Partial Fourier Reconstruction Through Data Fitting andConvolution in k-Space

Feng Huang,* Wei Lin, and Yu Li

A partial Fourier acquisition scheme has been widely adoptedfor fast imaging. There are two problems associated with theexisting techniques. First, the majority of the existing tech-niques demodulate the phase information and cannot provideimproved phase information over zero-padding. Second, seri-ous artifacts can be observed in reconstruction when the phasechanges rapidly because the low-resolution phase estimate inthe image space is prone to error. To tackle these two prob-lems, a novel and robust method is introduced for partial Fou-rier reconstruction, using k-space convolution. In this method,the phase information is implicitly estimated in k-space throughdata fitting; the approximated phase information is applied torecover the unacquired k-space data through Hermitian oper-ation and convolution in k-space. In both spin echo and gradi-ent echo imaging experiments, the proposed method consis-tently produced images with the lowest error level when com-pared to Cuppen’s algorithm, projection onto convex sets–based iterative algorithm, and Homodyne algorithm. Significantimprovements are observed in images with rapid phase change.Besides the improvement on magnitude, the phase map of theimages reconstructed by the proposed method also has signif-icantly lower error level than conventional methods. MagnReson Med 62:1261–1269, 2009. © 2009 Wiley-Liss, Inc.

Key words: partial Fourier; fast imaging; data fitting; convolu-tion; reconstruction

A partial Fourier (PF) acquisition scheme is an importantfast imaging method that has been widely adopted (1-6). APF acquisition scheme completely acquires one-half of thek-space while only a small low-frequency portion is col-lected on the other side of the k-space. The ratio betweenthe partially acquired k-space data size and the full k-space data size is known as PF fraction. Compared to a fullk-space acquisition, PF reconstruction techniques can pro-duce an image with a similar spatial resolution but areduced signal-to-noise ratio, using the partially acquireddata. PF reconstruction is based on the fact that if theobject is real in image space, its Fourier transform is Her-mitian. Hence only one-half of the k-space is needed toreconstruct a real object function. In reality, however, theinhomogeneous B0 field, unwanted phase shift, etc., makethe reconstructed images complex instead of purely real.To alleviate this problem, a small portion of the k-space issampled symmetrically around the origin and used to pro-vide a low-resolution phase estimate, which is used for thePF reconstruction. Based on the method of using the phase

estimate, the PF reconstruction algorithms can be dividedinto two categories. One category is to demodulate thephase information prior to exploiting the Hermitian prop-erty of the k-space for a true real value object. Most of theexisting techniques, such as the Margosian approach (7),Homodyne detection (8), finite impulse response (FIR) andmodified FIR (9), and projection onto convex sets–basediterative method (10), belong to this category. One obviousdrawback of these methods in this category is that thereconstructed phase information cannot be better thansimple zero-padding. The other category is to apply thephase estimate to recover the unacquired k-space data andhence to produce an image with improved magnitude andphase. Cuppen’s algorithm (11) belongs to this category.Based on the analysis and results in McGibney et al. (9),Cuppen’s algorithm produces the best reconstruction if theexact phase information is given. However, when the exactphase information is not available, it produces large am-plitude distortions in the regions of local rapid phasechange, which spread as low-frequency oscillations intoother areas.

To reduce artifacts in images reconstructed by PF algo-rithms and/or to improve reconstructed phase informa-tion, advanced PF reconstruction algorithms have beendeveloped for specific applications, such as functionalMRI (4) and water-fat decomposition (6). The applicabilityof these algorithms for general applications is unknown asyet. The aim of this work is to design a general PF recon-struction algorithm that produces improved magnitudeand phase information even when image phase changesrapidly.

All PF reconstruction algorithms rely on the estimatedphase information. How to estimate phase information iscrucial to the accuracy of reconstruction. As shown by Fig.8 in McGibney et al. (9), a carefully approximated phasemap, using polynomial or generalized series, can dramat-ically reduce the artifacts associated with the phase cor-rection operator. All existing algorithms (7-11) approxi-mate the phase information in image space. However,there are intrinsic difficulties when phase information isestimated in image space from the low-resolution image,due to noise and Gibbs’ ring artifact. Alternatively, thephase information can be implicitly approximated in k-space through data fitting, which is similar to the calcula-tion of convolution kernels in generalized autocalibratingpartially parallel acquisitions (GRAPPA) (12). In this way,the difficulties in image space can be avoided.

