a fast wavelet-based reconstruction method-3rd

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 9, SEPTEMBER 2011 1

A Fast Wavelet-Based Reconstruction Methodfor Magnetic Resonance Imaging

M. Guerquin-Kern*, M. Hberlin, K. P. Pruessmann, and M. Unser

Abstract In this work, we exploit the fact that wavelets canrepresent magnetic resonance images well, with relatively fewcoefcients. We use this property to improve magnetic resonanceimaging (MRI) reconstructions from undersampled data witharbitrary k-space trajectories. Reconstruction is posed as anoptimization problem that could be solved with the iterativeshrinkage/thresholding algorithm (ISTA) which, unfortunately,converges slowly. To make the approach more practical, wepropose a variant that combines recent improvements in convexoptimization and that can be tuned to a given specic k-spacetrajectory. We present a mathematical analysis that explains theperformance of the algorithms. Using simulated and in vivo data,we show that our nonlinear method is fast, as it accelerates ISTAby almost two orders of magnitude. We also show that it remainscompetitive with TV regularization in terms of image quality.

Index Terms Compressed sensing, fast iterative shrinkage/ thresholding algorithm (FISTA), fast weighted iterative shrinkage/ thresholding algorithm (FWISTA), iterative shrinkage/thresh-olding algorithm (ISTA), magnetic resonance imaging (MRI),non-Cartesian, nonlinear reconstruction, sparsity, thresholdedLandweber, total variation, undersampled spiral, wavelets.

I. INTRODUCTION

M AGNETIC RESONANCE IMAGING scanners providedata that are samples of the spatial Fourier transform(also know as k-space ) of the object under investigation. TheShannonNyquist sampling theory in both spatial and k-spacedomains suggests that the sampling density should correspondto the eld-of-view (FOV) and that the highest sampled fre-quency is related to the pixel width of the reconstructed images.However, constraints in the implementation of the k-space trajectory that controls the sampling pattern (e.g., acquisition du-ration, scheme, smoothness of gradients) may impose locallyreduced sampling densities. Insufcient sampling results in re-constructed images with increased noise and artifacts, particu-larly when applying gridding methods.

Manuscript received January 05, 2011; revised March 22, 2011; acceptedMarch 26, 2011. Date of publication April 07, 2011; date of current versionAugust 31, 2011. This work was supported by the Swiss National CompetenceCenter in Biomedical Imaging (NCCBI) and the Center for BiomedicalImaging of the Geneva-Lausanne Universities and EPFL. Asterisk indicatescorresponding author .

*M. Guerquin-Kern is with the Biomedical Imaging Group, cole Polytech-nique Fdrale de Lausanne, CH-1015 Lausanne, Switzerland.

M. Unser is with the Biomedical Imaging Group, cole PolytechniqueFdrale de Lausanne, CH-1015 Lausanne, Switzerland.

M. Hberlin and K. P. Pruessmann are with the Institute for Biomedical En-gineering, ETH Zrich, CH-8092 Zrich. Switzerland.

Color versions of one or more of the gures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identier 10.1109/TMI.2011.2140121

The common and generic approach to alleviate the reconstruction problem is to treat the taskas aninverse problem [1]. Inthis framework, ill-posedness due to a reduced sampling densityis overcome by introducing proper regularization constraintThey assume and exploit additional knowledge about the ob ject under investigation to robustify the reconstruction.

Earlier techniques used a quadratic regularization termleading to solutions that exhibit a linear dependence upon thmeasurements. Unfortunately, in the case of severe undersampling (i.e., locally low sampling density) and depending on thstrength of regularization, the reconstructed images still suffefrom noise propagation, blurring, ringing, or aliasing errors. is well known in signal processing that the blurring of edgecan be reduced via the use of nonquadratic regularization. Iparticular, -wavelet regularization has been found to outperform classical linear algorithms such as Wiener ltering in thdeconvolution task [2].

Indicative of this trend as well is the recent advent of Com- pressed Sensing (CS) techniques in MRI [3], [4]. These let usdraw two important conclusions.

The introduction of randomness in the design of trajectories favors the attenuation of residual aliasing artifacts be

cause they are spread incoherently over the entire image. Nonlinear reconstructionsmore precisely, -regularizationoutperform linear ones because they impose constraints that are better matched to MRI images.

Many recent works in MRI have focused on nonlinear reconstruction via total variation (TV) regularization, choosinnite differences as a sparsifying transform [3], [5][7]. Nonquadratic wavelet regularization has also received some attention [3], [8][11], but we are not aware of a study that comparthe performance of TV against -wavelet regularization.

