COMMUNICATION

Alias Subtraction More Efficient Than ConventionalZero-Padding in the Fourier-Based Calculation of theSusceptibility Induced Perturbation of the MagneticField in MR

Job G. Bouwman1 and Chris J. G. Bakker1,2*

Forward calculation of the susceptibility induced field shift byFourier-based procedures requires spatial zero-padding toprevent aliasing artifacts (periodic wrap-around). Paddingwith a factor of two gives accurate results, however, halvesthe maximal attainable resolution, and slows down the calcu-lation, which may hamper the feasibility of real-time calcula-tions. Herein is proposed to first perform the calculation atthe original resolution—allowing aliasing—and to remove ali-asing with an additional convolution in a lower resolution, toalleviate these restrictions regarding memory size and calcu-lation speed, a procedure we termed ‘‘virtual’’ zero-padding.Virtual zero-padding was numerically and experimentallytested and validated with conventional zero-padding and theanalytical solution (in the case of spheres) on several phan-toms. A demonstration of the increased efficiency is given byimplementing virtual zero-padding in a dynamic calculationprocedure. The improved efficiency is expected to be relevantregarding the ongoing increase in spatial and temporal reso-lution in ultra-high-field MRI. Procedures are presented forcircular convolution using the discrete Green’s function andk-space filtering using the continuous Green’s function. MagnReson Med 000:000–000, 2012. VC 2012 Wiley Periodicals, Inc.

Key words: susceptibility; field calculation; Fourier;convolution; aliasing; zero-padding

INTRODUCTION

Forward Fourier-based field shift calculations suffer fromaliasing

Fast and accurate, forward calculation of the susceptibil-ity induced field shift is an important tool for iterativeprocedures in quantitative susceptibility mapping (1–10),for modeling the BOLD effect (11,12) and for real-timecalculations during MR-guided thermal ablation (13–15)and cryoablation (16). The ongoing increase in spatial re-solution of ultra-high-field MRI, being sensitive to sus-

ceptibility differences of the microstructure (17,18) leadsto large matrices requiring efficient algorithms. Fourier-based convolution methods (19–22) have proven to befast and accurate (23), yet suffer from aliasing (periodicwrap-around) due to the decomposition into 3D sinu-soids (24). To prevent aliasing, the relative size of theobject must not exceed 50 or 60% of the field of view(FOV) in each dimension (19–21,25); however, theneeded spatial buffer depends on the susceptibility dis-tribution and the required accuracy (20). Especially for3D convolutions such a spatial buffer has dramatic con-sequences for the calculation time and the needed mem-ory capacity (25) since the fast Fourier transform requiresO(N3logN3) calculations for a matrix of N3(19), limitingboth the spatial and temporal resolution of thesecalculations.

Aliasing has a Smooth Spatial Distribution

The enormous increase in calculation time seems dispro-portional regarding the relatively smooth spatial distribu-tion of this aliasing artifact, being the demagnetizingfield of neighboring copies in the periodic susceptibilitygrid. This imbalance brought us to investigate if aliasingcan be eliminated more efficiently.

Removing Aliasing in a Lower Resolution

Herein is proposed to decouple the calculation proce-dure into two separate parts: one convolution withoutzero-padding and an additional convolution to calculatethe effect of aliasing, which can be used to correct theinitial outcome. Although the derived algorithms for this‘‘virtual’’ zero-padding (vZP) are mathematically equiva-lent to conventional zero-padding (cZP), it will be shownthat such a decoupled procedure has four advantages.First, calculation may be about four times faster if alias-ing is estimated in a lower resolution. Second, aliasingcan be removed afterwards, and the previously neededspace for cZP can now be used to perform the calcula-tion in a higher resolution, leading to smaller artifactsand more localized errors (22). Third, for dynamic fieldshift calculations, both parts of the calculation can beupdated separately, allowing real-time results. Moreover,due to the temporal smoothness of aliasing, calculationof the internal field can be prioritized. Fourth, with vZPthe user is exempt from choosing an appropriate padding

1Image Sciences Institute, University Medical Center Utrecht, TheNetherlands.2Department of Radiology, University Medical Center Utrecht, TheNetherlands.

