? = ?(?H?)−1?H. The following bound follows
readily:
????
dR(fB)
dfB
????≤ max
k
s(tk)2?
m,n
m?=n
?n,m2π(tn− tm) = B. [A1]
Given the discretization spacing ?fB = fB,l+1− fB,l, this
provides a bound on the maximum difference of the global
Page 9
Joint Estimation of Water/Fat Images and Field Map579
optimum of the continuous function R(fB) from the dis
cretized version {R(fB,t)}T
t=1:

min
fB∈[fB,MIN,fB,MAX]R(fB) −
min
l=1,2,...,L{R(fB,l)}L
l=1 ≤?fB
2
B [A2]
which produces a useful criterion to ensure that the cost
function R(fB) is smooth with respect to ?fB. For exam
ple, let us consider the signal from voxel (65,65) in the
brain acquisition shown in Fig. 3. Using field inhomo
geneity bounds ±400 Hz with L = 300 uniformly spaced
discretized values, Eq. [A2] guarantees that the difference
between the minimal residual in the discretized and con
tinuous problems will be at most 0.6% of the range of the
residual.
LIST OF SYMBOLS
am
LP amplitude corresponding to mth component
B0
main (static) magnetic field (T)
D
matrix for computing 2D finitedifferences for
regularization of field map
fB
offresonance frequency caused by field
inhomogeneity (Hz)
fB,v
singlevoxel LP estimate of field inhomogeneity (Hz)
fo
B
optimal estimate for field inhomogeneity (Hz)
fB,MIN
lower bound on the field inhomogeneity (Hz)
fB,MAX
upper bound on the field inhomogeneity (Hz)
fq
B
field inhomogeneity at voxel q (Hz)
fq,cur
B
current value of field inhomogeneity at voxel q (Hz)
fq,new
B
updated value of field inhomogeneity at voxel q (Hz)
fF
chemical shift between water and fat (Hz)
fB
vector containing the complete field map (Hz)
fB,v
vector containing the field map estimated
voxelby voxel (Hz)
ˆfB
gm
mth LP coefficient
g
LP coefficient vector
Gpolynomial associated with LP coefficient vector
I
identity matrix
L number of different field inhomogeneity values for
discretization in VARPRO
N number of acquisitions
P number of distinct coils in multicoil acquisitions
Qnumber of voxels in image
R maximumlikelihood cost function in the
VARPRO formulation
R0
original maximumlikelihood cost function
Rp,0
original maximumlikelihood cost function for
signal from pth coil
RMC
augmented cost function for multicoil acquisitions
(VARPRO formulation)
RMC,0
augmented cost function for multicoil acquisitions
sacquired signal at a given echo time
sp
acquired signal from coil p at a given echo time
s
vector of acquired signals at different times
tn
echo time of the nth acquisition (sec)
W
weighting matrix for field map regularization in
LP method
regularized field map (Hz)
zm
?
LP pole corresponding to mth component
matrix used in the derivation of the bound on the
accuracy of the VARPRO discretization
neighborhood of voxel q in MRF formulation
spacing between echo times of consecutive
acquisitions (sec)
threshold for stopping criterion in ICM iteration
regularization parameter for LP method
diagonal matrix applying effect of field
inhomogeneity into signal
regularization parameter for VARPRO method
with MRF prior
amplitude of water component
amplitude of fat component
amplitude of water component in pth image
(multicoil acquisitions)
amplitude of fat component in pth image
(multicoil acquisitions)
vector containing both water and fat amplitudes
vector containing both water and fat amplitudes
in pth image (multicoil acquisitions)
optimal estimate for water and fat amplitudes
estimated frequency of mth component detected
with LP (Hz)
estimated frequency of component assigned to fat
(Hz)
estimated frequency of component assigned to
water (Hz)
matrix mapping amplitudes into signal samples
in the absence of field inhomogeneity
matrix mapping component amplitudes to
signal samples
δq
?t
ε
λ
?
µ
ρW
ρF
ρW,p
ρF,p
ρ
ρp
ρo
φm
φF
φW
?
?
REFERENCES
1. Dixon
1984;153:189–194.
2. Glover GH, Schneider E. Threepoint Dixon technique for true
water/fat decomposition with B0 inhomogeneity correction. Magn
Reson Med 1991;18:371–383.
