IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014 1689

DynamicMR Image Reconstruction–Separation FromUndersampled ( )-Space via Low-Rank

Plus Sparse PriorBenjamin Trémoulhéac*, Nikolaos Dikaios, David Atkinson, and Simon R. Arridge

Abstract—Dynamic magnetic resonance imaging (MRI) is usedin multiple clinical applications, but can still benefit from higherspatial or temporal resolution. A dynamic MR image reconstruc-tion method from partial ( )-space measurements is introducedthat recovers and inherently separates the information in the dy-namic scene. The reconstruction model is based on a low-rank plussparse decomposition prior, which is related to robust principalcomponent analysis. An algorithm is proposed to solve the convexoptimization problem based on an alternating direction method ofmultipliers. The method is validated with numerical phantom sim-ulations and cardiac MRI data against state of the art dynamicMRI reconstruction methods. Results suggest that using the pro-posed approach as a means of regularizing the inverse problem re-mains competitive with state of the art reconstruction techniques.Additionally, the decomposition induced by the reconstruction isshown to help in the context of motion estimation in dynamic con-trast enhanced MRI.

Index Terms—Compressive sensing (CS), dynamicmagnetic res-onance (MR) imaging, low-rank, robust principal component anal-ysis, sparsity.

I. INTRODUCTION

M AGNETIC resonance imaging (MRI) is a medicalimaging technique that produces images of internal

structures of the body. Dynamic MRI, a magnetic resonancesignal with both spatial and temporal information, is used inmultiple clinical applications such as cardiovascular imaging ordynamic contrast enhanced MRI. However, MRI is inherentlya slow process due to a combination of different constraints in-cluding nuclear relaxation times, peripheral nerve stimulation,power absorption and signal to noise. This can limit spatialand temporal MR resolutions, yet they are critical to monitordynamic processes where events change on relatively small

Manuscript received February 08, 2014; revised April 04, 2014; acceptedApril 27, 2014. Date of publication April 30, 2014; date of current versionJuly 30, 2014. This work was supported by Engineering and Physical SciencesResearch Council (EPSRC) under Grant EP/H046410/1 and the National In-stitute for Health Research (NIHR) funded Comprehensive Biomedical Re-search Centre (CBRC) at University College London. Asterisk indicates cor-responding author.*B. Trémoulhéac is with the Centre for Medical Image Computing, Univer-

sity College London, WC1E 6BT London, U.K. (e-mail: b.tremoulheac@cs.ucl.ac.uk).N. Dikaios and S. R. Arridge are with the Centre for Medical Image Com-

puting, University College London, WC1E 6BT London, U.K.D. Atkinson is with the Centre for Medical Imaging, University College

London, WC1E 6JF London, U.K.Digital Object Identifier 10.1109/TMI.2014.2321190

scales (few millimetres and sub-second). Additionally, longscan durations can affect patient comfort and for that reasonincrease chances of motion artefacts.Hence, many approaches have been proposed to reduce ac-

quisition time since the development of MRI. Popular tech-niques include for example echo planar imaging [1], fast low-angle shot imaging [2], and parallel MR imaging [3] which usesmultiple receiver coils.In addition, complementary acceleration approaches that ex-

ploit information redundancy in the signal have been developed.In general, part of the -space measurements that would nor-mally be acquired are skipped. Doing so results in an ill-posedinverse problem that needs to be regularized by incorporatingprior information about the signal to provide physiologicallyrepresentative and accurate images, i.e., without artefacts intro-duced by the ill-posedness. The prior information may be anyvaluable assumption about the signal. Although this approachhas been used previously, it has recently received interest dueto compressed sensing (CS) [4], [5]. CS refers to the topic ofsignal acquisition and reconstruction from incomplete measure-ments yielding acceptable or perfect recovery using the factthat the signal of interest is sparse (either in its direct repre-sentation or after a transform to another domain). Intuitively,a signal is said to be sparse if only a small fraction of coeffi-cients are significant. CS has been successfully applied in MRIand dynamic MRI [6]–[8]. More recently, researchers have alsolooked at exploiting the low-rank property of matrices, insteadof simply sparsity of vectors. These techniques have started togain interest in dynamic MRI [9], [10] and have been com-bined with a sparsity prior [11], [12]. In addition, there hasbeen interest in the low-rank plus sparse decomposition model,also referred to as robust principal component analysis (RPCA)[13], [14]. Many results in the literature have reported that itis possible under some assumptions to recover both low-rankand sparse components from only a fraction of observations[13], [15]–[17]. The proposed approach in this paper is basedon previous investigations using the low-rank plus sparse de-composition model as both a regularization prior and a separa-tion method in dynamic MRI from partial measurements [18],which was itself inspired by the work of Gao et al. [19] in dy-namic computed tomography. Gao et al. have applied their ap-proach in cardiac cine MRI [20]–[22] and diffusion MRI [23],and recent work by Otazo et al. [24] has also highlighted therole of the low-rank plus sparse decomposition as a backgroundand contrast separation.