Various algorithms have been recently proposed for solvinggeneral linear inverse problems subject to -regularizationSome of them deal with an approximate reformulation othe regularization term. This approximation facilitates reconstruction sacricing some accuracy and introducing extrdegrees of freedom that make the tuning task laborious. Insteathe iterative shrinkage/thresholding algorithm [2], [12], [13(ISTA) is an elegant and nonparametric method that is mathematically proven to converge. A potential difculty that needto be overcome is the slow convergence of the method whethe forward model is poorly conditioned (e.g., low samplindensity in MRI). This has prompted research in large-scalconvex optimization on ways to accelerate ISTA. The efforts sfar have followed two main directions.

Generic multistep methods that exploit the result of pas

iterations to speed up convergence, among them: two-step0278-0062/$26.00 2011 IEEE


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1650 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 30, NO. 9, SEPTEMBER 2

iterative shrinkage/thresholding [14] (TwIST), Nesterovschemes [15][17], fast ISTA [18] (FISTA), and mono-tonic FISTA [19] (MFISTA).

Methods thatoptimize wavelet-subband-dependentparam-eters with respect to the reconstruction problem: multilevelthresholded Landweber (MLTL) [20], [21] and subband

adaptive ISTA (SISTA) [22].In this work, we exploit the possibility of combining and tai-loring the two generic types of accelerating strategies to comeup witha new algorithm thatcan speedup the convergenceof thereconstruction and that can accomodate for every given k-spacetrajectory. Here, we rst consider single-coil reconstructionsthat do not use sensitivity knowledge. In a second time, we con-rm the results with SENSE reconstructions [1].

We propose a practical reconstruction method that turns outto sensibly outperform linear reconstruction methods in termsof reconstruction quality, without incurring the protracted re-construction times associated with nonlinear methods. This isa crucial step in the practical development of nonlinear algo-rithms for undersampled MRI, as the problem of xing the reg-ularization parameter is still open. We also provide a mathemat-ical analysis that justies ouralgorithmandfacilitates the tuningof the underlying parameters.

This paper is structured as follows. In Section II, we describethe basic data-formation model for MRI and derive the dis-crete forward model.Therepresentation of theobjectby waveletbases is considered in Section II-C2; in particular, we legitimatethe use of a wavelet regularization term (which promotes spar-sity) to distinguish the solution from other possible candidates.In Section III, we propose a fast algorithm for solving the non-linear reconstruction problem and present theoretical arguments

to explain its superior speed of convergence. Finally, we presentin Section IV an experimental protocol to validate and compareour practical method with existing ones. We focus mainly onreconstruction time and signal-to-error ratio (SER) with respectto the reference image.

II. MRI AS AN INVERSE PROBLEMIn this section, we present the MR acquisition model and the

representation of thesignal that is used to specify thereconstruc-tion problem. Themain acronymsandnotationsaresummarizedin Table 1. We motivate the sparsity assumptions in the waveletdomain and the variational approach with regularization thatis used to solve the inverse problem of imaging.

A. Model of Data Formation

1) Physics: We consider MRI in two dimensions, in whichcase a 2D plane is excited. The time-varying magnetic gradientelds that are imposed dene a trajectory in the (spatial) Fourierdomain that is often referred to as k-space. We denote by thecoordinates in that domain. The excited spins, which behave asradio-frequency emitters, have their precessing frequency andphase modied depending on their positions. The modulatedpart of the signal received by a homogeneous coil is given by

(1)

It corresponds to the Fourier transform of the spin density thwe refer to as object . The measurements, concatenated in thevector , correspond to sampled values ofthis Fourier transform at the frequency locations along thk-space trajectory.

2) Model for the Original Data:

a) Spatial discretization of the object: From here on, weconsider that the Fourier domain and, in particular, thesamplinpoints , are scaled to make the Nyquist sampling intervaunity. This can be done without any loss of generality if thspace domain is scaled accordingly. Therefore, we model thobject as a linear combination of pixel-domain basis function

that are shifted replicates of some generating function , sthat

(2)

with

(3)

In MRI, the implicit choice for is often Diracs deltaDifferent discretizations have been proposed, for examplby Sutton et al. [23] with as a boxcar function or later byDelattre et al. [24] with B-splines. But it has not been workedout in detail how to get back the image for general that anoninterpolating, which is the case, for instance for B-splinof order greater than 1. The image to be reconstructed [i.e., thsampled version of the object ] is obtained by ltering thcoefcients with the discrete lter

(4)

where denotes the Fourier transform of .b) Wavelet discretization: In the wavelet formalism, some

constraints apply on . It must be a scaling function that saises the properties for a multiresolution [25]. In that case, thwavelets can be dened as linear combinations of the anthe object is equivalently characterized by its coefcients ithe orthonormal wavelet basis. We refer to Mallats book [26for a full review on wavelets. There exists a discrete waveltransform (DWT) that bijectively maps the coefcients to th

wavelet coefcients that represent the same object in a continuous wavelet basis. In the rest of the paper, we represent thDWT by the synthesis matrix . Note that the matrix multplications and have efcient lterbankimplementations.

