Journal of Magnetic Resonance 228 (2013) 136–147

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Journal of Magnetic Resonance

journal homepage: www.elsevier .com/locate / jmr

An efficient de-convolution reconstruction method for spatiotemporal-encodingsingle-scan 2D MRI

Congbo Cai a, Jiyang Dong b, Shuhui Cai b, Jing Li b, Ying Chen b, Lijun Bao b, Zhong Chen b,a,⇑a Department of Communication Engineering, State Key Laboratory for Physical Chemistry of Solid Surfaces, Xiamen University, Xiamen 361005, Chinab Department of Electronic Science, Fujian Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen 361005, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 June 2012Revised 26 December 2012Available online 11 January 2013

Keywords:Single-scan MRIImage reconstructionSpatiotemporal-encodingDe-convolution

1090-7807/$ - see front matter � 2013 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.jmr.2012.12.020

⇑ Corresponding author at: Department of Electronratory of Plasma and Magnetic Resonance, XiamenChina. Fax: +86 592 2189426.

E-mail address: chenz@xmu.edu.cn (Z. Chen).

Spatiotemporal-encoding single-scan MRI method is relatively insensitive to field inhomogeneity com-pared to EPI method. Conjugate gradient (CG) method has been used to reconstruct super-resolvedimages from the original blurred ones based on coarse magnitude-calculation. In this article, a new de-convolution reconstruction method is proposed. Through removing the quadratic phase modulation fromthe signal acquired with spatiotemporal-encoding MRI, the signal can be described as a convolution ofdesired super-resolved image and a point spread function. The de-convolution method proposed hereinnot only is simpler than the CG method, but also provides super-resolved images with better quality. Thisnew reconstruction method may make the spatiotemporal-encoding 2D MRI technique more valuable forclinic applications.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Accelerating the acquisition process, or equivalently increasingthe achievable resolution with a fixed acquisition time, is of majorinterest for magnetic resonance image (MRI) [1–3]. Ultrafast MRIplays an essential role in experiments demanding high temporalresolution like functional MRI [4,5], free-breath heart imaging[6,7] and high-dimensionality experiments such as diffusion tensorimaging [8–10]. Echo-planar imaging (EPI) has been a commonlyused single-scan MRI technique for a long time. However, EPI isvery sensitive to field inhomogeneity. In contrast to EPI’s relianceon signal contributions arising simultaneously from the entire ex-cited body, a single-scan MRI method recently proposed by Fryd-man and co-workers relies on local signal contributionsprogressively along the encoding direction [11–13]. This methodis based on spatiotemporal-encoding (SPEN) of MR spin interac-tions. At each instant, the intensity of acquired signal is propor-tional to a corresponding local spin density. This property allowsSPEN MRI to cope efficiently with field inhomogeneities and to re-solve image component of multiple chemical sites in the sample[14,15]. The unique sequential excitation of magnetization inher-ent in frequency sweep encoding promises to open an interestingvenue for novel image contrast generation and for designing newimaging methods such as for magnetic resonance angiography

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ic Science, Fujian Key Labo-University, Xiamen 361005,

[16,17]. Sequential excitation of magnetization has been provento be highly valuable in minimizing geometrical distortions in sin-gle scan imaging under severe B0 field inhomogeneity [17,18].Based on similar spatiotemporal-encoding principle, Chamberlainand co-workers also proposed a new ultrafast magnetic resonanceimaging method [19]. The technique is based on rapid acquisitionby sequential excitation and refocusing (RASER). RASER methodcan sequentially refocus magnetization with the same echo timefor all spins, and the effects of chemical shift and field inhomoge-neity can both be reduced [20].

The SPEN and RASER MRI methods are both based on the spatio-temporal-encoding mechanism, and introduce a quadratic phase tothe spin signal. Images acquired from spatiotemporal-encodingMRI without super-resolved (SR) reconstruction have low resolu-tion. Conjugate gradient (CG) method has been proposed to getSR images. It should be noted that the quadratic phase modulationhas also been introduced into MRI for some different purposes [1],such as improving dynamic range [21,22], or reducing aliasing arti-facts [23], or in a compressed sensing perspective [1].

CG method is effective for most cases. However, its effective-ness depends on the small condition number of the coefficient ma-trix, which may not always be satisfied. In this paper, a new de-convolution method is proposed to reconstruct the SR images ina more simple and efficient way.

2. Theory

For the sake of simplicity in the following theoretical derivation,all of the off-resonance effects are neglected. A one-dimensional

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 137

sample along y-axis with a field of view (FOV) of L is considered.The imaging scheme we focus upon relies on spatially dependentexcitation and directly detecting the spin density profile at sequen-tial locations along the y-axis. The following theoretical deductionis for SPEN sequence, and similar deduction can be performed forRASER sequence. The signal for the sequence shown in Fig. 1bcan be expressed as the following [12]:

sðtÞ /Z L=2

�L=2qðyÞe

i �O2

i2R�

c2G2e y2

2R þcGeTey2 þcGayt

� �dy; ð1Þ

For simplicity, Ga is assumed to be constantly applied over anacquisition time Ta with jGeTej ¼ jGaTaj [12]. In Eq. (1), Oi is the ini-tial sweep frequency of 90� chirp pulse (in, e.g., rad/s), c is thegyromagnetic ratio of inspected nucleus; Ge is the encoding gradi-ent amplitude and Te is its duration; Ga is the decoding gradientamplitude; R is the sweep rate of the chirp pulse; qðyÞ is the spindensity at position y, and t varies from 0 to Ta.

Eq. (1) can be discretized, rendering a linear set of equations [12]:

~sðtiÞ ¼XMSR

k¼1

bPðyk; tiÞ~qSRðykÞ; i ¼ 1; . . . ;N: ð2Þ

where N = Npe, ~sðtiÞ is the experimentally measured signal vectorcomposed of N complex spatially encoded points, f~qSRðykÞgk¼1;...;MSR

Fig. 1. MRI sequences. (a) Blipped spin-echo EPI sequence, (b) SPEN sequence, andTacq = 2(Tro + Tpe) � Npe (see text for more details). Abbreviations: e: spatially encoded excacq: acquisition, and Npe: number of phase encoding steps.

is a column vector of MSR real or complex numbers making up theSR enhanced image and to be reconstructed, and bPðyk; tiÞ is a com-plex coefficient matrix depending on the phase terms given in Eq.(1). In the previous paper [12], SR method was applied to solve thislinear set of equations to obtain the ‘‘super-resolved’’ f~qSRðykÞg vec-tor from ~sðtiÞ. A least-square error minimization of the unknowntarget function based on the CG method was chosen [12].

