Partial k-Space Recconstruction

John Pauly

September 29, 2005

1 Motivation for Partial k-SpaceReconstruction

In theory, most MRI images depict the spin den-sity as a function of position, and hence should bereal valued. If this were true, then by the sym-metry of the Fourier transform, only half of thespatial-frequency data will need to be collected.Since real functions have conjugate symmetry inspatial frequency space, the uncollected data couldbe synthesized by reflecting conjugate data acrossthe origin. Unfortunately, there are many sourcesof phase errors that cause the real-valued assump-tion to be violated. These include variations inthe resonance frequency, flow, and motion. As aresult, partial k-space reconstructions always re-quire some type of phase correction, to correct forthese sources of incidental phase variation. Thisallows real images to be reconstructed.

An example of a gradient-recalled axial head imageshown in Fig. 1 illustrates the problem. The mag-nitude reconstruction of a full k-space acquisitionis shown in Fig. 1a and, the phase in Fig. 1b. Thelinear component of the phase has been corrected,leaving only the non-linear components. The abso-lute values of the real and imaginary componentsare shown in Fig. 1c and d. Clearly, a significantamount of phase correction is required before theconjugate phase symmetry can be exploited.

The reason for this phase is that the precessionfrequency varies across the head. The image phase

a) Magnitude b) Phase

c) abs(Real) d) abs(Imag)

Figure 1: Axial gradient echo image acquired at 0.5Twith an echo time of 13.8 ms. At this time water andfat are rephased. The linear shim terms have been cor-rected, leaving the non-linear components due to sus-ceptabilty shifts.

is approximately

φ(x, y) = ω(x, y)TE (1)

where ω(x, y) is the local resonant frequency inrad/s, and TE is the echo time. This can be due toinhomogeneity of the main magnet. These changesin frequency vary slowly with spatial position, andcan in theory be calibrated out of the system withproper shimming. More fundamental, and moreproblematic, is the variation in frequency due to

1

kx

kyky,max

-ky,max

ky,s

-ky,s

SymmetricallySampled

AsymmetricallySampled

Figure 2: Partial k-space acquisition for reducing scantime by reducing the number of phase encodes required.

the magnetic susceptibility difference between tis-sue and air. At air-tissue interfaces it is commonto see local frequency shifts of several parts-per-million (ppm). These are seen in these images inthe brain tissue adjacent to the ears, which is di-rectly above the auditory canal, and around thenasal passages. An additional problem is thatthese shifts can occur over relatively short dis-tances, and this governs the amount of resolutionrequired for phase compensation, the amount ofcoverage in k-space that will be required, and ulti-mately how “partial” a partial k-space acquisitioncan be.

1.1 2DFT Applications

There are two general applications of 2DFT imag-ing where it is desirable to collect only a fraction ofthe full k-space data. The first is for reducing scantime by reducing the number of acquisitions thatare required to construct an image of a given reso-lution. This is illustrated in Fig. 2. Slightly morethan half of the complete k-space data is collected,allowing the scan time to be reduced by almost a

R F

G z

G y

G x

s(t)

kx

kya) b)

Figure 3: Pulse sequence with a reduced echo time,resulting in a partial echo readout.

kx

ky

kx,max-kx,max kx,s-kx,s

SymmetricallySampled

AsymmetricallySampled

Figure 4: Partial k-space acquisition for reducing echotimes by collecting only a fraction of the full echo.

factor of two.

The second application is for reducing echo times.Here the area of the readout dephasor gradientis reduced so that the echo comes earlier in thereadout window as is shown in Fig. 3. This canbe important for reducing flow dependent dephas-ing, and through plane susceptibility-induced sig-nal loss. This case is illustrated in Fig. 4.

2

1.2 Applications to Other k-Space Ac-quisition Methods

Partial k-space acquisitions are also of interest formost other k-space acquisition methods. For EPIa partial k-space acquisition reduces the echo time,which can be quite long for a fully symmetric ac-quisition. Reconstruction of this type of partialk-space EPI data is very similar to the reducedphase-encode 2DFT case.

