Denoising Sparse Images from GRAPPA Using theNullspace Method

Daniel S. Weller,1* Jonathan R. Polimeni,2,3 Leo Grady,4 Lawrence L. Wald,2,3

Elfar Adalsteinsson,1 and Vivek K. Goyal1

To accelerate magnetic resonance imaging using uniformlyundersampled (nonrandom) parallel imaging beyond what isachievable with generalized autocalibrating partially parallelacquisitions (GRAPPA) alone, the DEnoising of Sparse Imagesfrom GRAPPA using the Nullspace method is developed. Thetrade-off between denoising and smoothing the GRAPPA solu-tion is studied for different levels of acceleration. Several brainimages reconstructed from uniformly undersampled k-spacedata using DEnoising of Sparse Images from GRAPPA usingthe Nullspace method are compared against reconstructionsusing existing methods in terms of difference images (a quali-tative measure), peak-signal-to-noise ratio, and noise amplifi-cation (g-factors) as measured using the pseudo-multiplereplica method. Effects of smoothing, including contrast loss,are studied in synthetic phantom data. In the experiments pre-sented, the contrast loss and spatial resolution are competitivewith existing methods. Results for several brain images dem-onstrate significant improvements over GRAPPA at high accel-eration factors in denoising performance with limited blurringor smoothing artifacts. In addition, the measured g-factorssuggest that DEnoising of Sparse Images from GRAPPA usingthe Nullspace method mitigates noise amplification better thanboth GRAPPA and L1 iterative self-consistent parallel imagingreconstruction (the latter limited here by uniformundersampling). Magn Reson Med 68:1176–1189, 2012. VC 2011Wiley Periodicals, Inc.

Key words: sparsity; receive arrays; accelerated imaging;denoising

The time-consuming nature of image encoding inmagnetic resonance imaging continues to inspire novelacquisition and image reconstruction techniques andtechnologies. In conventional Cartesian Fourier trans-form-based imaging, 3D k-space is sampled along readout

lines uniformly spaced in the phase-encode directions.To reduce the scan time, accelerated parallel imaginguses multiple receiver coils to produce high-qualityimages without aliasing from k-space acquisitions thatare undersampled in the phase-encode directions.

To obtain superior image reconstructions from evenless data than is possible with multiple receiver coilsalone, we investigate combining sparsity-promoting regula-rization with the generalized autocalibrating partially paral-lel acquisitions (GRAPPA)-accelerated parallel imagingreconstruction method (1). GRAPPA directly estimates themissing k-space lines in all the coils from the under-sampled data using a set of specially calibrated reconstruc-tion kernels. The proposed method uses a combination ofa weighted fidelity to the GRAPPA reconstruction, a jointsparsity penalty function, and a data-preserving nullspaceformulation. This method is designed to accommodate uni-form Cartesian sampling patterns and can be implementedwith similar complexity to existing sparsity regularizationmethods; in addition, this method can be easily adapted torandom Cartesian subsampling by modifying only theGRAPPA reconstruction step, although this extension isnot studied here.

Accelerated Parallel Imaging

Given multiple receiver coils, postprocessing techniques,such as sensitivity encoding (SENSE) (2), simultaneousacquisition of spatial harmonics (SMASH) (3), GRAPPA,and SPIR-iT (4), synthesize unaliased images from multi-coil undersampled data in either k-space or the imagedomain. GRAPPA and SPIR-iT acquire several additionalk-space lines, called ACS lines, to form kernels for usein reconstruction. One important limitation of GRAPPAis the assumption of uniform undersampling. However,GRAPPA can be extended to nonuniform Cartesian sub-sampling at greater computational cost by using manykernels, one for each source/target pattern encountered.In contrast, SPIR-iT easily accommodates nonuniformsampling patterns, because SPIR-iT uses the kernel toenforce consistency between any point in k-space and itsneighborhood of k-space points, both acquired andunknown. All these methods successfully reconstructdiagnostically useful images from undersampled data;however, at higher acceleration factors, GRAPPA tendsto fail because of noise amplification or incompletelyresolved aliasing. Because of its indirect formulation,SPIR-iT is more computationally intensive than GRAPPAfor uniform undersampling.

Sparsity-Enforcing Regularization of Parallel Imaging

Because a wide variety of MR images are approximatelysparse (5), the compressed sensing (CS) (6–8) framework is a

1Research Laboratory of Electronics, Massachusetts Institute of Technology,Cambridge, Massachusetts, USA.2A. A. Martinos Center, Department of Radiology, Massachusetts GeneralHospital, Charlestown, Massachusetts, USA.3Department of Radiology, Harvard Medical School, Boston, Massachusetts,USA.4Department of Image Analytics and Informatics, Siemens CorporateResearch, Princeton, New Jersey, USA.

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

Grant sponsor: NSF CAREER; Grant number: 0643836; Grant sponsor:NIH; Grant number: R01 EB007942 and EB006847; Grant sponsor: NIHNCRR; Grant number: P41 RR014075; Grant sponsors: Siemens CorporateResearch and an NSF Graduate Research Fellowship

*Correspondence to: Daniel S. Weller, M.S., Research Laboratory ofElectronics at the Massachusetts Institute of Technology, Room 36-680, 77Massachusetts Avenue, Cambridge, MA 02139-4307. E-mail: dweller@mit.edu

Received 11 March 2011; revised 18 November 2011; accepted 20November 2011.

DOI 10.1002/mrm.24116Published online 28 December 2011 in Wiley Online Library(wileyonlinelibrary.com).