This research is supported by the Dutch Technology Foundation STW, whichis the applied science division of NWO, and the Technology Programme ofthe Ministry of Economic Affairs, Agriculture and Innovation (project number10712).*Correspondence to: Chris J. G. Bakker, Department of Radiology, RoomE01.132, University Medical Center Utrecht, Heidelberglaan 100, 3584CXUtrecht, The Netherlands. E-mail: c.j.g.bakker@umcutrecht.nl

Received 20 January 2012; revised 6 April 2012; accepted 27 April 2012.

DOI 10.1002/mrm.24343Published online in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 000:000–000 (2012)

VC 2012 Wiley Periodicals, Inc. 1

factor. It will be shown that the only adverse effect ofthis approach is a negligible accuracy loss in the periph-ery of the FOV, caused by the loss of high frequenciesduring downscaling, and caused by the fact that cZPameliorates Gibbs ringing in k-space filtering (KF)(23,25,26) whereas vZP lacks this beneficial effect.

Followed Approach

The general approach was to create a scale model of theartificial susceptibility environment, from which theinduced field shift could be calculated using the Fourier-based convolution. The rescaled field shift induced inthe central cavity was then used as the estimated alias-ing distribution. To this end, the exact parts of the peri-odic susceptibility distribution responsible for aliasingwere identified. This led to distinctive algorithms fortwo fundamentally different Fourier-based procedures,namely circular convolution (CC) based on the discretelysampled Green’s function (22,25), and KF based on theFourier transform of the continuous Green’s function(19–21). The derived algorithms were implemented eas-ily with ‘‘off-the-shelf’’ Fourier transform algorithms. Oursoftware will be made publically available on: http://www.mathworks.com/matlabcentral/fileexchange/.

Validation

The mathematical equivalence of vZP and cZP could eas-ily be demonstrated by executing them in the same reso-lution. The performance of low resolution vZP wasnumerically and experimentally tested, and validatedwith cZP on several phantoms and (in the case ofspheres) with the analytical solution as well.

Demonstration of Potential

To illustrate the potential of this approach, we imple-mented vZP in a dynamic field calculation scheme. Thetemporal accuracy of any such procedure will depend onboth the degree of change within the susceptibility dis-tribution and the latency, that is, the time needed to pro-cess each single (zero-padded) distribution. It will beshown that the reduced latency of vZP in combinationwith the decoupled implementation of updating the fieldand updating the aliasing correction increases the tempo-ral accuracy. We quantified this accuracy gain with a 4Dnumerical simulation mimicking how an external mov-ing applicator or a passive shimming device induces adynamic field shift within a certain region of interest(ROI).

THEORY

The Superposition Principle and Fourier-Based FieldCalculations

The relative field shift dð~rÞ of the z-component of themagnetic field Bzð~rÞ induced within a susceptibility dis-tribution xð~rÞ embedded in a medium of susceptibilityxeð~rÞ if placed in an external homogenous field B0 isdefined as:

dð~rÞ ¼ Bzð~rÞ � B0ð1þ xeð~rÞÞB0

½1�

In normal MR-experiments with small volume suscepti-bilities below 100 parts per million (ppm), the practicalinfluence of higher order magnetization is negligible andthe susceptibility induced perturbation can be very wellapproximated by the superposition of the fields inducedby the individual substructures of this susceptibility dis-tribution, known as the Born approximation (19–22,25).The forward calculation of the spatial distribution of thefield shift induced within a discrete susceptibility distri-bution X ¼ xð~rÞ � xe, where ve is assumed to be constant,is then simply the convolution of this distribution withthe field induced by an elementary substructure. As wasalready pointed out in (27), this convolution can beefficiently carried out by a multiplication in the Fourierdomain in the general form:

d ¼ F�1½FðXÞ � Gð~kÞ� ½2�

Here, F and F�1 are respectively the forward and inversediscrete Fourier transform (DFT) and Gð~kÞ is the Green’sfunction in the Fourier domain including the Lorentzcorrection.