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1690 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014

The work presented in this paper is particularly developedfor dynamic magnetic resonance imaging. Image reconstruc-tion is formulated as an inverse problem regularized by a low-rank plus sparse prior, with a Fourier transform as the sparsi-fying transform in the temporal direction. The alternating direc-tion methods of multipliers (ADMM) framework is proposed tosolve the minimization problem and derive an algorithm called- RPCA. Random sampling schemes for dynamic undersam-pled MRI are tested and compared. Experiments using com-plex-valued noise, numerical phantoms and in vivo MRI dataalong with comparisons against state of the art dynamic MRreconstruction methods are presented. Additionally, the useful-ness of the decomposition provided by the reconstruction modelis shown in the context of motion estimation in dynamic contrastenhanced MRI.The rest of the article is organized as follows. Section II

briefly reviews reconstruction methods for undersampleddynamic MRI and the robust principal component analysistechnique. Section III presents the proposed - RPCA ap-proach. Numerical simulations are shown in Section IV.Sections V and VI, respectively, present discussion and con-clusion of this study.

II. BACKGROUND

A. Dynamic MRI From Partial Measurements

The imaging equation in dynamic MRI can be written as

(1)

where represents the measured ( )-space signal,is the desired image function and represents

the noise. Considering complex-valued raw MR data, the noisecan reasonably be modelled by an additive white Gaussiandistribution on both real and imaginary components (with i.i.d.random variables) [25].Finding the closest representation of the true object

from a limited number of measurements is the inverseproblem of interest in this paper. Here, limited refers to under-sampled data or sub-Nyquist sampling in accordance with theShannon–Nyquist sampling theory.

B. Fast Imaging Methods Previously Proposed

Many techniques to tackle this inverse problem in dynamicMRI rely on the assumption that a Fourier transform along thetemporal dimension returns an approximately sparse signal,because the original 2-D images in time exhibit significantcorrelation and/or periodicity. This prior knowledge has beenused in techniques such as UNFOLD [26] and - BLAST[27]; the latter additionally exploits the compactness of thesignal distribution.Compressed sensing (CS) [4], [5] suggests that if the signal

of interest is sparse, it is possible under certain assumptions toreconstruct the signal exactly with high probability with fewersamples than the standard Shannon–Nyquist theory demands.In other words, CS proposes to directly acquire compressivemeasurements and reconstruct the signal from this reduced set

of samples. CS has been applied to MRI [28] and in particulartechniques have been developed specifically for dynamic MRI,such as - SPARSE [6] and - FOCUSS [7]. - FOCUSS firstestimates a low-resolution version of the ( )-space, prior to aCS reconstruction using the FOCUSS algorithm [29], a generalestimationmethod to find localized energy solution from limiteddata.

C. Low-Rank and Sparsity Methods

Consider images of dimension . Approachesbased on low-rank matrix completion are usually based on ama-trix which is formed so that each column represents a vectorizedMR image of the sequence,[30], [10]. This matrix, referred to as a Casorati matrix, is verylikely to be approximately low-rank, where only a few singularvalues are significant, because of the high correlation betweenimages.Generally, a finite-dimensional spatio-temporal MRI model

equivalent to (1) is adopted

(2)

where represents the stacked ( )-space measure-ments vector, is the MRI encoding oper-ator modelling both the sub-Nyquist sampling and Fourier trans-form with represents the matrix to recover,and is the noise vector. A common approach to recoverlow-rank matrices is based on rank minimization subject to adata fidelity term. However, similar to the norm in CS recon-struction, the rank operator makes the minimization computa-tionally intractable as the dimension of the problem increases.The rank penalty is often replaced by the nuclear norm, and theequality constraint relaxed. The nuclear norm (also known astrace norm or Schatten -norm with ) is known to be theconvex envelope of the rank operator [31], [32] and is definedas the sum of singular values, i.e., , wheredenotes the th singular value of . In its Lagrangian form, thisleads to a nuclear norm regularized linear least squares problemwhich can be solved efficiently using accelerated proximal gra-dient methods [33], [34], but it is also possible to solve variantsof the rank minimization problem without the use of the nuclearnorm, for example based on PowerFactorization [9], [35].Finally, there has been recent interest to explore the combi-

nation of both low-rank and sparsity penalties [11], [12]. Thesemethods formulate the problem as

