off-resonance correction of mr images - medical imaging, ieee...

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 6, JUNE 1999 481

Off-Resonance Correction of MR ImagesHermann Schomberg,Member, IEEE

Abstract—In magnetic resonance imaging (MRI), the spatialinhomogeneity of the static magnetic field can cause degradedimages if the reconstruction is based on inverse Fourier trans-formation. This paper presents and discusses a range of fastreconstruction algorithms that attempt to avoid such degra-dation by taking the field inhomogeneity into account. Someof these algorithms are new, others are modified versions ofknown algorithms. Speed and accuracy of all these algorithmsare demonstrated using spiral MRI.

Index Terms—Conjugate phase reconstruction, magnetic reso-nance imaging, off-resonance correction, simulated phase evolu-tion and rewinding.

I. INTRODUCTION

M OST MRI methods generate their images in two steps.First, the object to be imaged is subjected to an

MRI experiment of one kind or another and then the imageis reconstructed from the outcome of this experiment. Thereconstruction algorithm is designed to invert a mathematicalmodel of the experiment. Usually, the Fourier transform istaken as the model and the reconstruction is done by a discreteversion of the inverse Fourier transform [1]–[5].

In practice, the assumed Fourier transform relationship be-tween image and data is marred by a number of imperfections,resulting in various kinds of image artifacts if the reconstruc-tion is nevertheless based on inverse Fourier transformation.One of the major imperfections is the spatial inhomogeneityof the static magnetic field, which typically causes distortedor blurred images. This paper presents and discusses a rangeof fast reconstruction algorithms that attempt to avoid suchartifacts by inverting, approximately, an extended model ofthe MRI experiment that takes the spatial inhomogeneity of thestatic magnetic field into account. Some of these algorithmsare new, others are modified versions of algorithms devised byto Noll et al. [6], [7], Man et al. [8], and Kadah and Hu [9].Speed and accuracy of all these algorithms are demonstratedusing spiral MRI.

The paper is organized as follows: Section II presents theextended model of the MRI experiment. Section III discussesinversion strategies. Sections IV and V present the reconstruc-tion algorithms. Section VI focuses on spiral MRI. Section VIIsummarizes the test results. Section VIII concludes the paperwith a few remarks.

Notations: The symbols and C denote the sets of realand complex numbers, respectively. The-fold Cartesian

Manuscript received July 28, 1997; revised October 30, 1998. The AssociateEditor responsible for coordinating the review of this paper and recommendingits publication was X. Hu.

The author is with Philips Research, 22335 Hamburg, Germany (e-mail:[email protected]).

Publisher Item Identifier S 0278-0062(99)06611-2.

product of and C is denoted by and C , respectively.An interval of the form is abbreviated by

. The notation : indicates a mapping (function)with domain and range . The concatenation of two

mappings : and : is the mapping :defined by . The partial derivative

of a differentiable function : with respect to theth variable is written as . The Hilbert space of square-

integrable functions : C is denoted by .

II. A N EXTENDED MODEL OF THE MRI EXPERIMENT

Underlying the considerations in Sections II–V is a typ-ical two-dimensional (2-D) MRI method implemented on atypical MRI system [5]. We attach a right-handed, Cartesian

-coordinate system to the main magnet such thatthe origin of this system lies in the isocenter of the mainmagnet and the -axis is parallel to the main magnetic field.Any direction perpendicular to the -axis is referred to astransverse. The generalization to three-dimensional (3-D) MRImethods is conceptually straightforward.

Prior to the MRI experiment, the object to be imaged (inmedical applications, the patient) is placed inside the bore ofthe magnet. For simplicity we assume that the slice to beimaged corresponds to the transverse plane . The spatialvariable in this plane is written as . The timevariable is denoted by. The object is supposed to stay at restduring the experiment.

The experiment itself consists of a sequence ofsubexperiments. All subexperiments have the same duration(repetition time), the same structure, and follow immediatelyupon each other. Each subexperiment, in turn, consists of anexcitation phase, a preparation phase, an acquisition phase,and a recovery phase. When imaging a transverse slice, only atransverse gradient field is used during the acquisition phase.See, e.g., [5] for more information about MRI experiments andthe numerous variations possible.

The subexperiments are designed to be time-shift invariant.So we can pretend that each subexperiment occurs in the sametime interval. We denote the beginning and the end of theacquisition phase by and , respectively, so that

(1)

is the duration of the acquisition phase (acquisition time).The (demodulated) MR signal acquired during the acquisitionphase of the th subexperiment is represented by a function

: C. Each MR signal is sampled at the timepoints

(2)

0278–0062/99$10.00 1999 IEEE

482 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 6, JUNE 1999

The outcome of the MRI experiment consists of thecomplex numbers , , .

The transverse magnetization in the plane at thebeginning of the preparation phase is independent ofand

and conveniently described by a function : C,whose real and imaginary parts correspond to the- and

-components of the transverse magnetization, respectively.The support of is contained in some region in -space,such as the square

(3)

or the disk

(4)

for some known . The magnitude of is closelyrelated to the proton density of the object, but also depends onthe MRI system and the experiment. The phase ofdependson various factors, but is often independent of the object.

By tracking the time evolution of the transverse magnetiza-tion and modeling the signal reception process one can show,similarly to the standard -space description of MRI [1]–[5],that the functions and are related according to

(5)

where : C is a weighting function, : a“frequency map,” : a path or trajectory inso-called -space, and : C an error term. Thesefunctions are further explained below.