~qSR ¼ arg minXN

i¼1

XMSR

k¼1

bPðyk; tiÞ~qSRðykÞ �~sðtiÞ�����

�����2

: ð3Þ

In Eq. (3), if the condition number of bP matrix is too large, thereconstructed result will be very sensitive to noise and non-idealfactors, and the reconstruction may fail. If the reconstruction issuccessful, the limit of MSR = N can usually be reached in the CGmethod. This means that the SR images can be totally recoveredby the reconstruction.

However, according to the SPEN MRI principle, the signal ac-quired at time t in Eq. (1) is mainly from a limited region centeredon y0. Supposing t varies from 0 to Ta, and the chirp pulse is selectedwith sweep rate R > 0 and Ge > 0, y0 will vary from L/2 to �L/2 (ifdifferent signs of R and Ge are selected, the derivation will be sim-ilar), thus we have the relationship

t ¼ Ta

2� Tay0

L: ð4Þ

(c) RASER sequence. For the three sequences, the overall acquisition time isitation, se: spin-echo, ro: readout direction, pe: phase-/spatially encoded direction,

138 C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147

Substituting Eq. (4) into Eq. (1), the signal integral can be mod-ified into

sTa

2� Tay0

L

� �/ e�

iO2i

2R

Z L=2

�L=2qðyÞe

i �c2G2

e y2

2R þ12cGeTeyþcGay Ta

2 �Tay0

L

� �h idy: ð5Þ

Provided that A ¼ cGaTaL , the signal integral can be further mod-

ified into

sTa

2� Tay0

L

� �/ e�

iO2i

2R

Z L=2

�L=2qðyÞeið�

c2G2e y2

2R þ12cGeTeyþcGayTa

2 ÞeiA2ðy�y0Þdy: ð6Þ

In order to produce a convolution term, Eq. (6) should be rewrit-ten as

sTa

2� Tay0

L

� �/ e�

iO2i

2R

Z L=2

�L=2qðyÞe

i �c2G2

e y2

2R þ12cGeTeyþcGayTa

2 �Ay2

2 �Ay0

2

2

� �eiA2ðy�y0 Þ2 dy:

ð7Þ

Eq. (7) can be transformed into

s Ta2 �

Tay0

L

� �e�

iO2i

2R e�iAy02

2

/Z L=2

�L=2qðyÞe

i �c2G2

e y2

2R þ12cGeTeyþcGayTa

2 �Ay2

2

� �eiA2ðy�y0 Þ2 dy: ð8Þ

Considering the gradients relationship shown above for spatio-temporal-encoding between the excitation (encoding) and acquisi-tion (decoding) processes and the existence of the refocusing pulse,GaTa ¼ �GeTe should be satisfied. In combination with A ¼ cGa

TaL ,

we can get

� c2G2e y2

2Rþ 1

2cGeTeyþ cGay

Ta

2� Ay2

2¼ 0: ð9Þ

Therefore, Eq. (8) can be simplified into

s Ta2 �

Tay0

L

� �e�

iO2i

2R e�iAy02

2

/Z L=2

�L=2qðyÞeiA2ðy�y0Þ2 dy; ð10Þ

or described as

lðy0Þ /Z L=2

�L=2hðyÞmðy� y0Þdy; ð11Þ

where lðy0Þ ¼ s Ta2 �

Ta y0L

� �e�

iO2i

2R e�iAy02

2

, hðyÞ ¼ qðyÞ, and mðyÞ ¼ eiA2y2 . Eq. (11) denotes

a convolution relationship l ¼ h�m and can be discretized as

~lðyiÞ /XN

k¼1

~hðykÞ~mðyi � ykÞ; i ¼ 1; . . . ;N; ð12Þ

where~lðyiÞ is the demodulated signal vector composed of N com-plex spatially encoded points;~hðykÞ is a column vector of N complexpoints making up the SR enhanced image; ~mðyi � ykÞ is a columnvector of 2N complex points discretized from function mðyÞ.

In frequency domain, the convolution relation can be trans-formed into the product relation [24]

FT½~lðyÞ� / FT½~hðyÞ� � FT½~mðyÞ�: ð13Þ

Here, FT means the discrete Fourier series. Hence the following for-mula can be derived:

~hðyÞ / IFTfFT½~lðyÞ�=FT½~mðyÞ�g: ð14Þ

Here, IFT is the inverse discrete FT. In the presence of noise, Eq. (14)is not robust for regions of small FT½~mðyÞ� value [25]. However, be-cause the power spectrum of ~mðyÞ is rather stable inside the band-width, and the bandwidth of~lðyÞ is usually far smaller than that of~mðyÞ, it is safe enough to ignore FTð~hðyÞÞ value when the value ofFT½~mðyÞ� is small because the latter is outside the effectivebandwidth.

It can be seen from Eq. (11) that~lðyÞ has a much simpler wave-form than~sðtÞ (discrete sðtÞ in Eq. (1)). The~sðtÞ and~lðyÞ from a uni-form one-dimensional (1D) simulation model and from an in vivorat brain are shown in Fig. 2. It can be seen that~lðyÞ is not oscillat-ing as violently as ~sðtÞ. The quadratic phase component has beeneliminated by the transformation from~sðtÞ to~lðyÞ. The severe phasewrapping occurring in ~sðtÞ disappears in~lðyÞ. It should be notedthat the quadratic phase information from~sðtÞ does not carry anyinformation about the imaged object and is solely determined bythe parameters of pulse sequence. For this reason, the transforma-tion from~sðtÞ to~lðyÞ does not lose any useful information for imagereconstruction. The simplicity of~lðyÞ is favorable for reconstructioncalculation.