The majority of existing techniques apply the phasecorrection in image space. Fig. 3 in McGibney et al. (9)shows the drawback of the image space correction. Thephase correction in image space corresponds to a circularconvolution operation in the k-space. This introduces er-ror at the edge between acquired and unacquired k-space

Advanced Concept Development, Invivo Corporation, Gainesville, Florida,USA*Correspondence to: Feng Huang, Invivo Corporation, 3545 SW 47th Avenue,Gainesville, FL 32608, USA. E-mail: fhuang@invivocorp.comReceived March 9, 2009; revision received May 6, 2009; accepted June 2,2009.DOI 10.1002/mrm.22128Published online 24 September 2009 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 62:1261–1269 (2009)

© 2009 Wiley-Liss, Inc. 1261

(“edge effect”). A larger convolution kernel will producemore serious distortion when exact phase information isnot available. Hence, phase correction in k-space with asmaller convolution was suggested in McGibney et al. (9),which is named as FIR algorithm. However, FIR belongs tothe first category, i.e., it applies the approximated phaseinformation to demodulate the phase. Hence, the recon-struction by FIR cannot have phase better than zero-pad-ding.

In this work, we propose to implicitly estimate phaseinformation in k-space through data fitting and apply theestimation in k-space to reconstruct the unacquired k-space data through convolution. Hence, the proposedmethod has a unique approach for phase estimation andbelongs to the second category methods in the sense ofhow to apply the phase estimate. This method avoids theintrinsic difficulties with phase estimation in image spacewhile it reduces distortion due to the application of phasecorrection in image space. Moreover, because the full k-space data are reconstructed, the enhanced phase informa-tion is provided. In all of our experiments, the proposedmethod consistently performs better than conventionalmethods for images acquired using either spin echo orgradient echo sequences.

THEORY

Let K be the matrix of a two-dimensional k-space data set,KH be the Hermitian matrix of K, i.e., conjugating andreflecting around the zero frequency point. I and IH areimages corresponding to K and KH, respectively. From thededuction in Cuppen’s approach (11), it is known that I� IHej2�� x�, where ��x� is the phase map of I. TakingFourier transform on both sides, we have

K � KH � FFT�ej2��x��, [1]

where R is the convolution operator. The support ofFFT�ej2�� x�� is very small when the phase map containsonly low-frequency information. In this case, the correla-tion between K and KH is so strong that the data in K can bereconstructed using data in KH through convolution with asmall convolution kernel �. And Eq. [1] becomes

K � KH � �. [2]

If K is acquired using PF acquisition scheme, then KH hasvalues at these unacquired phase encoding lines, as shownin Fig. 1. Based on Eq. [2], these unacquired phase encod-ing lines in K can be determined from KH through convo-lution with �. Since the same convolution kernel � isapplicable in the whole k-space domain, � can be approx-imated using the overlapped section of K and KH throughdata fitting. Fig. 1 demonstrates the steps of the proposedmethod.

The only parameter in this algorithm is the size andshape of the convolution kernel. To reduce distortion andreconstruction time, a kernel with small size is suggested.In practice, � has size of 3 � 3 � 9 � 9. A larger kernel canproduce better results when the image phase changes rap-idly, but takes longer reconstruction time. When mul-

tichannel data are available, the proposed method can beapplied to each channel individually before the generationof the composite image. If coil sensitivity maps are avail-able, a composite image can be produced using sensitivitymaps and images from zero-padding using Roemer’smethod (13). Partial composite k-space can be produced bytruncating the FFT of the composite image. PF reconstruc-tion can then also be applied to the partial compositek-space to reduce reconstruction time. It should be noticedthat the sensitivity maps have to be calculated with bodycoil. In this way, only the phase introduced by coil sensi-tivity will be removed, and all other phase information canbe preserved.

MATERIALS AND METHODS

From the analysis and results in McGibney et al. (9), ithas been concluded that the iterative methods producelower root mean square error (RMSE) than noniterativemethods. Hence, the proposed method was comparedwith both Cuppen’s iterative method and projectiononto convex sets– based iterative method. In addition,because the homodyne method is a widely appliedmethod, the proposed method was also compared withhomodyne method. It is well known that PF reconstruc-tion algorithms produce more artifacts with images ac-quired with gradient echo sequence due to rapid phasechange. Hence, the proposed method was evaluatedwith images acquired with both spin echo sequence andgradient echo sequence.