The frequency map (also known as off-resonance mapor field map) is related to the strength of the static magneticfield according to

(6)

where is the gyromagnetic ratio of protons [MHz/T], the strength of the static magnetic field at,and the demodulation frequency of the receiver. Thespatial variation of and, hence, is due to the slightinhomogeneity of the applied static magnetic field and tothe spatially varying magnetic susceptibility of the object[10], [11]. In medical MRI, is typically belowsome 10 parts per minute (ppm) in. Moreover, variesslowly and smoothly with position, except perhaps across theboundaries between biological tissue and air [10]–[12]. Thefactor of the integrand in (5) describes the off-resonance effect resulting from the spatial inhomogeneity ofthe static magnetic field during the acquisition phase. Changing

outside the support of does not affect the value ofthe integral. We exploit this freedom to let quickly go tozero outside the support of , making sure that has abounded support. A condition of this kind is needed for somemathematical arguments at a later stage. For later purposes we

also define

(7)

(8)

The weighting function in (5) takes various system-,method-, and object-dependent factors into account. If the MRImethod under consideration does not employ apulse duringthe preparation phase, may be written as

(9)

where is a system- and method-dependent complex constant,the duration of the preparation phase (preparation time),

and a complex-valued function that is closely related tothe sensitivity pattern of the receive coil [13]. The factor

in (9) accounts for the off-resonance effect duringthe preparation phase. If the MRI method does employ apulse at time with , then the weightingfunction changes to

(10)

where the sign depends on the axis of thepulse. We regardthe function

(11)

as the image of the object that is to be reconstructed fromthe outcome of the experiment. The phase ofis usuallysmooth and the support of equals the support of .Most morphological information about the object is alreadyconveyed by .

The factor of the integrand in (5) captures theeffect of the transverse gradient field during the acquisitionphase of the th subexperiment. The trajectory is relatedto the waveform : of this gradient fieldby an equation of the form

(12)

where the initial value depends on the details ofthe preparation phase. These initial values and the gradientwaveforms are chosen such that the trajectories fill a centeredregion in space, such as the square

(13)

or the disk

(14)

for some known . [We let in analogyto .] The grid points or sampling pointsform a certain pattern in , the “sampling pattern.”

Finally, the function in (5) closes the gap between thefunction on the left and its postulated integral represen-tation on the right. Such a gap is caused by the numerousidealizations underlying the derivation of (5). For example,the derivation ignores relaxation effects, the chemical shiftbetween fat and water, the effects of eddy currents, and the

SCHOMBERG: OFF-RESONANCE CORRECTION OF MR IMAGES 483

-dependency of , , and . In addition, the MR signalsare cluttered by noise. (Yet is a function, not a randomvariable.) The error term is unknown, but expected to be small.

Omitting the term under the integral signin (5) yields the standard -space description of the MRIexperiment, but also increases the error term.

MRI methods may be classified by their trajectories andsampling patterns. In spin warp MRI [14] and its relatives,the trajectories are chosen as straight equidistantly spacedlines parallel to the -axis (for example) and the samplingpoints form a Cartesian grid in . Similar grids arisewith blipped echo planar imaging (EPI) [15]–[17]. We refer toall such methods as Cartesian MRI. In spiral MRI [18]–[20],the trajectories are chosen as interleaved archimedian spiralsstarting at the origin of -space, resulting in a spiral samplingpattern in . In radial MRI [21], [22], the trajectoriesare chosen as straight radial lines, resulting in a radial samplingpattern in .

To facilitate the reasoning about inversion strategies andreconstruction algorithms, it is advisable to extend the semi-continuous equation (5) to a fully continuous equation betweenfunctions in . As the first and major step toward thisend, we try to establish an operator in such that(5) may be rewritten as

(15)

The standard -space description of MRI suggests to decom-pose in the form

(16)

where is the 2-D Fourier transform as defined by

(17)

and is a perturbing operator that takes the off-resonanceeffect into account. An equivalent decomposition of is

where is the identity operator in.

To be able to define , we request that the MRI methodunder consideration admit a “time map”: suchthat

(18)

Such a time map exists if the trajectories do not intersect or, ifthey do, all trajectories intersecting at a point visit that point atthe same time (time shift of the subexperiments understood).Under these conditions, we can defineat the grid points

by (18) and then extend onto all of . The extensionis somewhat arbitrary, but usually there is a natural choice. Formathematical convenience at a later stage, we letquickly goto zero outside , making sure that has a bounded support.Without loss of generality we may assume that

(19)

All major MRI methods do admit a time map. Even more,many MRI methods, including the standard versions of Carte-sian, spiral, and radial MRI, have natural time maps that maybe expressed in closed form. For example, the natural timemap of spin-warp MRI is

ifelse.

(20)

In addition, the natural time maps of the standard versions ofCartesian, spiral, and radial MRI are continuous in. Othernatural time maps, in particular those of some segmentedversions of blipped EPI [16], [17] or of ring-segmented spiralMRI [23], are discontinuous in .

Assuming from now on the existence of a time map, wedefine by the formula

(21)

The bounded supports of and make a compact operatorof the Hilbert–Schmidt type [24, ch. VI]. The sumis then a linear continuous operator in and given by

(22)

It follows from (5), (18), and (22) that satisfies (15), asdesired.

As the second and minor step, we extend the MR data ontoall of -space by picking a function that satisfies

(23)

In addition, is to be well-behaved in between the samplingpoints and to vanish outside.

It now follows from (15) and (23) that

(24)

where closes the gap between and andsatisfies

(25)

Equation (24) is the wanted fully continuous companion of (5).The operator may be regarded as an extended model of theMRI experiment and will be referred to as the MR transform.Since is also compact, the Fredholm alternative holdsfor as well as for [24, ch. VI].Unlike the standard model, , the extended model dependson the MRI system and the MRI experiment (viaand )and even on the object (via). We write if we wantto emphasize this dependency. The values ofoutside haveno effect on the values of in so that our decision tolet go to zero outside does not affect the validity of themodel. Setting to zero outside is in line with our lack ofmeasurements outsideand the observation that , whichequals outside the support of, tends to zero as .

It is not strictly necessary that and have a boundedsupport. For example, if and are arbitrary real con-stants, then the function belongsto

(26)


In the standard approach to MRI, is taken as model andreconstructed by computing a discrete approximation toin , using the available samples ofin . It follows from(16) and (24) that

(27)

which shows that the accuracy of the reconstruction is affectedby the size of as well as by the size of . The termin (27) represents a continuous (as opposed to discrete) versionof the off-resonance artifact. The accuracy of the result is alsoaffected by the size of , the density of the sampling pattern in

, and the choice of the reconstruction algorithm. Henceforth,an MR image reconstructed by inverse Fourier transformationwill be called a standard image.