(3)

where is related to prior information about the rank andabout the sparsity. In - SLR [11], the regularization penal-ties are the nonconvex Schatten -normwith , and represents the spatio-temporal totalvariation norm, i.e., the norm of the gradient in directions

, and approximated by finite differences; and are theassociated regularization parameters. The authors in [12] havealso proposed to exploit both rank and sparsity a priori infor-mation using respectively the partial separability model and atemporal Fourier transform. A notable feature of this work lies

TRÉMOULHÉAC et al.: DYNAMIC MR IMAGE RECONSTRUCTION–SEPARATION FROM UNDER-SAMPLED ( )-SPACE 1691

in a single formulation of both constraints using a sparsity con-straint to regularize the partial separability model.

D. Robust Principal Component Analysis

Robust PCA (RPCA) [13], [14] is a mathematical methodthat decomposes a given matrix into a low-rank and sparsecomponents. Considering a Casorati matrix of dimensions

, RPCA describes the convex minimization problem

(4)

RPCA is solved efficiently via alternating direction methodsof multipliers (ADMM) that relies on the augmented La-grangian function

(5)

where denotes the trace inner product, is the Lagrangemultiplier of the linear constraint and is the penalty param-eter (a positive scalar). ADMM is an iterative scheme that min-imizes over and separately, then updates Lagrange mul-tipliers. There exist closed-form expressions for and

, respectively the singular value thresholding operator(where represents the

singular value decomposition) and the shrinkage operator de-fined element-wise as . Thepenalty parameter can be fixed as in [13], [14] although an-other strategy is to update it dynamically [36]. The iterative pro-cedure is described in Algorithm 1 where denotes the iterationnumber (and the last iteration). RPCA can be stopped whenthe quantity is small or whena maximum number of iterations is reached. Parameter canbe seen as a trade-off between how much the low-rank compo-nent gets “low-rank” and how much the sparse component gets“sparse”. The authors of [13] suggested the theoretically sup-ported value

(6)

where . In general, this choice offers a reasonable sep-aration, although by varying it can be tailored to a givenapplication.Note that the low-rank plus sparse decomposition is not

unique if the given matrix is both low-rank and sparse, sinceboth components can then be seen interchangeably as eitherlow-rank or sparse (for example, a matrix that has only one

Fig. 1. Schematic RPCA decomposition. Given a matrix that is neither low-rank nor sparse, RPCA estimates low-rank and sparse matrices such that

.

Fig. 2. RPCA on a breath-hold cardiac MRI sequence with .Algorithm 1 with was used to generate figures in this example.Decomposition resulted in a rank-1 matrix for the low-rank part as shown bythe only nonzero singular value, while the sparse component does not have alow rank because most of its singular values are not close to zero. It can beseen on the corresponding images and histograms that the sparse componentis much more sparse than the low-rank one. Physiologically, the low-rank partappears as a static component while the sparse component captures motion, inthis particular case mostly heartbeats.

nonzero element). In a dynamic MR imaging scenario, it isunlikely that the data to reconstruct would be both low-rankand sparse at the same time. Generally it will be approximatelylow-rank (because 2-D images exhibit significant correlationand/or periodicity in time) but not sparse in the image domainbecause either 2-D images represent anatomical sections thatare rarely sparse or the presence of noise makes images notsparse. A graphic representation of RPCA is shown in Fig. 1,and RPCA applied to MR images is presented in Fig. 2.

III. METHOD

A. Low-Rank Plus Sparse Prior

A dynamic MR image reconstruction method fromsub-Nyquist measurements based on an intrinsic separa-tion between low-rank plus sparse components is introduced.The method has strong connections with robust principal com-ponents analysis from partial entries [13], [15] and low-rankmatrix recovery framework in accelerated dynamic MRI [9],