Magnetic Resonance in Medicine 68:1176–1189 (2012)

VC 2011 Wiley Periodicals, Inc. 1176

reasonable alternative for recovering high-quality imagesfrom undersampled k-space. However, remaining faithful touniform undersampling destroys the incoherent sampling(with respect to the sparse transform domain) useful for CS.Applying CS or methods combining parallel imaging and CSsuch as L1 iterative self-consistent parallel imaging recon-struction (L1 SPIR-iT) (9,10) to uniformly undersampled dataresults in images with noticeable coherent aliasing artifacts.Because the noise amplified by parallel imaging methodssuch as GRAPPA remains unstructured in the sparse trans-form domain regardless of the k-space sampling pattern,sparsity-enforcing regularization, nevertheless, canimprove image quality in the uniformly undersampled casevia denoising. This combination of GRAPPA and sparsity-based denoising remains not fully explored. his work, ofwhich a preliminary account is presented in (11), succeedsCS-GRAPPA (12), L1 SPIR-iT, CS-SENSE (13), and otherjoint optimization approaches. It combines GRAPPA withsparsity regularization to enable quality reconstructionsfrom scans with greater acceleration where noise amplifica-tion dominates the reconstruction error.

CS-GRAPPA, CS-SENSE, and L1 SPIR-iT all combineparallel imaging methods (GRAPPA, SENSE, and SPIR-iT) with CS, but each differs from the proposed method(beyond undersampling strategies). CS-GRAPPA com-bines CS and GRAPPA, alternatingly applying GRAPPAand CS reconstructions to fill the missing k-space lines;the iterative structure of this method may limit the syn-ergy between sparsity and multiple coils. CS-SENSEsequentially applies CS to each aliased coil image andcombines the CS results using SENSE; like SENSE, CS-SENSE is highly sensitive to the quality of the estimatesof the coil sensitivity profiles. L1 SPIR-iT is most similarto the proposed method, as it jointly optimizes both forsparsity and for fidelity to a parallel imaging solution.However, L1 SPIR-iT iteratively applies the SPIR-iT ker-nel, updating the k-space data for consistency instead offilling in the missing k-space data directly. The presentedalgorithm uses a direct GRAPPA reconstruction andapplies sparsity as a postprocessing method for denoising.

The proposed method—DEnoising Sparse Images fromGRAPPA using the Nullspace method (DESIGN)—jointlyoptimizes a GRAPPA fidelity penalty and simultaneoussparsity of the coil images while preserving the data byoptimizing in the nullspace of the data observation matrix.The effects of adjusting the tuning parameter balancingsparsity and GRAPPA fidelity are evaluated. The resultingalgorithm is compared against GRAPPA alone and otherdenoising methods, to measure the additional improve-ment possible from combining these accelerated imagereconstruction techniques, as well as against L1 SPIR-iT,the prevailing method combining parallel imaging and CS(albeit here with uniform undersampling).

THEORY

GRAPPA

In this work, we assume that the 3D acquisition is acceleratedin only the phase-encode directions, so the volume can be di-vided into 2D slices in the readout direction and each slicereconstructed separately. Thus, GRAPPA and SPIR-iT arepresented for 2D accelerated data with 2D calibration data;

this formulation follows the ‘‘hybrid space coil-by-coil data-driven (CCDD) method’’ with ‘‘independent calibration’’ dis-cussed in Ref. 14; other formulations are certainly possible.

For a given slice, denote the 2D k-space value at fre-quency (ky,kz) encoded by the pth coil as yp(ky,kz), thefrequency spacing corresponding to full field-of-viewsampling (Dky, Dkz), the number of coils P, and the sizeof the correlation kernel (By,Bz). In GRAPPA, the missingdata are recovered using

yp ky þ ryDky ;kz þ rzDkz� �

¼XPq¼1

XBy

by¼1

XBz

bz¼1gðp;ry ;rzÞq ðby ; bzÞyq

� ky þ by �dBy

2

�� �RyDky ;kz þ bz �

Bz

2

� �� �RzDkz

� �; ½1�

where the kernel coefficients gðp;ry ;rzÞq ðby ; bzÞ are com-

puted from a least-squares fit using the ACS lines in thatslice. Let M and N represent the number of acquired andfull k-space data points in a coil slice, respectively; theM � P matrix D is the acquired data for all the coils, andthe N � P matrix Y is the full k-space for all the coils.Collecting the GRAPPA reconstruction equations for thefull k-space yields the full k-space reconstructionYGRAPPA ¼ G(D), where G is a linear system that can beimplemented efficiently using convolutions.

SPIR-iT

SPIR-iT uses a different kernel to form consistency equa-tions for each point in the full k-space with its neighborsand organizes the resulting consistency equations intothe linear system Y ¼ GS(Y); the complete derivation ofthe consistency kernel coefficients can be found in Ref.4. SPIR-iT minimizes the l2 consistency error using theconstrained optimization problem

YSPIR�iT 2 minimizeY

jjY�GSðYÞjj2F s:t: jjD� KYjj2F � e;

½2�

where K is the M � N subsampling matrix selecting theobserved k-space data points, ||�||F is the Frobeniusnorm, and e constrains the allowed deviation from thedata. When preserving the acquired data exactly (settinge ¼ 0 above), the SPIR-iT optimization problem is con-veniently expressed in the nullspace of the subsamplingmatrix K (15). Let Kc represent the (N � M) � N subsam-pling matrix selecting the missing k-space. The range of[KT,(Kc)T] is RN, and each row of Kc is in the nullspaceof K. Thus, if the missing k-space data X ¼ KcY, then Y¼ KTD þ (Kc)TX is a nullspace decomposition of Y.When the data are preserved with equality, the SPIR-iToptimization problem becomes

