Original Research

MRI Reconstruction From 2D Truncated k-Space

Jianhua Luo, PhD,1* Yuemin Zhu, PhD,2 Wanqing Li, PhD,3 Pierre Croisille, MD, PhD,2

and Isabelle E. Magnin, PhD2

Purpose: To shorten acquisition time by means of bothpartial scanning and partial echo acquisition and toreconstruct images from such 2D partial k-spaceacquisitions.

Materials and Methods: We propose an approach toreconstructing magnetic resonance images from 2D trun-cated k-space in which the k-space is truncated in bothphase- and frequency-encoding directions. Unlike conven-tional reconstruction techniques, the proposed approachis based on a newly developed 2D singularity functionanalysis (SFA) model and a sparse representation of animage whose parameters can be estimated from the 2Dpartial k-space data. Such a sparse representation leadsto an accurate recovery of the missing k-space data and,hence, an accurate reconstruction of the image.

Results: The proposed approach can reconstruct animage from as little as 20%–30% of the k-space data andthe quality of the reconstructed image is comparable tothe reference image that is reconstructed from the com-plete k-space data.

Conclusion: Despite the high asymmetry of a 2D trun-cated k-space, the proposed approach allows for accuratereconstruction without the need of phase correction and,thus, overcomes the assumption of slow phase variationsthat is usually required by the existing reconstructionmethods. It provides a new way of fast imaging for appli-cations that require a significant reduction of the acquisi-tion time.

Key Words: partial k-space; partial echo; reconstruction;fast imaging; magnetic resonance imagingJ. Magn. Reson. Imaging 2012;35:1196–1206.VC 2011 Wiley Periodicals, Inc.

MAGNETIC RESONANCE IMAGING (MRI) has been,owing to its powerful imaging function and noninva-sive characteristics, widely used in medicine for diag-nosis, monitoring, and treatment evaluation of diseasesand disorders. The potentialities of MRI are, however,conditioned by imaging speed. To meet the growing needfor functional imaging and moving organ imaging, fastacquisition techniques such as fast low-angle shot(FLASH) (1,2), echo planar imaging (EPI) (3), and rapidacquisition with relaxation enhancement (RARE) (4)have been developed and are used in clinics. However,the speed at which data can be measured in MRI is fun-damentally limited by physical and physiological con-straints. Therefore, partial k-space acquisition thatreduces the number of measured k-space samplesbecomes an additional mechanism to further shortenthe imaging time while maintaining image quality.

There are two basic approaches to partial k-space ac-quisition for reducing imaging time in MRI. The firstapproach is called the half-scan (or partial scan) tech-nique that acquires a little more than half of the completek-space in the phase-encoding direction. The half-scantechnique utilizes the symmetry of the k-space data. Thetime reduction can be nearly a factor of two (5). The sec-ond approach is called the partial echo technique thatshortens the echo time in a sequence and acquires partialechoes in the frequency-encoding direction. The reduc-tion of the echo time is possible because if the first part ofthe echo is not received, the dephasing lobe of the fre-quency-encoding gradient is not to be on for a quite longduration, and this saves TR time (6). However, the partialecho misses the partial data in the frequency (or read-out)-encoding direction. We refer to the partial k-spacedata acquired either in the phase-encoding direction orin the frequency direction as 1D partial k-space data.

MR image reconstruction from 1D partial k-spacedata has received much attention in the past two dec-ades. Much effort has focused on the reconstructionfrom the partial k-space truncated in the phase-encoding direction (7–19). To correctly reconstruct thespatial MR image from such 1D partial k-space data,a basic idea is to correct the phase drift. To do so, theacquisition is usually performed by setting the highertruncation frequency to uh, and the lower truncationfrequency to a negative integer �ul instead of zero.The central k-space data around the origin is thenused to estimate the phase. This idea constitutes thecore part of the phase-corrected reconstruction

1College of Biomedical Engineering, Shanghai Jiaotong University,Shanghai, P.R. China.2CREATIS, CNRS UMR 5220, Inserm U 1044, INSA Lyon, University ofLyon, France.3AMRL, ICT Research Institute, University of Wollongong, Australia.

Contract grant sponsor: Chinese NSFC; Contract grant number:30911130364; Contract grant sponsor: French ANR 2009; Contractgrant number: ANR-09-BLAN-0372-01; Contract grant sponsor:Region Rhone-Alpes of France; Contract grant number: CIBLE 2010.