CC Versus KF

Two classes of implementations are described in the lit-erature (25), according to the domain in which the ele-mentary substructures are defined, leading to a differentform of Gð~kÞ. The first, CC, is based in real space andthe convolution kernel is discretely sampled from thespatial Green’s function (22,25):

Gdð~kÞ ¼ F ½Gdð~rÞ� with

Gdð~rÞ ¼14p � 3z2�r2

r5 if |~r| 6¼ 00 if |~r| ¼ 0

� ½3�

The second implementation, KF, is based in k-space and isdone by multiplying the Fourier transform of the suscepti-bility distribution by the discretely sampled analytical Fou-rier transform of the continuous Green’s function (19–21):

Gcð~kÞ ¼13 � kz

2

kx2þky

2þkz2 if |~k| 6¼ 0

0 if |~k| ¼ 0

(½4�

Different Manifestation of Aliasing for CC and KF

Without a spatial buffer, both implementations sufferfrom aliasing artifacts, an inherent consequence of apply-ing the DFT of X. However, the manifestation of aliasingfor these two implementations is fundamentally differ-ent, which can be explained by the role of the DFT. InCC, the DFT of X is only used to speed up the calcula-tion by the convolution theorem, and aliasing onlyoccurs if the padding factor p is smaller than two. Thepenetration depth of aliasing is 2–p, so padding factor of1.6 makes that for each dimension the peripheral 40% ofthe resulting field is corrupted by aliasing. In KF, how-ever, the DFT of X is used to determine the elementarysubstructures in X, being the 3D infinite sinusoidal pat-terns. Consequently, the complete FOV is corrupted by aharmonic aliasing distribution, which can be ameliorated

2 Bouwman and Bakker

but never fully eradicated by zero-padding. One shouldtherefore not be misled about the general appearance ofEq. 2: CC is truly based in the spatial domain, and KF istruly based in the Fourier domain. Another consequenceof this fundamental difference is that only KF leads toGibbs ringing artifacts (25).

Aliasing in CC is Based in the Spatial Domain

With a spatial padding factor smaller than two, CC suf-fers from aliasing, mainly caused by the influence of theclosest artificial neighbors. Due to the finite nature of thekernel, only a sub-region of this close neighborhood isresponsible, illustrated by the A in Fig. 1b. The artifactis slightly ameliorated by the erroneous exclusion of asubregion of the original distribution illustrated by the Ein Fig. 1b. The distribution of this combined artifact ofCC is then given by:

derror ¼ dA � dE ¼ C½ZcðXÞ � Gnear� � C½ZðXÞ � Gfar�; ½5�

where Z is a zero-padding function, embedding X in thecenter of an empty matrix, with a padding factor of two,Zc creates a distribution of the environment of X, beingcomplementary to the zero-padded distribution, Gnear

and Gfar are the central and the peripheral part of thediscretely sampled Green’s function and C crops the cen-tral 50% of the matrix in each dimension.

Aliasing in KF is Based in the Frequency Domain

Regardless of the used padding factor, KF always suffersfrom aliasing, mainly caused by the influence of the clos-

est artificial neighbors. Due to the infinite nature of thekernel, the complete periodic pattern enclosing theseclose neighbors is responsible, illustrated by A1 in Fig.1e,f. The artifact is slightly increased by the erroneousinclusion of more distant grids of neighbors illustratedby A2 and A3 in Fig. 1e,f. The distribution of this arti-fact in KF is given by Eq. 6 for the first grid and Eq. 7 forthe mth grid:

dA1 ¼ CðF�1fF ½ZcðXÞ� � GgÞ ½6�

and

dAm ¼ CðF�1fF ½ZcðZm�1ðXÞÞ� � GgÞ; ½7�

where Z is a zero-padding function, embedding X in thecenter of an empty matrix, with a padding factor of two,Zc creates a distribution of the artificial environment ofX, being complementary to the zero-padded distribution,and C crops the central 50% of the matrix in eachdimension. However, the effect of the second and higherorder grids is negligible for practical purposes and willnot be corrected for in this article.

METHODS

Implementation of vZP

For both CC and KF, the proposed method was imple-mented with a fixed vZP padding factor of two. Thismeans that for KF the influence of higher order gridswas neglected. The overall procedure consisted of proc-essing the ‘‘unpadded’’ (UP) matrix, and then to subtractthe estimated effect of aliasing:

FIG. 1. Aliasing inducing substructures of the periodic susceptibility grid for CC based on the discrete Green’s function (a–d) and KF

based on the continuous Green’s function (e,f). a: The convolution kernel (the spatial Green’s function) slides over the circular distribu-tion. b: The unwanted influence of artificial neighbors (A) and the erroneously excluded influence of the original distribution (E). c: Theunwanted influence of A can be calculated by convolving the artificial environment with the central part of the Green’s function. d: The

erroneously excluded influence of E can be calculated by convolving the zero-padded distribution with the peripheral part of the Green’sfunction. e: In KF all artificial neighbors in the infinite susceptibility grid contribute to aliasing. To calculate this aliasing, the environment

must be decomposed into periodic subpatterns A1, A2, etc. f: Elementary subunit of A1 inducing the aliasing of this grid used to calcu-late its influence, which is complementary to the zero-padded distribution.