B. Matrix Representation of the Model

Since a FOV determines a nite number of coefcient, we handle them as a vector , keeping the discrete coord

nates as implicit indexing. By simulating the imaging of thobject (2), and by evaluating (1) for , we nd that thnoise-free measurements are given by

(5)


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GUERQUIN-KERN et al. : A FAST WAVELET-BASED RECONSTRUCTION METHOD FOR MAGNETIC RESONANCE IMAGING

where , the MRI system matrix, is decomposed as

(6)

There, is a space-domain vector such that .A more realistic data-formation model is

(7)or

(8)with and a residual vector representing the effect onmeasurements of noise and scanner imprecisions. The inverseproblem of MRI is then to recover the coefcients (or )from the corrupted measurements . Its degree of difcultydepends on the magnitude of the noise and the conditioning of the matrix (or ).

C. Variational Formulation

1) General Framework: The solution is dened as theminimizer of a cost function that involves two terms: the datadelity and the regularization that penalizes unde-sirable solutions. This is summarized as

(9)

where the regularization parameter balances the twoconstraints. In MRI, is usually assumed to be a realization of a white Gaussian process, which justies the choice

as a proper log-likelihood term. The ill-conditioning in-herent to undersampled trajectories imposes the use of the suit-able regularization term .

Standard Tikhonov regularization corresponds to thequadratic term and leads to the closed-formlinear solution that istractable both theoretically and numerically. When the recon-struction problem is sufciently overdetermined to make noisepropagation negligible, regularization is dispensable and theleast-squares solution, which corresponds to Tikhonov with

, is adequate. The approach is statistically optimal if theobject can also be considered as a realization of a Gaussianprocess. Unfortunately, this assumption is hardly justied fortypical MR images. The quality of the solution obtained by thismeans quickly breaks down when undersampling increases.

TV reconstruction is related to the sum of the Euclideannorms of the gradient of the object. In practice, it is dened as, where the operator returns pixelwise the

-norm of nite differences. The use of TV regularization isparticularly appropriate for piecewise-constant objects such asthe SheppLogan (SL) phantom.

2) Sparsity-Promoting Regularization: The main idea in thiswork is to exploit the fact that the object can be well repre-sented by few nonzero coefcients (sparse representation) in anorthonormal basis of functions . Formally, we write that

, (sparse support )and

(small error ).It is well documented that typical MRI images admit sparse

representation in bases such as wavelets or block DCT [3].

The -norm is a good measure of sparsity with interestinmathematical properties (e.g., convexity). Thus, among the candidates that areconsistentwith themeasurements,wefavor a solution whose wavelet coefcients have a small -norm. Speciically, the solution is formulated as

(10)

with(11)

This is the general solution for wavelet-regularized inversproblems considered by [13] as well as many other authors.

III. WAVELET REGULARIZATION ALGORITHMSIn this section, we present reconstruction algorithms tha

handle constraints expressed in the wavelet domain whilsolving the minimization problem (10). By introducingweighted norms instead of simple Lipschitz constants, we revisit the principle of the standard ISTA algorithm and simplifthe derivation and analysis of this class of algorithms. We enup with a novel algorithm that combines different acceleratiostrategies and we provide a convergence analysis. Finally, wpropose an adaptation of the fast algorithm to implement thrandom-shifting technique that is commonly used to improvresults in image restoration.

A. Weighted Norms

Let us rst dene the weighted norm corresponding to a poitive-denite symmetric weight matrix as .The requirement that the eigenvalues of denoted

must be positive leads to the norm property

B. Principle of ISTA Revisited

An important observation to understand ISTA is to see thathenonlinear shrinkageoperation,sometimescalled soft-thresholding, solves a minimization problem [27], with

(12)

Byseparabilityof norms, this applies component-wise to vectors of . Thismeans that the -regularized denoising problem (i.e., whenin (11) is the identity matrix) is precisely solved by a shrinkagoperation.

The iterative shrinkage/thresholding algorithm (ISTA) [2][13], also known as thresholded Landweber (TL), generates sequence of estimates that converges to the minimizerof (11) when it is unique. The idea is to dene at each step new functional whose minimizer will be thenext estimate

(13)


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Two constraints must be considered for the denition of .