*Address reprint requests to: J.H.L., Luo, 800 DongChuan Road, Col-lege of Biomedical Engineering, Shanghai Jiaotong University, Shang-hai, 200240, PR China. E-mail: jhluo@sjtu.edu.cn

Received June 2, 2011; Accepted November 18, 2011.

DOI 10.1002/jmri.23538View this article online at wileyonlinelibrary.com.

JOURNAL OF MAGNETIC RESONANCE IMAGING 35:1196–1206 (2012)

CME

VC 2011 Wiley Periodicals, Inc. 1196

techniques reported previously (5,7,9–13). Note thatall these phase-corrected methods or, more generally,the constrained reconstruction methods often use thephase encodings from �u1 to N/2 to reconstruct anN � N image. In other words, they only deal with thepartial k-space data whose k-space truncations occurat negative spatial frequencies and compensate forthe missing k-space data in the range of [�N/2, �u1]by using those in [u1 N/2]. In the case that thek-space is truncated at both negative (�u1) and posi-tive (uh) spatial frequencies, conventional phase-corrected reconstruction methods are not suitable,since they are not able to appropriately recover themissing k-space data from the acquired data in therange of [uh, N/2] or [�N/2, �uh]. Moreover, when thehigh (positive) spatial frequency part of the k-space isnot acquired, discontinuities between the acquiredand nonacquired k-space data could arise; the con-ventional methods cannot suppress the thus gener-ated Gibbs artifacts in the image domain without alter-ing spatial frequency components. To overcome thisproblem, a method was proposed (19) that first compen-sates for the missing negative spatial frequencies in theregion [�uh, �u1] using phase correction and Hermitiansymmetry property and then recovers the nonacquiredk-space data beyond the region [�uh, uh] by means of amathematical model based on singularity function anal-ysis (SFA). Another approach, called the total variation(TV) method, which is mainly used in computerized to-mography (CT) reconstruction, was also employed forreconstructing MR images from undersampled k-space(20). The TV method aims to find a solution that opti-mizes the total variance of the reconstructed image.Recently, compressive sensing (CS) theory has also beenused to reconstruct images from incomplete k-spacedata (21–24). However, truncation of k-space introducesan extra spatial phase and the CS-based reconstructionmethods do not resolve this problem, although the spar-sity can be made separately on the real and imaginarycomponents of the complex spatial image and optimiza-tion can be done jointly.

In principle, it is feasible to employ both partialscanning and partial echo acquisition in order tofurther shorten the acquisition time. We call suchacquisition 2D partial k-space acquisition, in whichk-space is truncated in both the frequency- andphase-encoding directions and data are missing inboth directions. Reconstructing images from such 2Dtruncated k-space data is a new problem and existingreconstruction methods are not suitable for this kindof highly asymmetric k-space data. This article pro-poses a novel approach to reconstructing MR imagesfrom 2D truncated k-space data. Unlike classicalmethods, the proposed approach is based on a newlydeveloped 2D SFA model and a sparse representationof an image whose parameters can be estimated fromthe 2D partial k-space data. Such a sparse represen-tation leads to an accurate recovery of the missingk-space data and, hence, an accurate reconstructionof the image. In addition, the approach does not relyon the assumption that the spatial phase varies slowlyacross the image because it does not require phasecorrection. The performance of this approach was eval-

uated using both simulated and real MR brain imagesand compared with the zero-filling (ZF) and the projec-tion-onto-convex sets (POCS) techniques.

MATERIALS AND METHODS

2D SFA Model

The SFA model reported previously (17,19) is 1D andcan be applied to 2D discrete images in a line-by-linemanner if the corresponding k-space data have beentruncated only in one direction. For instance, Fig. 1aillustrates a case of partial scan acquisition, wherethe k-space is truncated in the phase-encoding direc-tion. In Fig. 1b, the k-space is truncated in the fre-quency-encoding direction, which corresponds to par-tial echo acquisition. The 1D SFA model, however, isnot applicable when the k-space is truncated in bothphase- and frequency-encoding directions (Fig. 1c,d).To deal with the 2D partial k-space data, we devel-oped a 2D SFA model in this study.