Alias Removal in Fourier-Based Field Calculation 3

dvZP � dUP � d̂alias ¼ F�1½FðXÞ � G� � d̂alias ½8�

For both procedures, vZP could be implemented by twoadditional calculations involving the downscaled FOV,one without padding, the other with conventional pad-ding. The difference between both outcomes could thenbe rescaled, resulting in the estimated aliasing distribu-tion:

d̂alias ¼ UnðF�1½FðDnðXÞÞ � G�� CfF�1½FðZfDnðXÞgÞ � G�gÞ; ½9�

where the elementary downscale operator D was imple-mented by a factor of two, C crops the central 50% of amatrix, and U up-scales a distribution by factor of two.The integer n (here either 1 or 2), determines the down-sampling factor (either 1=2 or 1=4).

However, specific alterations of this general procedureoptimized the efficiency for the two distinctive Fourier-based approaches. For CC this resulted in:

d̂CCvZP ¼F�1½FðXÞ � G� � C �Un½RðF�1ðFfDnðXÞg � GÞÞ� F�1ðFfZ½DnðXÞ�g � GÞÞÞ�; ½10�

in which R is the repetition operator, embedding a distri-bution in a larger matrix surrounded by copies of thisdistribution. For KF this resulted in:

d̂KFvZP ¼ F�1½FðXÞ � G� � C �Un½F�1ðFfZc½DnðXÞ�g � GÞ�½11�

Implementation of the Operators

The operators C (cropping the center), Z (zero-padding),ZC (complement of zero-padding), and R (repetition)were implemented straightforwardly. To illustrate themathematical relationship between the operators we notethat R ¼ Z þ ZC, and that C�Z is the identity operator.The scaling operators D and U were both implementedin the spatial domain using a factor of two. Down-sam-

pling was done by taking the mean value of each corre-sponding group of 2 � 2 � 2 voxels, the upscale operatorU scales the matrix by a factor of two by linear interpola-tion, keeping the exact center of the matrix (being the‘‘index’’ 1/2 þ N/2) preserved under the affine transfor-mation. To optimize speed, upscaling, and croppingwere integrated in one combined operation. All methodswere implemented in MATLAB R2009b, invoking thebuilt-in forward and inverse Fourier transform, and sim-ulations were carried out on a IntelVR CoreTM2 Quad CPUof 2.50 GHz, with 8 GB RAM on a 64-bit Operatingsystem.

Validation Using Numerical Simulations

Several numerical phantoms were used to demonstratethe equivalence of vZP and cZP in the same resolution,and to quantify and investigate the minor accuracy losswhen vZP is performed in a lower resolution. Thesephantoms included spheres of tissue-like susceptibilityand a 3D checkerboard pattern to test the robustness forobjects fully occupying the FOV.

Validation by Experimental Data

An experimental phantom (Fig. 2a) was constructed con-sisting of two agar-filled spheres allowing validationwith the analytical solution. The large sphere (3% agar;diameter 77 mm) representing the ROI, was placed inthe center of an MRI-head coil fixed in a scaffold ofexpanded polystyrene. Depending on its relative posi-tion, the smaller sphere (3% agar, diameter 47 mm)induced a field shift within this ROI. A prior referencescan of this ROI was followed by five scans, in whichthe smaller sphere was consecutively placed in five dif-ferent positions in the coronal plane, at 0� (inferior tothe ROI), 22.5�,45�,67.5�, and 90� (on the right of theROI). The same FOV was chosen for the complete proce-dure, enclosing both the ROI as well as all five positionsof the smaller sphere. This was done to localize thespheres, to create the numerical susceptibility distribu-tion and to calculate the induced field by vZP and theanalytical solution. The size of the FOV (136 � 136 � 88

FIG. 2. A small sphere moves around a larger sphere inducing a varying field shift within the larger sphere, being the ROI. a: Experi-mental set-up: the ROI (dark sphere) was centrally placed in a head coil. The small sphere is repositioned in five discrete steps in the

coronal plane, covering an orbit of 90� from the inferior position toward the position in the right of the ROI. b: Same setup was used forthe application of vZP in a dynamical scheme. After calculating the unpadded FOV, the aliasing correction is calculated in a lower reso-

lution. Priority can be given in updating the unpadded FOV.

