PROMISE: Parallel-Imaging and Compressed-SensingReconstruction of Multicontrast Imaging Using SharablEInformation

Enhao Gong,1,2 Feng Huang,3* Kui Ying,2,4 Wenchuan Wu,2 Shi Wang,4 and

Chun Yuan2,5

Purpose: A typical clinical MR examination includes multiplescans to acquire images with different contrasts for comple-

mentary diagnostic information. The multicontrast schemerequires long scanning time. The combination of partially par-

allel imaging and compressed sensing (CS-PPI) has been usedto reconstruct accelerated scans. However, there are severalunsolved problems in existing methods. The target of this

work is to improve existing CS-PPI methods for multicontrastimaging, especially for two-dimensional imaging.

Theory and Methods: If the same field of view is scanned inmulticontrast imaging, there is significant amount of sharableinformation. It is proposed in this study to use manifold shara-

ble information among multicontrast images to enhance CS-PPI in a sequential way. Coil sensitivity information and struc-ture based adaptive regularization, which were extracted from

previously reconstructed images, were applied to enhance thefollowing reconstructions. The proposed method is called Par-

allel-imaging and compressed-sensing Reconstruction Of Mul-ticontrast Imaging using SharablE information (PROMISE).Results: Using L1-SPIRiT as a CS-PPI example, results on

multicontrast brain and carotid scans demonstrated that lowererror level and better detail preservation can be achieved by

exploiting manifold sharable information. Besides, the privilegeof PROMISE still exists while there is interscan motion.Conclusion: Using the sharable information among multicon-

trast images can enhance CS-PPI with tolerance to motions.Magn Reson Med 73:523–535, 2015. VC 2014 Wiley Periodi-cals, Inc.

Key words: parallel imaging; compressed sensing; multicon-trast; multichannel; sharable information

INTRODUCTION

MRI is a powerful and widely applied imaging modalityin both clinical and research settings. One of the mostimportant features of MRI is that it can provide imageswith multiple contrasts for complementary diagnosticinformation. Therefore, a typical clinical MR examina-tion is composed of several scans to acquire images withdifferent contrasts, such as T1 weighted (T1w), T2weighted (T2w), and Proton density weight (PDw)images. The multicontrast scheme provides more infor-mation for diagnosis than single contrast imaging withlonger acquisition, which may result in higher cost andhigher sensitivity to motion artifacts. Therefore, reducingtotal acquisition time of multicontrast imaging is highlydesired.

Partially parallel imaging (PPI) (1,2), which exploitsthe data correlation among multiple coil elements forreconstruction with partially acquired data, has beenwidely used clinically for fast imaging. However, PPItechniques are limited by channel encoding capabilityand it reduces acquisition time at the cost of an increaseof noise and artifact level. Compressed sensing (CS) (3)recovers undersampled data based on the assumption ofimage sparsity in certain transformed domain(s).Because these two categories of methods use differentauxiliary information for reconstruction, the combina-tion of these two methods, called CS-PPI in this work,can result in impressively better recovery than eachindividual method which have been shown in litera-tures (4–8).

However, there are still several problems unsolved inCS-PPI. The most difficult problem is how to get accu-rate coil sensitivity information, which is essential forthe accuracy of parallel imaging based reconstruction.The calculation of coil sensitivity uses either low reso-lution prescan (1) or self-calibration signal (2). Coil sen-sitivity estimation from prescan may suffer fromprolonged acquisition time or misregistration due topatient motion. The self-calibration scheme needs suffi-cient autocalibration signal (ACS) for accurate coil sen-sitivity while the acquisition of ACS limits the netacceleration factor. Another problem is how to balanceartifact-reduction and detail preservation using the pre-defined regularization parameter. An improper regulari-zation may result in either residual noise/artifact orover-smoothed image.

The scans for multiple contrasts share significantamount of redundant information, due to the same fieldof view (FOV) is imaged in the same system using thesame radiofrequency coil. In this study, we propose touse manifold sharable information to tackle the existing

1Magnetic Resonance System Research Lab, Department of ElectricalEngineering, Stanford University, Stanford, California, USA.2Center for Biomedical Imaging Research, Department of Biomedical Engi-neering, Tsinghua University, Beijing, China.3Philips Research China, Shanghai, China.4Key Laboratory of Particle and Radiation Imaging, Department of Engi-neering Physics, Tsinghua University, Beijing, China.5Department of Radiology, University of Washington, Seattle, Washington,USA.

*Correspondence to: Feng Huang, Ph. D., Philips Research China, No.1,Building 10, Lane 888, Tianlin Road, Shanghai, China, 200233. E-mail:f.huang@philips.com

Additional Supporting Information may be found in the online version ofthis article.

Received 28 March 2013; revised 29 December 2013; accepted 2 January2014

DOI 10.1002/mrm.25142Published online 25 February 2014 in Wiley Online Library (wileyonlinelibrary.com).

Magnetic Resonance in Medicine 73:523–535 (2015)

VC 2014 Wiley Periodicals, Inc. 523

problems and hence improve CS-PPI reconstruction formulticontrast imaging.