Whether the off-resonance artifact present in a standardimage is acceptable depends on its size and nature. As is wellknown, the off-resonance artifact manifests itself mainly asdistortion in Cartesian MRI [25] and mainly as blurring inspiral and radial MRI [7]. While a moderate distortion may betolerable, blurring is generally not.

The size of the off-resonance artifact grows with bothand. More precisely, the value of the expression equals

the number of extra turns made during the acquisition phase bythe transverse magnetization at a point with ,relative to the transverse magnetization at a point with

. This number of extra turns can be significant.For example, when T, ppm, and

ms, then .It is always possible to reduce the size of by reducing

. However, according to (12), if the shape of the trajectoriesand the size of are not to be altered, the reduction of

must be compensated for by an increase of the gradientstrength, which is possible only to a limited extent. Whenthis possibility is exhausted, one must shorten the lengths ofthe trajectories in , which means that more trajectories areneeded to cover with the required density. In other words,one is left with increasing the number of subexperiments. This,however, increases the total duration of the MRI experiment,which is generally undesirable.

Another approach for reducing the off-resonance artifactsconsists of computing a discrete approximation to

(28)

Since is compact, the spectrum of contains onlyisolated eigenvalues with at most one accumulation point at

[24, Th. VI.15]. Therefore, exists unless 1happens to be an eigenvalue of . Also, as a consequenceof the Fredholm alternative and the inverse mapping theorem,

is continuous if it exists [24]. Thus, as long as noeigenvalue of is close to 1, the term in (28)will not blow up and will be close to . Ideally, animage reconstructed in this way would have no off-resonanceartifact.

For this approach the frequency and time maps must beknown. While the time map is known at least at the gridpoints , an accurate estimate of the frequency map isdifficult to obtain. Fortunately, the -dependency of and

thus of [cf. (9)–(11)] makes it possible to extract a firstguess of from two standard images obtained with slightlydifferent preparation times [25]. The resulting frequency mapwill be degraded by the off-resonance artifacts present in thetwo standard images, but as long as these artifacts are small,a frequency map obtained in this way may yet be usable. Ifone is willing to spend the effort, one can reconstruct the twoimages again, this time based on (28) and the current guessof the frequency map, and extract an iterated frequency mapfrom the two newly reconstructed images. The process maybe continued. Nevertheless, in practice only some more orless accurate estimate of the frequency map will be available.As a result, an image reconstructed via (28) may still showartifacts caused by the imperfections of the frequency map.

Once has been estimated andreconstructed, one canalso recover using (9) and (11) or (10) and (11), asappropriate. Under a wide range of circumstances, the phaseof is independent of the object and once it has beendetermined, one can exploit (9) and (11) or (10) and (11) toextract an estimate of even from a single image.

III. I NVERTING THE MR TRANSFORM

In this section we study how one might design recon-struction algorithms that compute a discrete approximation to

.If were known explicitly, one could perhaps design

a reconstruction algorithm by discretizing the formula for. Unfortunately, is not known in closed form. An

alternative approach consists of computing a discrete versionof where is an explicitly known near inverse of .

One such near inverse is . Approximating byis akin to the standard approach to MRI. To discuss

the accuracy of this approach, we determine the point spreadfunction (PSF) of the operator , i.e., the function

: C such that

(29)

Using the explicit representations of and , one caneasily show that is given by

(30)Inserting the series expansion

(31)

into the right-hand side (RHS) of (30) and interchanging theorder of summation and integration gives

(32)

The interchange is justified by the Lebesgue dominant con-vergence theorem [24], which is applicable because, , andthe support of are bounded. Inspection of the RHS of (32)suggests that as . On the other


hand, when , then will generally be smallonly in regions where . As a result, will beclose to in regions where is small, but in regions where

is not so small, will generally not be close to .While the influence of on is local, the influence of isnonlocal and more complicated. Still we can say that unlessor are very small, is not a good approximation to .

A much better approximation to is provided by theconjugate operator given by the formula

(33)

The properties of are analogous to those of . Note thatif . The PSF of can be found like

that of and evaluates to

(34)

In contrast to , this PSF depends on the difference betweenand . As a result, will be close to in

regions where varies slowly, even if is not close to zerothere. The influence of on is still local. If varies rapidlyin some region, this will spoil the quality of the approximationonly in that region. The influence of on is the same asthat on . We refer to computing as conjugate phasereconstruction (CPR). This term was coined in [6] as an aliasfor the weighted correlation method proposed by Maedaet al.[26]. While the weighted correlation method was derived andanalyzed in discrete terms, it may also be seen as a methodfor computing a discrete approximation to by means of amatrix-vector product. However, since the matrix involved isdense and has entries when the image haspixels, the weighted correlation method is rather slow exceptfor small images.

In Section IV we shall present a range of fast CPR algo-rithms. All these algorithms make use of an approximation ofthe form

(35)

with suitable functions and . Substituting for andfor and setting

(36)

turns (35) into

supp (37)

Inserting (37) into (33) gives the approximation

(38)

where we have also used the fact that the function issensitive to the values of and only when supp(because and the influence of is local) and when

(because was set to zero outside).

A discrete version of the RHS of (38) can serve as a CPRalgorithm. Different discretization strategies for the RHS of(38) or different functions and in (35) will lead to differentCPR algorithms. Algorithms of this type can be arranged totake advantage of the fast Fourier transform (FFT), whichmakes them potentially faster than the weighted correlationmethod.

If and are arbitrary real constants, we obtain a variantof (38) by rewriting (33) as [cf. (26)]

(39)

The expression on the right may be approximated similarlyas the expression on the left, but now the approximation(35) needs to be good for and

.Another interesting near inverse of is given by

with . Computing a discreteapproximation to was proposed by Kadah andHu under the name simulated phase evolution and rewinding(SPHERE) [9]. The PSF of is analyzed in[9]. (Strictly speaking, the term SPHERE implies the usageof an estimate of obtained from two standard images inthe usual way.)