Other trajectories are more interesting. Projec-tion acquisitions can exploit partial k-space sym-metries, as can spiral acquisitions. The proper wayto reconstruct these data sets is still an open re-search question. We will briefly describe the issuesand possible solutions below.

1.3 Approaches to 2DFT Partial k-Space Reconstruction

We are going to consider two different types of ap-proaches to partial k-space reconstruction. Firstare direct methods. These operate by constructinga real image in a single pass. One example is thehomodyne algorithm [1]. These methods have lim-itations due to the interaction of phase compensa-tion and synthesis of the conjugate data. The sec-ond type of approach addresses these limitationsby using an iterative algorithm. One example ofan iterative algorithm is called POCS for “projec-tion onto convex sets” [2-4]. This operates by it-eratively synthesizing the missing data that wouldbe consistent with the data that was collected. Westart with the direct algorithms, describe their lim-itations, and then move on to iterative algorithms.

2 Direct Partial k-Space Recon-struction

2.1 Trivial Reconstruction by ZeroPadding

The simplest way to reconstruct a partial k-spacedata set is to simply fill the uncollected data(phase-encodes or readout samples) with zeroes.Then, perform the 2DFT and display the magni-tude. This works acceptably if the collected k-space fraction is close to 1, and works poorly asthis fraction approaches 0.5. This is illustrated inFig. 5 for a k-space fraction of 9/16ths. The recon-struction of the full k-space data is shown in (a),and the reconstruction of the zero-padded partialk-space data in (b). The result is significant blur-ring in the phase-encode direction. Clearly thisis unacceptable, and this motivates the search forother solutions.

The reason for the blurring can be identified byconsidering the data set to be the product of a fullk-space data set multiplied by a weighting func-tion. In this case an offset step function, wherethe offset corresponds to the k-space fraction. Thiswill be denoted W (ky), and is illustrated in Fig. 6.The inverse Fourier transform of this function isthe impulse response that produces the blurring.If we look at the real component, we see a sharpimpulse at the desired resolution plus a broadercomponent that corresponds to the width of thesymmetrically acquired data. There is also a sig-nificant undesired imaginary component.

2.2 Phase Correction and ConjugateSynthesis

In order to correct for the blurring from the trivialreconstruction we need to fill in the missing uncol-lected data. From Fig. 1 it is clear that in gen-

3

a)

b)

Figure 5: Comparison of a reconstruction of a full k-space data, and a trivial partial k-space reconstructionof the same data set where only 144 of 256 phase en-codes have been used, and the remaining 112 have beenreplaced by zeros. Note the significant blurring in thephase-encode (left-right) direction.

eral phase correction must be applied before thek-space symmetry can be exploited to synthesizethe missing data. In order to do the phase correc-tion, we will use the narrow strip of data for whichwe have symmetric coverage. The phase of thislow resolution image is then used to phase correctthe partial k-space data. After inverse transform-ing the phase correction, the image reconstructedfrom the partial k-space data is transformed back

W(ky)

ky

y

y

y

abs{w(y)}

real{w(y)}

imag{w(y)}

Figure 6: A k-space weighting function W (ky)thattruncates a full k-space data set into a partial k-spacedata set. In this case the k-space fraction is 9/16ths, asin Fig. 5. The blurring evident is Fig. 5 is due to theconvolution of the inverse transform of W (ky), which isplotted here in absolute value, and real and imaginarycomponents.

to the spatial frequency domain, where the datacorresponding to the missing data is synthesizedby conjugate symmetry,

M(kx, ky) = M∗(−kx,−ky). (2)

This process is illustrated in Fig. 7. The partialk-space data is Mpk(kx, ky), Ms(kx, ky) is the nar-row strip of symmetric data, and mpk(x, y) andms(x, y) are the corresponding images producedby a inverse Fourier transform. The phase cor-rection function is a unit amplitude image with aphase that is the conjugate of mx(x, y),

p∗(x, y) = e−i6 ms(x,y). (3)