In literatures, sharable information among multicontrastimages has been used to improve either PPI or CS. In2006, Larkman et al (9) proposed to use joint histogramentropy among images with different contrasts to enhancePPI. Recently, Li et al (10–12) proposed a method referredas “Correlation Imaging” that used the sharable sensitivitybased information, called “correlation function,” to recon-struct the multicontrast images in the context of PPI. Bil-gic et al (13) uses a Bayesian based method to exploit thesimilarity of sparsity among images with different con-trasts in the context of compressed sensing (3). These pre-vious methods do not fully take the advantages of thesharable information. Both Larkman’s method for PPI andBilgic’s method for CS only uses the sharable anatomicalinformation. Li’s method on correlation imaging only usesthe similarity of coil sensitivity information. These previ-ous methods also have some potential issues. Mutualinformation and Bayesian based method take high compu-tational costs and Correlation Imaging mainly works withuniform sampling. Another potential problem of theseexisting techniques is the sensitivity to interscan motions.If there are interscan motions, the structural informationwill be mismatched between images with different con-trasts. Strong constraints in the reconstruction model, onthe similarity of structure [like in Larkman et al (9) andBilgic et al (13)] or the coil sensitivity maps in the imagedomain (as in Li and Dumoulin) (10), could result in arti-facts, which may mislead the diagnosis.

Different from previous methods, in this study, it isproposed to use manifold sharable information toimprove CS-PPI. Also, all constraints using the sharableinformation are more heuristic and softer than the exist-ing methods to avoid side effects due to motions. Sameas previous methods, this study also focus on multicon-trast two-dimensional (2D) imaging because multicon-trast 2D imaging is more intensively used clinically than3D imaging (10) and can be more time consuming, Theproposed method is called Parallel-imaging andcompressed-sensing Reconstruction of MulticontrastImaging using SharablE information (PROMISE).

The rest of this study is organized as follows. InTheory section, how to use the sharable information tosolve the existing problems in CS-PPI is introduced. Theimplementation details and the design of experimentsare described in the Methods section. The validation ofthe method is presented in the Result section. In the Dis-cussion section, the advantages of PROMISE and furthergeneralization of PROMISE are included. Finally, weprovide concluding remarks in the Conclusion section.

THEORY

Sharable Information

Using spin echo sequence as an example, the MRI signalS can be defined by the equation below (a simplifiedversion):

S ¼ B�1 rFðBþ1 ;B0Þð1� e�TR=T1Þe�TE=T2 [1]

where r is the proton density which is a function of mainmagnetic density B0, transmit magnetic density Bþ1 and

several properties of the scanned subject according toBoltzmann distribution (14). B�1 is the sensitivity map ofthe receive coil. F •ð Þ denotes a function of variables inparentheses, T1 and T2 are the T1 and T2 maps of thescanned subject, TR is the repetition time, and TE is theecho time. For T1 weighted, T2 weighted, and proton den-sity weighted images, only TR and TE are different inacquisition. If the same location is scanned, these imagesshare the same structural information in r, the same mag-netic information B0, B�1 , as well as T1 and T2 maps.

In this work, it is proposed to take advantages of theset of sharable information, including coil sensitivityinformation and image structural information, to improveCS-PPI. The list above demonstrates a practical exampleof the set of sharable information in multicontrast fastimaging. For different applications, it is possible to useadditional or alternative sets.

Combination of PPI and CS

The combination of PPI and CS is the trend of fast imag-ing because it can produce better images than those byeach individual method (4–7,15). In this study, L1-SPI-RiT (7), an iterative reconstruction method using L1

norm and k-space data self-consistency as regularization,is used as a specific example of the combination method.The objective function E(x) to minimize is defined as

EðxÞ ¼ jjDx � kjj22 þ l0jjGx � xjj22 þ l1jjCðF�1xÞjj1þ l2jjrðF�1xÞjj1 [2]

where x is the reconstructed k-space of all channels, k isthe partially acquired data, D is the operator for data under-sampling, G is a general convolution operator for SPIRiT, C

is the wavelet transform, and r is the gradient operator.The nonnegative regularization parameters l0, l1, and l2,balance these four terms. The last two terms are sparsityconstraints; they enforce the sparsity of the reconstructedimage in these transformed domains. In Eq. [2], two spar-sity constraints are used as an example. The number ofsparsity constraints can be more or less than 2. F�1ð•Þ isdefined as below and iFFT is inverse fast Fourier transform

F�1x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNch

j¼1

ðiFFTðxjÞÞ2vuut [3]

The reconstruction process is to minimize the objectivefunction to solve x ¼ arg min EðxÞ. The accuracy of the CSPPI reconstruction using Eq. [2] relies on several factors,including the accuracy of the convolution kernels whichimplicitly uses the coil sensitivity and the regularizationparameters. In the proposed method, sharable informationamong multicontrast images is used to improve these fac-tors to solve the existing problems in L1-SPIRiT.

Sequential Reconstruction and Consideration ofAcquisition Order

The usage of the sharable information can be in either asequential or a joint way. In the sequential way, the sharableinformation is extracted from previously reconstructed oracquired data and applied in the reconstruction of the fol-lowing images. Images of each contrast are reconstructed

524 Gong et al.

individually. In the joint way, the multicontrast images arereconstructed jointly with constraints on the similarity ofsharable information (13). Currently, a clinical imaging pro-tocol includes several scans processed sequentially andimage of each scan is presented immediately after the acqui-sition. Hence, we propose to use the sequential way for theusage of sharable information because it follows the clinicalconvention. If a joint reconstruction scheme is used, theimages of two or more scans need be reconstructed and pre-sented together. The presentation, as well as the possiblefeedbacks from the acquired scans, has to be delayed. Theclinical acceptance of joint-reconstruction is uncertain.