In Section V we shall present a range of fast SPHEREalgorithms. The essential step of these algorithms is thecomputation of with . The approximation(37) now leads to

(40)

which is formally similar to (38).A very accurate approximation to can be derived

from the work of Norton [27]. Assuming that and aredifferentiable, this approximation reads

(41)where

(42)

is the Jacobian of the transformation : definedby . It follows from (33),(41), and (42) that

(43)

Thus, computing can be reduced to three CPR’s. Ignoringthe derivative terms in (43) leads back to ordinary CPR.

Whenever we know a good near inverse of, we cancompute iteratively [28, Sec. 2.5]. With as nearinverse, the iteration reads

(44)

(45)

The initial step once again amounts to ordinary CPR. Eachsubsequent step requires one CPR and one evaluation of

. This term may be computed similarly as in


(40). Convergence is guaranteed if , wheredenotes the operator norm in . We refer to the

algorithm (44), (45) as CPR with post-iteration. The algorithmspresented in Sections IV and V may also be used to implementCPR with post-iteration.

A different class of algorithms for computing isobtained by discretizing the equation and solving theresulting linear systems of equations. It is also possible to solvea discrete version of the equivalent equation

, which is a Fredholm integral equation of the secondkind for [29]. Methods of this kind are described in [30]. Itseems, however, that the resulting algorithms are slower thanthe algorithms presented in this paper.

IV. FAST CPR ALGORITHMS

A. Preliminaries

From now on we assume thathas its support infor some (large) integer and some (small) number .For simplicity, is to be an “FFT-friendly” integer thatadmits an FFT algorithm. We also assume thatis a subsetof . The grid formed by the sampling pointsneed not be Cartesian and, unless stated otherwise, we assumethat it is not. We can compute a standard image on aCartesian grid in with grid spacing using thegridding method [31]–[33]. The procedure may be formulatedas

(46)

where C is the vector with components, a discrete convolution operator, an FFT-

based discrete version of , a discrete weightingfunction, and C the resulting standard image. Weregard the vectors occurring in (46) as “grid functions” andand as operators acting on grid functions. Specifically,the operator transforms , which is a discrete approximationto on the non-Cartesian grid in, into a discrete approx-imation to on a Cartesian grid inwith grid spacing where is an auxiliary windowfunction with a small support. The operator transformsthe output of this “gridding step” into a discrete approximationto on the Cartesian gridin . The grid function is a discrete approximationto on the same grid. The juxtaposition ofand in (46) is understood as pointwise multiplication.

Finally, we assume that the region is so large and thesampling pattern in so dense that the standard image isnot seriously degraded by truncation and aliasing errors. Also,the errors in the data and the off-resonance effect should be sosmall that the standard image is at least a crude approximationof .

We wish to compute a grid function C that is adiscrete approximation to the RHS of (38) (or its variant for

) on the Cartesian grid in . Anestimate of the frequency map is needed on the same Cartesiangrid. Also needed is a safe estimate of the support of .If the standard image is available and not too degraded, it

may be used to find an that is only a little larger thansupp . In the worst case, may have to bechosen. The available estimates of suppand are then usedto estimate the numbers , , and defined in (7) and (8).For simplicity, we shall use the symbols , , and alsofor the respective estimates. The time map may or may notbe known in closed form.

B. Discretization Strategies

First we discuss three discretization strategies for the RHSof (38) (or one of its variants), assuming that the integerand the functions and in (35) and (36) have already beenchosen. Each discretization strategy leads to a different classof algorithms.

The critical step is the discretization of the terms .Since the functions and are certainly known onthe non-Cartesian grid in , we may use the griddingmethod to compute . The resulting algorithms havethe general form

(47)

Here, C is a discrete version of , sampled onthe non-Cartesian grid in , and C is adiscrete version of , sampled on the Cartesian grid in

. Again, the juxtaposition of two grid functions onthe same grid in (47) is understood as pointwise multiplication.The summation in (47) is also understood pointwise. (Theseconventions also apply to similar formulas below.) Since

, the time map is notexplicitly needed. CPR algorithms based on this discretizationstrategy were proposed by Nollet al. [6], [7].

To avoid all but one of the time consuming gridding steps,Man et al. [8] suggested replacing in (47) by ,where C is a discrete version of sampled on theCartesian grid in . The resulting algorithms have thegeneral form

(48)

(49)

It is now necessary to evaluate on the Cartesiangrid in . This is no problem if is known in closedform. Otherwise, some form of interpolation must be used toresample or from the non-Cartesian grid to the Cartesiangrid in . If the time map is discontinuous in, thisinterpolation may incur an extra error.

The replacement of by creates some error,too. As the gridding step is, in essence, a discrete versionof a convolution with a narrow window function, this erroris negligible when is smooth. However, is not alwayssmooth. For example, the natural time maps of spiral andradial MRI (cf. Section VI) are not differentiable atand this behavior carries over to. Some time maps are evendiscontinuous in .


To avoid this extra error, we suggest resamplingonto theCartesian grid in by first computing via (46) andthen

(50)

(51)

In (50), represents an FFT-based discrete version of.Together, (46) and (50) effect a resampling of the MR datafrom the non-Cartesian grid to the Cartesian grid in[33]. The general algorithm (50), (51) acts on the standardimage and involves only Cartesian grids. When combinedwith (46), it requires two 2-D FFT’s more than the generalalgorithm (48), (49). In many situations, the standard image isalready available or needs to be computed anyway, and thenthe extra cost reduces to a single 2-D FFT. Ifis continuous in

, then will also be continuous there and the resamplingof via (46) and (50) will be accurate. Conversely, ifisdiscontinuous in , then will also be discontinuous thereand the resampling may of be inaccurate.

Discontinuous time maps (on non-Cartesian grids) remaindifficult to cope with. In the case of ring-segmented spiral MRIor other segmented non-Cartesian MRI methods, one can stillapply an instantiation of one of the general algorithms (48) and(49) or (50) and (51) to the data in each segment separatelyand add the results. If there are many segments, algorithmsbased on (47) may be faster.