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[10]. The proposed approach assumes that the Casorati matrixcan be expressed as a linear combination of a low-rank plussparse component, and at the same time assumes that this priorinformation is strong enough to be able to reconstruct imagesfrom partial samples. At this point, it is important to distinguishbetween methods that consider the signal to reconstruct asbeing simultaneously low-rank and sparse (as presented inSection II-C) and the approach presented in this paper thatconsiders the signal to reconstruct as being the sum of low-rankplus sparse components.Image reconstruction is formulated as a minimization

problem with the convex objective functiondefined as

(7)

where represents the MRI encoding operator modelling boththe sub-Nyquist sampling and Fourier transform as described inSection II-C, denotes the Fourier transform operator alongthe temporal dimension, is a regularization parameter andis the decomposition parameter1 as defined in (6). The inclu-sion of those priors in the reconstruction problem makes the as-sumption that the dynamic imaging data have the property ofbeing separable into an approximately low-rank and approxi-mately sparse components. The additional operator can bejustified by the fact that the proposed method deals with the re-construction of undersampled dynamic imaging data: a Fouriertransform along the temporal dimension as a sparsifying trans-form has been shown to improve sparsity in many dynamic re-construction methods (e.g., [6], [7]). This is beneficial since itis assumed in such studies that the more the signal is sparse, thehigher the undersampling ratio can be. An illustration is pro-vided in Fig. 3.The method is named - RPCA, since the first term in the

objective function represents a data fitting criterion enforcingconsistency between reconstructed data and partial acquired( )-space samples, and the second term encloses a low-rankplus sparse decomposition.

B. Complex-Valued Data

Although RPCA has been studied for real-valued matrices, itcan be shown empirically that complex-valuedmatrices can alsobe separated into the sum of low-rank plus sparse components.The RPCA algorithm can readily handle a complex-valued ma-trix since it is based on operators that are easily extensible tocomplex-valued data. Indeed, the singular value thresholdingoperator (SVT) is a shrinkage on singular values that are alwaysnonnegative real numbers (even for a complex-valued matrix).Hence, SVT can be simply generalized and defined as

, where denotes the Hermitian transpose of. The shrinkage operator can also be easily extended to com-

plex-valued numbers. The reason for interest in complex-valued

1The decomposition parameter can be seen as either or interchangeably.In the following, we will mainly refer to it as because it is easier to interpretsince it can be seen as a scaling parameter that does not depend on the matrixdimensions.

Fig. 3. Justification of the additional Fourier transform operator along the timedimension using different decomposition parameter . The norm of the sparsecomponent transformed using the Fourier operator along the time dimension(gray) is generally lower than the norm of the sparse component on its own(black). This figure has been generated using Algorithm 1 and breath-hold car-diac MRI dataset of Fig. 2.

decomposition is that MR data are inherently complex-valued,although usually only magnitude images are displayed.

C. Image Reconstruction

To minimize (7), an algorithm is derived based on ADMM,which can be interpreted as a variable splitting scheme com-bined with the augmented Lagrangian [37], [38]. Variable split-ting is introduced

(8)

and the associated augmented Lagrangian function

(9)

where are Lagrangian multipliers and denotes the real part.Ignoring constants irrelevant to optimization, (9) can also bewritten as

(10)

ADMM minimizes over and separately leadingto sub-problems that have closed-form solutions. The nuclearnorm minimization problem is solved analytically via singularvalue thresholding [39]; the solution of the norm problem is

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found by soft thresholding [13], [14], and the other sub-prob-lems are quadratic resulting in a linear system of equations

(11)

(12)

(13)

(14)

Based on these analytical solutions, an image reconstructionprocedure is derived (Algorithm 2), which can be seen as RPCAfor undersampled ( )-space MRI data. Penalty parametersand are both set and fixed to 1, although as in standard RPCAthey could be updated dynamically. To ensure convergence, the- RPCA algorithm is stopped when a maximum number of200 iterations is reached, or when

.

D. Sampling Strategies

Low-rank matrix completion and compressed sensing havestrong parallels [31]. Hence, the sampling strategy adoptedin this paper is similar to compressed sensing MRI methods.One of the requirements for a successful CS reconstruction inMRI is the incoherence of undersampling artefacts. This canbe achieved by undersampling randomly in -space. However,sampling strategies must also satisfy hardware and physiolog-ical constraints, which generally means that trajectories mustfollow relatively smooth lines and curves [8].Since a conventional strategy to acquire Fourier samples in