Due to the sequential usage of sharable information, theorder of acquisition should be decided by the speed andsensitivity to motion of these sequences to achieve highertotal acceleration factor and more consistent image quality.For the first scan, there is no or very limited prior informa-tion. The faster and/or motion insensitive sequences shouldbe used with low acceleration factor to provide sufficientlyaccurate sharable information without long acquisitiontime. After the reconstruction/acquisition of the first image,a set of full k-space with high resolution anatomical infor-mation is available. Hence, manifold sharable information(coil sensitivity maps and structural information) can beextracted as prior information for the following reconstruc-tions with slower sequences at higher reduction factors.For example, T1w sequence is faster than T2w sequence inbrain imaging. Hence, the T1w image can be acquired withlow acceleration factor, say 1 to 2. Slower and motion sen-sitive T2w sequences are then acquired with higher acceler-ation factors, say 4 to 5, because extracted sharableinformation is available to enhance the reconstruction.

Parallel-imaging and compressed sensing Reconstructionof Multicontrast Imaging Using SharablE Information(PROMISE)

Based on the ideas above, a general framework, PROMISE,is proposed to exploit the sharable information among

images with different contrasts for fast imaging. Figure 1shows the flowchart of the framework. Without loss of gen-eralization, it is assumed that the examination is composedof three scans: Prescan for initial coil sensitivity maps,T1w scan, and T2w scan. As shown in Figure 1, the T1wimage is acquired with equally spaced 1D-undersamplingscheme at an acceleration factor of 2. The T2w image isacquired with higher acceleration factors with randomly1D-undersampled trajectory. There are four major proc-esses in the reconstruction: (i) reconstruction of the T1wwith coil sensitivity maps from prescan; (ii) extraction ofsharable information (implicit coil sensitivity informationand spatially adaptive regularization parameters definedwith structure information) from the reconstructed T1wimage; (iii) partial reconstruction of the T2w imageacquired using GRAPPA operator; and (iv) final recon-struction of the T2w image using a modified version of Eq.[2] with the information extracted from step 2. The imple-mentation details are provided in the Methods section.

METHODS

Extraction of Sharable Coil Sensitivity Information

In this specific example, low-resolution sensitivity mapsfrom prescan were used for T1w scan reconstruction.After the reconstruction of the T1w image, a full k-spaceof each individual channel was produced by projectingthe sensitivity weighted image back to k-space. The 64center k-space lines from the T1w image were used tocalibrate convolution kernels for SPIRiT (16) andGRAPPA operator (17) to implicitly use the coil sensitiv-ity information in k-space, which were shown to bemore robust to interscan motion than sensitivity maps inthe image domain. Because sufficient k-space data areused for the calibration, the convolution kernels aremore accurate than those using a few self-calibrationdata. In addition, great numbers of ACS lines are no lon-ger required for future scans.

FIG. 1. Flow chart of PROMISE.

Improving CS-PPI Methods for Multicontrast Imaging 525

Extraction of Sharable Structure Information as SpatiallyAdaptive Regularization

The sharable structure information was extracted frompreviously acquired/recovered images using statisticalestimation. The statistical probability of representingboundaries/edges of each component in sparsity trans-form, p, is computed using a Hidden Markov Tree basedExpectation Maximization (EM) (18):

p ¼ pðfS contributes to boundariesgjCm;QðCmÞÞ [4]

where p �ð Þ is the operator for probability calculation, Cmis the wavelet transform of image m, S is the coefficientof the wavelet transforms, QðCmÞ is the superparametercharacterizing the properties of the wavelet transformcoefficients. The details of derivation are included in theAppendix, and more explanation can be found in Gonget al (19) and Duarte et al (20).

To reduce the sensitivity to motion, the extracted edgeinformation was not directly used to enforce the similar-ity of the edges between reference image and the to-bereconstructed image. Instead, it was used to define a spa-tially adaptive regularization parameter.

The regularization parameter l (l1, l2, or both) in Eq.[2] balances data fidelity and the regularization. A largerl emphasizes regularization more, but may causereduced sharpness and loss of fine structures; a smallerl emphasizes data fidelity more but may result in resid-ual noise/artifacts. It has been shown in references (21–23) that a spatially adaptive regularization based on priorimage or noise distribution, rather than using a constantregularization parameter, can result in better balancebetween maintaining signal to noise ratio (SNR) and per-serving boundaries. Based on similar but not the sameidea, spatially adaptive regularization in the sparsitytransform domains was used in this work, to preserveboundaries and fine structures while sufficiently removenoise/artifacts. The adaptive regularization implicitlyand softly enforces the sparsity supports from the fuzzystructure information estimated from previously recon-structed images considering the possible existence ofminute mismatches of structures.

Hence, the objective function in Eq. [2] can be modi-fied with estimated sharable information.

EðxÞ ¼ jjDx � kjj22 þ l0jjGx � xjj22 þ l1jjWnCðF�1xÞjj1þ l2jjWIrðF�1xÞjj1 [5]

Here the L1 norms of sparsity transforms are used asregularization with spatially adaptive weights Wn andWI , for wavelet and Total-Variance transform respec-tively. At locations without boundaries/edges orsignificant-value coefficients, a larger parameter is usedto sufficiently suppress noise/artifacts; at locations nearboundaries/edges or contains significant-value compo-nents, a smaller parameter is used to avoid blurring. Theweighted Total-Variance transform is a nonquadratic ana-log to the smoothness based functional proposed in previ-ous study (22), while the wavelet transform characterizesstructures similarity robustly in multiple scales of resolu-tions. The weights are defined by a set of point-wiseweighting functions (24).

Wn ¼ ðpn þ dÞ�q;1 �Wn � d�q

Wi ¼ ðpi þ dÞ�q; 1 �Wi � d�q[6]

pn and pi are the statistical probabilities of being boun-daries/edges in wavelet and image space, which are com-puted using Eq. [3]. The parameter d is for overflowprevention; and parameter q is for controlling the orderof regularization. Constant parameters, k1 and k2 inEq. [2] and d and q in Eq. [4], are defined empirically (k1

and k2 are empirically set to be 0.001, d is set to be 10�10

and q is �0.02).]