The problem with this approach is due to the ef-fects of the phase compenstation step near theboundary of the acquired data. The multiplica-tion by the phase compensation function in theimage domain is a convolution in the frequency

4

domain, and the size of this convolution functioncan be significant. The fact that the convolutionis operating on zero data for the uncollected phaseencodes produces errors near the boundary. Be-low we will describe a conjugate synthesis methodbased on k-space weighting that reduces this prob-lem, and present some examples of reconstructedimages.

Before proceeding, it is interesting to considerwhat sorts of features we are likely to loose ifwe rely on a phase correction function with lim-ited spatial resolution. Figure 8 compares a fullk-space reconstruction mf (x, y), left, with thereal part of the phase corrected full-k-space dataRe{mf (x, y)p∗(x, y)}. The difference image isshown on the right. This shows which featureswill tend to be lost in a partial k-space reconstruc-tion. One of the main effects is loss of vessel signal.This is to be expected since the vessels are toosmall to be resolved by the phase compensationfunction, and because motion through the slice se-lect and imaging gradients produces velocity de-pendent phase shifts. The other areas of signalloss are near the air-tissue boundaries such as thesinuses.

The difference image has interesting noise charac-teristics. The backround in the phase correctedimage has lower noise because one component ofthe complex noise has been supressed. Where theimage has significant signal, the difference betweenthe two images is very close to zero. The reasonfor this is illustrated in Fig. 9, which shows a vec-tor diagram for the sum of two complex numbers,S the signal from some voxel, and n, the noisecomponent from that voxel. If |S| >> |n|, onlythe ni, the component of the noise that is in-phasewith the signal, contributes to |S + n|. Hence,|S+n| ≈ |S+ni|, and the magnitude operation hassuppressed the noise component that is in quadra-ture with the signal. A more detailed descriptionof this effect, as well as a discussion of the inter-mediate case where |S| is on the same order as themagnitude of |n| is given in [1].

Initial partial k-spacedata set Mpk(kx,ky)

DFT-1

SymmetricData DFT-1

Xp*(x,y)=exp(-i angle{ms(x,y)})

ms(x,y) mpk(x,y)

p*(x,y)mpk(x,y)Phase CorrectedPartial Image Data

DFT

Phase CorrectedPartial k-space Data

Conjugate Symmetry

DFT-1

m(x,y)

Desired Image

Ms(kx,ky)

Figure 7: Summary of the phase correction and con-jugate synthesis algorithm.

5

abs(mf(x,y)) Re{mf(x,y)p*(x,y)} difference*16

Figure 9: A full k-space reconstruction (left), and the real part of the phase corrected image. The phasecorrection function was computed using ±1/16th of the k-space data (corresponding to a 9/16ths k-space re-construction). The difference image shows vessels, as well as areas of rapid local change in phase, such as theair-tissue interfaces in the sinuses.

S+n

S

n

ni

Re

Im

Figure 8: Magnitude operator suppression of the noisecomponent that is in quadrature with the signal vectorS. The magnitude of the sum |S + n| is approximately|S + ni|, where ni is the component of the noise that isin-phase with S.

2.3 Homodyne Reconstruction

The drawback with the previous method is that,after phase correction in image space, the datamust be transformed back to the frequency do-

main in order to fill in the missing data, and thenan inverse transform back to the image domain isrequired to reconstruct the final image. Anotherapproach, called homodyne, eliminates these lasttwo transforms. The ways this is done is based onthe symmetry properties of the Fourier transform.

The real part of an image corresponds to the conju-gate symmetric component of the transform. Theimaginary part corresponds to the conjugate anti-symmetric component. The weighting functionthat truncates the full k-space data set to producethe partial k-space data set shown in Fig. 6 canbe decomposed into symmetric and anti-symmetriccomponents, as shown in Fig. 10.