XSPIR�iT 2 minimizeX

KTDþ ðKcÞTX�GSðKTDþ ðKcÞTXÞ��� ������ ���2

F;

� YSPIRI�iT ¼ KTDþ ðKcÞTXSPIR�iT: ½3�

Augmenting the SPIR-iT optimization problem withthe hybrid l2,1 norm of the sparsifying transform of thecoil images and utilizing random undersampling of k-space yield the L1 SPIR-iT method. Denote the

DESIGN 1177

regularization parameter l, the sparsifying transform C,and the inverse discrete Fourier transform F�1, the aboveoptimization problem becomes

XL1SPIR�iT 2 minimizex

jjKTDþ ðKcÞTX�GSðKTDþ ðKcÞTXÞ

jj2F þ lXNn¼1jj CF�1ðKTDþ ðKcÞTXÞh i

n; : jj2: ½4�

DESIGN: Combining GRAPPA and Sparsity

To regularize GRAPPA with sparsity, the objective func-tion consists of a least-squares term to favor fidelity tothe GRAPPA k-space result and a function to promote si-multaneous sparsity across the coil images in the sparsetransform domain. Using the notation G(D) to denote theGRAPPA-reconstructed full k-space, l for the tuning pa-rameter trading-off fidelity to the GRAPPA solution forsparsity, k�ks for the simultaneous sparsity penalty func-tion, we have

YDESIGN 2 minimizeY

jjY�GðDÞjj2F þ ljjCF1Yjjss:t D ¼ KY: ð5Þ

Denote the sparse transform representation of a singlevector w. To construct the simultaneous sparsity penaltyfunction on the sparse transform matrix W, we first con-sider a single vector w and a separable approximation tothe l0 ‘‘norm’’

PNn¼1 sðwnÞ. For example, s(wn) ¼ |wn| for

the l1 norm in Refs. 6 and 8, s(wn) ¼ |wn|p for the lp (p

< 1) penalty in Ref. 16, or s(wn) ¼ log(1 þ |wn|2) for the

Cauchy penalty function (17). In this article, the l1 normis chosen because of its convexity and suitability forapproximately sparse signals. Now, let W represent thesparse transform coefficients for a collection of vectorsY; W ¼ WF�1Y. To extend this sparsity penalty functionto simultaneous sparsity across coil images, we considers(wn) applied to the l2 norm of the vector of sparse trans-form coefficients [Wn,1, …, Wn,P]:

jjWjjs ¼XN

n¼1 sðjj½Wn;1 � � � Wn;P �jj2Þ: ½6�

By enforcing simultaneous sparsity across the magni-tudes of the sparse representations of all the coil images,we target the sparsity inherent in the combined image.

The GRAPPA reconstruction operations in k-spacenonuniformly amplify the additive noise in the imagedomain. Coupled with correlation across channels, theGRAPPA reconstruction noise power varies in each voxeland across coils, so we replace the GRAPPA fidelityterm with a weighted l2 norm. Here, we weight the devi-ation from the GRAPPA result for each voxel in each coilby its contribution to the final combined image, with thenotion that voxels with B1

� attenuation (due to thereceive coils’ sensitivities) or more noise amplification(due to GRAPPA) should be allowed to deviate more andcontribute less to the final image. The multichannelreconstructed images are combined as a postprocessingstep using per-voxel linear coil-combination weights Cthat are computed from an estimate of the coil sensitivity

profiles. Because of the known smoothness of coil sensi-tivity profiles, this estimate is computed from low-reso-lution data. If the ACS lines are situated at the center ofk-space (as is the case for the datasets evaluated in thisarticle), this calibration data are suitable for estimatingthe coil sensitivity profiles due to the known smoothnessof those profiles. Otherwise, a sum-of-squares combina-tion of the reconstructed coil images can be performedand the GRAPPA fidelity term is reweighted by just thecoil noise covariance. Based on the formulation of themultichannel SNR equation in (18) and SENSE, the coilcombination weights are computed to be ‘‘signal normal-ized,’’ i.e., the gain is unity. In particular, we use theweights that combine the data across channels such thatthe resulting SNR is optimized (if the sensitivities areexact):

½C1ðx; y; zÞ _s CPðx; y ; zÞ�

¼

S1ðx; y; zÞ...

SPðx; y ; zÞ

2664

3775H

L�1

S1ðx; y; zÞ...

SPðx; y ; zÞ

2664

3775

0BB@

1CCA�1

S1ðx; y ; zÞ...

SPðx; y ; zÞ

2664

3775H

L�1

½4�

where [�]H is the conjugate transpose, Sp(x,y,z) are thecoil sensitivity estimates, and L is the measured coilnoise covariance matrix. Each voxel of the combinedimage is formed by multiplying the vector [C1(x,y,z), . . .,CP(x,y,z)] by the stacked vector of the correspondingpixel in all of the coil images. This combination corre-sponds to performing SENSE on unaccelerated data withlow-resolution coil sensitivity estimates. Using these coilcombination weights, DESIGN becomes

YDESIGN 2 minimizeY

jjC � F�1ðY�GðDÞÞjj2F þ ljjCF�1Yjjss:t D ¼ KY: ð8Þ

Although not investigated here, one could also substi-tute analytical GRAPPA per-coil g-factors (based on thekernel) for the coil combination weights. However, thecoil combination weights may already be available, aswe use them to combine the full reconstructed k-spaceinto a single combined image.