MRI is along parallel equispaced -space lines onto a Cartesian

grid, a convenient way to achieve incoherent undersampling isto randomly select fewer lines. However, since the energy dis-tribution of MR images in -space is known to be concentratedclose to the center, a common strategy consists of densely sam-pling central -space lines and randomly selecting lines else-where. Although selection of random lines can be drawn froma simple uniform probability distribution, a better approach isto give lower probabilities to the selection of lines nearer to the-space edges in order to take into account the energy distri-bution and also because it may overcome coherence problemsat low spatial frequencies for some sparsifying transforms [40].This is often referred to as polynomial variable density sam-pling [28]. In this paper, a similar sampling strategy is adaptedto dynamic imaging, where for each acquisition time frame thesampling density of -space lines is one near -space center anddecreases towards the edges of -space.Note that it has been reported in the literature that radial sam-

pling provides better reconstruction results from a CS point ofview since resulting artefacts more closely resemble noise com-pared to Cartesian undersampling [8]. Hence, a radial schemeis also tested with acquisition being made by 2-D projectionsat different angles. Equi-angular spacing projections are usedand incoherency in time is achieved by applying a random ro-tation between on the whole pattern across eachacquisition frame as done in [11]. Note the radial sampling ishere directly approximated to the closest Cartesian trajectory,hence it is referred to as “pseudo-radial” since it does not in-clude important steps that would have to be included in a realradial-basedMRI acquisition (i.e., a density compensation func-tion and a gridding procedure).An illustration of the sampling patterns used in this study

is shown in Fig. 4. These undersampling strategies can beachieved by omitting readouts from conventional Cartesian orradial acquisitions. This makes them particularly suitable andinexpensive from an MR acquisition point of view, since onlyminor pulse sequence modifications are required.

IV. EXPERIMENTS AND RESULTS

A. Framework

1) Preliminaries: Experiments were run in MATLAB ona Linux platform. Intensity of data were normalized betweenvalues 0 and 255 prior to any processing. Simulated datawere created directly in the image domain and in vivo datawere based on magnitude-reconstructed images from an MRscanner. Datasets were then undersampled retrospectivelyusing a polynomial variable density or pseudo-radial samplingschemes as shown in Fig. 4. In all experiments, white Gaussiannoise was added explicitly on each real and imaginary channelof the undersampled data with a standard deviation toobtain more realistic simulations.Performance of reconstruction methods are quantified with

the following metric:

(15)

where (resp. represents the ground truth fully-samplednoiseless matrix (resp. estimated matrix). This quantity is

1694 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014

Fig. 4. (a) One time frame acquisition pattern for polynomial variable densitysampling and (b) the ( )-space sampling pattern function of time (left) withits associated probability density function (right). (c) One time frame acquisitionpattern for pseudo-radial sampling and (d) the angles of rotation function of time(left) with the associated uniform probability density function (right).

expressed in decibels for convenience. Since (15) provides aglobal measure, the normalized mean square error (NMSE) ateach time frame will be shown for some experiments

(16)

where (resp. represents the ground truth fully-samplednoiseless image (resp. estimated image) at time frame .2) Comparisons With Other Reconstruction Algorithms: A

zero-filled inverse Fourier transform and a sliding window re-construction using a zeroth-order hold technique [41] are in-cluded, mainly to illustrate the level of undersampling. Compar-isons with dynamic MR reconstruction methods - FOCUSS[7] and - SLR [11] are provided. Literature review suggeststhat they are arguably very efficient state of the art techniquesrespectively in CS, and low-rank and sparsity methods for dy-namic MRI reconstruction.

- FOCUSS is implemented with 40 inner iterations (con-jugate gradient step), two outer iterations (FOCUSS step), andweighting matrix power factor of 0.5. The low-resolution initialestimate is important to guarantee a good performance of -FOCUSS and is obtained by using a zero-filled inverse Fouriertransform using the low-frequency samples. In - SLR, thenonconvex Schatten -norm with is used. There areparameters to tune related to the continuation strategy of the op-timization algorithm that are used to improve the convergencerate. These parameters are set at suggested values provided inthe - SLR package (penalty parametersfor Schatten and TV norms; penalty parameters incrementationboth set to 25 in the outer loop; maximum number of 50 innerand nine outer iterations).

Fig. 5. Modelling local intensity changes (showing here pixel intensity valuesin time) as the uptake and washout of a contrast agent using the modified Toftsmodel [42].

3) Regularization Parameters: In - FOCUSS, one regu-larization parameter can be tuned that controls the stability ofthe solution under noisy conditions. Here, reconstructions witha different regularization parameter selected from a range ofvalues are computed and the best one is selected in accordancewith (15). In - SLR, different regularization parametersand for respectively the Schatten norm and spatio-temporalTV norm are tested, and the best reconstruction is selected ac-cording to (15). A similar strategy for - RPCA is employedby varying both the regularization and decomposition pa-rameters. Note that if one is looking for a specific type of de-composition, can be fixed accordingly, although it should benoted that it may also affect the reconstruction results, which isdiscussed in Section V-A.