The imaginary part of the impulse response shownin Fig. 6, due to the antisymmetric componentin spatial frequency space in Fig 10, will be sup-pressed by retaining the real part of the image.The real part of the impulse response is the inversetransform of the symmetric component in Fig. 10,and shows two readily identifiable elements. Oneis the desired impulse at the origin. The other isa much broader sinc, due to the overweighting ofthe low spatial frequencies.

6

W(ky)

ky

(W(ky)+W*(-ky))/2

ky

(W(ky)-W*(-ky))/2

ky

Symmetric

Antisymmetric

Figure 10: The weighting function that truncates afull k-space data set to a partial k-space data set canbe decomposed into symmetric and antisymmetric com-ponents, corresponding to the real and imaginary com-ponents of the impulse response, shown in Fig. 6.

The key idea in the homodyne algorithm is topreweight the k-space data so that when we takethe real part of the image data, it correspondsto a uniform weighting in k-space. The simplestapproach is illustrated in Fig. 11. Here the am-plitude of the high spatial frequencies have beendoubled relative to the symmetrically acquired lowfrequency data.

The only problem with this approach is that thephase correction in image space corresponds to aconvolution in k-space. The weighting of Fig. 11has sharp discontinuities that produce transientsfrom this convolution, and this can produce im-age artifacts. As a result, the weighting shownin Fig. 12 is preferred. The central k-space datais weighted linearly from zero up to 2. Again,the symmetric component is uniform. The anti-

W(ky)

ky

(W(ky)+W*(-ky))/2

ky

(W(ky)-W*(-ky))/2

ky

Symmetric

Antisymmetric

2

Figure 11: Doubling the high spatial frequencies rel-ative to the central k-space data results in a uniformweighting for the symmetric component of the weight-ing.

symmetric component is also smoother, and thisresults in the imaginary component of the im-pulse response having fewer oscillations, which canalso reduce image artifacts. Other, even smootherweightings can also be used.

A comparison of homodyne reconstructions of thegradient recalled echo data from Fig. 1 is shown inFig. 13. On the left is a full k-space reconstruc-tion. In the middle is a homodyne reconstruc-tion of 9/16ths of the data using a step weight-ing. Note the ghost of the scalp interfering withthe brain. Using a ramp weighting, shown on theright, effectively eliminates this artifact. However,it produces an additional artifact above the audi-tory canal, where the phase is changing rapidly asa function of space.

The homodyne algorithm is summarized in Fig. 14.The central, symmetrically sampled data is recon-structed as a low resolution image, ms(x, y). Aphase correction image p∗(x, y) is computed as inEq. 3. The partial k-space data is preweighted in

7

Full k-Space Step Weighting Ramp Weighting

Figure 13: Comparison of the performance of different homodyne k-space weighting functions for a 9/16thsdata set. This is aggressive for phase gradients in this data set. On the left is the full k-space reconstruction. Inthe middle a step k-space weighting has been used. Note the distinct ghosts from the subcutaneous fat in thescalp (arrow). Using a ramp weighting, right, effectively eliminates this artifact. However, an additional artifactappears above the auditory canal, where the image phase is rapidly changing (arrow).

W(ky)

ky

(W(ky)+W*(-ky))/2

ky

(W(ky)-W*(-ky))/2

ky

Symmetric

Antisymmetric

2

Figure 12: Using a linear ramp weighting over thecentral symmetrically sampled strip in k-space reducesthe transients at the boundaries of the different k-spaceregions, and still produces a uniform k-space weightingfor what will be the real component of the image.

the partial k-space direction. The weighted partialk-space data is inverse Fourier transformed to pro-duce an image mpk(x, y) ∗ w(x, y). This is phasecorrected by multiplying by p∗(x, y). The final im-

age is obtained by taking the real part of the result

mhd(x, y) = Re{p∗(x, y) (mpk(x, y) ∗ w(x, y))}(4)