To solve this optimization problem, we first convert itto an unconstrained form using the aforementioned null-space method:

XDESIGN 2 minimizex

jjC � F�1ðKTDþ ðKcÞTX�GðDÞÞjj2Fþ ljjCF�1ðKTDÞ þ ðKcÞTXjjs: ½9�

The iteratively reweighted least squares (IRLS) methodused in Refs. 19 and 20 is applied in conjunction withleast-squares solver least-squares minimum residual(LSMR) (21), available online at http://www.stanfor-d.edu/group/SOL/software/lsmr.html. The IRLS methodtransforms the above problem into a succession of least-squares ones:

Xtþ1 2 minimizex

jjC � F�1ðKTDþ ðKcÞTX�GðDÞÞjj2F

þ ljjD12ðXtÞCF�1ðKTDþ ðKcÞTXÞjj2F ½10�

1178 Weller et al.

where the diagonal matrix [D(X)]n,n ¼ qs(wn)/qwn/wn forwn ¼ kWn,:k2, and W ¼ CF�1(KTD þ (Kc)TX); using thischoice of reweighting matrix with IRLS is equivalent tousing half-quadratic minimization (22–24). The gradient(or when s(�) is not differentiable, the subgradient) of theobjective function is as follows:

2KcF�HðC� � C � F�1ðKTDþ ðKcÞTX�GðDÞÞÞþ 2lKcF�HC�HðDðXtCF�1ðKTDþ ðKcÞTXÞÞ; ½11�

where [�]* is the element-wise complex conjugate. Settingthe gradient equal to zero yields the linear system AHAx ¼AHb, where x ¼ vec(X), diag(�) constructs a diagonal ma-trix from a vector, � is the Kronecker product, and

A ¼diagðvecðCÞÞffiffiffi

lpðIP�P � D

12ðXtÞCÞ

" #IP�P � F�1ðKcÞT; and

b ¼vecðC � F�1ðGðDÞ � KTDÞÞ�

ffiffiffilp

vecðD12ðXtÞCF�1KTDÞ

" #: ½12�

The IRLS method consists of fixing the diagonalweight matrix, solving the resulting linear system usingLSMR, recomputing the weight matrix for the new vectorx, and repeating until convergence. Solving the normalequations AHAx ¼ AHb using LSMR only requires thatthe matrix–vector multiplications Ax and AHy be effi-cient to compute for arbitrary vectors x and y.

Design Choices

Before applying DESIGN denoising, a couple importantdesign choices must be considered. Based on the class ofimages, an appropriate sparsifying transform must bechosen. Empirical evidence presented in Ref. 5 suggeststhat a wavelet or finite-differences transform yieldsapproximately sparse representations for many types ofmagnetic resonance imaging data, including brainimages. Multiscale multiorientation transforms such asthe curvelet (25), contourlet (26), or shearlet (27) mayimprove sparsity; however, the extension of DESIGN toovercomplete transforms is not discussed here. In addi-tion, the optimal choice of tuning parameter should bedetermined, based on the expected noise amplificationand sparsity of the desired image.

METHODS

In practical applications of DESIGN, data are acquiredusing a uniform subsampling of Cartesian k-space. A smallset of ACS lines is acquired and is used to compute the 3� 3 GRAPPA kernel (the 2D kernel is the size of threeneighboring k-space blocks in each direction) to generatecoil combination weights and to include as known data inthe reconstruction. Before computing the sensitivity pro-files, the ACS lines are apodized using a Blackman win-dow that reduces the effects of low-resolution blurring onthe profile estimates. The noise covariance matrix L ismeasured via a noise-only acquisition (no RF excitation)collected before the main acquisition; the noise covariancematrix is computed using the sample covariance acrossthe coils of all the k-space samples.

Once the data are acquired, slices orthogonal to the read-out direction are reconstructed individually usingGRAPPA and the DESIGN algorithm, using design parame-ters appropriate for the acquisition. The optimization prob-lem is solved using IRLS, combined with LSMR for invert-ing linear systems. Pseudocode for the complete algorithmis shown in Table 1. Each inner iteration of LSMR requires2P fast Fourier transform (FFT)s, P fast sparsifying trans-forms, and its transpose, as well as multiplication by diag-onal weight matrices and subsampling matrices. Althoughb (on the right side of the normal equation) only needs tobe recomputed when the weight matrix changes, comput-ing b has similar complexity. Thus, the overall algorithm’scomplexity is comparable to existing sparsity-baseddenoising methods. Because the time-consuming opera-tions (FFT, wavelet, etc.) can be parallelized on a GPU, anefficient implementation suitable for practical use ispossible.

For the experiments that follow, several full-field-of-view 3D datasets are acquired using unacceleratedsequences on Siemens Tim Trio 3-T systems (SiemensHealthcare, Erlangen, Germany) with the vendor 32-channel head-coil array. Two T1-weighted magnetizationprepared rapid gradient echo (MPRAGE)s (both 256 �256 � 176 sagittal, 1.0 mm isotropic) and a T2-weightedturbo spin echo (TSE) (264 � 256 � 23 axial, 0.75 mm �0.78 mm � 5 mm) all were collected, with acquisitiontimes of 8, 8, and 4 min, respectively. From each of thesedatasets, a 2D axial slice is extracted (and for the firsttwo datasets, cropped); sum-of-squares combinations ofthese cropped images are used as the ground-truth. Sli-ces from these three datasets and a synthetic phantomfor studying contrast loss are shown in Fig. 1. Multiplechannels were generated for the synthetic data using theBiot–Savart B1 simulator written by Prof. Fa-Hsuan Linavailable online at http://www.nmr.mgh.harvard.edu/�fhlin/tool_b1.htm. To generate the reduced-field-of-viewdata, each slice is undersampled in both directions in k-space. A 36 � 36 block (not undersampled) at the centerof k-space is retained as the ACS lines used in the recon-structions; the total (effective) acceleration R accountsfor this additional block of data. Based on the empiricalresults in the literature (5), a four-level ‘‘9-7" discretewavelet transform (DWT) is selected as an appropriatesparsifying transform for these images. This data areprocessed using the different reconstruction and denois-ing algorithms implemented in MATLAB. Each recon-structed image is evaluated qualitatively by generating adifference image and quantitatively by computing thepeak-signal-to-noise ratio (PSNR):