4 Bouwman and Bakker

mm) was chosen such that a spatial buffer of 5 mm waspresent on each side. The scan matrix was (136 � 136 �88), resulting in isotropic voxels of 1 � 1 � 1 mm3. Thespatial settings of the numerical computation were cho-sen accordingly. Scans were acquired on a 1.5 T clinicalscanner (Achieva 1.5T, Philips Healthcare, Best, TheNetherlands) , using an RF-spoiled coronal 3D dual-echogradient-echo sequence with a repetition time (TR) of 10ms, echo times (TE1, TE2) of 3 and 6 ms, a flip angle of10�, and one signal average, resulting in a scan durationof 2 min per position. A large readout bandwidth of 868Hz per pixel was chosen to minimize geometric distor-tion in the read direction.

Shimming was done on the central sphere, in absenceof the smaller sphere. To obtain a model for the inputsusceptibility distribution, the larger sphere was local-ized by fitting a 77 mm sphere on the magnitude imageof the reference scan, the smaller spheres were localizedsimilarly from the five consecutive scans. The relative

susceptibility was set to xagar � xair ¼ (�9.0 � þ0.36) ¼�9.36 ppm. These susceptibility models were used as aninput distribution, from which the field shift was calcu-lated, using the analytical solution, and the vZP algo-rithms. The experimental field shift was calculated by:

dexpð~rÞ ¼Dfexpð~rÞ

gB0ðTE2 � TE1Þ ; ½12�

with g being the gyromagnetic ratio (2p � 42.57 MHz/T)and the phase step Dfexp ¼ arg(s2/s1) being the argumentof the complex division of the measured signal at TE1

and TE2. The echo spacing was sufficiently small tokeep phase jumps between both echoes within (�p, p)almost everywhere except at the boundary close to theperturber; however, they could easily be unwrappedusing a straightforward region growing procedure. Alloutcomes (experimental, vZP, cZP, and the analyticalsolution) were pair wise compared.

FIG. 3. Coronal cross-sections of calculated field shift distributions and their differences, induced by a spherical numerical phantom (diam-

eter ¼ 84) of tissue-like susceptibility (dx ¼ �9.0 ppm) embedded in a cubical FOV (963). The six subtraction images were created by sub-tracting the calculated fields in the left column from those in the upper row. The three subtraction images involving the calculation without abuffer show the same aliasing pattern. Grey values of the other three subtraction images were multiplied by 10 to visualize the boundary

artifacts, caused by the discretization of the sphere. Still there is no visible difference between cZP and vZP. In the left lower corner, theroot mean squared (rms) deviation is given over the complete FOV of these subtraction images. Given here are the results of CC. Similar

results were found for KF although small Gibbs ringing artifacts were observed, which are slightly ameliorated by cZP, but not by vZP. [Colorfigure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Alias Removal in Fourier-Based Field Calculation 5

Validation by the Analytical Solution for Spheres

For numerical and experimental phantoms composed ofspheres, the calculated result was compared with the an-alytical solution. The analytical solution for a sphere ofradius R including the Lorentz correction is given by (1):

dð~rÞ ¼ðxi�xeÞR3

3� 3z2�r2

r5 if |~r| > R0 if |~r| < R

�½13�

It should be emphasized that all numerical field calcula-tions will fail close to the boundary of dichotomousspherical phantoms, due to the discrete nature of theobjects (20,21,25). Here, spheres were generated withpartial volume boundaries, to ameliorate these ‘‘simula-tion artifacts.’’

Demonstration of the Potential of vZP Implemented in aDynamical Field Shift Procedure

The dynamical field shift procedure was implemented asillustrated in Fig. 2b. A numerical 4D dynamic phantomwas generated, similar to the experimental phantomdescribed before. During the simulation, the smallersphere was assumed to move in a uniform circularmotion from the inferior position (0�) toward the posi-tion on the right (90�), in a time span of twenty seconds.For both cZP and vZP, the calculated outcome was com-pared with the analytical solution corresponding to thealready changed position of the smaller sphere. Thesetup chosen here simulates how the field shift within acertain ROI is perturbed by a moving external object,representing, for instance, a HiFU-applicator or a cryo-probe. Other examples of dynamic susceptibility distribu-tions are related to breathing, cardiac motion, the change

in susceptibility of fat during MR-guided thermal therapyor the formation of air bubbles in thermal therapy.