Figure 1. Different 1D and 2D truncated k-spaces and 2Dsingularity function w(x � 30, y � 50). a: 1D truncated par-tial k-space in the phase-encoding direction. b: 1D truncatedpartial k-space in the frequency-encoding direction. c,d: 2Dtruncated k-spaces in both the phase-encoding direction andthe frequency direction. e: 2D singularity function w(x � 30,y � 50).

MRI Reconstruction by 2D Truncated k-Space 1197

We define the following 2D singularity functions(Fig. 1e):

wðx � xj;y� yjÞ ¼ dðx � xjÞuðy� yjÞ; ½1�with

dðx � xjÞ ¼ 1; x ¼ xj0; x 6¼ xj

�½2�

and

uðy� yjÞ ¼ 1; y � yj

0; y < yj

�; ½3�

where d(x � xj) is a Kronecker function, and u(y � yj)is a step function.

Thus, any discrete image f(x,y) can be expressed asthe weighted sum of 2D singularity functions:

f ðx ;yÞ ¼XQj¼1

ajwðx � xj;y� yjÞ; x ;y ¼ 1;2; . . .N ; ½4�

where {(x1,y1),(x2,y2),. . .,(xQ,yQ)} are called the singularpoints, {a1,a2,. . .aQ} are the singularity degrees and Qis the number of singular points. Singular points(xj,yj) is a point at which f(xj,yj) � f(xj,yj �1) = 0. Thesingularity degree aj corresponds to the differencevalue of the neighboring pixels at the correspondingsingular point, that is, aj ¼ f(xj,yj) � f(xj,yj �1).

Designating the discrete Fourier transform (DFT) byDFT[�], the 2D singular spectral function is defined as:

Wjðkx ;kyÞ ¼ DFT ½wðx � xj;y� yjÞ�; j ¼ 1;2; . . . ;Q: ½5�

Taking the DFT of both sides of Eq. [4] yields:

F ðkx ;kyÞ ¼XQj¼1

ajWjðkx ;kyÞ ½6�

with F(kx,ky) ¼ DFT[f(x,y)].We say that Eqs. [4–6] define a 2D SFA model. Note

that this is a rather general discrete model of represent-ing any discrete images, which does not rely on anyassumptions (low-frequency phase, piecewise constantvariation, smoothness variation, etc.) on the image. Ifthe image is a piecewise constant image, the number ofsingular points will be much smaller than the number ofimage pixels. On the contrary, in the extreme case thatthe image contains purely noise the 2D SFA model canstill exactly represent it, and, in this case, the number ofsingular points is equal to the number of image pixels.In practice, a physical image always presents spatialcorrelation. As a result, the number of singular points isalways much smaller than that of image pixels.

Taking the finite difference along y-direction of bothsides of Eq. [4], we have:

Df ðx ;yÞ ¼ f ðx ;yÞ � f ðx ;y� 1Þ

¼XQj¼1

ajd x � xj;y� yj

� �: ½7�

Equation [7] shows that the finite difference of animage along y-direction can be expressed as the

weighted sum of 2D Kronecker functions located at thesingular points. Due to the property of Kronecker func-tions, the right-hand side of Eq. [7] is not zero only atthe singular points. This implies that the singularpoints and their corresponding singularity degrees canbe obtained from the finite difference of the image in ydirection. In Eq. [1], if the line scan is changed to col-umns, that is, w(x � xj, y � yj) ¼ u(x � xj)d(y � yj) thefinite difference is then calculated in the x direction.

Reconstruction Using the 2D SFA Model

As mentioned, a 2D SFA model is determined by a setof singular points and the associated singularitydegrees. The problem of applying the 2D SFA model toreconstruct an MR image from a 2D partial k-spacetruncated in both phase- and frequency-encodingdirections becomes how to estimate the singularpoints and the corresponding singularity degrees fromthe partial k-space data.

Let I(kx,ky) be the scanning matrix whose elements areeither 1 or 0 corresponding respectively to the acquiredand nonacquired k-space points (Fig. 2a), that is:

Iðkx ;kyÞ ¼ 1; ðkx ;kyÞ 2 acquiredk � space0; ðkx ;kyÞ =2 acquiredk � space

�: ½8�

If the complete k-space is scanned, ie, I(kx,ky) ¼ 1,kx,ky ¼ 1,2,. . .,N, then the corresponding spatial func-tion is given by d(x,y) ¼ IDFT[I(kx,ky)]. In the partial scan-ning case, the inverse Fourier transform of I(kx,ky) is:

iðx ;yÞ ¼ IDFT ½Iðkx ;kyÞ�: ½9�

In general, i(x,y) is a complex-valued function sincek-space is highly asymmetric due to the 2D trunca-tion. It is in fact a deteriorated version of the 2D Kro-necker d(x,y) because of the missing of some spatialfrequencies. Therefore, the modulus of i(x,y) can beconsidered as an approximate version of the 2D Kro-necker d(x,y) using incomplete k-space data. An exam-ple of real(i(x,y)) and imag (i(x,y)) is shown in Fig. 2b,c,respectively. Let F(kx,ky) represent the complete k-spacedata of the original image f(x,y). The image ~f ðx ;yÞ whichis reconstructed from a 2D truncated k-space usingzero-filling method (that is, filling up the nonacquiredk-space data with zeros) can be written as:

~f ðx ;yÞ ¼ IDFT ½F ðkx ;kyÞIðkx ;kyÞ�¼ f ðx ;yÞ�iðx ;yÞ; ½10�

where ‘‘*’’ denotes convolution.Equation [10] means that ~f ðx ;yÞ is the resultant

image of the original image f(x,y) being distorted byi(x,y) due to the 2D k-space truncation. In order toestimate the singular points and the singularitydegrees of f(x,y) from ~f ðx ;yÞ, we compute the finite dif-ference ~f ðx ;yÞ in the y direction:

D~f ðx ;yÞ ¼ ~f ðx ;yÞ � ~f ðx ;y� 1Þ ¼ Df ðx ;yÞ�iðx ;yÞ: ½11�

Equation [11] indicates that the finite difference off(x,y) is also distorted by i(x,y) in the same way as the

1198 Luo et al.

original image is distorted because of the 2D k-spacetruncation.

Substituting Eq. [7] into Eq. [11] yields:

D~f ðx ;yÞ ¼XQj¼1

aj~d x � xj;y� yj

� �; ½12�

where ~d x � xj;y� yj

� � ¼ d x � xj;y� yj

� � � iðx ;yÞ ¼i x � xj;y� yj

� �. So:

D~f ðx ;yÞ ¼XQj¼1

aji x � xj;y� yj

� �: ½13�

Hence, the y-directional difference image D~f ðx ;yÞ of~f ðx ;yÞ is the weighted sum of i(x � xj, y � yj), i ¼1,2,. . .Q, each term is a shifted version of i(x,y) fromthe original point (0,0) to the singular point (xj,yj). i(x� xj, y � yj) is a deteriorated version of the 2D Kro-necker function d(x � xj, y � yj), and would usuallyhave a peak at singular point (xj,yj). Figure 2d–f illus-trates respectively Df(x,y) and the real and imaginaryparts of D~f ðx ;yÞ.

According to Eq. [13], the difference image D~f ðx ;yÞis the superposition of Q oscillating functions aji(x �

xj, y � yj), centered at (x1,y1),(x2,y2),. . ., and (xQ,yQ)respectively. All oscillating functions aji(x � xj, y � yj)will interfere with each other. Singular points havinghigher singularity degrees will have larger oscillationand stronger influence on their neighboring singularpoints. As a result, singular points having smaller sin-gularity degrees may be buried. When the largestoscillating component is removed from D~f ðx ;yÞ, thesecond largest oscillating component, which may beburied by the largest oscillating one, will become thelargest. This leads us to extract the singular pointsfrom D~f ðx ;yÞ in a layered manner. Assuming that thesingular points are rearranged according to the abso-lute value of their singularity degrees, that is,ðxj;yjÞ; jajj > jajþ1j; j ¼ 1;2; . . . ;Q. We then havejD~f ðx1;y1Þj ¼ maxx ;y2f1;2;...;NgfjD~f ðx ;yÞjg (ie, jD~f ðx ;yÞjreaches the maximum value at the first singularpoint). Therefore, the point having the maximumjD~f ðx ;yÞj value is taken as the first singular point(x1,y1). Once the first singular point is detected, thecomponent a � iðx � x1;y� y1Þ corresponding to thefirst singular point with a ¼ D~f ðx1;y1Þ Iðkx ;kyÞ

�� ��=N2 issubtracted from D~f ðx ;yÞ, yielding the residueD~f ðx ;yÞ � a � iðx � x1;y� y1Þ. The point at which theabsolute value of this residual image reaches

Figure 2. a: The scanning matrixI(kx,ky). b,c: The real and imagi-nary parts of i(x,y) ¼ IDFT(I(kx,ky)),respectively. d: The finite differ-ence Df(x,y) in y direction of theShepp Logan image. e,f: The realand imaginary parts of D~f ðx ;yÞ ¼Df ðx ;yÞ � iðx ;yÞ, respectively.