1) It is sufcient for the convergence of thealgorithm thatis an upper bound of and matches it at

; this guarantees that the sequence ismonotonically decreasing.

2) The inner minimization (13) should be performed by asimple shrinkage operation to ensure the rapidity andaccuracy of the algorithm.

In accordance with Constraint (1), can take the genericquadratically augmented form

(14)

with the constraint that is positive denite, wherethe weighting matrix plays the role of a tuning parameter.

Then, ISTA corresponds to the trivial choice ,with the value of chosen to be greater or equal to theLipschitz constant of the gradient of , so that

.Let us dene , , and(15)

Then, using standard linear algebra, we can write(16)

(17)

This shows that Constraint (2) is automatically satised.Note that both the intermediate variable in (15) and the

threshold values will vary depending on .

Algorithm 1 : ISTA

Repeat ;

Beck and Teboulle [18, Thm. 3.1] showed that this algorithmdecreases the cost function in direct proportion to the numberof iterations , so that . Here, wepresent a slightly extended version of their original result whichis valid for any reference point during iterations.

Proposition 1: Let be the sequence generated by Al-gorithm 1 with . Then, for any ,

(18)

Proof: [18, Thm. 3.1] gives the result for . Con-sider a sequence such that . As the iterationdoes not depend on , we get . The result fol-lows immediately.

Selecting as small as possible will clearly favor the speedof convergence. It also raises the importance of a warm startingpoint.

Among the variants of ISTA, FISTA, proposed by Beck andTeboulle [18], ensures state-of-the-art convergence propertieswhile preserving a comparable computational cost. Thanks toa controlled over-relaxation at each step, FISTA quadraticallydecreases the cost function, with .

C. Subband Adaptive ISTA (SISTA)

SISTA is an extension of the ISTA that was introduced bBayram and Selesnick [22]. Here, we propose an interpretatioof SISTA as a particular case of (14) with a weighting matrithat replaces advantageously the step size in (15). The ideis to use a diagonal weighting matrix if it is

not diagonal, Constraint (2) would not be fullledwith coecients that are constant within a wavelet subband.1) SISTA: In the same fashion as for ISTA, (15) and (17) ca

be adapted to subband-dependent steps and thresholds.Accordingly, SISTA is described in Algorithm 2.

Algorithm 2 : SISTA

Repeat ;

2) Convergence Analysis: By considering the weightedscalar product instead of , we can adapt theconvergence proof of ISTA by Beck and Teboulle (see Proposition 1). This result is new, to the best of our knowledge.

Proposition 2: Let be the sequence generated by Al-gorithm 2 with . Then, for any ,

(19)

Proof: We rewrite the cost function (11) with the changeofvariable . We then apply ISTA to solvethat problemwith , , and (Note that

is positive-denite if is positive-denite). Theiteration can be rewritten,

in terms of the original variable, as. The latter is an iteration of SISTA (see Algorithm 2)According to Proposition 1, we have

, which translates directly into theproposed result.

Therefore, by comparing Propositions 1 and 2, an improveconvergence is expected. The main point is that, for a warmstarting point or after few iterations , the weightednorm in (19) can yield signicantly smaller values than the onweighted by in (18).

3) Selection of Weights: Bayram and Selesnick [22] providea method to select the values of for SISTA. To presenthis result, let us introduce some notations. We denote b

an index that scans all the wavelet subbands, coarsscale included, by the corresponding weight constantand by the corresponding block of . We also dene

. The authors of [22]show that, for each subband, the condition

(20)

is sufcient to impose the positive deniteness of that is required in (14). In the present context, we propose tcompute the values by using the power iteration methodonce for a given wavelet family and k-space sampling strateg


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Fig. 1. Reference images from left to right: in vivo brain, SL reference, and wrist.

D. Best of Two Worlds: Fast Weighted ISTA (FWISTA)

Taking advantage of the ideas developed previously, we de-rive an algorithm that corresponds to the subband adaptive ver-sion of FISTA. In the light of the minimization problem (11),FWISTA generalizes the FISTA algorithm using a parametricweighted norm. We give its detailed description in Algorithm3, where the modications with respect to FISTA is the SISTAstep in the loop.

Algorithm 3 : FWISTA

input : , , , and ;

Initialization: , , ;repeat

(SISTA step);

;

;

;

until desired tolerance is reached ;

return ;

In the same fashion as for SISTA, we revisit the convergenceresults of FISTA [18, Thm. 4.4] for FWISTA.