C. The Algorithms

We now discuss possible choices for the functionsandin (35). In principle, each choice can be combined with any ofthe three discretization strategies described in Section IV-B.The CPR algorithms presented in this paper, however, are allbased on (50) and (51).

In practice, one also needs some way of estimating thenumber of terms, , from the data. For the moment, we suggestchoosing via a parameter in the form

(52)

the hope being that might become an algorithm-dependentconstant.

1) Discrete Frequency Exponential Approximation:A nat-ural choice for the functions , proposed by Nollet al. [7],are the discrete frequency exponentials

(53)

The frequencies and the companion functions are yet tobe determined.

One of the simplest choices for and , also indicatedin [7], is

(54)

if

else.(55)

The associated functions are discontinuous anddefine a partitioning of into the disjoint segments

(56)

The effect of the choice (55) is a sort of nearest neighbor in-terpolation in (51). The algorithm (50), (51) with as in (53),

as in (54), and as in (55), will be referred to as algorithmCPR-DFE-NN. (DFE stands for discrete frequency exponentialapproximation, NN for nearest neighbor interpolation.)

The image quality offered by algorithm CPR-DFE-NN maybe hampered by the discontinuity of the functions. (SpiralMRI is an important exception, see Sections VI and VII). Thefollowing choice, due to Manet al. [8], avoids this problem.Let and be two parameters subject to

(57)

Also let

(58)

(59)Similarly, let

(60)

(61)Then for each define : C by

(62)

where : is a window function such as

if

if

if .

(63)

Note that when . For eachwe wish to approximate in by a trigono-

metric polynomial of the form

(64)

Suitable coefficients can be found by trigonometricinterpolation, i.e., by requesting that

(65)

In view of (57)–(62) and (64), this may be rewritten as

(66)


The coefficients are then given by [34]

(67)

In practice, one calculates the coefficients for a finelyspaced set of frequencies and uses interpolationto find them at intermediate values of. It can be deducedfrom (67) that

(68)

(69)

It therefore suffices to compute only for the in oneof the frequency bins . Unless happens to be FFT-friendly, the sum (67) cannot be evaluated via an FFT but, inpractice, is so small that a direct evaluation via (67) is fairlyfast. To achieve a good approximation in (35) with a small

, the choice of and is both essential and critical (cf.Section VII). The algorithm (50), (51), with as in (53),as in (59), and as in (67), will be referred to as CPR-DFE-T.(T stands for trigonometric interpolation.)

2) Discrete Time Exponential Approximation:We obtaindual algorithms if we interchange the roles ofand . Thefunctions are then discrete time exponentials,

(70)

and the times and the companion functions are stillto be determined. If we choose them in analogy to (54)and (55), we obtain algorithm CPR-DTE-NN (DTE standsfor discrete time exponential approximation.) A versionof algorithm CPR-DTE-NN based on (47) was proposedby Noll et al. [6]. The analogs of the choices (57)–(67)lead to algorithm CPR-DTE-T.

3) Polynomial Approximation:It was pointed out in [8]that the functions might also be chosen as polynomials in,although suitable companion functionswere not exhibited.It was also demonstrated that discrete frequency exponentialsare nearly optimal. Nevertheless, polynomial approximationcan lead to an attractive CPR algorithm, as we shall now show.

The algorithm to be presented turns out to be fastest when itworks with centered versions of and . These are defined by

(71)

with and as in (58) and (60), respectively. With thefurther definitions

(72)

we have

(73)

as a special case of (39). Due to the centering, we further haveif and if

. As a result, we now need to find functionssuch that

(74)

We propose to let with real coefficientsand yet to be determined. This choice makes the

function a polynomial in the single variable .Moreover, due to the centering, when

and . So now we seek realcoefficients and such that

(75)

To find such coefficients, we determine the real polynomialthat interpolates the function

at the zeroes of the transformed Chebyshev poly-nomial in the intervalwhere is the standard Chebyshev polynomial of degreeand . The polynomialis easy to compute and affords nearly the same accuracyin as the minimax polynomial of the samedegree [28, Secs. 5.8–5.10]. The coefficientsare foundsimilarly by interpolating . Since the interval

is centered and the function is even,the coefficients are zero when is odd. Similarly, thecoefficients are zero when is even. The number of termsrequired to achieve a prescribed accuracy in (75) increaseswith the size of the interval in which the approximation mustbe good. Centering and minimizes the size of this intervaland also centers it.

The resulting algorithm may be stated as follows:

(76)

(77)

(78)

(79)

Here, , , , and are discrete versions of thefunctions , , , and , respectively, sampled on thepertinent Cartesian grids. The algorithm (76)–(79) with theabove choice of the coefficients will be referred to as algorithmCPR-P (P stands for polynomial approximation).

For large values of and , the summations and multipli-cations in (77) and (78) involve numbers of grossly differentorders of magnitude. In such cases, algorithm CPR-P maysuffer from a loss of accuracy due to rounding errors.

Interchanging the roles of and in (74) does not lead toa new algorithm.

D. Miscellaneous Remarks

The accuracy of the above algorithms depends on theaccuracy of the approximation in (35) or (75). This accuracy,in turn, depends on the choice of the functionsand andwill generally increase with . The number of terms necessaryto achieve a prescribed accuracy will also grow with .Different algorithms may well produce different artifacts. Theaccuracy of the resulting images will also be affected by theaccuracy of the available frequency map.


The computational effort for computing one of the termsin the sums (51) or (77) and (78) is dominated by the effortfor computing a 2-D complex FFT. Thus, the computationalcomplexity of these algorithms is mainly determined by thenumber of terms required for the desired accuracy. The op-eration count may be reduced a little by means of a binarymask indicating the region . Such a mask is also useful tosuppress artifacts that may arise when the estimated frequencymap is grossly false and nonzero outside.

In principle, the CPR algorithms presented in Section IV-C may be applied to Cartesian MRI as well. The initialresampling of the MR data onto a Cartesian grid inis then unnecessary. Also, there is no need to resample thetime map when it is not known in closed form. The standardimage will be distorted and, if the frequency map is obtainedin the usual way, it will be distorted as well. Algorithm CPR-DFE-NN is not usable with Cartesian MRI, as it shifts thesegments defined in (56) by different amounts, thus leavingthe boundaries between the segments visible. The fact that thetime map of a Cartesian MRI method depends only on(or ), may be used to speed up the computation of the 2-DFFT’s involved.