PSNR ¼ 20log10maxðjref:imagejÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1N

Pjjrecon:imagej � jref:imagejj2

q dB:

½13�

Both the difference images and the PSNR are com-puted from magnitude images. As PSNR does notaccount for the importance of specific image features orthe preservation of certain anomalies such as tumors, theincluded difference images are used to understand thealgorithms’ relative performances.

DESIGN 1179

Optimizing the Tuning Parameter

To denoise a dataset using DESIGN, one must select avalue of l that achieves the desired trade-off betweenGRAPPA fidelity and sparsity. The sparsity penalty can beviewed as regularization, as it denoises the GRAPPAreconstruction. In this experiment, the effect of varying l

is studied for a single dataset and different accelerationfactors. As tuning parameter selection is common to manyregularization problems, inspiration for choosing an appro-priate value can be drawn from the literature, includingcross validation for the lasso (28) or reference image-basedL-curve selection for parallel imaging (29). Because thenoise amplification of GRAPPA, and accelerated parallelimaging in general, is tied to the undersampling factor, thenoise in the data, and the sparsity of the image, the choiceof tuning parameter, l, varies from image to image and ac-quisition to acquisition. For each acceleration, DESIGN isrun for a coarse range of l from 10a, for a ¼ �5, �4, . . ., 5,and 6, and repeated for densely spaced l around the opti-mal coarse value to determine the best value. The optimal

choice of l is determined for each dataset independentlyin the performance comparisons to follow.

Performance Comparisons

To evaluate the performance of DESIGN, the implementa-tion using the DWT (4-level 9-7 wavelet) sparsifyingtransform with the l1 norm is compared against GRAPPA,GRAPPA with multichannel adaptive Wiener filter-baseddenoising, and L1 SPIR-iT (admittedly limited by uniformundersampling). The multichannel Wiener filter-baseddenoising is approximated by (i) estimating a global noisevariance on the combined image using median absolutedeviation (30), (ii) estimating the local signal mean andvariance for each pixel of the combined image, (iii) gener-ating signal and noise covariances across coils using low-resolution coil sensitivities estimated from the ACS lines,and (iv) denoising each pixel of the uncombined datausing the resulting multichannel Wiener filter. The imple-mentation of L1 SPIR-iT used in this article is developedin Ref. 10, which is available online at http://

FIG. 1. Sum-of-squares magnitude images of 2D axial slices from three fully sampled brain (T1-weighted, normalized; T1-weighted, nor-

malized; and T2-weighted, unnormalized) datasets and a synthetic noise-free contrast phantom. These images are used as referenceimages (ground truth) for the difference images and magnitude-PSNR calculations shown throughout this article.

Table 1Pseudocode for Proposed Method, DESIGN

1. Estimate coil sensitivities from low-resolution data and compute C.2. Compute GRAPPA kernel and run GRAPPA reconstruction on data D to

yield result G(D).3. Initialization:

X / initial guess

(possibly from GRAPPAresult or previous run of method).

4.wn jjCnF

�1ðKTDþ ðKcÞTXÞjj2 ; ½D�n;n 1

wn

@sðwnÞ@wn

5.A ¼ diagðvecðCÞÞffiffiffi

lpðIP�P � D

12ðXtÞCÞ

IP�P � F�1ðKcÞT; b ¼ vecðC � F�1ðGðDÞ � KTDÞÞ

�ffiffiffilp

vecðD12ðXÞtCF�1KTDÞ

;

X / LSMR(A, b)6. Repeat steps 4 and 5 until convergence.7. Full k-space result: Y KTDþ ðKcÞTX:8. Combined image: I

PPp¼1½C:F�1Y�;p:

1180 Weller et al.

www.cs.berkeley.edu/�mjmurphy/l1spirit.html. Varioussizes of the SPIR-iT kernel are simulated, but little differ-ence is observed between them; a 7 � 7 kernel (SPIR-iTkernel size refers to k-space points, not blocks) is usedhere. These performance experiments are repeated for var-ious accelerations for a representative slice from eachdataset. For L1 SPIR-iT, the regularization parameter l ischosen for each acceleration factor via the same parametersweep approach as for DESIGN for each dataset.

Eliminating Noise Covariance Estimation

To study the potential for eliminating the requirement ofestimating L from an additional noise-only acquisition,which requires additional time, the performance compar-isons for moderate levels of uniform undersamplingshown in Fig. 4 are repeated for the slices of the threehuman datasets used previously, substituting the iden-tity matrix for the measured noise covariance matrices.This approximation affects the coil combination weights,reweighting both the GRAPPA fidelity term of theDESIGN algorithm and the contributions of the recon-structed coil images to the combined image.