RESULTS

Mathematical Equivalence of Virtual and cZP

The correctness of Eqs. 10 and 11 was confirmed by allnumerical simulations in which the decoupled aliasingcalculation was performed in the original resolution. Forall phantoms, vZP led to exactly the same results as cZPwith a factor of two.

Effect of Estimating Aliasing in Half the OriginalResolution

Figure 3 shows the spatial distributions of the field shiftinduced by a numerical spherical phantom (diameter ¼84, FOV ¼ 963, and Dx ¼ �9.0 ppm) described by the an-alytical solution, and three outcomes of CC, namely,without alias prevention, with cZP (padding factor ¼ 2),and with vZP (down-sampling factor ¼ 1=2). Both vZPand cZP effectively reduced the error (rms) comparedwith the analytical solution from 0.5 to 0.04 ppm, andthis residual error is mainly due to the discrete nature ofthe phantom. However, the mean difference betweenvZP and cZP is only 0.0004 ppm, which is negligible formost practical purposes.

Figure 4 shows the robustness of low-resolution alias-ing estimation for a numerical phantom fully occupyingthe FOV. It illustrates the similarity of the estimated andthe ‘‘true’’ aliasing distribution (removed by cZP), andthe different manifestation of aliasing for CC and KF.Small deviations can be seen at the periphery of theFOV, caused by the loss of high frequencies duringdownscaling, and Gibbs ringing in the case of KF.

FIG. 4. Demonstrating the robustness of vZP for a numerical phantom fully occupying the FOV, consisting of a 3D checkerboard pat-

tern (shown in the background of the right column), for CC (a–c) and KF (d–f). The phantom is a cubical FOV (1283, dx ¼ �9.0 ppm)with cubical air cavities (323, dx ¼ 0.36 ppm). The phase images show the aliasing distribution removed by cZP (a,d) and estimated byvZP (b,e), calculated with B0 ¼ 1.5 T, Dt ¼ 40 ms. The contours (c,f) represent the difference between vZP and cZP, each contour line

represents a phase difference of 0.05 radians, corresponding with a field off-set of 0.003 ppm. Close inspection of the aliasing distribu-tions reveals subtle differences between vZP and cZP in the peripheral regions. This figure also shows that the aliasing distribution of

CC differs from that of KF. The contours of KF (f) show that vZP does not ameliorate Gibbs ringing.

6 Bouwman and Bakker

The Spatial Distribution of the Differences BetweenConventional and vZP

Since the accuracy loss of downscaling is mainly expectedto be located at the periphery of the FOV, the mean andthe maximal absolute deviation were calculated againstthe distance to the boundary of the FOV. Figure 5 showsthat small differences are seen at the border, although theyfall off rapidly to negligible values toward the center.

Experimental Validation

Figure 6 shows the measured and calculated field shift,with the perturbing sphere placed in the axial positionrelative to the ROI. Numerical calculations were per-formed with CC, both without alias correction and withvZP. For this most perturbing position, the need for zero-padding is clear: without alias correction the mean error

over the ROI is 0.42 ppm, with a maximum of 4.1 ppm.With vZP this mean error is reduced to 0.024 ppm, whichis in the order of the noise in the Dfexp images. This isconfirmed by the fact that the rms between the measure-ments and the analytical solution is also 0.024 ppm. Thedifference between vZP and the analytical solution is0.0013 ppm. These values were observed for the otherpositions as well; however, with the perturber in the diag-onal (least perturbing) position antialiasing is less needed,as the maximal error without padding is then 0.98 ppm.