For single-scan 2D SPEN MRI, usually only dozens of echoes canbe acquired before the signal is attenuated greatly by transverserelaxation. However, if the FOV is L, ~mðyÞ is approximately bandlimited at its maximum instantaneous frequency AL=ð4pÞ. Accord-ing to the Nyquist sampling theorem, the sampling rate should beat least fsampling ¼ AL=ð2pÞ ¼ cGaTa=ð2pÞ, so N P cGaTaL=ð2pÞshould be satisfied, where N is the number of spatially encodedpoints. In usual cases, N is far bigger than the acquirable number.Therefore, to avoid the reconstruction artifacts from the frequencyfolding of ~mðyÞ, the acquired signal should be linearly interpolated.Conducting linear interpolation to original signal~sðtÞ would resultin serious problem because the phase of~sðtÞ is wrapped seriously,while with a less-phase-wrapped~lðyÞ, the problem will be allevi-ated greatly.

For SR reconstruction method, the SR enhancement factor (SRF)can be calculated according to the following equation [26,30]:

SRF ¼Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

DO � Tep ; ð15Þ

where DO is the bandwidth of the chirp pulse. We can use SRF torepresent the efficiency of SR reconstruction for spatiotemporal-encoding MRI method under different experiment conditions.

3. Materials and methods

To evaluate the algorithms described above, the SR results fromthe de-convolution reconstruction method are compared withthose from the CG reconstruction method. The images obtainedwith the blipped EPI sequence are also provided for comparison.The single-scan spatiotemporally encoded 2D MR images are ob-tained with SPEN and RASER methods unless otherwise specified.The sequences shown in Fig. 1 were applied in simulations, phan-tom experiments, and in vivo rat brain experiments. The experi-ments were performed at 298 K on a 7.0 T Varian MRI systemwith a horizontal-bore Magnex magnet, equipped with 10 cm boreimaging gradients (40 G/cm) (Varian Associates, Inc., Palo Alto, CA).In in vivo rat brain experiments, the rats were anaesthetized andmaintained in an animal bed. All the operations were handled un-der protocols approved by the Animal Care and Use Committee inour university. The simulations and post-experimental data pro-cessing were all carried out with the SPROM software developedby our group on a personal computer (Intel Core i7, 12 GB memo-ries, 64 bit version Windows 7 operation system) [27]. (All theimaging sequences and post-processing codes used in this studyare available upon request.)

In the 2D spatiotemporal-encoding MRI, spatiotemporal-encoding and conventional k-encoding were acted along theorthogonal y and x axis, respectively. This means that the de-con-volution reconstruction and CG reconstruction were both appliedalong the y axis only. For the CG reconstruction method, the num-ber of iterations was 30 and no regularization was used becausethe condition number of P matrix was relatively low (usually

Fig. 2. (a and b) Are~sðtÞ and~lðyÞ from a uniform 1D simulation model, respectively. The simulation parameters are L = 6.0 cm, Te = 3 ms, Ge = 0.025 T/m, Ga = 0.11 T/m. (c andd) Are~sðtÞ and~lðyÞ from an in vivo rat brain. The parameters are L = 4.5 cm, Ge = 0.033 T/m, Te = 4 ms, Ga = 0.035 T/m.

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 139

smaller than 6). In the multi-scan MRI and spin-echo EPI, the imagematrix size was 512 � 512 after 2D FT. For a fair comparison, theSR images from CG reconstruction and de-convolution reconstruc-tion had the same matrix size. It should be pointed out that the SRimages from de-convolution reconstruction usually have morepoint number than the acquired one along the spatiotemporal-encoding axis by the interpolation of ~lðyÞ. For example, 64 datapoints were acquired, and 512 data points were got after de-convo-lution reconstruction in this paper. However, for CG reconstruc-tion, the interpolation of ~sðtiÞ is difficult because it is seriouslywrapped. Therefore, to reconstruct SR images with the same ma-trix size as the one used in the de-convolution reconstruction,the interpolation was conducted to ~qSR after the CG reconstructionwas done. Both CG reconstruction and de-convolution reconstruc-tion conducted a FT with zero padding along the frequency dimen-sion to 512 points.

4. Results

4.1. Numerical simulation

Figs. 3–5 show the different behaviors of the de-convolutionreconstruction method and CG reconstruction method under dif-ferent conditions. A matrix size of 512 � 512 grids and a FOV of6 � 6 cm2 were used for these 2D models in the simulation. InFig. 3, the model is some circles of different radiuses, as shownin Fig. 3a. It can be seen that the images obtained with pre-reconstructed magnitude-processing possesses a poor definition(Fig. 3c). Here the magnitude processing is to perform FFT alongthe readout-direction to achieve the signal magnitude. The de-convolution method (Fig. 3d–f) and CG method (Fig. 3g–i) can bothimprove the fidelity of images with different chirp pulse settingand echo number. To give a more qualitative comparison between

the de-convolution method and CG method, the region circum-scribed with a green square in Fig. 3a is amplified for Fig. 3a, b, dand g, and shown in Fig. 3j–m correspondingly. From Fig. 3l andm, we can see that the de-convolution method can give an SR im-age with higher digital resolution and better quality than CG meth-od. After the SR reconstruction, the quality of image is comparableto that from the EPI sequence under a similar sequence pattern andparameters. However, there are still some weak artifacts in the SRreconstructed image (Fig. 3d).

By analyzing Eq. (8) we can conclude that the convolution inFourier domain induced by the chirp modulation implies that thebandwidth of the modulated signal is the sum of the individualbandwidths of the original signal qðyÞ and the chirp modulation[1]. It should be noted that the bandwidth of the original signalqðyÞ does not mean that the signal is band-limited and some en-ergy may remain beyond it [1]. The artifacts are produced by un-der-sampling the images. When the bandwidth of the chirpmodulation is reduced from 64 kHz/3 ms to 32 kHz/3 ms, the arti-facts are reduced accordingly (Fig. 3e and h). The artifacts are alsoreduced when the acquisition number of spatially encoded pointsincreases to 128 (Fig. 3f and i). The peripheral profile of the imagedobject corresponds mainly to the high frequency components in k-space, therefore the artifacts arise mainly from it. In Fig. 4, thesharpness of the circles increases from (a to d) step by step. Whenthe sharpness of the peripheral profile increases, the folding arti-facts become more obvious under the same simulation condition(Fig. 4e–h and i–l).