MRI Reconstruction by 2D Truncated k-Space 1199

maximum is then taken as the second singular point.Repeating this two-stage process until the maximumabsolute value of the residue is smaller than a prede-fined threshold T, the Q singular points can bedetected one by one.

The selection of the threshold T is subject to thenoise level in the image. If it is too large, there is a riskof having false negative detection of the singular points.If it is too small, false positives are likely to occur dueto the noise. Therefore, there exists an optimal valuefor the threshold T. Empirically, the optimal T value isaround the value of four times the standard deviationof the noise. According to Eq. [13], we can define thefollowing algorithm for determining singular points:

• Step 1: Obtain the spatial image ~f ðx ;yÞ from theacquired 2D partial k-space data using the zero-filling method and compute D~f ðx ;yÞ ¼ ~f ðx ;yÞ�~f ðx ;y� 1Þ, and let j ¼ 1.

• Step 2: Search for the singular point (xj,yj) atwhich the maximum a ¼ D~f ðx1;y1Þ Iðkx ;kyÞ

�� ��=N2

is reached. Add (xj,yj) to the singular point set Q,and construct i(x � xj, y � yj).

• Step 3: Compute rðx ;yÞ ¼ D~f ðx ;yÞ � a � iðx�xj;y� yjÞ, and let D~f ðx ;yÞ ¼ rðx ;yÞ.

• Step 4: If max1�x ;y�N ð rðx ;yÞj jÞ > T (where T is pre-defined so that the singular points with smallerD~f ðx ;yÞ��� ��� are omitted), then j ¼ j þ 1 and go toStep 2.

• Step 5: Output the singular point set{(x1,y1),(x2,y2),. . .,(xQ,yQ)} and end.

Once the singular points {(x1,y1),(x2,y2),. . .,(xQ,yQ)}are obtained, the singular spectral function Wj(kx,ky)is constructed, and the singularity degrees areobtained by solving the following equation system forall acquired k-space data:

F ðkx ;kyÞ ¼XQj¼1

ajWj kx ;ky

� �; I kx ;ky

� � ¼ 1: ½14�

Since Eq. [14] may be under- or overdetermined,pseudo inverse is used to obtain the solution a1,a2,. . .,aQ.

The nonacquired k-space data F ðkx ;kyÞ can then berecovered using the singularity spectrum functionsWj(kx,ky) and the singularity degrees a1,a2,. . .,aQ

F ðkx ;kyÞ ¼XQj¼1

ajWj kx ;ky

� �; I kx ;ky

� � ¼ 0: ½15�

The final spatial image f(x,y) is reconstructedthrough inverse Fourier transform of the acquired andthe recovered k-space data:

f ðx ;yÞ ¼ IDFT ðF ðkx ;kyÞIðkx ;kyÞ þ F ðkx ;kyÞÞ ½16�

We refer to the above reconstruction method as the2D SFA method.

RESULTS

2D Partial k-Space Acquisitions

We evaluated the proposed reconstruction method onsynthetic, phantom, and human brain images. Thesynthetic images are Shepp Logan images of 256 �256 (Fig. 4a), which were corrupted by an additivewhite Gaussian noise of zero-mean and various stand-ard deviations (SDs) 1,2,. . .,9, and were modulatedby the phase illustrated in Fig. 3. The scanning matrixof a 2D partial k-space was simulated from the com-

plete k-space using Iðkx ;kyÞ ¼ 1; �20 � kx ;ky < 1280; otherwise

�

as shown in Fig. 4f.The phantom images were collected from a commercial

phantom used to assess the resolution of an MRI system.They were acquired on a 7T Bruker system using a multi-slice multiecho (MSME) sequence (512 � 512, TR ¼ 600msec, TE ¼ 10 msec, slice thickness ¼ 0.6 mm, NEX ¼ 1,FOV ¼ 80 mm) (T1-weighted). The scanning matrix

was Iðkx ;kyÞ ¼ 1; �88 � kx < 88;�88 < ky < 2560; otherwise

�,

as shown in Fig. 8f.The human brain images were acquired on a 1.5T

whole body Siemens Sonata system (Erlangen, Ger-many), using a 3D sagittal T1 MPR sequence (matrixsize of 256 � 256 � 176, spatial resolution of 1 � 1 �1 mm3, TR ¼ 1970 msec, TE ¼ 4.69 msec, NEX ¼ 1).The scanning matrix was the same as that for SheppLogan images.