The Increased Calculation Speed

Absolute calculation times of cZP and vZP (padding fac-tor ¼ 2) are given in Table 1 for 3D cubical susceptibilitydistributions, ranging from 1283 to 5123. For matrices ofthis scale, the calculation of a (2N)3-matrix should take

FIG. 6. Experimentally measured field shift (c,d) versus the field calculated with vZP (e,f) and without zero-padding (g,h), with the per-turber placed in the inferior position (a, b, c, e, and g) and in a position in 45� relative to the ROI (d, f, and h). The left-right (a) and infe-

rior-posterior (b) profile correspond to the marked lines in (c). The noisy signal is the experimental measurement, the dashed line is thecalculation field without zero padding, and the continuous smooth line is the calculated field shift with vZP (c). Comparison of (g) and (h)shows that the necessity of aliasing prevention depends on the relative position of the perturber. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

FIG. 5. Difference between the calculated field shift using conventional (cZP) and vZP, plotted against the distance to the boundary of

the FOV, as induced by (a) a numerical spherical phantom (diameter ¼ 252) of tissue-like susceptibility (dx ¼ �9 ppm) embedded in acubical FOV (2563) and (b) the checkerboard phantom of Fig. 4. The dashed lines are the mean and maximum absolute difference forCC; the continuous lines represent these measures for KF.

Alias Removal in Fourier-Based Field Calculation 7

roughly nine times longer than a N3-matrix, which fol-lows directly from the O[N3log(N3)] relation mentionedin the introduction. If the values within the tables aresignificantly higher, then the used fast Fourier trans-form-algorithm faced problems with memory allocation.The vZP algorithm for KF (factor 1=4) had no such prob-lems in processing alias-free matrices up to 5123. Justbefore such problems arise, vZP is roughly three to fourtimes faster than cZP. The tabulated values of CC do notinclude the DFT of the Green’s function, since this canbe done in advance and retrieved from memory duringsubject-specific field calculations.

Capacity to Process Larger Matrices

Figure 7 shows the accuracy if a matrix of 5123 issequentially filled with a series of spheres of increasingradius. This large FOV represents a matrix size for whichadditional cZP is not feasible anymore, due to memoryrestrictions. The figure clearly demonstrates that vZPexpands the size range of distributions for which alias-ing-free results can be obtained.

Illustration of Potential in Dynamic Field Calculations

In Fig. 8, the temporal accuracy of cZP and vZP is depictedfor the dynamic numerical phantom illustrated in Fig. 2.Figure 8a shows that the optimal conventional padding fac-tor for this phantom was about 1.25. Figure 8b shows thatthe vZP implementation could be optimized by calculatingaliasing in a resolution of 1=4 and by giving priority inupdating the internal field relative to the aliasing correction.Figure 8c compares the optimal vZP with the optimal cZPsetting. Contrary to the static case, KF performed better ondynamic phantoms, because its speed (see Table 1) becamemore decisive than its small Gibbs ringing artifacts.

DISCUSSION

It was investigated whether alias removal can be more ef-ficient than cZP in the Fourier-based calculation of themagnetic induction field. To this end, the spatial distri-bution of aliasing was estimated in a lower resolutionusing a scale model of the alias-inducing environment in

the periodic susceptibility grid, and then subtracted fromthe field that was calculated without aliasing prevention.This combines two ideas previously reported in the liter-ature: the principle of background field shift removal(28) and the observation that aliasing has a smooth spa-tial distribution, which was exploited before in the con-text of MR image reconstruction (29). To implement this‘‘virtual’’ zero-padding (vZP), distinctive algorithms werederived (see Fig. 1) for CC based on the discrete Green’sfunction (Eq. 10) and for KF using the continuousGreen’s function (Eq. 11).

It was demonstrated that vZP is practically as accurateas cZP, by several numerical simulations and an experi-mental setup mimicking conditions to be encounteredin, for example, MR-guided hyperthermia (30) or passive

FIG. 7. The effect of aliasing on the accuracy if a large FOV isfilled with a centrally embedded sphere (dx ¼ �9 ppm) with a di-ameter ranging from 0 to 100% of the FOV size. For each calcula-

tion, the rms deviation from the analytical solution is given. Thedashed lines represent the results of CC, the continuous lines

those of KF. The upper dashed and continuous lines are theresults without zero padding, the lower lines the results of vZP.[Color figure can be viewed in the online issue, which is available

at wileyonlinelibrary.com.]