V. FAST SPHERE ALGORITHMS

Every CPR algorithm presented in Section IV-C has aSPHERE counterpart. The general form of these algorithmsis [cf. (40)]

(80)

(81)

Like the CPR algorithms of Section IV-C, these algorithms acton and work with Cartesian grids throughout. The functions

and may be chosen in the same way as in Section IV-C. Specifically, if , , and are chosen as in (53)–(55),respectively, we obtain algorithm SPH-DFE-NN, the SPHEREcounterpart of CPR-DFE-NN. Algorithms SPH-DFE-T, SPH-DTE-NN, SPH-DTE-T, and SPH-P arise similarly. AlgorithmSPH-DFE-NN leaves the boundaries between the segmentsdefined in (56) visible and is not usable. Algorithm SPH-DTE-NN was first described in [9].

Assuming the same number of terms, the computationalcomplexity of all these SPHERE algorithms is similar to that oftheir CPR counterparts. Most remarks made in Section IV-Dcarry over.

After a change of the sign of and without the final inverseFourier transformation, these SPHERE algorithms are alsosuited to compute the term in (45).

VI. A PPLICATION TO SPIRAL MRI

The trajectories of spiral MRI [18]–[20] are interleavedarchimedian spirals covering the disk . The trajec-tories may be written as

(82)

Fig. 1. (a) The function � defined in (85) and its inverse, �1�

. In thisexample,� = 0:125. (b) The associated time map, as defined in (84).

where : C is defined by

(83)

and : is a smooth monotonically increasingfunction with and . The natural time map is

ifelse.

(84)

A good choice for is [5, Sec. 3.6]

(85)

in which case the sampling density in is adequatewhen . Fig. 1(a) illustrates the functionsand for . Fig. 1(b) shows the resulting naturaltime map.

The functions and are real and rotationally sym-metric. In regions where varies slowly, the rotational sym-metry of shapes the PSF (32) of in a waythat makes a blurred version of , where the amountof blur near grows with and . The blurring of thestandard image makes it possible to obtain a good yet safeestimate of supp.

The algorithms described in Sections IV and V may be usedto deblur standard spiral MRI images. Despite the discontinuityof its coefficient functions, algorithm CPR-DFE-NN is appli-cable to spiral MRI, because the small amount of remainingblur tends to obscure the boundaries between the segmentsdefined in (56).

Furthermore, algorithm CPR-P may be speeded up consid-erably, provided and the phase of are reasonably smooth(which they usually are). We can formulate the requirementfor as

(86)

where the functions , , , and are real and

(87)

for some low-pass filter , such as

(88)


For the fast version of algorithm CPR-P, let, , , , andbe defined as in Section IV-C-3. The algorithm exploits

the fact that the function

(89)

is computable from and satisfies

(90)

We first show that is computable from . Sincenear the origin of space, we have

(91)

(92)

(93)

So is computable from . All other terms in (89)are either known or also computable.

To demonstrate (90), we note that

(94)

(95)

(96)

(97)

(98)

The crucial step here is the transition from (95) to (96),which is based on an interchange of the order of the mul-tiplication with and the application of the operator

. The error resulting from this interchange is smallbecause is smooth by assumption and the PSF of

(99)

(100)

is concentrated where .Now we show that can be computed efficiently. Like

any other complex-valued function, admits the representa-tion

(101)

with the Hermitian functions

(102)

Exactly as described in Section IV-C3, we can find an integerand real coefficients , such that when is

odd, when is even, and

(103)

Substituting (101) into the RHS of (103) gives

(104)

with

(105)

(106)

The functions and are real because the coefficientsand are real, is real, and are Hermitian, and

is real and rotationally symmetric (and hence Hermitian).Since must be nearly real, and must be nearlyzero and

(107)

Combining all these observations leads to the followingalgorithm for computing a discrete approximationtofrom :

(108)

(109)

(110)

(111)

(112)

(113)

(114)

(115)

Here, and are discrete versions of and ,respectively, is an FFT-based discrete version offorreal input data and Hermitian output data [28], andis an FFT-based discrete version of for Hermitian inputdata and real output data. In (110), denotes the complexconjugate of , denotes pointwise magnitude, isan approximation to , and is an approximation to

. We refer to this variant of algorithm CPR-P as CPR-P-r,as it relies on the rotational symmetry of the time map.

The advantage of algorithm CPR-P-r over the originalversion is that can be made nearly two times faster than

. Part of this gain, however, is lost by the computationaleffort for (109)–(112). Due to the additional approximationsinvolved, algorithm CPR-P-r may not be quite as accurateas algorithm CPR-P. It has been found experimentally thatthe range of acquisition times for which algorithm CPR-P-rprovides good results can be extended by subtractingfrom the in (110) and adding to the in (115).

To some extent, it is possible to deblur spiral MRI im-ages without explicit knowledge of a frequency map. Thestarting point for such methods is algorithm CPR-DFE-NN.The functions required for this algorithm are not knowninitially, but can be guessed by an iterative procedure basedon the assumption that in (86) should be real. Deblurringalgorithms of this kind are described in [7] and [35].


Fig. 2. (a) and (e) Standard spiral MR images of a phantom. (b) and (f) Estimated field maps. (c) and (g) Best possible CPR images. (d) and (h) Bestpossible SPHERE images. The associated acquisition times are 20 ms (top row) and 40 ms (bottom row).

Algorithm CPR-P-r and the mentioned deblurring algo-rithms are also applicable to radial MRI. The natural timemap of radial MRI is given by (84) with .