Proposition 3: Let be the sequence generated by Al-gorithm 3. Then, for any

(21)

Proof: In the spirit of the proof of Proposition 2, weconsider the change of variable and applyFISTA to solve the new reconstruction problem. The step

is equivalent to. The convergence results of FISTA

[18, Thm. 4.4] applies on the sequence , which leads to

.This result shows the clear advantage of FWISTA compareto ISTA (Proposition 1) and SISTA (Proposition 2). Moreovewe note that FWISTA can be simply adapted in order to imposa monotonic decrease of the cost functional value, in the samfashion as MFISTA. The same convergence properties apply[19, Thm. 5.1].

E. Random Shifting

Wavelet bases perform well the compression of signals bucan introduce artifacts that can be attributed to their relativlack of shift-invariance. In the case of regularization, this can bavoided by switching to a redundant dictionary. The downsidehowever, is a signicant increase in computational cost. Altenatively, the practical technique referred to as random shiftin(RS) [2] can be used. Applying random shifting is much simpleand computationally more efcient than considering redundantransforms and leads to sensibly improved reconstruction.

Here, we propose a variational interpretation that motivateour implementation of FWISTA with RS (see Algorithm 4). Wconsider the DWT , with , where

represent the different shifting operations required to get translation-invariant DWT. The desired reconstruction would bedened as the minimizer of

(22)

In 1D, this formulation includes TV regularization, inother words a single-level undecimated Haar WT withoucoarse-scale thresholding.

Rewriting (22) in terms of wavelet coefcients, we get

(23)

with

(24)


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For a current estimate, we select a transform and performa step in the minimization of the cost with respect to whilekeeping , for xed. A SISTA step is used, as the mini-mization subproblem (24) takes the form (11). The minimizersof the functionals are expected to correspond to images closeto each other and to the minimizer of (22). In the rst iterations

of the algorithm, the minimization steps with respect to anyare functionally equivalent (i.e., the modication is mostly ex-plained by the gradient step). This motivates the use of FWISTAiterations at rst and the switch to ISTA steps afterwards.

As the scheme is intrinsically greedy, we do not have a theo-retical guarantee of convergence. Yet, in practice, we have ob-served that the SER stabilizes ata muchhighervalue than it doeswhen using ISTA schemes with no RS (cf Fig. 4).

Algorithm 4 : FWISTA with RS

input : , , , , ;

Consider the sequence of DWT with RS: ;Initialization: , , , , ,

;

repeat

;

;

(cost-function evaluation);

if then

;

if then

;

;

;

;

;

;

until stopping condition is met ;

IV. EXPERIMENTS

A. Implementation Details

Our implementation uses Matlab 7.9 (Mathworks, Natick,MA). The reconstructions run on a 64-bit 8-core computer,clock rate 2.8 GHz, 8 GB RAM (DDR2 at 800 MHz), MacOS X 10.6.5. For all iterative algorithms, a key point is thatmatrices are not stored in memory. They only represent op-erations that are performed on vectors (images). In particular,

is computed once per dataset. Matrix-to-vectormultiplication with , specically, , have anefcient implementation thanks to the convolution structure of the problem [28], [29]. For these Fourier precomputations, we

Fig. 2. Time evolution of the SER with respect to the minimizer for seveISTA algorithms. Times to reach 35 dB are delineated.

Fig. 3. Time evolution of the difference in cost function value with respectthe minimizer for several ISTA algorithms.

used the NUFFT algorithm [30] that is made available onlineFor wavelet transforms, we used the code provided online[21]. This Fourier-domain implementation proved to be fastthan Matlabs when considering reconstructed images smallethan 256 256 and the Haar wavelet. It must also be noted ththe 2D-DFT were performed using the FFTW library whicefciently parallelizes computations.

For Tikhonov regularizations, we implemented the classicaconjugate gradient (CG) algorithm, with the identity as the reularization matrix. ForTV regularizations, we considered the iterativelyreweighted least-squares algorithm (IRLS), whichcorresponds to the additive form of half-quadratic minimizatio[31], [32]. We used 15 iterations of CG to solve the linear innproblems, always starting from the current estimate, which crucial for efciency. For the weights that permit the quadratapproximation of the TV term, we stabilized the inversion overy small values.

We implemented ISTA, SISTA, and FWISTA as described iSection III,with theadditionalpossibility to userandomshiftin(see Section IV-C2). For the considered reconstructions usinour method, described in Algorithm 4, was a reasonable choice. The Haar wavelet transform was used, with thredecomposition levels when no other values are mentioned. As i

1http://www.eecs.umich.edu/~fessler/code/ 2http://bigwww.ep.ch/algorithms/mltldeconvolution


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TABLE IGLOSSARY

usual for wavelet-based reconstructions, the regularization wasnot applied to the coarse-level coefcients.