Note that ideally, the weighting convolution andthe image domain phase correction should occurin the other order. If the image phase is varyingrapidly in space, phase correction at one pixel canresult in incomplete suppression of the tails of theimaginary component from a nearby pixel. Thiseffect is illustrated in Fig. 15. The signal from twovoxels separated by five voxels is plotted for thecase where both voxels are on resonance, and inphase, and for the case where the frequency differ-ence between the two voxels produces π/2 phaseshift. On resonance the two voxels are resolved asexpected. Off resonance, the tail of the quadra-ture component of one voxel interferes with the in-phase component of the other voxel. After phasecorrection, this interference remains. Hence, forthe homodyne algorithm to work well, the gradi-ent of the image phase must be small compared tothe length of the tails of the imaginary componentof the impulse response.

8

Initial partial k-spacedata set Mpk(kx,ky)

SymmetricData

Xp*(x,y)=exp(-i angle{ms(x,y)}

ms(x,y) mpk(x,y)*w(x,y)

m(x,y) Desired Image

X

Real part

Pre-WeightingFunction W(ky)

DFT-1 DFT-1

Figure 14: Summary of the homodyne algorithm.

2.4 Conjugate Synthesis by k-SpaceWeighting

In the phase correction conjugate synthesis algo-rithm, the unknown data is generated by copyingdata from the conjugate symmetric location in k-space. The boundary between the acquired dataand the synthesized data is a potential source of ar-tifacts. These can be reduced by using the same k-space weighting method as in the homodyne algo-rithm. Examples are shown in Fig. 16. In this casethe k-space fraction has been increase to 5/8ths ofk-space because the algorithm didn’t produce ac-ceptable results at 9/16ths.

Voxel a

Voxel b

a+b

On Resonance p/2 Phase shift in 5 voxels

Voxel a

Voxel b

a+b

PhaseCorrected

Figure 15: A limitation of the homodyne algorithm.Two voxels are separated by five voxels. If both areon resonance, suppressing the imaginary component re-solves the two voxels as desired, shown on the left. Ifthe difference frequency between the two produces aπ/2 phase shift, the tails of quadrature component fromone voxel interferes with the in-phase component of theother voxel. Phase correction rotates both interferingcomponents into the real channel, producing artifacts.

2.5 Summary of Direct Methods

Both the homodyne algorithm, and the phase cor-rected conjugate synthesis approaches work wellif the rate of change of image phase is limited.The problems with the homodyne approach arethe result of performing phases correction afterconjugate synthesis. The problems with the phasecorrected conjugate synthesis approach are due toperforming the conjugate synthesis after the phasecorrection.

3 Iterative Partial k-Space Re-construction

The methods of the previous section performthe reconstruction in one pass. Problems arise

9

Full k-Space Step Weighting Ramp Weighting

Figure 16: Comparison of the performance of different k-space weighting functions for a 5/8ths data set,for a phase compensated conjugate synthesis reconstruction where the conjugate synthesis is done using thesame k-space weighting technique as is used in the homodyne reconstruction. On the left is the full k-spacereconstruction. In the middle a step k-space weighting has been used On the right a ramp has been used. Eachof these results in reasonable reconstructions. At 9/16ths ghosting is produced with either weighting function(not shown), similar to the step weighted homodyne example of Fig. 13.

from the interaction between phase correction andthe conjugate synthesis method, as was describedabove. Another approach is to estimate the miss-ing k-space data by iteratively applying phase cor-rection and conjugate synthesis. In the image do-main, the image phase is constrained to be that ofthe low resolution estimate. In the frequency do-main, the k-space data is constrained to match theacquired data when available. Iterating producesan estimate that approximately satisfies both setsof constraints.

There are several variations on this idea, depend-ing on how the constraints are applied, and howthe iteration is performed. We describe here asimple version of the POCS (for Projection ontoConvex Sets) algorithm [3]. It is closely related tothe earlier Cuppen algorithm [2].