In this article, the linear interpolation was conducted in de-convolution method. We should admit that the linear interpolationis an imperfect interpolation and may be one source of the aliasingartifacts. Other interpolation method may give results with fewerartifacts, which deserves further study. However, we can see thatthe conventional CG method also produces artifacts because ofthe under-sampling (Figs. 3 and 4).

Fig. 3. Simulated MR images (6 � 6 cm2) obtained from single-scan EPI and SPEN MRI under homogeneous field. The total scan time is 36 ms with image matrixsize = 64 � 128 along the y and x axis respectively if not specified, sw = 256 kHz. Here vertical axis is phase/spatiotemporally-encoded axis and horizontal axis is frequency-encoded axis. (a) Simulation model; (b) spin-echo EPI image (Te = 2 ms, Tse = 2 ms), (c) SPEN MR image before SR reconstruction collected under a condition similar to (b).Te = 3 ms; R = 64 kHz/3 ms, Ge = �0.025 T/m, Ga = �0.107 T/m, SRF = 4.6; (d) SR reconstruction image of (c) using de-convolution method; (e) SR reconstruction image under acondition similar to (c) except for the chirp pulse which has R = 32 kHz/3 ms, Ge = �0.0125 T/m, SRF = 6.5; (f) SR reconstruction image under a condition similar to (c) exceptthat the echoes number is 128, SRF = 9.2, and the total scan time is 67 ms; (g–i) are reconstruction images corresponding to (d–f) respectively using CG method; (j–m) areexpanded views of (a, b, d and g) of the region circumscribed in the green square of (a).

140 C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147

The greatest advantage for spatiotemporal-encoding single-scanMRI over the conventional single-scan EPI is that it possesses a high-er built-in immunity to the field inhomogeneity. The acquired sin-gle-scan data possess spatial and temporal resolutions comparableto those achieved by EPI, but have a much higher immunity to fre-quency-dispersing artifacts. The comparison of Fig. 5b with c showsthat when the frequency bandwidth of the chirp pulse is reducedfrom 64 kHz/3 ms to 32 kHz/3 ms, the ability of spatiotemporal-encoding MRI in resisting the background field inhomogeneity isalso weakened, though smaller frequency band limit of the chirppulse can lead to less serious folding artifacts.

4.2. Phantom validation

In phantom experiments, a polyethylene bottle with a diameterof about 4.5 cm was filled with water and six smaller polyethylenebottles with a diameter of about 1.3 cm, three of which were filledwith water and others were empty. FOV was 6 � 6 cm2, and theslice thickness was 0.2 mm. Multi-scan gradient echo sequencewas used to give a reference imaging. EPI, SPEN and RASER se-quences (Fig. 1) were implemented under the same experimentconditions except that RASER sequence had a longer echo time topreserve self-refocusing capability.

Fig. 4. Reconstructed SR images by SPEN sequence using models with different sharpness of peripheral profile. The simulation parameters are the same as those for Fig. 3c.Here vertical axis is spatiotemporally-encoded axis and horizontal axis is frequency-encoded axis. (a–d) The original models; (e–h) SR reconstructed images of (a–d) using de-convolution method, respectively; (i–l) SR reconstructed images of (a–d) using CG method, respectively.

Fig. 5. EPI and spatiotemporally-encoded MRI images under an inhomogeneous field with about 600 Hz line-width along the x and y axes. The model and the simulationparameters are the same as the counterparts of Fig. 3. Here vertical axis is spatiotemporally-encoded axis and horizontal axis is frequency-encoded axis. (a) Image from EPI;(b) SR reconstructed image from SPEN MRI with R = 64 kHz/3 ms; (c) SR reconstructed image from SPEN MRI with R = 32 kHz/3 ms; (d) SR reconstructed image from RASERMRI with R = 6.4 kHz/30 ms and SRF = 4.6.

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 141

It can be seen from Fig. 6 that the images obtained with magni-tude-processed method from spatiotemporal-encoding MRI arerather blurry. Both the de-convolution method and CG methodcan produce SR images. The de-convolution method provides abetter result with less deviation from the reference multi-scan gra-dient echo image than CG method (Fig. 6f vs. i). The chirp pulseswith excitation bandwidth and duration of 64 kHz/4 ms and32 kHz/4 ms respectively produce images with similar quality.The spatiotemporal-encoding MRI gives better results than EPI,especially around the interface of water and air where strongsusceptibility heterogeneities exist, as shown in Fig. 6b.

In Fig. 7, the background field was poorly shimmed both alongthe spatially encoded axis and the frequency encoded axis. It canbe seen that the images reconstructed by the de-convolutionmethod retain far less distortion caused by the field inhomogeneity

compared to the images reconstructed by CG method. The largersweep rate of the chirp pulse (R = 64 kHz/4 ms) offers a higherinsensitivity to the field inhomogeneity than the smaller one(R = 32 kHz/4 ms) (Fig. 7f vs. g). It is coincident with the conclusionfrom Fig. 5. RASER can provide significantly higher immunity tooff-resonance effects, like those caused by inhomogeneous back-ground field and chemical-shift offsets, and can give a pure T2-weighted contrast mechanism [12]. In Fig. 7, RASER sequence pro-duces the best results under the inhomogeneous field.