VII. T EST RESULTS

The various reconstruction algorithms presented inSections IV–VI were tested on standard images of a phantom.The phantom consisted of a cylindrical plastic containerfilled with NaCl- and CuSO-doped water. A 2-D array ofthin plastic rods served to make “holes” in the water. Across-sectional slice of this phantom was imaged in a seriesof MRI experiments. The MRI system was a 1.5 T PhilipsGyroscan NT equipped with shielded gradient coils. Thephantom was positioned in the center of the bore with itslongitudinal axis parallel to the axis of the magnet. The MRdata were acquired with an ordinary spiral pulse sequence,using a standard excitation pulse and nopulse during thepreparation phase. In all cases, the flip angle of the excitationpulse was 45, the slice thickness 8 mm, and the repetitiontime 100 ms. The acquisition time,, was either 10, 20, 40,or 60 ms. The number of spirals, , was set to 36, 18, 9,or 6, respectively. The spirals were chosen as described inSection VI with mm, , and . In allcases, the maximum gradient strength and the maximal slewrate were 16.40 mT/m and 31.48 mT/m/ms, respectively. Thenumber of samples per acquisition,, was set to inall cases. For each acquisition time, two MRI experimentswith preparation times ms and ms were made.

Using a phantom has the advantage that its true image isknown. The above experimental setup also ensures that theerror term in (5) is relatively small. As the true frequencymap is the same in all cases, the size of the off-resonanceeffect is solely determined by the size of.

All postprocessing of the MR data was done on a SunMicrosystems Sparcstation 20. The computer programs werewritten in C, using single precision floating point arithmetic

throughout. Computation times were measured by calling thestandard C-library function clock() at the beginning and theend of the program. The resulting computation times includethe overhead caused by the operating system. A 2-D 256complex FFT takes about 0.5 s on this platform.

In all cases, the image size was and the pixelsize mm. Standard images of the phantom werereconstructed using an adaptation of the general griddingmethod described in [33] to spiral MRI. The implementationachieves a reconstruction time of about 3.7 s per image.Fig. 2(a) and (e) shows the standard images at ms andat ms, respectively. As expected, the amount of blurgrows with . At ms there is also some distortion nearthe top of the phantom. The standard images at msand ms continue the trend in the respective directionsand are not shown here. To obtain a sharp standard image ofthis phantom with this MRI system, an acquisition time of5 ms or less is required.

Frequency maps were extracted from the two standard im-ages acquired at each acquisition time. The procedure involvesa phase unwrapping step, a masking step to set the frequencymap to zero outside the estimated support of the image, anda 2-D median filter. The computation of a frequency maptakes about 2.4 s, excluding the reconstruction time for thetwo standard images. Fig. 2(b) shows the estimated frequencymap at ms. The minimum and maximum frequencieswithin the estimated support of the image are

Hz and Hz, resulting in a frequencyspan of Hz. The estimated frequency map at

ms is shown in Fig. 2(f) and has a frequency spanof Hz. The two frequency maps are almostidentical inside the phantom but differ near its boundary.Again, the frequency maps estimated at ms and

ms continue the trend in the respective directionsand are not shown here.

For each acquisition time, the standard image obtained with


a preparation time of 2 ms was postprocessed with each ofthe CPR algorithms presented in Sections IV and VI, usingthe frequency map associated with the same acquisition time.Here is a summary of the findings.

As expected, the images produced by the various CPRalgorithms become better as increases. At each acquisitiontime, there is even a best possible CPR image that most ofthese algorithms are able to produce if onlyis large enough.At ms and ms, the best possible CPR imagesare practically free of blur and distortion. At ms and

ms, some blur and distortion remains near the topand the bottom of the phantom. Fig. 2(c) and (g) shows thebest possible CPR images at ms and at ms,respectively. As usual, the best possible CPR images atms and ms continue the trend in the respectivedirections. However, at ms the remaining artifactsnear the top of the phantom are so large that one is inclinedto call them unacceptable.

On the other hand, there are substantial differences betweenthe various CPR algorithms with respect to their behavior as

grows from small to large. For example, algorithm CPR-DFE-NN, when is small, produces images that are stilla bit blurred. As increases, the images converge slowlybut steadily toward the best possible CPR image. AlgorithmCPR-P exhibits a totally different behavior. Whenis belowsome threshold value (which depends on and presumablyon ), the images produced by this algorithm show low-frequency wavy artifacts. As grows beyond the threshold

, these artifacts disappear completely and the resultingimages converge very rapidly to the best possible CPR image.Algorithm CPR-DFE-T exhibits a similar but less pronouncedthreshold behavior. The proper choice of the parametersand in (57)–(61) is critical for the performance of thisalgorithm. As and are related via (57), it suffices tochoose only one of them. A good recipe, found experimentally,is to set where is either or

and chosen such that is odd. (Amongother things, this choice ensures thatand are among the

.) Algorithm CPR-DTE-NN requires extremely many termsto produce images that come close to the best possible CPRimage. Algorithm CPR-DTE-T behaves roughly like algorithmCPR-DFE-T. However, a simple rule for deriving good choicesof the parameters or from or could not be found.Finally, algorithm CPR-P-r produces good magnitude imagesat ms and ms, but not at ms and

ms. Fig. 3(a) and (b) shows the best possible imagesproduced by this algorithm at ms and at ms,respectively. The choice of the parameter in (88) is notcritical: a good choice is .

The four SPHERE algorithms SPH-DFE-T, SPH-DTE-NN,SPH-DTE-T, and SPH-P were also tested. Again, there isa best possible SPHERE image for each acquisition time.However, except at ms, the image quality of thebest possible SPHERE images is noticeably inferior to that ofthe best CPR images obtained at the same. Fig. 2(d) and (h)shows the best possible SPHERE images at ms and at

ms, respectively. Otherwise, the SPHERE algorithmsbehave roughly like their CPR counterparts.

In practice, one wants to compute a reasonably good CPR or

Fig. 3. (a) Best possible image produced by algorithm CPR-P-r atT = 20

ms. (b) Best possible image produced by algorithm CPR-P-r atT = 40 ms.(c) Iterated frequency map atT = 40 ms. (d) Best possible CPR image atT = 40 ms, recomputed with the iterated frequency map shown in (c).