Reconstructions were limited to the pixels of the ROI for all

algorithms. The regularization parameter was systematicallyadjusted such that thereconstruction mean-squared error (MSE)inside the ROI was minimal. For practical situations where theground-truth reference is not available, it is possible to adjustby considering well-established techniques such as the discrep-ancy principle, generalized cross validation, or L-curve method[33].

B. Spiral MRI Reconstruction

In this section, we focused on the problem of reconstructingimages of objects weighted by the receiving channel sensitivity,given undersampled measurements. This problem, which in-volves single-channel data and hence differs from SENSE, ischallenging for classical linear reconstructions as it generates

Fig. 4. Time evolution of the SER for several algorithms for the SL simulatiusing Haar wavelets.

artifacts and propagates noise. We considered spiral trajectoriewith 50 interleaves, with an interleave sampling density reducedby a factor compared to Nyquist for the highest frequencies and an oversampling factor 3.5 along the trajectorySpiral acquisition schemes are attractive because of their versatility and the fact that they can be implemented with smootgradient switching [34], [35].

We validate the results with the three sets of data that wepresent below. The corresponding reference images are showin Fig. 1.

1) MR Scanner Acquisitions: The data were collected on a3T Achievasystem (Philips Medical Systems, Best, TheNetherlands). A eld camera with 12 probes was used to monitor th

actual k-space trajectory [36]. An array of eight head coils provided the measurements. We acquired in vivo brain data from ahealthy volunteer with parameters ms and

ms. Theexcitation slice thickness was 3mm with a ip anglof 30 . The trajectory was designed for a FOV of 25 cm with pixel size 1.5 mm. It was composed of 100 spiral interleavesThe interleaf distance for the highest sampled frequencies dened a fraction of the Nyquist sampling density .

The subset used for reconstruction corresponds to half of th100 interleaves. The corresponding reduction factor, dened athe ratio of the distance of neighboring interleaves with thNyquist distance, is .

2) Analytical Simulation: We used analytical simulations of the SL brain phantom with a similar coil sensitivity, followinthe method described in [37]. The values of these simulated datawere scaled to have the same mean spatial value (i.e., the samcentral k-spacepeak) as thebrain reference image.A realizationof complex Gaussian noise was added to this synthetic k-spacdata, with a variance corresponding to 40 dB SNR. The 17

176 rasterization of the analytical object provided a reliablreference for comparisons.

3) Simulation of a Textured Object: A second simulationwas considered with an object that is more realistic than thSL phantom. We chose a 512 512 MR image of a wristhat showed little noise and interesting textures. We simulateacquisitions with the same coil prole and spiral trajector(176 176 reconstruction matrix), in presence of a 40 dB SN


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TABLE IIVALUES OF THE OPTIMAL SER AND CORRESPONDING REGULARIZATION PARAMETERS ARE SHOWN FOR THE DIFFERENT WAVELET BASES

TABLE IIIRESULTS OF THE PROPOSED WAVELET METHOD FOR DIFFERENT WAVELET DECOMPOSITION DEPTHS. VALUES OF THE REGULARIZATIONPARAMETER, THE FINAL SER, THE RELATIVE MAXIMAL SPATIAL DOMAIN ERROR, AND THE TIME TO REACH0 0.5 dB OF THE FINAL SER

TABLE IVRESULTS OF THE ALGORITHMS CG (LINEAR), IRLS (TV), AND OUR METHOD (WAVELETS) FOR DIFFERENT DEPTHS. VALUES OF THE REGULARIZATION

PARAMETER, THE FINAL SER, THE RELATIVE MAXIMAL SPATIAL DOMAIN ERROR, AND THE TIME TO REACH0 0.5 dB OF THE FINAL SER

Fig. 5. Result of different reconstruction algorithms for the three experiments. For each reconstruction, the performance in SER with respect to th(top-left), the reconstruction time (top-right), and the number of iterations (bottom-right) are shown.

Gaussian complex noise. The height of the central peak wasalso adjusted to correspond to that of the brain data. The refer-

ence image was obtained by sinc-interpolation, by extractingthe lowest frequencies in the DFT.

C. Results

In this section, we present the different experiments we conducted. Thetwomain reconstruction performance measures thawe considered are as follows.


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Reconstruction duration, which excludes all aforemen-tioned precomputations and the superuous monitoringoperations.