Performance Evaluation

The images reconstructed from the complete k-spacedata using inverse Fourier transform were used asthe references. The conventional ZF and POCSmethods were applied to the 2D partial k-space dataand compared with the proposed method. Thereconstruction quality was assessed by visuallyobserving the difference in the modulus image, thephase image, the spectra (ie, k-space data)log(1þ|F(kx,ky)|), and the error image between thereference image and the reconstructed one, and byquantitatively calculating the error SD and the his-togram of the error image.

Figure 3. The spatial phase used to modulate the syntheticimage.

1200 Luo et al.

Reconstruction of Synthetic Images

Figure 4 shows the reconstruction results on the syn-thetic Shepp Logan images corrupted by an additivewhite Gaussian noise of zero-mean and SD 5 andmodulated by the phase illustrated in Fig. 3. Theimage reconstructed by the 2D SFA method (Fig. 4d)is visually very close to the reference image (Fig. 4a),which is not the case with the ZF (Fig. 4b) or POCSmethod (Fig. 4c). Artifacts are perceptible in both ZFand POCS images, while they are not noticeable in the2D SFA image. The difference between the k-space

(Fig. 4e) of the reference image and that (Fig. 4h) ofthe 2D SFA reconstructed image is very small. On thecontrary, many frequency components are lost or vis-ually distorted in the k-space corresponding to the ZF(Fig. 4f) and POCS (Fig. 4g) methods. More quantita-tively, the SD curves of error images for the threemethods are plotted in Fig. 5 as a function of thenoise levels, that is, the Gaussian SDs, ranging from1 to 9. The error images are the differences betweenthe reference image and the reconstructed images.Regardless of the noise level, the 2D SFA reconstruc-tion always induces smaller SD values than both theZF and POCS methods.

Figure 6, which gives both gray-level (right column)and graphical (left column) representations of thesame error images, further visually demonstrates thereconstruction quality of the ZF, POCS, and 2D SFAmethods. The error images were calculated by sub-tracting the reference image (Fig. 4a) from each of theimages reconstructed by the ZF (Fig. 4b), POCS (Fig.4c), and 2D SFA (Fig. 4d) methods. The ZF and POCSreconstructions (Fig. 6a,b) exhibit much larger errorvalues at the edges than the 2D SFA reconstruction(Fig. 6c). In particular, the error images resulting fromthe ZF and POCS methods present very visible ana-tomical patterns compared with that of the 2D SFAmethod. This verifies that ZF and POCS reconstruc-tions are not able to recover the nonacquired k-spacedata well, whereas the 2D SFA method can.

Figure 7 gives more quantitative comparisonbetween the ZF, POCS, and 2D SFA methods. Thecurves represent the histograms of the error imagesbetween the reference and the reconstructed images.In the ideal reconstruction case, in which all the non-acquired k-space data are perfectly recovered, the his-togram is a Dirac impulse at the origin. The histogramwould spread out if the reconstruction has errors. Theflatter the histogram, the larger the error. The histo-gram corresponding to the 2D SFA method is the

Figure 4. Results of reconstruction on synthetic images.a: The reference image. b–d: The images reconstructed fromthe 2D truncated k-space data using respectively the ZF,POCS, and 2D SFA methods. e–h: The spectra of the imagesin (a–d), respectively.

Figure 5. Reconstruction errors of different methods as afunction of noise levels. The horizontal axis represents theSD of the added noise, and the vertical axis is the SD of theerror images between the reference image and the imagesreconstructed by the ZF, POCS, and 2D SFA methods,respectively. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

MRI Reconstruction by 2D Truncated k-Space 1201

narrowest, thus demonstrating that the reconstruc-tion errors induced by the 2D SFA are the smallest.