PPI Enhanced Initialization for Data Fidelity Term

Similar but not the same to the method proposed in (25),a generalized autocalibrating partial parallel acquisition(GRAPPA) operator (17) was applied to estimate a PPIbased initialization for the recovery of partial k-space,which has the distance not larger than Dky to theacquired data. There are two reasons for this implemen-tation. First, when random trajectory is used, GRAPPAoperator is applicable because it does not require specificsampling patterns and is faster than other k-space meth-ods due to the small convolution kernel. Second, theresult of GRAPPA operator is accurate enough to be usedas reference when the distance between the extrapolateddata and the acquired data is not larger than Dky (26).Hence the result of GRAPPA operator was also used inthe fidelity term in Eq. [5], in addition to initialization.

Data Acquisition

Three sets of 2D multicontrast images were acquired ona Philips 3 Tesla (T) system (Philips Healthcare, Best,Netherland). Two sets of the images are multislice axialbrain images acquired with an eight-channel head coil(Invivo Corporation, Gainesville, FL). Each set was com-posed of three scans of brain: low-resolution T1w pre-scan, high resolution T1w and T2w scans. For thedataset 1, the volunteer was asked to keep still duringthe scan. To validate the proposed method with inter-scan rigid motions, the volunteer was asked to movehead between the T1w and T2w scans when dataset 2was acquired. The two T1w datasets used a fast fieldecho (FFE) sequence (FOV 230 � 230 mm2, repetitiontime (TR) 170 ms, echo time (TE) 3.9 ms, flip angle 80�,slice thickness 5 mm). The matrix sizes for the low andhigh resolution images are 64 � 64 and 256 � 256,respectively. The T2w dataset also used an FFE sequence(FOV 230 � 230 mm2, TR 3000 ms, TE 80 ms, flip angle90�, slice thickness 5 mm). The matrix size is 256 � 256.For all datasets, the PE direction was left–right. Theacquisition time for the 16 slices of T1w images andT2w images were 86 seconds and 108 s, respectively.

To validate the sensitivity of the proposed method tononrigid interscan motion, dataset 3 composed of twocarotid MR scans was used. The two scans were 2Dquadruple-inversion-recovery T1-weighted (QIR T1w)scan and 2D double-inversion-recovery T2-weighted (DIRT2w) scan. Both scans used turbo spin echo (TSE)sequence. A homemade 36-channel neurovascular coil(27), which provides 11 channels for carotid imaging

526 Gong et al.

with two elements in phase encoding directions (ante-rior–posterior), was used in the experiments. The 16axial slices are acquired (FOV 160�160 mm2, voxel size0.6 � 0.6 mm2, slice thickness 2 mm). The total acquisi-tion time for QIR T1w scan (TR 800ms, TE 10ms, TSEfactor 10) was 6 min 11 s and the total acquisition timefor DIR T2w scan was 3 min 40 s (TR 4800 ms, TE 50ms, TSE factor 12).

Both datasets were fully acquired, and retrospectivelyundersampled in the experiments. For T1w data, uni-form 1D undersampling along PE direction was usedwith reduction factor of 2. For T2w data, a random 1Dundersampling scheme along PE direction was used forreconstruction schemes (shown in Table 1). The same 1DVariable Density random undersampling trajectory wasused L1-SPIRiT, Joint Bayesian CS and PROMISE, whichwas generated based on point spread function whichwas proposed in literature (3) to optimize the incoher-ence of the undersampling pattern.

Design of Experiments

Four sets of experiments were designed to evaluatePROMISE. The first set of experiments was used to dem-onstrate the advantages of using each kind of sharableinformation. The set of extracted information, coil sensi-tivity information and structure based spatially adaptiveregularization parameters, were added to a L1-SPIRiTbased model one-by-one. In the second set of experi-ments, PROMISE was compared with existing methodslisted in Table 1. Both sets of experiments used in vivodataset 1.

Two multicontrast brain imaging datasets with simu-lated and actual interscan motion, respectively, wereused in the third set of experiments to validate the toler-ance to rigid interscan motion. Dataset 1 was used tosimulate the in-plane motion. One slice of compositeT2w image was first synthesized from multichannel fullysampled T2w scan data. The composite T2w image wasartificially displaced from the original position by two-by-two pixels (two pixels down along both FE directionand two pixels right along PE direction equivalent toapproximately 3 mm displacement) and rotated by 5�

counterclockwise around the center of slice. Then a sen-sitivity map is multiplied to the single image to synthe-size a set of multichannel T2w image with the simulatedinterscan motion. Dataset 2 contains actual interscan

motion. Both in-plane and through-plane motionsbetween the T1w and T2w scans can be clearly observed.In addition to the reconstruction using PROMISE, L1-SPIRiT, and Joint Bayesian based CS was also used forcomparison.

The carotid dataset was used to test the robustness tononrigid interscan motion in the fourth set of experi-ments. In this experiment, it is assumed that the T2wimage which needs much shorter acquisition time wasscanned first with full k-space. The T1w image wasscanned with acceleration and reconstructed using shara-ble information extract from T2w scan. L1-SPIRiT andJoint Bayesian based CS were also used for comparison.