Noise Amplification and Geometry Factors

Geometry factors (g-factors) describe the noise amplifica-tion of a reconstruction method due to the geometry of

the coils beyond the factor of vR loss of SNR due toundersampling k-space (2). For a linear algorithm suchas GRAPPA or SENSE, the noise amplification shouldalso be linear, so the g-factors only depend on the sam-pling pattern/acceleration factor. Also, the g-factors canbe computed analytically for GRAPPA, as is done in Ref.31. For nonlinear algorithms, the SNR loss depends onthe input noise level, so g-factors are valid only in asmall range around that noise level. Note that althoughthe PSNR is sensitive to most sources of error, includingblurring and aliasing, g-factors are only indicative ofnoise amplification in the reconstruction. To computethe g-factors for GRAPPA, GRAPPA with multichanneladaptive Wiener filter-based denoising, L1 SPIR-iT, andDESIGN, 400 Monte Carlo trials are used. The noiseamplifications from each simulation are averaged, andthe resulting values are combined across coils using thecoil combination weights used to form the combinedimage. The parameters for each of the reconstructionalgorithms are identical to those used in the precedingperformance evaluations. In each trial, zero-meanadditive white complex-valued gaussian noise is addedto the k-space data before performing reconstructions;the noise covariance LAWGN is chosen to be the same asthe noise covariance matrix L observed for the acquisi-tion. This method for computing g-factors is described inRef. 32.

FIG. 2. DESIGN denoising results and difference images as l increases for the l1 norm with (a) R ¼ 8.7, (b) R ¼ 10.5, and (c) R ¼ 12.0uniform undersampling, all with the four-level ‘‘9-7" DWT. The value of l increases from left to right, and the results for the PSNR-opti-mal choice of l for this dataset are shown in the center column. When l is small, DESIGN resembles GRAPPA, and as l increases,

denoising and smoothing affect the result more noticeably. At high acceleration, these results suggest choosing a value of l thatdenoises without excessive blurring becomes more challenging.

DESIGN 1181

Oversmoothing and Loss of Contrast

When denoising images, one must be careful not to over-smooth, as the reduction of contrast or resolution canhide important features. To explore the impact ofDESIGN denoising on tissue contrast, a synthetic 32-channel contrast phantom (based on the phantomdescribed in Ref. 33) is generated, complex gaussiannoise is added, and the various reconstruction anddenoising methods are compared on R ¼ 12.1 uniformundersampled data from this phantom using the noise-free magnitude image as ground truth. This experimentis carried out for noise with 0.25 and 0.1% standarddeviation (before amplification due to undersampling) todepict the extent of contrast loss at different noise levels.Parameter sweeps for l are carried out for both DESIGNand L1 SPIR-iT in each experiment.

RESULTS

Several experiments are performed exploring designchoices and comparing DESIGN to existing methods.

Optimizing the Tuning Parameter

For DESIGN, the optimal choice of l is expected toincrease with R because the noise amplification due toGRAPPA and undersampling worsens with greater accel-eration. In Fig. 2, a representative region in DESIGNreconstructions of a slice of the first T1-weightedMPRAGE dataset depicts the trend in denoising andoversmoothing as l increases, with optimal values of l

generally increasing at higher accelerations. The noisepresent in the leftmost column and the oversmoothing inthe rightmost column together demonstrate the

FIG. 3. Performance comparisons with R ¼ 8.7, 8.6, and 9.9 uniform undersampling, respectively, for these three datasets. For the rep-resentative slices from each dataset, the magnitude and difference images of a representative region are shown, along with PSNR, for(a) GRAPPA, (b) GRAPPA with multichannel adaptive local Wiener filter-based denoising, (c) L1 SPIR-iT, and (d) DESIGN denoising. The

PSNR-optimal choices of l are shown for both L1 SPIR-iT and DESIGN. Note that L1 SPIR-iT is limited here by uniform undersampling.

1182 Weller et al.

significance of tuning parameter selection in obtainingdesirable results.

Performance Comparisons

Results and difference images using GRAPPA, GRAPPAwith multichannel adaptive local Wiener filter-baseddenoising, L1 SPIR-iT, and DESIGN denoising are shownfor representative slices of all three datasets in Figs. 3–5for increasing levels of uniform undersampling (see thefigures for the total accelerations R of the three datasets),respectively. The magnitude image PSNR values forthese different reconstruction methods are plotted as afunction of R in Fig. 6 for each of these images, over abroad range of total accelerations.

Several trends are evident from the images and plots.First, the noise power in the GRAPPA result increases

significantly as R increases. As expected, the PSNR gainfrom denoising is more evident at higher accelerations.In addition, the noise amplification in L1 SPIR-iTappears to increase far more slowly than with GRAPPA,so using the SPIR-iT parallel imaging result in place ofthe GRAPPA result may yield improved performance athigh accelerations with only the additional computationalcost up front to compute the SPIR-iT result. However,unlike GRAPPA, the L1 SPIR-iT result appears far worseat unaliasing without the help of random undersampling,as coherent aliasing is clearly visible in magnitude anddifference images for the last two datasets in Figs. 4 and5. PSNR is insensitive to this aliasing because the effectsare localized in the image. Blotches are clearly visible inthe multichannel Wiener filter-denoised results; both L1

SPIR-iT and DESIGN denoising produce more consistentimages. The multichannel Wiener filter also appears in

FIG. 4. Performance comparisons with R ¼ 10.5, 10.5, and 12.4 uniform undersampling for these three datasets, respectively. For therepresentative slices from each dataset, the magnitude and difference images of a representative region are shown, along with PSNR, for(a) GRAPPA, (b) GRAPPA with multichannel adaptive local Wiener filter-based denoising, (c) L1 SPIR-iT, and (d) DESIGN denoising. The

PSNR-optimal choices of l are shown for both L1 SPIR-iT and DESIGN. Note that L1 SPIR-iT is limited here by uniform undersampling.