Reconstruction of Phantom Images

Reconstruction results on physical phantom imagesare shown in Fig. 8, where Fig. 8a represents the ref-erence image corresponding to the complete k-space.The images reconstructed using the ZF (Fig. 8b) andPOCS (Fig. 8c) methods present many artifacts, whichare particularly visible at the edges and in the stripedarea that experiences rapid variations. To better visuallyassess the quality difference among the three recon-struction methods, we also show in Fig. 8 a zoomedregion, in which artifacts in the region between theblack disc and the striped area (or between the blackring and the striped area) are clearly visible in the ZFand POCS reconstructed images, whereas in the 2DSFA reconstructed image, no artifacts are noticeable

Figure 7. Histograms of the error images between the refer-ence and reconstructed images. [Color figure can be viewed inthe online issue, which is available at wileyonlinelibrary.com.]

Figure 6. Graphical (left column)and gray-level (right column) rep-resentations of the same errorimages. a–c: The ZF, POCS, and2D SFA methods, respectively.

1202 Luo et al.

(Fig. 8d). The reconstruction difference between thethree methods can also be seen in k-space. As expected,the ZF method does not recover any frequency compo-nents and the nonacquired k-space data are simplyreplaced by zeros. Although the POCS method hasrecovered the nonacquired k-space data, the recoveredk-space data present larger errors than those recoveredfrom the 2D SFA method. This can be readily observed

when comparing the reference k-space (Fig. 8e) with therecovered k-spaces shown in Fig. 8g,h.

Reconstruction of Human Brain Images

Reconstruction results on a human brain volume areshown in Fig. 9, where reconstruction quality isassessed in terms of the SD of error images as a

Figure 8. Reconstruction results on physical phantom images. a: The reference phantom image. b–d: The images recon-structed from the 2D truncated k-space data using respectively the ZF, POCS, and 2D SFA methods. e–h: The spectra cor-responding to the images in (a–d), respectively. i–l: Respectively the zoomed versions of the highlighted rectangular regionsin (a–d).

MRI Reconstruction by 2D Truncated k-Space 1203

function of image slices ranging from 50 to 130. Forall slices the 2D SFA method always induces thesmallest SD values compared with the ZF and POCSmethods. For some slices (at the beginning or end ofthe brain volume), the POCS method generated largerSD values than the ZF method. An example of thereconstructed images (88th slice) is illustrated in Fig.10, where the left column represents the modulusand the right column is the phase of the reconstructedimages. The image reconstructed by the 2D SFAmethod is less blurred and has more details than theimages reconstructed by the ZF and POCS methods.This can also be clearly observed in the zoomed region.The difference in reconstruction quality between thethree reconstruction methods is particularly visible inthe phase images, in which the phase reconstructedwith the 2D SFA method (Fig. 10h) is much closer tothe reference phase (Fig. 10e) than those obtained withthe ZF (Fig. 10f) and POCS (Fig. 10g) methods. It canbe seen that the phase generated by the POCS methodis smoother than that generated by the ZF method.

DISCUSSION

The essential aspect of the proposed method is torecover, using the 2D SFA model, the nonacquiredk-space data from the 2D partial k-space acquisition,in which the k-space data are truncated in both thephase-encoding direction and the frequency-encodingdirection. Although the 2D SFA can represent exactlyany type of image without making any assumptions(low-frequency phase, piecewise constant variation,smoothness variation, etc.) on the image, the effi-ciency of this model to reconstruct images fromincomplete k-space data depends on how much spa-tial correlation is presented in the image. If the imageis highly piecewise constant, the number of singular

points will be small and the proposed reconstructionwill be very efficient. In the extreme case that all pix-els are singular (which corresponds to a noise image),although the 2D SFA can still represent the image,accurate reconstruction will not be possible if thek-space under investigation is incomplete. Neverthe-less, spatial correlation always exists in MR images ofhuman organs. Therefore, in practice, the number ofsingular points is always much smaller than the num-ber of image pixels, and the proposed reconstructioncan always work more or less efficiently.

In addition, the performance of the proposed 2DSAF model is also influenced by the size of thescanned k-space data block. The larger the size, thehigher the reconstruction accuracy, but with lessreduction of the acquisition time. When the size is toolarge, although reconstruction accuracy is very highin this case, the reduction of the scanning time willnot be significant. Likewise, the smaller the size, thefaster the scanning speed, but the reconstructionerrors become nonnegligible and distortions mayoccur in the reconstructed image. Reconstructionaccuracy can be further improved if the noise corrup-tion on the singularity degrees can be reduced. Onthe other hand, the choice of the threshold used fordetermining singular points could be a source of thereconstruction errors. The greater the threshold,the higher risk there is of having false negatives of thesingular points. The smaller the threshold, the higherrisk there is of having false positives of the singularpoints. Both false positive and false negative detectionof the singular points could degrade reconstructionaccuracy of the 2D SFA method. For a given image,when its k-space data are undersampled too much,errors in determining singular points as well as singu-larity degrees will be introduced. Missing k-space datawill not be exactly recovered in this case, thus causingtruncation artifacts in the reconstructed image.