Implementation of Existing Methods for Comparison

For comparison, L1-SPIRiT and Joint Bayesian CS wereimplemented. Because the target of this work is toimprove CS-PPI, specifically taking L1-SPIRiT as a basicCS-PPI framework, we implemented L1-SPIRiT accordingto the references (7,16), and the example codes (http://www.eecs.berkeley.edu/�mlustig/Software.html) werereferred. A 7 � 7 convolution kernel, 20 ACS lines andTikhonov regulation with weight of 0.01 were used forkernel calibration. The regularization parameter l wasempirically optimized to be 0.001. The wavelet soft-threshold parameter for L1-SPIRiT was empirically opti-mized to be around 0.002 to achieve a balance of aliasingreduction and block-artifact reduction. Nonlinear itera-tive reconstruction methods were used for L1-SPIRiT andPROMISE. The number of iterations follows the sampledcode of L1-SPIRiT mentioned above to guarantee theminimum root mean square error (RMSE) was reachedand the result with minimum RMSE was selected.

Because Bayesian CS uses sharable structural informa-tion to improve CS, we also implemented it for compari-son. The code was downloaded from online softwarepackage (http://web.mit.edu/berkin/www/software.html).We implemented the joint reconstruction function forcomplex image of each channel and then synthesizedinto one image using root-sum-of-square as in equation[3].

Image Quality Assessment

For accuracy assessments, both the magnitude and thedistribution of the error were measured. The images

Table 1

Reconstruction Methods Used in Comparison

Acquisition Reconstruction Sharable Information

Chosenmethod

2D Multislice 1Dtrajectory

3D Isotropic 2Dtrajectory

PPI CS Sharable sensitiv-ity information

Sharable structureinformation

L1-SPIRiT(16,34)

Randomundersampling

Poisson disk 3 3

Joint Bayesianbased CS(13)

Randomundersampling

3 3

Proposedmethod:

PROMISE

Randomundersampling

Poisson disk 3 3 3 3

Improving CS-PPI Methods for Multicontrast Imaging 527

reconstructed using the full k-space data were used asreference images. The normalized RMSE in a region-of-interest (ROI) was used to evaluate the magnitude oferror:

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXjIrecon � Iref j2

.XjIref j2

r[7]

where Iref is the reference image reconstructed with fullyacquired k-space data and Irecon is the reconstructionwith partially acquired k-space data. Error maps illustrat-ing the spatial distribution of the differences in magni-tude between the reconstruction and the reference werealso presented. The ROI for error computing is definedby the image support thresholding out the background ofthe reference image.

All the algorithms mentioned before were imple-mented using MATLAB (MathWorks Inc., Natick, MA)with Intel Core 2.4 GHz Duo CPU and 64-bit OperatingSystem.

RESULTS

Figure 2 shows one example of the extracted boundaryinformation. Figures 2c and 2d are the extracted bound-ary information from T1w brain image in Figure 2a forthe reconstruction of T2w image in Figure 2b. Figures 2cand 2d are in the wavelet and image domains, respec-tively. It can be clearly seen that the extracted bounda-

ries are banded sparse, which are defined by theprobability as boundaries.

Figure 3 demonstrates the performance improvementof different components in PROMISE. By adding differ-ent component of manifold sharable information (shownin the table below the figures) one-by-one to conven-tional L1-SPIRiT, the reconstructed results at reductionfactor of 5 are shown: L1-SPIRiT with random trajectory(Fig. 3b), L1-SPIRiT using shared sensitivity informationfrom T1w data (Fig. 3c), L1-SPIRiT with spatially adapt-ive weight estimated from structure information (Fig. 3d)and PROMISE (with both sharable sensitivity informa-tion and structure information) (Fig. 3e). Correspondingerror maps are five-times brightened and shown in Fig-ure 3g–j. The detail of the methods and normalizedRMSEs are shown in the table below.

Figures 4 to 6 demonstrate the comparison of recon-structions using the 2D multislice in vivo brain dataset,which compares the proposed algorithm PROMISE withother algorithms listed in Table 1. Dataset 1 with netreduction factor 5 was used. Figure 4a is the referenceimage reconstructed with full k-space data. Figure 4bshows the 1D random undersampling trajectory used forall three methods (L1-SPIRiT, Joint Bayesian and the pro-posed method). The lower three rows of Figure 4 showthe reconstructions (left) and the corresponding errormaps (right) of L1-SPIRiT (Fig. 4c,d), Joint Bayesianbased CS (Fig. 4e,f), and the proposed PROMISE (Fig.

FIG. 2. Demonstration of the extractedstructural information. a,b: The T1w (a)

and T2w (b) images. The statisticalboundary information is extracted fromthe T1w image reconstructed with

reduction factor of 2. c: The extractedboundary information in the wavelet

domain. d: the extracted boundaryinformation in the image domain.

528 Gong et al.

4g,h), respectively. The errors maps were brightened fivetimes for better visibility. The values at the right-bottomof the error maps are the corresponding normalizedRMSE. Figure 5 demonstrates the correspondingzoomed-in regions, which is defined in Figure 4a. Figure6 shows the comparison of RMSE at different net reduc-tion factors from 2 up to 6. In Figure 6a, the dotted line,dashed line, and solid lines are for L1-SPIRiT, JointBayesian CS, and PROMISE. In Figure 6b, reconstructionRMSEs while exploiting different sharable informationare shown with labeling. The combinatory usage of shar-able sensitivity and structure information significantimproves the reconstruction.