3. Reeder SB, Wen Z, Yu H, Pineda AR, Gold GE, Markl M, Pelc NJ. Multi
coil Dixon chemical species separation with an iterative least squares
estimation method. Magn Reson Med 2004;51:35–45.
4. Reeder SB, Pineda AR, Wen Z, Shimakawa A, Yu H, Brittain JH, Gold
GE,BeaulieuCH,PelcNJ.Iterativedecompositionofwaterandfatwith
echo asymmetry and leastsquares estimation (IDEAL): Application
with fast spinecho imaging. Magn Reson Med 2005;54:636–644.
5. Bernstein MA, King KF, Zhou XJ. Handbook of MRI pulse sequences.
Burlington, MA: Elsevier Academic Press; 2004.
6. PinedaA,ReederS,WenZ,PelcNJ.CramérRaoboundsforthreepoint
decomposition of water and fat. Magn Reson Med 2005;54:625–635.
7. Golub G, Pereyra V. Separable nonlinear least squares: The variable
projection method and its applications. Inverse Prob 2003;19:R1–
R26.
8. Boada F, Liang ZP, Haacke EM. Improved parametric reconstruction
using variable projection optimization. Inverse Probab 1998;14:19–
27.
9. Haldar JP, Anderson J, Sun SW. Maximum likelihood estimation of
T1 relaxation parameters using VARPRO. In Proceedings of the 15th
Annual Meeting of ISMRM, Berlin, Germany, 2007 (Abstract 41).
10. Yu H, Reeder SB, Shimakawa A, Brittain JH, Pelc NJ. Field map esti
mation with a region growing scheme for iterative 3point waterfat
decomposition. Magn Reson Med 2005;54:1032–1039.
11. Bhagwandien R, Moerland MA, Bakker CJG, Beersma R, Lagendijk JJW.
Numerical analysis of the magnetic field for arbitrary magnetic suscep
tibility distributions in 3D. Magn Reson Imaging 1994;12:101–107.
WT. Simple protonspectroscopicimaging.Radiology
Page 10
580Hernando et al.
12. Schomberg H. Offresonance correction of MR images. IEEE Trans Med
Imaging 1999;18:481–495.
13. Fessler JA, Yao D, Noll DC. Regularized fieldmap estimation in MRI.
In Proceedings of the 3rd IEEE International Symposium on Biomed
ical Imaging: Macro to Nano, Arlington, VA, USA, 2006, pp 706–
709.
14. Geman S, Geman D. Stochastic relaxation, Gibbs distribution and the
Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intell
1984;6:712–741.
15. Besag J. On the statistical analysis of dirty pictures. J R Stat Soc Ser B
1986;48:259–302.
16. Kellman P, McVeigh ER. Image reconstruction in SNR units: a general
method for SNR measurement. Magn Reson Med 2005;54:1439–1447.
17. RoemerPB,EldelsteinWA,SouzaSP,HayesCE,MuellerOM.TheNMR
phased array. Magn Reson Med 1990;16:192–225.
18. Hayes CE, Roemer PB. Noise correlations in data simultaneously
acquired from multiple surface coil arrays. Magn Reson Med
1990;16:181–191.
19. Bernstein MA, Grgic M, Brosnan TJ, Pelc NJ. Reconstructions of
phase contrast, phased array multicoil data. Magn Reson Med
1994;32:330–334.
20. Kumaresan R. On the zeros of the linear predictionerror filter for
deterministic signals. IEEE Trans Acoust 1983;31:217–220.
21. Ulrych TJ, Clayton RW. Time series modelling and maximum entropy.
Phys Earth Planet Int 1976;12:188–200.
22. Vogel CR. Computational methods for inverse problems. Philadelphia,
PA, USA: SIAM; 2002.
23. van Trees HL. Detection, estimation and modulation theory, part I.
New York: Wiley; 1968.
24. Yu H, McKenzie A, Shimakawa A, Pelc NJ, Brittain JH, Reeder SB.
IDEAL waterfat separation with simultaneous T2∗estimation. In
Proceedings of the 14th Annual Meeting of ISMRM, Seattle, WA, USA,
2006 (Abstract 624).
25. Vanhamme L, Sundin T, van Hecke P, van Huffel S. MR spectroscopy
quantitation: A review of timedomain methods. NMR Biomed
2001;14:233–246.