For all the synthetic, phantom, and human brainimages studied in the present research, the k-spacedata used were chosen to be about 20%–30% of thecomplete k-space, the threshold in the 2D SFA modelequal to four times the SD of the noise in the recon-structed image. Under these conditions, the modulus,phase, and spectral images reconstructed by the 2DSFA method were fairly close to the reference images.The ZF method showed the worst performance. Thisis not surprising, because the ZF method does notrecover any nonacquired k-space data. The POCSmethod can recover some missing k-space data, butthe reconstruction errors were not negligible and thequality of the recovered data is not comparable to thatrecovered by the 2D SFA method. This can beexplained by the fact that the POCS method is basedon the assumption that the phase of the entire imagecan be replaced by the phase estimated from the cen-tral part of the k-space. Therefore, the POCS methodcan never give exact reconstruction unless all the fre-quency components of the image to be reconstructedare totally limited to the central part of its k-space,which is not the case for physical MR images.

Like the TV method, the proposed method alsomakes use of neighborhood correlation of image

Figure 9. Assessment of reconstruction quality of a humanbrain volume. The horizontal axis represents the slice num-ber and the vertical axis is the SD of the error imagesbetween the reference slice and the corresponding recon-structed slices reconstructed using the ZF, POCS, and 2DSFA methods. [Color figure can be viewed in the online issue,which is available at wileyonlinelibrary.com.]

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points, which exists in all images except purely noiseimages. The TV method exploits neighborhood correla-tion through minimizing the total variation. It neitheraccounts for the location of edges, nor seeks to findthe true solution corresponding to the completek-space acquisition, but just to find a solution opti-mizing the objective function among the infinite num-

ber of solutions. In contrast, the proposed methodaims to find the true solution. The 2D SFA modelallows an exact representation of an image, in a con-tent-independent manner (regardless of the content ofthe image). If the image presents spatial correlation,the proposed reconstruction enables us to find anaccurate solution from the incomplete k-space data.

Figure 10. Illustration of reconstruction results on the 88th slice of the human brain volume. a: The reference slice. b–d: The sli-ces reconstructed from the 2D truncated k-space data using respectively the ZF, POCS, and 2D SFA methods. e–h: The phaseimages corresponding respectively to (a–d). i–l: The zoomed versions of the highlighted rectangular regions in (a–d), respectively.The scanning matrix is the same as in Fig. 4f and the percentage of data sampled is 33%.

MRI Reconstruction by 2D Truncated k-Space 1205

The proposed method also shares some similaritywith the CS method in the sense that both methodsexploit spatial correlation of image pixels throughsparse representation of the image to be recon-structed, and turn the underdetermined mathemati-cal problem due to k-space undersampling into awell-defined one. However, the approaches to obtain-ing the parameters of the sparse representation arevery different in these two methods. The CS methoduses optimization techniques to obtain the sparsestrepresentation of the image. In contrast, the proposed2D SFA method extracts explicitly (in a layered man-ner) singular points (edge information), and then esti-mates the singularity degrees by solving a system ofequations; it does not seek the sparest representation,but aims to obtain exact reconstruction of the originalimage. Finally, the work of Gottlieb and Shu (25) givesa very interesting mathematical basis on how torecover a function from a finite number of its Fouriercoefficients, which could benefit further developmentof the proposed method.

In conclusion, the proposed approach offers aneffective way to tackle the problem of image recon-struction from a 2D partial k-space, in which the k-space data are missing in both the phase-encodingdirection and the frequency-encoding direction. De-spite the high asymmetry of such a k-space, with arelatively small quantity of k-space data, the approachcan still reconstruct the image with comparable qual-ity to the reference image that is reconstructed fromthe complete k-space. In addition, the approach needsneither phase estimation nor phase correction, andthus overcomes the assumption of slow phase varia-tions. It provides a new way of fast imaging for appli-cations that require a significant reduction of the ac-quisition time.

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