Data Acquisition

To produce images with smooth phase, two sets of axialbrain data were acquired on a 3.0-T Achieva scanner (Phil-ips, Best, Netherlands), using an eight-channel head coil(Invivo Corp., Gainesville, FL) with the following scanparameters: field of view 230 � 230 mm2, matrix size256 � 256, pulse repetition time/echo time � 2000/20 ms.Inverse recovery sequence was used for both data sets.Two different inversion times were used to suppress graymatter (inversion time � 800 ms) or fat (inversion time �180 ms), respectively. Phase encoding direction was ante-rior-posterior.

To produce images with rapid phase change, one car-diac data set and one abdomen data set were acquired.The short-axis cardiac data set was acquired on a 1.5-TGE system (GE Healthcare, Waukesha, WI) using a GEfour-channel cardiac coil. A fast imaging sequence em-ploying steady-state acquisition was used (field of view280 mm; matrix size 192 � 224; pulse repetition time4.510 ms; echo time 2.204 ms; flip angle 45°; slice thick-ness 6 mm; number of averages two). Phase encodingdirection was anterior-posterior. The axial abdomendata set was acquired on a 3-T Philips system using a32-channel cardiac coil (Invivo Corp.). A breath holddual fast field echo sequence was used (field of view375 mm; matrix size 204 � 256; pulse repetition time180 ms; echo time 1/echo time 2 2.3/5.8 ms; flip angle80°, slice thickness 7 mm; number of averages one).Phase encoding direction was anterior-posterior.

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For all data sets, full k-space data were acquired. PFacquisition was produced artificially by setting a portionof k-space to be 0. The simulated PF fraction was 0.625 inall experiments.

Implementation of Reconstruction Algorithms

Except for the proposed method, a low-pass filter is usedfor the approximation of the phase information in imagespace. To reduce Gibb’s rings, the low-pass filter suggestedby Eq. [13.97] in Bernstein et al. (14) was adopted in thiswork:

L�k�

� �1, �k� � k0 � w/2

cos2����k� � �k0 � w/2��

2w �, k0 � w/2 � �k� � k0 � w/2

0, �k� � k0 � w/2

,

[3]

where k-space data are acquired from � k0 to kmax, and wis a positive parameter, which is fixed to be 6 in this paper.The estimated phase is the phase of the image correspond-ing to the low-pass–filtered k-space. Cuppen’s method(10), projection onto convex sets–based iterative method(11), and Homodyne method (14) were implemented inMatlab (MathWorks Inc., Natick, MA). For iterative meth-ods, the reconstruction accuracy depends on the numberof iterations. The results of iterations 1 to 10 were recordedand the one with lowest RMSE was used for comparison.Fig. 1 and Eq. 2 were used to implement the proposedmethod. The size of the convolution kernel was 5 � 5 forimages acquired with spin echo sequences and 7 � 7 forimages acquired with gradient echo sequences. To reducethe dominant effect of the very central k-space data, thehigh-pass filter defined by Eq. 2 in Huang et al. (15) wasadopted with c � 12 and w � 3. The convolution kernelwas calculated with the filtered k-space data. All PF re-construction algorithms were applied to each channel in-dividually before the generation of composite imagethrough square root of sum of squares.

FIG. 1. Steps of the proposed method.

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Evaluation Criteria

To evaluate the image quality of the reconstructed images,the difference map and RMSE are used. The differencemap depicts the difference in magnitudes between thereconstructed and reference images at each pixel andshows the distribution of error. The RMSE quantitativelymeasures the global error.

All data were processed on an HP xw4100 workstationwith dual 3.2-GHz processors and 2-GB random accessmemory (Hewlett-Packard Company, Palo Alto, CA).

RESULTS

Comparison Using Images Acquired by Spin EchoSequence

The brain data sets were used in this experiment. Since thephase map in this experiment was smooth, a convolutionkernel with size 5 � 5 was used for the proposed method.Fig. 2 shows the comparison. The number at the rightlower corner of each image is RMSE. For these two datasets, all algorithms produced low RMSE. This is reason-able since the image was acquired with a spin echo se-quence, which results in a smooth phase map and thelow-resolution phase estimate is sufficient. The proposedmethod has slightly lower RMSE than all other algorithms.