4.3. In vivo rat brain imaging

Multi-scan spin echo imaging was conducted for reference. EPI,SPEN imaging and RASER imaging were conducted under the sameexperimental conditions in single scan. The resulting images dealt

Fig. 6. Phantom MRI experiments for different reconstruction methods under a relatively homogeneous field. The image matrix size = 64 � 64, sw = 250 kHz. Here verticalaxis is phase/spatiotemporally-encoded axis and horizontal axis is frequency-encoded axis. (a) Image obtained with multi-scan gradient echo sequence; (b) image obtainedwith EPI sequence (Te = 4 ms, Tse = 4 ms, total scan time = 50.5 ms); (c and d) images obtained with SPEN sequence after magnitude processing: (c) Te = 4 ms, Ge = 0.025 T/m,Ga = 0.026 T/m, R = 64 kHz/4 ms, SRF = 4.0, and total scan time = 51.1 ms; (d) Te = 4 ms, Ge = 0.0125 T/m, Ga = 0.013 T/m, R = 32 kHz/4 ms, SRF = 5.7, and total scantime = 51.1 ms; (e) image obtained with RASER sequence after magnitude processing. Te = 26 ms, Ge = 0.0043 T/m, Ga = 0.029 T/m, R = 11 kHz/26 ms, SRF = 3.8, and totalscan time = 94.3 ms. (f–h) SR reconstructed MR images of (c–e) using de-convolution method, respectively; (i–k) SR reconstructed MR images of (c–e) using CG method,respectively. The representative regions for SNR calculations are marked in (a), green for signals and red for noises. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

142 C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147

with magnitude processing method, de-convolution and CG SRreconstruction methods are all demonstrated in Fig. 8. The band-selective excitation feature of the spatiotemporal-encoding MRImakes it suitable for reduced FOV MRI. Reduced FOV MRI is veryuseful for imaging local parts of a large body because it can reducethe imaging time for a given spatial resolution or enhance the spa-tial resolution with the same imaging time [28,29]. For spatiotem-poral-encoding MRI, two methods can be used to achieve reducedFOV MR images. One is reducing the excitation range of chirp pulseso that only part of the imaged object is modulated by the qua-dratic phase. The other is encoding the whole FOV while decodingonly part of excited or modulated range. In practical experiments,these two methods can be applied solely or collectively. The de-convolution reconstruction method for the reduced FOV is the

same as that for the full FOV reconstruction except that thereconstruction parameters may be different. To show thatde-convolution method can also be effective for reduced FOVMRI, the reduced FOV rat brain images obtained with the firstmethod are also shown in Fig. 8.

It can be seen that for the spatiotemporal-encoding MR images,the reconstructed SR images from de-convolution method havehigher quality than those from CG method (Fig. 8). Comparingthe SR reconstructed images from spatiotemporal-encoding MRIwith EPI images, we can see that the spatiotemporal-encodingMRI can provide images with less distortions caused by suscepti-bility heterogeneities. However, the signal intensity on the left sideof the images from SPEN MRI is lower than that on the right side.This is because the signals are sequentially acquired from one side

Fig. 7. Phantom MR images for different reconstruction methods under a relative inhomogeneous field. The experimental parameters are the same as the counterparts ofFig. 6, respectively. Here vertical axis is phase/spatiotemporally-encoded axis and horizontal axis is frequency-encoded axis. (a) Image obtained with multi-scan gradientecho sequence; (b) image obtained with EPI sequence; (c and d) images obtained with SPEN sequence after magnitude processing: (c) R = 64 kHz/4 ms, (d) R = 32 kHz/4 ms;(e) image obtained using RASER sequence with R = 11 kHz/26 ms after magnitude processing; (f–h) SR reconstructed images of (c–e) using de-convolution method,respectively; (i–k) SR reconstructed images of (c–e) using CG method, respectively. The representative regions for SNR calculations are marked in (a), green for signals and redfor noises. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 143

to another of the images and the different echo time gives rise todifferent signal attenuation level on the images. This is differentfrom EPI. Though in EPI each echo also has different echo time, dif-ferent echo corresponds to different k-space position, not the imag-ing space position. One of the possible solutions to correct thesignal intensity deviation is to reduce the total acquisition time,but this will put a higher demand on the hardware because thefield gradients for imaging need to be switched more quickly,and will introduce stronger eddy effects. Post-processing theimages is a second choice. The RASER sequence, in which each echohas a same evolution time, is another choice. However, becauseRASER sequence needs a relatively long echo time, the relativelylow signal-to-noise ratio (SNR) of RASER image may be a obstaclefor an object with a relatively short T2 relaxation time, such asin vivo tissue.

4.4. SNR comparison

To compare the different behaviors of SR reconstruction meth-ods quantitatively, the SNR of SR reconstructed images from phan-tom experiments by de-convolution method and CG method areshown in Tables 1 and 2. The SNR values are measured as the ratiobetween the average of signals and the root-mean-square ofnoises. The signals are from the representative region marked witha green square as shown in Figs. 6a and 7a. It should be noted thatthe noises calculated for Table 1 are from the whole area outsidethe imaged object, so they are mainly from the artifacts after theSR reconstructions, and these SNR are denoted as aSNR. Therefore,the aSNR shown in Table 1 can be seen as a relatively quantitativedescription about the reconstruction artifacts. It can be seen fromTable 1 that de-convolution method can provide higher aSNR than

Fig. 8. Single-scan in vivo images of a mouse brain (FOV = 4 � 4 cm2, 2 mm slice thickness, image matrix size = 64 � 64 if not specified), sw = 250 kHz. Here vertical axis isfrequency-encoded axis and horizontal axis is phase/spatiotemporally-encoded axis. (a) Multi-scan spin echo image; (b) EPI image (Tse = 4 ms, Te = 4 ms, total scantime = 50.5 ms); (c–f) magnitude-processed spatiotemporal-encoding MR images: (c) SPEN imaging, Te = 4 ms, Ge = 0.038 T/m, Ga = 0.039 T/m, R = 64 kHz/4 ms, SRF = 4.0, totalscan time = 51.1 ms; (d) SPEN imaging, Te = 4 ms, Ge = 0.019 T/m, Ga = 0.020 T/m, R = 32 kHz/4 ms, SRF = 5.7, total scan time = 51.1 ms; (e) RASER imaging, Te = 13 ms,Ge = 0.0115 T/m, Ga = 0.0783 T/m, R = 19 kHz/13 ms, SRF = 4.1, total scan time = 94.3 ms, (f) SPEN imaging, Te = 4 ms, Ge = 0.019 T/m, Ga = 0.039 T/m, R = 16 kHz/4 ms, reducedFOV = 2 � 4 cm2, image matrix size = 32 � 64, only 32 echoes were acquired, SRF = 4.0, total scan time = 31.6 ms; (g–j) reconstructed images of (c–f) using de-convolutionmethod; (k–n) reconstructed images of (c–f) using CG method. The representative regions for SNR calculations are marked in (a), green for signals and red for noises. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

144 C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147

CG method for different sequences and background field homoge-neities. The SR reconstructed images from RASER sequence havelargest aSNRs under the different cases even though RASER se-quence has longest echo time. It means that RASER sequence canprovide images with least artifacts after SR reconstruction. This isbecause the acquired signal in each echo is entirely refocused forRASER sequence, while SPEN sequence does not have such prop-erty since Te – Tacq is utilized.