SPHERE image as fast as possible. Thus, for each algorithmand acquisition time, an attempt was made to find out thenumber of terms required for a reasonably good CPR orSPHERE image, respectively. Table I lists the results for eachacquisition time and for each algorithm that could produce areasonably good image at this acquisition time. The columnslabeled list the number of terms required for the reasonablygood image. The associated computation times are listed in theadjacent columns. The goodness of an imagewas measuredby the relative error where denotespointwise magnitude, denotes the Euclidean norm inC , and is the pertinent best possible CPR or SPHEREimage. Table I also lists the relative error in each case. Thebest possible CPR and SPHERE images were computed byalgorithms CPR-DFE-T and SPH-DFE-T, respectively, using avery large number of terms. As the above measure of goodnessis sometimes misleading, it was combined with a subjectivejudgment based on visual inspection. The step behavior ofthe P-class algorithms makes it easy to identify the numberof terms required for a reasonably good CPR image. For theT-class algorithms, which have a less marked step behavior,this number of terms is less well defined and remains a littleambiguous. For the table, the smallest possible number wasselected. For the best image quality, one or two more termsare needed. For the two DFE-T algorithms the parameterwas chosen as described above. For the two DTE-T algorithmsgood values for were determined by trial and error. Forthe NN-class algorithms, which exhibit no step behavior atall, the number of terms was chosen via (52) withfor algorithm CPR-DFE-NN and for the two DTE-NN algorithms. These choices provide a good compromisebetween image quality and computation time.

At ms and ms, the table contains the resultsfor the three algorithms CPR-DFE-NN, CPR-DFE-T, and


TABLE ITEST RESULTS

CPR-P only. At these acquisition times, algorithm CPR-P-rand all SPHERE algorithms were unable to produce acceptableimages at all, algorithm CPR-DTE-NN was definitely too slow,and algorithm CPR-DTE-T was discarded because it lacksa known practicable way to determine good values for itsparameters.

With respect to image quality per number of terms, algo-rithms CPR-DFE-NN, CPR-DFE-T, and CPR-P are clearly thebest among the tested CPR algorithms. Algorithm CPR-P-r isthe fastest, but restricted to images with small and mediumsize off-resonance artifacts. At ms, algorithm CPR-Pbegins to suffer from rounding errors, but in this regime theCPR approach, as such, is already of limited value. Amongthe SPHERE algorithms, algorithm SPH-P is the clear winner.

The table also lists the quantity , which is themaximum value of the parameterthat would produce via(52). The results suggest that might be chosen as in (52),where depends on the algorithm and is otherwise constant ora slowly varying function of that may be predeterminedand tabulated.

An attempt was made to improve the best possible CPRimage at ms by using an iterated frequency map, asindicated at the end of Section II. Fig. 3(c) and (d) shows theiterated frequency map and the recomputed best possible CPRimage, respectively. The distortion near the top of the phantomhas disappeared, but some ripples remain.

CPR with post-iteration was also tried, but did not yieldsatisfactory results. In hindsight, this may be understood. Theiteration attempts to change the image at places where CPR byitself is not very accurate, but these are also the places wherethe frequency map is likely to be inaccurate.

The better of the CPR algorithms were also applied to astandard spiral MR image of a transverse cross section of ahuman brain. To obtain this image, the excitation phase of thepulse sequence was augmented by a standard fat suppressiontechnique [36]. The flip angle of the excitation pulse was now60 , the slice thickness 7 mm, the acquisition time ms,and the repetition time 1000 ms. The number of spirals was

Fig. 4. (a) Standard spiral MR image of a human brain obtained atT = 20

ms. (b) Associated frequency map. (c) Best possible CPR image. (d) Bestpossible SPHERE image. (e) Iterated frequency map. (f) Best possible CPRimage recomputed with the iterated frequency map shown in (e).

set to , resulting in a maximum gradient strength ofabout 20 mT/m. Again, the image size was andthe pixel size mm. The standard image is shown inFig. 4(a). The associated estimated frequency map is shownin Fig. 4(b). Its frequency span is Hz so that

. The variation of the frequency map in and


near the nasal cavities is extreme. The best possible CPRand SPHERE images associated with this frequency map areshown in Fig. 4(c) and (d), respectively. The improvement ofthe CPR image near the eyes and at the back of the headis noticeable. Again, the appearance of the SPHERE imageis less satisfactory than that of the CPR image. Both imageshave also changed significantly in and near the nasal cavities,but given the strong variation of the frequency map there,the accuracy of the images in this region may be questioned.Fig. 4(e) and (f) shows the iterated frequency map and therecomputed best possible CPR image, respectively. The imagehas changed in and near the nasal cavities, but its accuracythere remains doubtful.

VIII. C ONCLUDING REMARKS

The results reported in this paper suggest that the approx-imate inversion of the MR transform via CPR is a viablemethod to remove off-resonance artifacts from standard spiralMR images. The approach allows one to reduce the numberof spirals by a factor between four to eight and still obtainsharp spiral MR images.

The three CPR algorithms CPR-DFE-NN, CPR-DFE-T,and CPR-P can be recommended. Algorithm CPR-P-r isappropriate when speed counts most.

The results also suggest that the approximate inversionof the MR transform by SPHERE is less accurate than anapproximate inversion by CPR, at least in the case of spiralMRI. For Cartesian MRI the situation may be different, asSPHERE may be able to make a better use of distortedfrequency maps [9].

The recommendable CPR algorithms are so fast that theymay be executed in real time on a suitable embedded parallelcomputer comprised of a small number of state-of-the-artmicroprocessors or digital signal processors.

The required frequency map can be estimated from twostandard images in the usual way. In favorable cases, itis conceivable to extract the frequency map from a singlestandard image. Improved frequency maps may be obtainediteratively, although the computational cost is high. Theoreti-cally, the approximation underlying the CPR approach couldbe overcome in various ways, but in practice the difficulty ofobtaining an accurate frequency map limits the value of suchattempts.

ACKNOWLEDGMENT

The author thanks B. Aldefeld for stimulating discussioins,P. Bornert for providing the MR data, and M. Fuderer and theanonymous reviewers for their comments and suggestions.

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