Comparison Using Images Acquired by Gradient EchoSequence

The cardiac data set and the abdomen data set were usedin this experiment. Figure 3 demonstrates the phase mapof one channel image from each data set respectively. Itcan be seen that there are some rapid changes in the phasemaps. In this scenario, the phase estimate from the low-resolution image has significant errors and it results inserious artifacts in the images reconstructed with conven-tional PF algorithms. Figures 4 and 5 demonstrate thecomparison of different methods with one time frame ofthe cardiac images and one slice of the abdomen images,respectively. The results by the proposed method have

FIG. 2. Comparison with images acquired by spin echo sequence.These images are difference maps between the reference and thereconstructed images. The two columns are for the images withsuppressed fat and suppressed gray matter respectively. The num-ber at the right lower corner is the RMSE.

FIG. 3. Phase map of one channel from the cardiac data set (a) andthe abdomen data set (b), respectively. Rapid phase change can beobserved in these images acquired with gradient echo sequences.

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significantly lower RMSE than all other algorithms. Asdiscussed in McGibney et al. (9), given an inaccurate phaseestimate, the two iterative methods had large amplitudedistortions in the regions of local rapid phase change,which spread as low frequency oscillations into areaswhere the phase estimate had been correct. The arrows onthe left column of Figs. 4 and 5 show the large amplitudedistortions in the regions of local rapid phase change. Thearrows in difference maps in Figs. 4 and 5 show the spreadof the distortions. Clearly, the results by the proposedmethod have a reduced level of distortion. Similar to theresults in McGibney et al. (9), Homodyne algorithm pro-duced images with the highest RMSE.

Improvement on Phase

Since algorithms in the first category (e.g., homodyne) cannotprovide more phase information than zero-padding, thesemethods were not included in this comparison. Instead, theresults using zero-padding method were used. Cuppen’s al-

gorithm and the proposed method reconstruct the unac-quired k-space data and hence can provide additional phaseinformation. To compare the proposed method and Cuppen’salgorithm on the contribution of phase improvement, phasemaps of the reconstructed image from one channel are shownin Fig. 6. The left column demonstrates the results when thephase map is smooth; the right column shows the resultswith rapid phase change. In both cases, Cuppen’s algorithmand the proposed method produced phase maps with higherspatial resolution and lower ringing artifacts than zero pad-ding. When phase map has rapid changes, the proposedmethod produced lower artifacts than Cuppen’s algorithm.The white arrows in Fig. 6 show the regions in which theCuppen’s algorithms produced more artifacts. The enhance-ment of phase map by the proposed method is important forapplications that are sensitive to phase information.

FIG. 4. Comparison with the cardiac image. Difference maps werebrightened 10 times for better visualization. The number at the rightlower corner is the RMSE. The white arrows in the left column showthe regions where conventional methods have large amplitude dis-tortions because of the local rapid phase change. The arrows in theright column show the regions with low-frequency oscillations dis-tributed from the regions with local rapid phase change.

FIG. 5. Comparison with the abdomen image. Difference mapswere brightened five times for better visualization. The number atthe right lower corner is the RMSE. The white arrows in the leftcolumn show the regions where conventional methods have large-amplitude distortions because of the local rapid phase change. Thearrows in the right column show the regions with low-frequencyoscillations distributed from the regions with local rapid phasechange.

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DISCUSSION

Advantages of k-Space Data Fitting and ConvolutionMethod

As mentioned in the introduction, the proposed methodhas two major differences from existing techniques. One ishow to approximate phase information; the other one ishow to apply the approximated phase information. Allexisting techniques explicitly approximate phase informa-tion ��x� in image space using low-resolution image. Theproposed method implicitly approximates phase differ-ence between I and IH (2��x�) through data fitting in k-space. The implicit approximation scheme can approxi-mate the phase information more accurately than the im-age space explicit methods because of the following three

reasons. First, as shown in Figs. 3 and 6, significant noiseis present in the image-space phase estimation. In addi-tion, it is hard to completely remove ringing artifacts in theimage reconstructed using only the central k-space data.Calculating the convolution kernel in k-space through datafitting avoids these difficulties. Second, in some applica-tions, 2��x� could be smoother than ��x�. Therefore, theapproximation of 2��x� could be easier. One specific ex-ample is out-phase water-fat imaging. Since in this casethe phase change between water and fat is approximately�, 2��x� becomes 2� (i.e., 0) at the boundaries betweenwater and fat. It is very difficult to accurately approximatesuch a sharp change � in image space from a low-resolu-tion image. However, the proposed method can easily