If the noises are from the representative regions marked with ared square, different SNR values would be obtained, as listed in Ta-ble 2. It should be noted that the representative regions of noises

are not those with relatively serious artifacts, so the noises mea-sured are mainly thermal noises. We can see from Table 2 thatde-convolution reconstruction method can also provide SR recon-structed images with higher SNR than CG method when the arti-facts are not considered as noises. On the other hand, a smallerfrequency bandwidth of the chirp pulse (32 kHz/4 ms) can providehigher SNR. Intuitively, this can be understood as an increase insignal averaging. Because the extra signals coming from the re-gions where the parabolic phase do not oscillate as fast as with ahigher bandwidth, they do not average to zero. It should be pointedout that smaller frequency bandwidth of the chirp pulse will also

Table 1aSNR of SR reconstructed images from phantom experiments with different recon-struction methods, background field homogeneities and sequences.

aSNR Homogeneous Inhomogeneous

CG De-convolution CG De-convolution

SPEN (R = 64 kHz/4 ms) 27 32.5 20.5 31.2SPEN (R = 32 kHz/4 ms) 20.9 30.4 18.8 28.2RASER (R = 19 kHz/13 ms) 26.3 35.1 26.0 34.0

Table 2SNR of SR reconstructed images from phantom experiments with different recon-struction methods, background field homogeneities and sequences.

SNR Homogeneous Inhomogeneous

CG De-convolution CG De-convolution

SPEN (R = 64 kHz/4 ms) 53.6 56.8 49.6 54.9SPEN (R = 32 kHz/4 ms) 74.4 81.4 68.9 72.0RASER (R = 19 kHz/13 ms) 48.7 54.2 45.7 51.8

Table 3SNR of SR reconstructed images from in vivo experiments with different reconstruc-tion methods and sequences.

SNR CG De-convolution

SPEN (R = 64 kHz/4 ms) 68.2 74.5SPEN (R = 32 kHz/4 ms) 92.1 101.7RASER (R = 19 kHz/13 ms) 24.2 25.7Reduced FOV (R = 16 kHz/4 ms) 85.6 92.4

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 145

weaken the ability of spatiotemporal-encoding MRI to resist the ef-fect of inhomogeneous field. Therefore, the selection of the fre-quency bandwidth of chirp pulse depends on the tradeoffbetween the two factors mentioned above. Note that the SNRs ofSPEN sequence and RASER sequence cannot be compared becausethe experiments were performed under different conditions.RASER sequence had longer echo time to realize self-refocusing,while SPEN sequence could not reach self-refocusing becauseTe – Tacq has been selected.

Table 3 shows that de-convolution reconstruction method canprovide SR reconstructed images with higher SNR under differentconditions for in vivo rat brain imaging. Reduced FOV method withSPEN sequence (R = 16 kHz/4 ms, SRF = 4) can provide higher SNRin a smaller imaged region compared to the full FOV method(R = 64 kHz/4 ms, SRF = 4) under the same SRF, i.e. under the sameinherent magnitude-processing resolution before SR reconstruc-tion. For Fig. 8f, only 32 echoes are acquired, therefore the signalattenuation is smaller than the attenuation of signal with 64 ech-oes acquired with full FOV method, and the signal intensity devia-tion caused by the difference in echo time is also weakened.

It should be pointed out that the gridding matrix size for the SRimages from de-convolution method is the same as that for the SRimages from CG method. Therefore, the same regions are selectedfor SNR calculation for different reconstruction methods underthe same sequence. For different sequences, similar regions withsimilar areas (considering different distortion styles) are selectedfor SNR calculation, and the regions have relatively uniform signalintensity distributions. Therefore, the calculated SNR value is sta-ble and reliable for comparison of SPEN sequences with differentR. For the reduced FOV case, the region for SNR calculation is cho-sen with similar area and position as the full FOV image.

5. Discussion

Except for the commonly used CG algorithm, some researchesfor new reconstruction algorithm have also been conducted. One

of which is the application of Fourier decoding [17]. In the Fourierdecoding method proposed by Shen and co-workers, Fourier trans-form is performed on the spatially encoded time domain imagevector which is modulated by a quadratic phase term and usuallyhas a high bandwidth. Therefore when the acquisition rate is highenough to satisfy the Nyquist condition, the Fourier decodingmethod can work well. However, the Nyquist condition is usuallybeyond the ability of the single-shot imaging sequence. So multi-shot imaging sequences must be applied for the Fourier decodingmethod [30].

To overcome the limitation of Fourier decoding method, a par-tial Fourier transform method was proposed recently [30]. The par-tial Fourier transform algorithm utilizes a segment of the acquiredsignals (not all signals) to reconstruct a voxel. The center of thesegment corresponds to the instant that the reconstructed voxelis at the vertex of the quadratic phase profile, thus it owns majorcontribution to the acquired signals or spatially encoded point. Be-cause only a segment of the acquired signals, which can be seen aslinearly modulated, is utilized to reconstruct a voxel, partial Fou-rier transform can be performed directly. This treatment is verysimple and efficient, and it brings two effects. One is that it will de-crease the resolution of the SR reconstructed images compared tothe CG method because some information outside the selected seg-ment is ignored in the reconstruction; the other is that it will re-duce the artifacts caused by the under-sampling because somehigh-frequency signal component is eliminated. Therefore, if theimaged body contains sharp texture or edges such as the waterphantom and simulation model used in this article, partial Fouriertransform is helpful to reduce the reconstruction artifacts. How-ever, if the imaged body contains smooth profile such as rat brain,de-convolution algorithm is a better choice.