Table 1Absolute and Relative Calculation Times of Conventional and vZP

Procedure Padding Down-sampling 1283 2563 5123 Relative Reduction

Circular convolution No buffer – 0.53 4.7 133 1cZP – 4.8 113 – 9a/24b

vZP 1/2 1.47 12.4 476 2.7a,b 69%a/89%b

vZP 1/4 1.14 9.2 370 2.0a,b 76%a/92%b

k-space filtering No buffer – 0.31 2.7 34.5 1

cZP – 2.82 35.4 – 9a/13b

vZP 1/2 0.97 7.7 165 3a,b 66%a/78%b

vZP 1/4 0.93 7.3 78 2.8a,b 67%a/79%b

Measured calculation times (in seconds) for Fourier-based field calculations on matrices of size 1283, 2563, and 5123, without zero-pad-ding, for cZP and vZP, both implemented with a padding factor of two. Excluded is the Fourier Transformation of the Green’s function inCC, since this can be done in advance and retrieved from memory during calculation. For vZP the speed is given for down-sampling

with factor 1/2 and 1/4. The fourth column shows the relative calculation time compared to the calculation without aliasing-prevention,the fifth column shows the reduction in calculation time if cZP is replaced by its virtual counterpart. The relative high calculation timesof cZP of matrices of size 2563 reflect that the program starts having memory problems and starts reallocating.aFor matrices of size 1283.bFor matrices of size 2563.

8 Bouwman and Bakker

shimming (31,32), where, respectively, an applicator orparamagnetic and diamagnetic objects are placed close tothe body. However, it was also demonstrated that vZP isabout three to four times faster than cZP (Table 1), andenables to process matrices up to eight times larger with-out aliasing artifacts (Fig. 7), since there is no need toreserve half of the space for aliasing prevention in allthree dimensions. Only minor accuracy loss is seen inthe periphery of the FOV (Fig. 5), caused by the loss ofthe highest frequency components in the susceptibilitydistribution during downscaling. Restricting the objectfrom the two outermost layers of voxels of the processeddistribution, will minimize this error.

The potential of vZP was demonstrated by an efficientdynamic field shift procedure, mimicking conditionswhere an external perturber continuously moves aroundthe ROI. For this type of dynamic applications, it wasshown that also—in addition to spatial smoothness—thetemporal smoothness of aliasing can be exploited, byupdating the internal field in a higher frequency thanthe aliasing correction. Even if an optimal conventionalpadding factor (1 � p � 2) is chosen, vZP improves thedynamic accuracy in real-time field calculations (Fig. 8),which renders the user exempt from choosing an optimalcZP-factor. Other fields in which the dynamic potentialof vZP may be exploited may be MR-thermometry, suchas is performed during MR-guided HIFU (13–15), hyper-thermia (30) and cryotherapy (16). Here, temporal accu-racy is important, since therapy is real-time controlledby the estimated temperature map. MR thermometry(with the PRFS method), however, is biased by anyphase shift other than the temperature dependent protonfrequency shift of 0.01 ppm per degree Celsius. Accuratereal-time modeling of field changes caused by changes inthe susceptibility distribution due to moving applicators(30), moving organs, heating of fat (15) or freezing of ma-lignant tissue (16) is, therefore, crucial to arrive atunbiased temperatures. Interestingly, the problem of per-iodic aliasing has recently also been recognized in solv-ing the inverse problem as well (10).

We have studied and compared vZP for both CC andKF. From the results, we conclude that the CC-imple-

mentation (Eq. 10) is superior for static phantoms, sincethe KF-implementation does not ameliorate its Gibbsringing artifacts. For dynamic phantoms, however, werecommend the KF-implementation (Eq. 11) since itsspeed (Table 1) makes up for its Gibbs ringing.

In summary, our findings demonstrate the feasibilityto remove efficiently the influence of the artificial peri-odic environment, expanding the spatiotemporal resolu-tion of field shift calculations. We expect this to bemostly beneficial in (1) the field of biophysical model-ing, (2) speeding up iterative processes in quantitativesusceptibility mapping, and (3) speeding up dynamicfield shift calculation in MR thermometry.

CONCLUSIONS

• vZP is more efficient than cZP, being up to fourtimes faster, with practically no accuracy loss.

• vZP allows processing matrices up to eight timeslarger than cZP.

• vZP implemented for CC (Eq. 10) is superior to KF(Eq. 11), since CC does not lead to Gibbs ringing.

• Dynamic distributions can best be processed withthe derived equations for KF (Eq. 11), since its la-tency is significantly smaller than the latency of CC.

• vZP is easily implemented using off-the-shelf Fou-rier transform algorithms; MATLAB-scripts are pub-lically available on http://www.mathworks.com/matlabcentral/fileexchange/.

ACKNOWLEDGMENT

An abstract of this work has been accepted at theISMRM conference in Melbourne (33).

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