FIG. 6. Comparison of reconstructed phase informa-tion. The left column is for a brain image acquired withspin echo sequence. The right column is for a cardiacimage acquired with gradient echo sequence. Thewhite arrows show regions that Cuppen’s algorithmproduced more serious errors than the proposedmethod.

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approximate the smooth phase map, which has no changebetween water and fat, using a small convolution kernel. Itis therefore ideal to use the proposed method to improvethe sharpness at the water-fat boundaries. Fig. 7 demon-strates an example using true PF acquisition with PF frac-tion 0.625. Third, the convolution kernel calculatedthrough data fitting will produce non-constant magnitudein image space, i.e., it contains information beyond lowfrequency phase information. In the data-fitting step, thelinear relationship among conjugate symmetric k-spacesignals is optimally calculated in the sense of least squareerror. To minimize the fitting error, the convolution kernelautomatically compensates the low-frequency phase infor-mation due to small kernel size by using nonconstantmagnitude. A similar observation can be found inGRAPPA. Convolution kernels with size as small as 4 � 1are sufficient to provide equivalent information containedin sensitivity maps though data fitting. Due to these ad-vantages, the proposed method is especially useful whenan image has rapid phase change. As shown in Fig. 3,where the phase change is slow, the improvement by theproposed method is limited. However, in Figs. 4 and 5,where the phase change is fast, the improvement in recon-struction is more significant.

The majority of the existing techniques apply the ap-proximated phase information for phase demodulation.The proposed method applies the approximated phaseinformation to calculate the unacquired data through con-volution in k-space. As shown in step 3 of Fig. 1, theconvolution does not use the unacquired data when theconvolution kernel size is smaller than the width of thecentral calibration signal. Since the convolution kernelsize used in this paper is smaller than 9 � 9, there is noedge effect issue in the proposed method. On the otherhand, as explained in McGibney et al. (9) using Fig. 3 in

that reference, image space phase correction will causeedge effect, which manifests as ringing artifacts. As shownin Figs. 4 to 6 in this paper, the proposed method pro-duced lower ringing artifacts than Cuppen’s algorithm dueto convolution in k-space.

Relationship With Cuppen’s Algorithm and FIR

Cuppen’s algorithm produces the perfect reconstructionwhen the exact phase information is given. As shown inthe analysis and results in McGibney et al. (9), Cuppen’salgorithm produced the lowest global error among thevarious methods compared even if the phase informationwas not exact. The underlying theory of the proposedmethod is very similar to that of Cuppen’s algorithm. Thedifferences are the implementation schemes. One is ink-space, and the other one is in image space with iteration.The k-space implementation makes the proposed methodmore robust than Cuppen’s algorithm when the exactphase information is not available. Moreover, the pro-posed method has no concerns about the number of itera-tions. To reduce distortion, McGibney et al. (9) suggestedFIR that also apply the phase information in k-spacethrough convolution. However, FIR has two differencesfrom the proposed method. First, they have different tar-gets. The target of FIR is to demodulate the phase infor-mation to make the reconstruction real instead of complex.The target of the proposed method is to reconstruct theunacquired data and hence to enhance both magnitudeand phase information. Second, the convolution kernelused in FIR is the truncated Fourier transform of the ex-plicit phase map approximated in image space. The pro-posed method implicitly approximates phase informationin k-space.

FIG. 7. Advantage of the pro-posed method for out-phase wa-ter-fat imaging. An out-phase wa-ter-fat image was acquired, withPF ratio 0.625, by dual fast fieldecho sequence on a Philips 3-Tsystem. The left column showsthe improvement of magnitude.The right column shows the im-provement of phase. For bettervisualization, the left lower quarterof the phase from channel 1 ispresented. The white arrowsshow the improvement in defini-tion of water-fat boundaries.