Compared to the Fourier decoding method, de-convolutionmethod demodulates the quadratic phase from the time domainimage vector at first, therefore the bandwidth of the vector is re-duced greatly and the Nyquist condition is easily met if the spindensity of imaged body varies more slowly than the quadraticphase. The demodulated signal can be transformed into a convolu-tion form, as is shown in Eq. (11).

In principle, the SR reconstructed images have resolutions com-parable to those achieved by EPI, but a much higher immunity tofrequency-dispersing artifacts [12]. However, because of the un-der-sampling of spatiotemporal-encoding method, the artifactsfrom the sharp edges will reduce the quality of SR reconstructionimages, as can be seen from Figs. 3–7. Fortunately, because of thetransverse relaxation attenuation of signal at the edges of imagedbody, the edges of in vivo tissue, such as in vivo rat brain, are notas sharp as those of the water phantom and simulation model. Thatis the reason why there are relatively less artifacts in the SR recon-struction images of Fig. 8. In such case, we can say that the SRreconstructed images have EPI-like spatial resolution. It shouldbe pointed out that EPI images and SPEN super-resolution imageshave similar smearing of edges, as can be seen from Fig. 3k and l.

Though until now spatiotemporal-encoding MRI was applied toonly one dimension and the other dimension is still encoded byconventional k-space encoding, two-dimensional (2D) spatiotem-poral-encoding MRI has its unique virtue. For 2D spatiotemporal-encoding, the pulse sequence will maintain similar architectureto conventional k-space encoded counterpart. There is great poten-tial for many robust k-space encoded sequences to be modified tospatiotemporal-encoding ones. Such modification may upgrade theoriginal sequence in its immunity to off-resonance effects. For CGmethod, it will be more difficult to reconstruct an image encodedspatially along two dimensions, because the coefficient matrix inEq. (2) will be very huge in this case (for a MR image with64 � 64 matrix, the coefficient matrix will be 64 � 64 � 64 � 64).However, the de-convolution method proposed herein can work

146 C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147

with an efficiency comparable to the 2D fast Fourier transform(FFT) for conventional k-space encoded MRI. Further work isunderway.

De-convolution method can also well maintain the phase pro-file of the reconstructed images for SPEN method. When the effectsof background inhomogeneous field are taken into consideration inEqs. (8) and (9), Eq. (10) can be modified into

lðy0Þ /Z L=2

�L=2qðyÞeihðyÞeiA2ðy�y0Þ2 dy; ð16Þ

where hðyÞ is the phase of signal at position y. In this case,hðyÞ ¼ qðyÞeihðyÞ. Because different echoes correspond to differentpositions of the imaged object and echo times, the hðyÞ afterde-convolution reconstruction needs a further calibration in thephase deviation caused by the disparity in echo time at differentspatially encoded position on the image. With the calibration ofhðyÞ, SPEN MRI may be applicable in a wider range, such as in sin-gle-scan susceptibility-weighted imaging (SWI), where EPI is toosensitive to inhomogeneous field and is not suitable in most cases.

It is possible to obtain information of different chemical sites inthe imaged body from the spectrum of lðyÞ in Eq. (16). This meansthat SPEN MRI method has a potential to be used for single-scanmagnetic resonance spectroscopy imaging (MRSI) in principle[12]. Because the spectrum of lðyÞ contains not only the informa-tion of different chemical sites, but also the spatial frequency infor-mation of the imaged object, if the profile of the imaged object isrelatively simple and the chemical sites are well-separated, the cal-culation needed for resolving chemical sites composition would berelatively simple. However, more works need to be done to achievea successful chemical sites composition analysis for more complexsample such as in vivo tissue.

It has been shown in Fig. 2 that because the original signal ismodulated by quadratic phase, and sparsely under-sampled, thephase of the original signal is seriously wrapped. By removingthe quadratic phase, the simplified signal still maintains the infor-mation of the imaged object since the quadratic phase is onlydecided by the pulse sequence itself. Liner interpolation can be ap-plied to the simplified signal to diminish the folding effect causedby the under-sampling of mðyÞ, and to avoid the ringing effectscaused by too small amount of data involved for the FFT (usually664 points). The numerical simulations under homogeneous fieldand inhomogeneous field show that the bandwidth of the originalimage and that of chirp pulse both determine the appropriate sam-pling rate for spatiotemporal-encoding MRI. When the samplingrate is too low, folding artifacts will appear in the reconstructedimages.

The phantom and in vivo experiments show that de-convolutionmethod can work under different situation, such as homogeneousfield, inhomogeneous field, reduced FOV, SPEN sequence andRASER sequence. Compared to CG method, the de-convolutionmethod can provide higher digital resolution and SNR with higherreconstruction efficiency. It is simpler and can be applied moreeasily.

6. Conclusion

In this article, a novel reconstruction method based on de-con-volution has been proposed. Compared to the CG method, thismethod can provide SR images with better quality. The new meth-od is simpler and more precise. By removing the quadratic phasemodulation in the original signal, the blurred or low-definition im-age obtained by magnitude processing can be expressed as a con-volution of the SR image and its point spread function. This methodcan help to reduce the complexity of the original signal, and makethe post-processing of the original signal much easier. The phase

information of the reconstructed images may also be preserved,which may make the spatiotemporal-encoding MRI methodapplicable in a wider range, such as SWI and MRSI. The methodprovided herein benefits the applications of spatiotemporal-encoding MRI in some important fields, such as blood oxygen leveldependent (BOLD) functional MRI, Cardiac imaging, and diffusiontensor imaging.

Acknowledgments

This work was supported by the NNSF of China under Grants(81171331, 11174239, and 11074209) and the Fundamental Re-search Fund for the Central Universities under Grant(2010121101).

References

[1] G. Puy, J.P. Marques, R. Gruetter, J.P. Thiran, D.V.D. Ville, P. Vandergheynst, Y.Wiaux, Spread spectrum magnetic resonance imaging, IEEE Trans. Med.Imaging 31 (2012) 586–598.

[2] K.J. Jung, T.J. Zhao, Parallel imaging with asymmetric acceleration to reducegibbs artifacts and to increase signal-to-noise ratio of the gradient echo echo-planar imaging sequence for functional MRI, Magn. Reson. Med. 67 (2012)419–427.