Signal to error ratio with respect to a reference, de-ned as and

. Practically,

the references are either the ground-truth images or theminimizer of the cost functional. It is known that SER isnot a foolproof measure of visual improvement but largeSER values are encouraging and generally correlate withgood image quality.

1) Convergence Performance of IST-Algorithms: In this rstexperiment, we compared the convergence properties of the dif-ferent ISTA-type algorithms, as presented in Section III, withthe Haar wavelet transform. The data we considered are thoseof the MR wrist image. The regularization parameter was adjusted to maximize the reconstruction SER with respect to theground-truth data. The actual minimizer of the cost functional,which is the common xed-point of this family of algorithms,was estimated by iterating FWISTA 100 000 times.

The convergence results are shown in Figs. 2 and 3 for thesimulation of the MR wrist image. Similar graphs are obtainedusing the other sets of data.

For a xed number of iterations, FISTA schemes (FISTAand FWISTA) require roughly 10% additional time comparedto ISTA and SISTA. In spite of this fact, their asymptoticsuperiority appears clearly in both gures. The slope of thedecrease of the cost functional in the log-log plot of Fig. 3reects the convergence properties in Propositions 1, 2, and3. When considering the rst iterations, which are of greatestpractical interest, the algorithms with optimized parameters(SISTA and FWISTA) perform better than ISTA and FISTA(see Fig. 3). The times required by each algorithm to reacha 30 dB SER (considered as a threshold value to perceivedchanges) are 415 s (ISTA), 53 s (SISTA), 12.7 s (FISTA), and4.4 s (FWISTA). With respect to this criterion, SISTA presentsan eight-fold speedup over ISTA, while FWISTA presents a12-fold speedup over SISTA and nearly a three-fold speedupover FISTA. It follows that FWISTA is practically close to twoorders of magnitude faster than ISTA.

2) Choice of the Wavelet Transform and Use of RandomShifting: The algorithms presented in Section III apply for anyorthogonal wavelet basis. For the considered application, wewant to study the inuence of the basis on performance. In thisexperiment we considered the BattleLemari spline wavelets[26] with increasing degrees, taking into account the necessarypostlter mentioned in (4).

We compared the best results for several bases. They wereobtained with FWISTA after practical convergence and are re-ported inTable II. Fig.4 illustrates the time evolution of the SERusing ISTA and FWISTA in the case of the SL reconstruction.Similar graphs are obtained with the other experiments.

It is known that the Haar wavelet basis efciently approxi-mates piecewise-constant objects like the SL phantom, whichis consistent with our results. On the other hand, splines of higher degree, which have additional vanishing moments, per-form better on the textured images (upper part of Table II).

Fig. 6. Evolution of theperformanceof thealgorithms. From topto bottom: SLsimulation, wrist simulation, and brain data. Times required to reach0 0.5 dBof the asymptotic value are indicated.

We present in the lower part of Table II the performanceobserved when using ISTA with RS. We conclude that, in thcase of realistic data, it is crucial to use RS as it improveresults by at least 0.7 dB, whatever the wavelet basis is. Thremarkable aspect there is that the Haar wavelet transformwith RS consistently performs best. Two important things cabe seen in Fig. 4: FWISTA is particularly efcient duringthe very rst iterations, while SISTA with RS yields the besasymptotic results in terms of SER and stability. Our methocombines both advantages.

In Table III, we present the results obtained using differendepths of the wavelet decompositions. Our reconstructiomethod is used together with the Haar wavelet transform anRS. The performances are similar but there seems to be an advantage in using several decomposition levels both in terms o


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Fig. 7. Reconstructions anderrormapsof differentIST-algorithmswith RS for the wristexperiment. Foreach reconstruction, the performance in SERwito the reference (top-left), the reconstruction time (top-right), and the number of iterations (bottom-right) are shown.

SER and reconstruction speed. The FWISTA scheme seems torecoup the cost of the wavelet transform operations associatedto an increase in the depth of decomposition.

3) Practical Performance: We report in Table IV the re-sults obtained for different reconstruction experiments usingstate-of-the-art linear reconstruction, TV regularization,and ourmethod. The images obtained when running the different algo-rithms after approximately 5 s, and after practical convergenceas well, are shown in Fig. 5. We display in Fig. 6 the time evolu-tion of the SER for the different experiments. In each case, weemphasize the time required to reach 0.5 dB of the asymptoticvalue of SER. Finally, we present in Fig. 7 the reconstruction

and error maps of the different IST-algorithms at different mo-ments of reconstruction. This was done with the wrist simulatedexperiment using the Haar wavelet basis and RS.