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Size and Shape of the Convolution Kernel in the ProposedMethod

A small convolution kernel size is used in the proposedmethod. One may argue whether such a small convolutionkernel works for images with rapid phase change. First ofall, it has to be clarified that the proposed method does notproduce the perfect reconstruction, but it produces betterresults than conventional methods with lower artifactlevel. Hence, it is only necessary to explain why the pro-posed method works better than conventional methodswhen image has rapid phase change.

It is true that a small convolution kernel is not sufficientto describe phase with rapid change. It is also true that anyimage space–based method cannot accurately estimatephase with rapid change, either, because there is no suffi-cient phase information in the acquired data. However, thek-space–based method gives lower noise and artifacts inthe phase estimate than those image space–based meth-ods. On one hand, as shown in Table 1, a larger convolu-tion kernel does work better. Table 1 shows the RMSEusing different convolution kernel sizes. Results with im-ages acquired with both spin echo and gradient echo arepresented. On the other hand, it can be seen that theimprovement on image quality is insignificant when thekernel size is larger than 7 � 7. Moreover, a larger convo-lution kernel means longer reconstruction time. Therefore

kernel size larger than 7 � 7 is not suggested. For imagesacquired with spin echo sequences, a kernel with size 5 �5 is sufficient. In this work, only square kernels are used.It is possible to adaptively optimize the kernel shape.Some exhaustive searching-based methods for GRAPPAconvolution kernel optimization (16,17) are potentiallyapplicable for the proposed method.

Furthermore, as pointed by McGibney et al. (9) andmentioned previously in this paper, image space–basedPF reconstruction algorithms introduce distortions associ-ated with phase correction because of the edge effect. Thesmall convolution kernel used in this work effectivelysuppresses the image distortion associated with phase cor-rection in image space, as shown in Figs. 4 and 5.

Other Implementation Considerations

There are two choices for when to apply PF reconstructionwhen there are multichannels, either before or after thegeneration of the composite image. One drawback of ap-plying PF after the generation of partial composite k-spaceis that extra information, complex coil sensitivity maps, isnecessary. However, if coil sensitivity maps are available,applying PF after the generation of the composite imageconsumes less reconstruction time, especially if there aremany channels and the convolution kernel is large. Ifphase information is expected from the composite image,the coil sensitivity maps have to be calculated using im-ages from phase array coil and the body coil. Figure 8a and8b shows one example of the application of PF after thegeneration of the composite image using the abdomenimage acquired with 32-channel coil.

As mentioned in the Materials and Methods section,the convolution kernel used in this paper was calculatedwith the high-pass–filtered k-space. The proposedmethod works well without using the high-pass filter.

FIG. 8. Results by applying PF re-construction after generation ofpartial composite k-space usingsensitivity maps. The PF fractionis 0.625. a,b: Magnitude andphase, respectively. c: The lowerright quarter of (a). d: The lower-right quarter of the reconstructionwithout using a high-pass filter.The white arrow shows theslightly blurred edge.

Table 1RMSE Comparison With Different Sizes of Convolution Kernels

3 � 3 5 � 5 7 � 7 9 � 9

Fat suppressed 3.3% 3.0% 3.0% 3.0%Gray matter suppressed 3.3% 2.8% 2.8% 2.8%Cardiac 7.3% 6.6% 6.4% 6.2%Abdomen 16.1% 15.8% 15.8% 15.8%

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However, the application of the high pass filter canreduce the dominant effect of the very central k-spacedata and hence improve the accuracy of the convolutionkernel. More discussions are available in Huang et al.(15). However, unlike the previous high pass-GRAPPA,the high-pass filter for PF reconstruction can be fixed inthis work. Figure 8c and d shows one comparison with(Fig. 8c) and without (Fig. 8d) high-pass filtering. Aslight improvement in spatial resolution and signal-to-noise ratio can be observed in the reconstruction byusing high-pass filtering.

CONCLUSION

A robust and efficient novel method is introduced forgeneral PF reconstruction. The proposed algorithm esti-mate and apply the phase information in k-space throughdata fitting and convolution. The unacquired k-space dataare recovered by the convolution with the Hermitian ma-trix of the partially acquired data. Experiments demon-strate that the advantage of the proposed method overprevious method in recovering magnitude and phase in-formation. The advantages of the proposed method aremore significant when the image has rapid phase change.

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