DESIGN 1183

Fig. 5 to hit its limit in denoising capability at very highaccelerations. Finally, the evidence of similar levels ofimprovement from denoising using DESIGN overGRAPPA alone in each of the three datasets supports thebroad applicability of the proposed method, at least foranatomical brain images.

Eliminating Noise Covariance Estimation

Using the identity matrix in place of the measured noise

covariance matrix, DESIGN is compared for moderately

undersampled data against GRAPPA, GRAPPA with mul-

tichannel Wiener filter denoising, and L1 SPIR-iT in Fig.

7 as is done in the previous performance comparisons.

The ability of DESIGN to improve upon the GRAPPA

reconstructions for these three datasets appears ham-

pered by using an identity matrix in place of the meas-

ured noise covariance matrix. In most cases, the images

denoised with DESIGN still appear preferable to both

GRAPPA alone or with multichannel Wiener filtering,

reducing noise and producing few artifacts. Thus, if the

measured noise covariance matrix is not available, the

DESIGN method may still be used to suboptimal results.

Noise Amplification

To further understand the denoising capabilities of

DESIGN, the retained SNR (1/vR/g-factors) is shown for

the first dataset with LAWGN ¼ L in Fig. 8. GRAPPA,

multichannel adaptive Wiener filter-based denoising, L1

SPIR-iT, and DESIGN denoising are displayed below the

FIG. 5. Performance comparisons with R ¼ 12, 12.1, and 14.6 uniform undersampling, respectively, for these three datasets. For therepresentative slices from each dataset, the magnitude and difference images of a representative region are shown, along with PSNR,for (a) GRAPPA, (b) GRAPPA with multichannel adaptive local Wiener filter-based denoising, (c) L1 SPIR-iT, and (d) DESIGN denoising.

The PSNR-optimal choices of l are shown for both L1 SPIR-iT and DESIGN. Note that L1 SPIR-iT is limited here by uniformundersampling.

1184 Weller et al.

reconstructed images reproduced from Fig. 4, using

exactly the same parameters as in the preceding perform-

ance comparisons, for R ¼ 10.5. The substantially lower

noise amplification present in the DESIGN result con-firms the noise suppression from regularizing theGRAPPA result with sparsity; the noise suppressionfrom DESIGN far exceeds that of the conventionaldenoising method based on multichannel local Wienerfiltering. Because of the nonlinearity of the DESIGN algo-rithm, these results would be expected to change whenthe added noise power described by LAWGN is varied.

The present implementation of L1 SPIR-iT does notappear to mitigate noise amplification nearly as much asDESIGN does for this acceleration factor. Thus, DESIGNmay be combined with L1 SPIR-iT in situations whennoise suppression is a priority. More study is needed tounderstand this apparent advantage.

Contrast Loss and Oversmoothing

The apparent downside of denoising using a methodlike DESIGN is the smoothing or blurring that results

FIG. 6. Magnitude image PSNRs plotted versus total acceleration R for representative slices of (a and b) two T1-weighted MPRAGE

datasets and (c) the T2-weighted TSE dataset for GRAPPA, GRAPPA with multichannel adaptive local Wiener filter-based denoising, L1SPIR-iT, and DESIGN denoising. The inset regions display the PSNR trends for smaller R, where the methods’ performances are moresimilar. Note that L1 SPIR-iT is limited here by uniform undersampling. For a given acceleration, the tuning parameter is determined for

each dataset individually.

DESIGN 1185

from aggressively using a sparsity-based penalty; thisblurring can reduce both contrast and spatial resolu-tion. To quantify the loss of contrast due to denois-ing, we process the synthetic contrast phantom attwo different noise levels and identify the circularcontrast regions lost in the noise in the reconstruc-tion. Upon visual inspection, we observe from Fig. 9that despite many of the contrast regions being lostin the noise in the GRAPPA reconstructions for bothnoise levels, those contrast regions are visible in thedenoised DESIGN result. At the 0.25% noise level,the contrast of the central circles in the bottom rowdiminishes from 610% to 68%. At the 0.1% noiselevel, the contrast decreases to 69%. Thus, contrastdegradation does not appear to be significant at thesenoise levels.

DISCUSSION

The primary design choices left to the user involve

selecting an appropriate sparsifying transform and choos-

ing the value of the tuning parameter. Throughout this

article, we achieve reasonable results using a four-level

‘‘9-7" DWT; although not specifically tuned for brain

images, a different transform may be necessary to apply

DESIGN successfully to other types of data. The first

experiment depicts the effect of choosing different values

for the tuning parameter l on DESIGN. As the primary

design choice left to the user, the tuning parameter may

be selected via either a parameter sweep, as is done in

this article, or a method like cross validation.The images depicting the results of reconstruction

using DESIGN demonstrate that the proposed method

FIG. 7. Performance comparisons with R ¼ 10.5, 10.5, and 12.4 uniform undersampling, for these three datasets, respectively, repeatedusing the identity matrix in place of the measured noise covariance. For the representative slices from each dataset, the magnitude anddifference images of a representative region are shown, along with PSNR, for (a) GRAPPA, (b) GRAPPA with multichannel adaptive local

Wiener filter-based denoising, (c) L1 SPIR-iT, and (d) DESIGN denoising. The choices of l were established based on coarse, then fine,parameter sweeps for each dataset independently.