[3] J. Tsao, Ultrafast imaging: principles, pitfalls, solutions, and applications, J.Magn. Reson. Imaging 32 (2010) 252–266.

[4] R.C. Charms, Applications of real-time fMRI, Nat. Rev. 9 (2008) 720–729.[5] O. Afacan, W.S. Hoge, F. Janoos, D.H. Brooks, I.A. Morocz, Rapid full-brain fMRI

with an accelerated multi shot 3D EPI sequence using both UNFOLD andGRAPPA, Magn. Reson. Med. 67 (2012) 1266–1274.

[6] N. Seiberlich, G. Lee, P. Ehses, J.L. Duerk, R. Gilkeson, M. Griswold, Improvedtemporal resolution in cardiac imaging using through-time spiral GRAPPA,Magn. Reson. Med. 66 (2011) 1682–1688.

[7] C. Brinegar, Y.J.L. Wu, L.M. Foley, T.K. Hitchens, Q. Ye, C. Ho, Z.P. Liang, Real-time cardiac MRI without triggering, gating, or breath holding, Conf. Proc. IEEEEng. Med. Biol. Soc. (2008) 3381–3384.

[8] P.J. Basser, D.K. Jones, Diffusion-tensor MRI: theory, experimental design anddata analysis: a technical review, NMR Biomed. 15 (2002) 456–467.

[9] A.T. Van, D. Hernando, B.P. Sutton, Motion-induced phase error estimation andcorrection in 3D diffusion tensor imaging, IEEE Trans. Med. Imaging 30 (2011)1933–1940.

[10] P. Hagmann, J.P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder, R.Meuli, DTI mapping of human brain connectivity: statistical fibre tracking andvirtual dissection, NeuroImage 19 (2003) 545–554.

[11] Y. Shrot, L. Frydman, Spatially encoded NMR and the acquisition of 2Dmagnetic resonance images within a single scan, J. Magn. Reson. 172 (2005)179–190.

[12] N. Ben-Eliezer, M. Irani, L. Frydman, Super-resolved spatially encoded single-scan 2D MRI, Magn. Reson. Med. 63 (2010) 1594–1600.

[13] N. Ben-Eliezer, L. Frydman, Spatiotemporal encoding as a robust basis for fastthree-dimensional in vivo MRI, NMR Biomed. 24 (2011) 1191–1201.

[14] A. Tal, L. Frydman, Spectroscopic imaging from spatially-encoded single-scanmultidimensional MRI data, J. Magn. Reson. 189 (2007) 46–58.

[15] N. Ben-Eliezer, Y. Shrot, L. Frydman, High-definition single-scan 2D MRI ininhomogeneous fields using spatial encoding methods, Magn. Reson. Imaging28 (2010) 77–86.

[16] J.G. Pipe, Spatial encoding and reconstruction in MRI with quadratic phaseprofiles, Magn. Reson. Med. 33 (1995) 24–33.

[17] J. Shen, Y. Xiang, High fidelity magnetic resonance imaging by frequencysweep encoding and Fourier decoding, J. Magn. Reson. 204 (2010) 200–207.

[18] N. Ben-Eliezer, E. Solomon, E. Harel, N. Nevo, L. Frydman, Fully-refocusedspatiotemporally-encoded MRI: robust MR imaging in the presence of metallicimplants, Proc. Int. Soc. Mag. Reson. Med. 20 (2012) 223.

[19] R. Chamberlain, J.Y. Park, C. Corum, E. Yacoub, K. Ugurbil, C.R. Jack Jr., M.Garwood, RASER: a new ultrafast magnetic resonance imaging method, Magn.Reson. Med. 58 (2007) 794–799.

[20] U. Goerke, M. Garwood, K. Ugurbil, Functional magnetic resonance imagingusing RASER, NeuroImage 54 (2011) 350–360.

[21] A.A. Maudsley, Dynamic range improvement in NMR imaging using phasescrambling, J. Magn. Reson. 76 (1988) 287–305.

[22] V.J. Wedeen, Y.S. Chao, J.L. Ackerman, Dynamic range compression in MRI bymeans of a nonlinear gradient pulse, Magn. Reson. Med. 6 (1988) 287–295.

[23] S. Ito, Y. Yamada, Alias-free image reconstruction using Fresnel transform inthe phase-scrambling Fourier imaging technique, Magn. Reson. Med. 60 (2008)422–430.

[24] D. Idiyatullin, S. Suddarth, C.A. Corum, G. Adriany, M. Garwood, ContinuousSWIFT, J. Magn. Reson. 220 (2012) 26–31.

[25] J.G. Pipe, Analysis of localized quadratic encoding and reconstruction, Magn.Reson. Med. 36 (1996) 137–146.

C. Cai et al. / Journal of Magnetic Resonance 228 (2013) 136–147 147

[26] A. Tal, L. Frydman, Single-scan multidimensional magnetic resonance, Prog.NMR Spectrosc. 57 (2010) 241–292.

[27] C.B. Cai, M.J. Lin, Z. Chen, X. Chen, S.H. Cai, J.H. Zhong, SPROM-an efficientprogram for NMR/MRI simulations of inter- and intra-molecular multiplequantum coherences, C. R. Phys. 9 (2008) 119–126.

[28] B.J. Wilm, J. Svensson, A. Henning, K.P. Pruessmann, P. Boesiger, S.S. Kollias,Reduced field-of-view MRI using outer volume suppression for spinal corddiffusion imaging, Magn. Reson. Med. 57 (2007) 625–630.

[29] E.U. Saritas, C.H. Cunningham, J.H. Lee, E.T. Han, D.G. Nishimura, DWI of thespinal cord with reduced FOV single-scan EPI, Magn. Reson. Med. 60 (2008)468–473.

[30] Y. Chen, J. Li, X.B. Qu, L. Chen, C.B. Cai, S.H. Cai, J.H. Zhong, Z. Chen, PartialFourier transform reconstruction for single-shot MRI with linear frequency-swept excitation, Magn. Reson. Med., doi: http://dx.doi.org/10.1002/mrm.24366.