Figures 7–9 show the performance of PROMISE whenthere were different types of interscan motions. Dataset 1with simulated in-plane motion, dataset 2 with both in-plane and through-plane mismatches, and dataset 3 withnonrigid motion was used for Figure 7 to Figure 9,respectively. For comparison, the results of L1-SPIRiTand Joint Bayesian CS are also provided. Reduction fac-tor of 4 was used in all three Figures. Both the recon-structed image and the corresponding error maps werepresented in lower rows for L1-SPIRiT, Joint Bayesian CSand PROMISE respectively. All three methods shared thesame 1D random undersampling trajectory. The errormaps were brightened five times for better visibility. Thenormalized RMSEs were shown at the right bottom ofeach error maps.

DISCUSSION

Advantages of Using Sharable Information

This study proposes to sequentially use the sharableinformation to enhance fast multicontrast imaging andresolve currently existing problems in CS-PPI. As onespecific example, sharable sensitivity and structure infor-mation from previous reconstructions was used toimprove the reconstruction of multicontrast images usingCS-PPI, specifically L1-SPIRiT.

Figures 4�6 demonstrated that PROMISE resulted inlower RMSE and better structure preservation than exist-ing methods when there was no interscan motion. Fig-ures 7–9 demonstrated that using sharable informationstill resulted in better reconstruction when there weredifferent kinds of interscan motions.

PROMISE is superior to the existing methods becausethat PROMISE used manifold prior information shared inscans. In the specific implementation, both sensitivityand structure information was estimated and exploited inPROMISE. The contribution of individual kind and theircombinations were shown in Figure 3. The normalizedRMSE was reduced from 11.5% to 7.6% by using mani-fold sharable information, and each component (sharablesensitivity and structural information) of PROMISE hascontributions to the improvement.

As Figures 3–6 show, PROMISE consistently resulted inlower RMSE than L1-SPIRiT and Joint Bayesian CS for the

FIG. 3. The improvements of T2w scan reconstruction by using manifold sharable information at reduction factor of 5. a: The referenceimage. b: L1-SPIRiT. c: L1-SPIRiTþshared coil sensitivity information (in k-space). d: L1-SPIRiTþshared structure information used for

adaptive weight. e: L1-SPIRiTþshared coil sensitivity informationþ shared structure information (PROMISE). f: the T1w image used formulticontrast information extraction. g–j: Corresponding difference maps which are five-times brightened. Different components of

PROMISE are used as the table shows below and the resulting RMSE.

Improving CS-PPI Methods for Multicontrast Imaging 529

same undersampled data, and the superiority increasedwith the net acceleration factor.

Sensitivity to Interscan Motion

As shown previously, the usage of sharable informationcan improve the reconstruction dramatically when thereis no interscan motion. However, if there are interscanmotions, then the coil sensitivity and the structure infor-mation would be misregistered. Models in the reconstruc-

tion strongly enforcing the similarity of misregisteredinformation would result in artifacts (Figs. 8e,f and 9e,f).Other previously proposed algorithms working in theimage domain also suffer the modeling error. PROMISE,however, uses two approaches to moderate the sensitivityto motion. The first one is to share implicit coil sensitivityinformation; the other one is to softly use the sharedboundary information as spatially adaptive regularizationparameters.

In PROMISE, coil sensitivity is implicitly used as con-volution kernels in GRAPPA operator and SPIRiT. Thesesmall size convolution kernels correspond to low-resolution coil sensitivity maps. Moderate motion does notchange the sensitivity convolution kernel much. Therefore,adjacent slices or time frames can share the same set of

FIG. 4. Comparison to the existing algorithms (L1SPIRiT, JointBayesian based CS, and the proposed method PROMISE) for the

reconstruction of simulated T2w brain image at net reduction fac-tor of 5. The left column shows the image reconstructed with fullk-space (a), L1-SPIRiT (c), Joint Bayesian based CS (e), and

PROMISE (g). An ROI for zooming-in is shown in a. The same 1Drandom undersampling trajectory for all the methods is shown inb. The images in the right column are the corresponding error

maps (d–h). The error maps were brightened five times.

FIG. 5. The corresponding zoomed-in images from Figure 4. Thezoomed in region is defined in Figure 4a. a: Corresponds to Figure

4a. b–g: Correspond to (c–h) in Figure 4. RMSEs in the zoomed-inROI are shown at the right bottom at each zoomed-in error map.

530 Gong et al.

convolution kernels. This feature has been used in litera-tures, such as zGRAPPA (28) and TGRAPPA (29).

Second, PROMISE does not enforce that the multicon-trast images explicitly share the same supports of spar-sity or locations of boundaries. The shared structureinformation is only used to define regularization parame-ters. Similar ideas have been proposed to achieve robustrecovery by using fuzzy prior information in weightedregularization (23). Hence, if there are interscan motions,even though the improper regularization parameter willresult in nonoptimal balance of detail preservation andartifact reduction, no change of contrast or other artifactswill be introduced by using sharable informationbetween scans. To further moderate this potential prob-lem robustly, multilevel resolution approach is used inthe extraction and implementation of structure informa-tion in both the wavelet domain and image domain. Asshown in Figure 2, instead of just a single pixel widthcurve of deterministic binary values, an edge in imagecould result in bands of weighted pixels in multipleresolution levels.

These two approaches help the proposed algorithm torobustly exploit the fuzzy structure information, andhence moderate motions in pixel level cannot change theresult noticeably. Due to the usage of these twoapproaches, PROMISE consistently resulted in betterreconstruction in our experiments than L1-SPIRiT andJoint Bayesian CS (Figs. 7–9) even when there wereinterscan motions. On the other hand, it is also true thatthe advantages of PROMISE over L1-SPIRiT, for exploit-ing structure information, will be gradually reduced withthe increase of motion level. However, an efficient singlestep registration operation after the first few iteration ofPROMISE can resolve the problem because PROMISE isrobust for moderate interscan displacement as discussedbefore. The results for PROMISE with registration will bereported separately.