The algorithm operates by iteratively transform-ing between the image domain and the spatialfrequency domain. In the spatial frequency do-main, the phase encodes that were actually ac-quired replace those of the current k-space esti-mate Mi(kx, ky). This updated data set is inverse

Fourier transformed to produce the new estimatedimage mi(x, y). The phase of this image is forcedto conform to the phase of the symmetrically ac-quired image ms(x, y) by computing

mi,pc = |mi(x, y)| p(x, y) (5)

wherep(x, y) = ei6 ms(x,y) (6)

The corresponding Fourier data Mi,pc(kx, ky) iscomputed by Fourier transform, and the entriescorresponding to the uncollected phase are propa-gated forward to Mi+1(kx, ky). The output imageis mi(x, y) on the last iteration. The algorithm issummarized in Fig. 17. This is a complex image.Either the magnitude or the real part of the phasecorrected image Re{mi(x, y)p∗(x, y)} can be used.

Typically the algorithm converges very rapidly, re-quiring four to five iterations before the changesfrom one iteration to the next are an order of mag-nitude below the noise floor of the MRI data.

Examples of both POCS and homodyne recon-structions of the data set of Fig. 1 are shown inFig. 18, along with difference images computed

10

Initial partial k-spacedata set Mpk(kx,ky)

DFT-1

SymmetricData

Xp(x,y) = exp(i arg{ms(x,y)})

ms(x,y)

mi(x,y)

p(x,y)abs(mi(x,y))

Phase ConstrainedImage Data

DFT

Zero Matrix

abs(mi(x,y))

Replace DataReplace Data

Data Constrainedk-Space DataMi(kx,ky)

DFT-1

Figure 17: Summary of the POCS algorithm.

with respect to the full k-space reconstruction.With either method, at 5/8ths data sets, bothmethods perform well. At 9/16ths, the POCSreconstruction has fewer artifacts than the step-windowed (shown) or ramp windowed homodynereconstructions.

4 Conclusions

All of these algorithms work well for the case wherethe image phase variations are smooth. When theimage phase changes rapidly, the homodyne algo-rithm produces ghosting. The POCS algorithmperforms somewhat better as the k-space fractiondecreases.

5 References

1. D.C. Noll, D.G. Nishimura, and A. Macovski,IEEE Transactions on Medical Imag., MI-10(2), 154, (1991).

2. J. Cuppen and A. van Est, Magn. Reson.Imaging, 5, 526, (1987).

3. E.D. Lindskog, E.M. Haacke,and W. Lin, emJ. Magn Reson., 92,126 (1991).

4. Z.-P. Liang and P.C. LauterburPrinciples ofMagnetic Resonance Imaging: A Signal Pro-cessing Approach, , IEEE Press, 2000.

5. G. McGibney, M.R. Smith, S.T. Nichols, andA. Crawley, Magn. Reson. Med, 30(1), 51,(1993).

11

HD 5/8

HD 9/16

POCS 5/8

POCS 9/16

Image error*4

Figure 18: Comparison of partial k-space reconstructions of the axial head data of Fig. 1. The top twoare homodyne reconstructions at 9/16ths and 5/8ths k-space, and the lower two are POCS reconstructions forthe same k-space fractions. Difference images are computed relative to the full-k-space reconstructions. Thehomodyne reconstructions at 9/16ths has clear ghosting. The POCS reconstruction at the same k-space fractionproduces fewer artifacts. Either works well at 5/8ths. 12

Phase Correct Full abs(PCF - Full)

Trivial PK abs(Triv - Full)

Homodyne abs(HD-PCF)

POCS abs(POCS-PCF)

Figure 19: Comparison of partial k-space reconstructions for a gradient recalled echo phantom data set with9/16ths k-space coverage. Here the homodyne algorithm and the POCS algorithm perform similarly. Both failin the vicinity of the “GE” logo, where the the phase compensation function p∗(x, y) doesn’t have the resolutionto track the rapid local changes in phase.

13