Firstly, we observe that TV and our method achieve similarSER (Table IV) and image quality (Fig. 5). They both clearlyoutperform linear reconstruction, with a SER improvementfrom 1.5 to 3 dB, depending on the degree of texture in the orig-inal data.Moreover, the pointwise maximal reconstruction errorappears to always be smaller with nonlinear reconstructions.Due to the challenging reconstruction task, which signicantlyundersamples of the k-space, residual artifacts remain in thelinear reconstructions and at early stages of the nonlinear ones.Although the k-space trajectory is exacly the same in the threecases, artifacts are less perceived in the in vivo reconstructions,while they stand out for the synthetic experiments.

Secondly, it clearly appears that the linear reconstructionimplemented with CG, leads to the fastest convergence, unfortunately with suboptimal quality. For a reconstruction timoneorder of magnitude longer, ouraccelerated methodprovidebetter reconstructions. Thisis illustrated in Fig. 5 for reconstruction times of the order of ve seconds (columns 1, 2, and 4).

Finally, we observe in Fig. 7 the superiority of the proposeFWISTA over the other types of algorithms. For the given rconstruction times, it consistently exhibits better image qualias can be seen in both reconstructions and error maps.

D. SENSE MRI Reconstruction

Our reconstruction method is applicable to linear MRimaging modalities. In this section, we report results obtaineon a SENSE reconstruction problem. The data were acquirewith the same scanner setup as in Section IV-B1. This timthe data from the eight receiving channels were used for recostruction, as well as an estimation of the sensitivity maps. Ain vivo gradient echo EPI sequence of a brain was performedwith T2 contrast. The data was acquired with the followinparameters: excitation slice thickness of 4 mm, ms

ms, ip angle of 80 , and trajectory composed o13 interleaves, supporting a 200 200 reconstruction matriwith pixel resolution 1.18 mm 1.18 mm. The oversamplinratio along the readout direction was 1.62.

The reference image was obtained using the complete set odata and performing an unregularized CG-SENSE reconstruc


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Fig. 8. Reconstructions (left column) and error maps (right column) for theSENSE EPI experiment using CG (rst row), IRLS-TV (second row), and ourmethod (third row). For each reconstruction, the performance in SER withrespect to the reference (top-left corner), the reconstruction time (top-rightcorner), the number of iterations (bottom-right corner), and a magnication of the central part (bottom-left) are shown.

Fig. 9. Evolution of theperformanceof thealgorithms.Times required to reach0 0.5 dB of the asymptotic value are indicated.

tion. The reconstruction involved 3 of the 13 interleaves, repre-senting a signicant undersampling ratio .

The images obtained using regularized linear reconstruction(CG), TV (IRLS), and our method are presented in Fig. 8. InFig. 9, the SER evolution with respect to time is shown forthe three methods. The times to reach 0.5 dB of the asymp-totic SER value are 5.9 s (CG), 49.1 s (IRLS), and 22.8 s (ourmethod).

With this high undersampling, the errors maps show that reconstructions suffer from noise propagation mostly in thecenteof the image. It appears that TV and our method improve qualtatively and quantitatively image quality over linear reconstruction (cf. Fig. 8). As it was observed in Section IV-B with thspiralMRI reconstructions, thisin vivo SENSEexperiment con-

rms that our method is competitive with TV. In terms of reconstruction duration, our method proves to converge in a time this of the same order of magnitude as CG (cf. Fig. 9).

V. CONCLUSIONWe proposed an accelerated algorithm for nonlinear wavele

regularized reconstruction that is based on two complementaracceleration strategies: use of adaptive subband thresholds plumultistepupdaterule. We provided theoretical evidence that thialgorithm leads to faster convergence than when using the accelerating techniques independently. In the context of MRI, thproposed strategy can accelerate the reference algorithm up ttwo orders of magnitude. Moreover, we demonstrated that, busing the Haar wavelet transform with random shifting, we arable to boost the performance of wavelet methods to make thecompetitive with TV regularization. Using different simulationand in vivo data, we compared the practical performance of ourreconstruction method with other linear and nonlinear ones.

The proposed method is proved to be competitive with TVregularization in terms of image quality. It typically convergewithin ve seconds for the single channel problems considereThis bringsnonlinearreconstruction forward to anorder of magnitude of the time required by the state-of-the-art linear reconstructions, while providing much better quality.

ACKNOWLEDGMENTThe authors would like to thank J. Fessler and C. Vonesch tmake their implementations of NUFFT and multidimensionawavelet transform available. We acknowledge C. VoneschI. Bayram, and D. Van De Ville for fruitful discussions. The authors would also like to thank P. Thvenaz for useful commenand careful proofreading.

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