Clinical Applicability

Besides interscan motion, another issue of existingmethods using sharable information is long reconstruc-

tion time (9,13). Compared with these existing meth-ods, PROMISE has lower computational cost. PROMISEonly has one additional step than L1-SPIRiT to extractsharable information. The extraction of boundary infor-mation took approximately 10 s in our implementationusing Matlab. The remaining steps for the calibration ofconvolution kernels and optimizing function in Eqs.[2,5] took the same computational cost as L1-SPIRiT.When the target of the acquisition acceleration is toreduce motion artifacts, or breath-hold time, it isworthwhile to spend extra seconds for the informationextraction between the scans. Moreover, the calculationcan also be processed in parallel with the preparationsteps for the following scans.

Generalization of PROMISE

This work focuses on 2D applications because multi-slice 2D imaging is widely used for multicontrastimaging clinically. Moreover the acquisition time couldbe even longer than 3D acquisition, as shown in Sup-plementary Table S1, which is available online. Faster2D acquisition does not only save acquisition time, butalso reduce motion artifacts, moderate geometry distor-tion artifacts in EPI acquisition (30) and improve thepatient comfort. However, the concept of the proposedmethod can also be easily extended to 3D. One exam-ple of the application in 3D imaging was demonstratedin Supplementary Figure S1 in the supplemental mate-rial. The set of T1w and T2w images were acquired ona Philips 3T system with an eight-channel coil. Pois-son disk distributed undersampling trajectory wasapplied for accelerated 3D sampling and the perform-ance of L1-SPIRiT and PROMISE were compared. InPROMISE, the sensitivity kernels were computed basedon T1w scan and the structural information wasextracted using the proposed scheme in each plane ofthe 3D dataset. PROMISE resulted in lower RMSE(6.1%) than L1-SPIRiT (7.2%) at reduction factor of 12.Thus, the usage of sharable information helped thepreservation of boundaries in 3D multicontrast imagingas well.

FIG. 6. a: Plot of RMSEs in image support of the T2w scan of dataset 1 at reduction factors from 2 up to 6. The lines represent thereconstructions using L1-SPIRiT (dotted), Joint Bayesian CS (dashed), and the proposed PROMISE (solid) respectively. b: Shows the

plots of the proposed method PROMISE with different sharable information used (circle, sensitivity only; triangle, structure only; star,proposed usage of both sharable information), at different reduction factor from 2 up to 6. The ROI is defined by the image support

extracted from reference T2w image.

Improving CS-PPI Methods for Multicontrast Imaging 531

Limitation and Future Work

This study reported the preliminary results of PROMISE.The major target of the study is to explain the advantagesof using manifold sharable information (sensitivity,structure, etc.) in a sequential approach. More detailexplanation of the boundary extraction step is providedin appendix.

The manuscript only provides a specific example set ofsharable information between multicontrast scans. Modifi-cations, especially the usage of additional sharable infor-mation, can be adopted for acquisition and reconstructionfor different applications. For example, with previousreconstructed k-space as a reference, undersampling tra-jectory can probably be optimized (31,32) to improve theacquisition for multicontrast imaging with sharable infor-mation as well. To focus on the major target of this study,we will present a clinical applicable trajectory optimiza-tion scheme for multicontrast imaging in a separate work.

FIG. 7. Comparison of the sensitivity to rigid interscan motion(simulated in-plane motion of dataset 1) of L1-SPIRiT, Joint Bayes-

ian, and PROMISE. The T2w image (a) contains simulated dis-placement from the T1w image (b). The T2w image was artificially

translated by two pixels to the right and two pixels to the bottomand rotated by 5 degrees counterclockwise and then synthesizedwith sensitivity maps. Reconstructed T2w image with net reduc-

tion factor of 4 and the corresponding error maps are shown inc,d (L1-SPIRiT), e,f (Correlation Imaging), and g,h (PROMISE),

respectively. The error maps are five times brightened and theRMSE in image support based on reference T2w image is labeledat the right-bottom of each error-map.

FIG. 8. Comparison of the sensitivity to rigid interscan motion(actual in-plane and out-of-plane motions in dataset 2) of L1-SPI-

RiT, Joint Bayesian CS, and the proposed PROMISE method. Thefully sampled and reconstructed T2w image is shown in a. PROM-ISE exploits sharable information extracted from the T1w scan (b).

Reconstructed image with reduction factor of 4 and the corre-sponding five times brightened error maps are shown in c,d (L1-

SPIRiT), e,f (Joint Bayesian CS), and g,h (PROMISE), respectively.RMSE in image support based on reference T2w image is labeledat the right bottom of each error-map.

532 Gong et al.

In this work, we proposed a statistical algorithm as a spe-cific example to extract structural information in asequential approach. There are other schemes for thesame purpose, such as Distance Transform (33) of CannyBoundary operation results. Based on simulation andexperiments, we found that the proposed method results

in the better balance of detail preservation and noise reduc-tion than other tested methods (RMSE 7.6% for statisticalmethod and 8.9% for Distance Transform approach whenR¼ 5). Due to the limitation of space, further discussionsand experimental results have to be presented separately.

CONCLUSIONS

In this study, the idea of sequentially exploiting sharableinformation among multicontrast scans is introduced toresolve the challenges in multicontrast MRI reconstruc-tion using Parallel Imaging and Compressed Sensing. Aspecific embodiment, PROMISE, shows significant andstable improvement for L1-SPIRiT and demonstratesadvantages compared with previously proposed meth-ods, when sharable information with or without inter-scan motion was used.