1186 Weller et al.

mitigates noise amplification due to both undersamplingand GRAPPA, improving PSNR and supporting thenotion of sparsity-enforcing regularization as an effectivedenoising method. However, improvements in PSNR,like mean-squared error, are not indicative of improveddiagnostic quality, and care must be taken to avoid over-smoothing. The included visual comparisons suggestthat the DESIGN denoising method may be beneficial atmoderately high accelerations, especially at a fieldstrength of 3 T with a receive coil array with 32 chan-nels. In addition, despite the sparsity regularization com-ponent of the joint optimization method, the algorithmfunctions properly with uniform undersampling, relyingon GRAPPA to mitigate the aliasing; this feature obviatesthe need to dramatically redesign pulse sequences usedfor acquisition; existing noisy GRAPPA reconstructionsmay be improved by postprocessing with this method.

However, the oversmoothing observed at very highaccelerations suggests that this method may not be suita-ble for generating diagnostically useful images withextreme undersampling. The results from simulationsinvolving synthetic 1.5-T data (with SNR degraded byadded noise and gray/white matter contrast reducedaccording to T1 values from Ref. 34) are depicted in Sup-porting Information Fig. S1. The reduction in imagequality across all methods suggests that the feasiblerange of acceleration with this method is not as great asfor 3 T; similar degradation is probable when far fewercoil channels are available. In addition, fewer channelsreduce the ability of GRAPPA to resolve aliasing at highaccelerations, and residual aliasing may remain inimages denoised using DESIGN, reducing the proposedmethod’s utility at high accelerations. Further study isrequired to understand the limits of DESIGN in terms of

aliasing and SNR loss from using 12- or 16-channel sys-tems, and lower acceleration factors may be necessarywith such systems.

The analysis using a synthetic contrast phantom sup-ports that although DESIGN blurs low-contrast elementsslightly, the degradation is not significant. Analyses ofeffective resolution conducted using both synthetic(based on the phantom described in Ref. 33) and realSiemens multipurpose resolution phantoms are depictedin Supporting Information Figs. S2–S4. Although thereconstructed image quality varies significantly amongmethods, the effective resolutions estimated using thefull width at half maximum of horizontal and verticalcuts of the 2D point-spread function are all on the orderof one voxel. However, this analysis is limited to simplefeatures and relatively low noise levels and does not pre-dict the contrast or texture degradation or loss of resolu-tion that would result with applying DESIGN to denoisehuman images acquired at high accelerations; such over-smoothing may hinder the isolation of small features orlow-contrast regions. Further tests on images with path-ologies are necessary before the effects of oversmoothingcan be ascertained definitively.

DESIGN denoising exhibits several distinctive charac-teristics when compared with L1 SPIR-iT, the state-of-the-art method for combining CS and parallel imaging.First, when random undersampling is not possible,GRAPPA, and hence DESIGN, is far more effective atmitigating coherent aliasing than the underlying SPIR-iTapproach; these artifacts are clear in several instances athigh accelerations. Furthermore, according to theretained SNR maps, DESIGN is much more effective atdenoising than L1 SPIR-iT, ignoring the uniform sam-pling constraint. Moreover, not having to convolve the

FIG. 8. The retained SNR in dB is compared across (a) GRAPPA, (b) GRAPPA with multichannel adaptive local Wiener filter-based

denoising, (c) L1 SPIR-iT, and (d) DESIGN denoising using the first dataset and the pseudo-multiple replica method with 400 MonteCarlo trials, for the four-level ‘‘9-7" DWT with R ¼ 10.5 uniform undersampling. The mean and minimum retained SNR correspond to theaverage and minimum gain/attenuation over the image, excluding the region without signal. The corresponding magnitude images for

these reconstructions from Fig. 4 are shown above for reference. The retained SNR maps are plotted on the same color scale to facili-tate comparison.

DESIGN 1187

SPIR-iT (or GRAPPA) kernel with the k-space data in ev-ery iteration simplifies the implementation of the algo-rithm. In situations where random undersampling isused, DESIGN can denoise the L1 SPIR-iT result, mitigat-ing SNR loss at high accelerations.

In conclusion, the proposed method successfully com-bines GRAPPA and sparsity-based denoising. Adjustingthe framework to accommodate nonuniform or non-Cartesian sampling patterns, using SPIR-iT or anothernonuniform GRAPPA-like operator, and/or gridding tech-

niques, would enable applicability to a greater numberof acquisition schemes, including radial and spiral tra-jectories. In addition, the proposed algorithm can benefitgreatly from implementation on a GPU, because the domi-nant computational operations (FFT and DWT) are allhighly parallelizable. Such an implementation would ena-ble cost-effective real-world application on clinical scan-ners. Further work includes testing DESIGN denoising ona variety of other types of MR images, carefully examiningthe effect of denoising on low-contrast anomalies such as

FIG. 9. Synthetic 32-channel contrast phantom, with 0.25 and 0.1% complex gaussian noise added, R ¼ 12.1 uniformly undersampled,

and reconstructed using (a) GRAPPA, (b) GRAPPA with multichannel adaptive local Wiener filter-based denoising, (c) L1 SPIR-iT, and (d)DESIGN denoising, with a four-level ‘‘9-7" DWT sparsifying transform. The magnitude difference images shown and the PSNR are rela-tive to the original noise-free synthetic image.

1188 Weller et al.

small tumors or lesions, and exploring more sophisticatedsparsifying transforms or penalty functions suggested by aBayesian analysis of the image denoising problem.

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