APPENDIX

Here, we present methods to quantify and determine theprobabilities of representing boundaries for componentsin the wavelet domain (pn) and image domain (pi).

1. Extraction of Boundary Information in Wavelet Domain

In the wavelet domain, boundaries are encoded as blocksof significant wavelet coefficients in multiple scales ofresolutions. In contrast, piecewise constant componentsand additional noises are encoded as small wavelet coef-ficients in most scales and impulses in high resolutionfrequency. Thus, exploiting the properties of the waveletdomain can help to distinguish boundaries from back-ground and noises. We implemented and improved theestimation method in the wavelet domain (24) to extractthe statistical structure information.

A Hidden Markov Tree (HMT) model was used tomodel the probability density function p of each waveletcoefficient un ¼ ðCmÞn as a Gaussian mixture densitywith a hidden binary state Sn, which in our applicationis either “Negligible” (Sn ¼ N) or “Significant” (Sn ¼ S).

pð•Þ 2 ½0; 1�;pðSn ¼ SÞ þ pðSn ¼ NÞ ¼ 1 [A1]

Extraction of structures at multiple levels of resolutioncan be achieved inherently due to the properties ofwavelet transform (24). There are three properties of theHidden Markov Quad-Tree model based on the proper-ties of 2D wavelet transform.

First, the values of wavelet coefficients obey a Gaus-sian mixture distribution. Second, the hidden states ofcoefficients in different scales in 2D wavelet quad-treesare correlated. The state that changes to daughter coeffi-cient from a mother coefficient n can be formulatedusing a Transition Matrix Tn. Third, both the Gaussianmixture distribution and Transition Matrices betweenstates are scale-dependent and can be represented byexponential functions of the scale J in wavelet-trees.These properties can be characterized by followingequations.

The mixture distribution function forms dual-Gaussiandistribution with hidden state and scale dependentvariance:

FIG. 9. Comparison of the sensitivity to nonrigid interscan motion

(actual in-plane motions and through-plane motions in carotiddataset) of L1-SPIRiT, Joint Bayesian, and the proposed PROMISE

method. Reference T1w image (a) and reconstructed T2w image(b) with interscan motion are shown in the first row. PROMISEexploits sharable information from the T2w scan to reconstruct

the undersampled T1w scan. A reconstructed T1w image withreduction factor of 4 and the corresponding five times brightenederror maps are shown in c,d (L1-SPIRiT), e,f (Joint Bayesian CS)

and g,h (PROMISE), respectively. The RMSE in image supportbased on the T2w image (from which sharable information is

extracted) is labeled at the right bottom of each error-map.Zoomed-in images of the left artery, at the position indicated inthe yellow square in a, are two times brightened and shown in the

left bottom of each subfigure.

Improving CS-PPI Methods for Multicontrast Imaging 533

f ðunÞ ¼ f ðunjSn ¼ SÞpðSn ¼ SÞ þ f ðunjSn ¼ NÞpðSn ¼ NÞ

f ðunjSn ¼ SÞ ¼ Nð0;sS;n2Þ ¼ Nð0;CsS

4�JaSÞ

f ðunjSn ¼ NÞ ¼ Nð0;sN ;n2Þ ¼ Nð0;CsN

4�JaN Þ[A2]

The transition of the hidden states is modeled usingtransition matrix with scale dependent transitionprobability:

pðSn ¼ SÞ

pðSn ¼ NÞ

24

35 ¼Tn

pðSm ¼ SÞ

pðSm ¼ NÞ

24

35 ¼

pnS!S pn

N!S

pnS!N pn

N!N

24

35

pðSm ¼ SÞ

pðSm ¼ NÞ

24

35

Tn ¼pn

S!S pnN!S

pnS!N pn

N!N

24

35

¼

1

4þ CAS4�JgS CAN 4�JgN

3

4� CAS4�Jgs 1� CAN4�JgN

26664

37775

pnS!S þ pn

S!N ¼ 1

pnN!S þ pn

N!N ¼ 1

pn�!� 2 ½0; 1�

[A3]

Thus, there are several superparameters of the HMT:

Q ¼ faN ;aS;gN ; gS;CdN;CdS

;CAN ;CASg: [A4]

Using a HMT based Expectation Maximization (EM)(18), we optimize the superparameters based on the mea-surement (wavelet transform of the reconstructed imagem) and estimate the statistical probabilities as a represen-tation of structure information in wavelet transform.

pn ¼ pðSn ¼ SjCm;QðCmÞÞ: [A5]

The details can be found in Gong et al (19) and Durateet al (20).

2. Extraction of Boundary Information in Image Domain

To extract the boundary information in the imagedomain, we extend conventional TV transform to amultiple-scale TV transform.

rXkði; jÞ ¼ ðXkði; jÞ � Xkði; j � 1Þ;Xkði; jÞ � Xkði � 1; jÞÞ

Xkði; jÞ ¼ Xði; jÞ FLPkði; jÞ

[A6]

Multiscale Total Variance (MSTV) is analog to a hardwavelet and is defined in [A6], in which Xði; jÞ is pixel-wise value in the image and FLPk

ði; jÞ is a low-pass filterin the image domain for level k. By using filters with dif-

ferent levels of resolution, MSTV can be generated usingEq. [A6]. MSTV shares similar properties as wavelet andcan be used to extract structure information and estimatestatistical regulation weights, as discussed in previoussession. Then the estimated probability is used as regula-rization weights in total variance transform.

In this way, we extracted boundary information in theimage domain. In addition, the proposed method issupreme to conventional edge operators because it is effi-cient and able to correct false edges using multiresolu-tion information.

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