B. Reconstruction Results

This section presents reconstruction results where the separa-tion is not of particular interest, but proves to be a strong enougha priori information to remain competitive against state of theart methods. In these experiments, in - RPCA is automati-cally selected to return the best reconstruction.1) Phantom Simulations: First experiments are conducted

on a numerical phantom of dimensions, as shown in Figs. 6 and 7. This phantom is created to

model typical dynamic MRI sequences with different types oftime-varying components. Specifically, it can include periodiclocal and global motion, and localized changes of intensity.Local motion simulates moving organs (such as the beatingheart) while global motion simulates respiratory-like move-ment imitating free-breathing imaging. Motion is modelledusing trigonometric functions with varying frequencies andamplitudes. Local intensity changes mimic a contrast enhanced(CE) signal, i.e., the uptake and washout of a contrast agentusing the modified Tofts model [42] as shown in Fig. 5. This istypical in dynamic contrast enhanced MRI studies. While sim-plistic, the major advantage of this phantom is the full controland availability of ground truth. Fine adjustments can be madesuch as adding specific levels of noise, creating motion-freeor CE-free dynamic sequences. To evaluate the performanceof the different reconstruction algorithms, the same phantomwith different time-varying elements is used. In particular,reconstruction methods are tested when the phantom has onlyintensity changes (no motion), periodic motion (no intensitychanges) and with a combination of the two. Table I providessome characteristics of the three phantoms in the noiseless

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Fig. 6. Qualitative results for phantom with a combination of intensity and motion (Cartesian sampling). (a) Magnitude images. (b) Zoom-in magnitude images(corresponding to the red square on the ground truth image). (c) Phase images. (d) - temporal profiles. (e) - temporal profiles (according to the dotted lines onthe ground truth image). Time frames shown in the first three rows correspond to the frames selected on the dotted lines on the temporal profiles. Left colormapsrefer to magnitude images. Right colormap refers to phase images.

case, i.e., the rank, norm of the Fourier transform along thetime dimension and norm of the gradient in directionsapproximated by finite differences. Note that noiseless (groundtruth) matrices are of dimensions with various ranksdepending on whether intensity/motion is present. However,noisy matrices are full-rank (80) because of the presence ofnoise, although they remain approximately low-rank with anumber of significant singular values about equivalent to theirnoiseless counterpart.In these experiments, the acceleration factor is approximately

10 (about 10% of acquired samples). Quantitative results arereported for Cartesian sampling and pseudo-radial sampling inTables II and III. Reconstruction errors are shown in decibelsand have been computed as in (15) with associated regulariza-tion parameter(s) for the different methods in brackets. For -SLR, they refer to and for - RPCA to . Visual eval-uations are provided in Figs. 6 and 7. Fig. 6 shows magnitude

images, phase images and temporal profiles for the phantomwith a combination of intensity and motion. Fig. 7 presents timeprofiles of reconstructions for the phantom with only intensityand with only motion.Different types of dynamic imaging are tested, but from these

results there is no indication in which - RPCA might have apreference for a certain one. In fact, it can be observed that -RPCA has a similar behavior as - SLR, although - SLR con-sistently provides better reconstructions than - RPCA itself.Both - SLR and - RPCA outperform - FOCUSS whenmotion is present, and when a combination of intensity and mo-tion is present. When only intensity is present however, - FO-CUSS seems to have a slight advantage over both - SLR and- RPCA.The general good performance of - SLR over other

methods can possibly be attributed to the fact that the phantomis a piecewise constant signal. A spatio-temporal total variation

1696 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014

Fig. 7. (a, c) - temporal profiles and (b, d) - temporal profiles of variousreconstruction methods for (a, b) intensity only phantom and (c, d) motion onlyphantom (Cartesian sampling).

prior is particularly efficient for this type of signal, since thetotal variation model penalizes highly oscillatory solutionswhile allowing jumps in the regularized solution. In otherwords, this means that the solution obtained by - SLR israther due to the sparsity prior than the low-rank one. For ex-ample, it can be seen that no prior information about the rank ofthe matrix was used in the reconstruction of the phantom withonly intensity for Cartesian sampling [Fig. 7(a) and (b)], aswas selected equal to zero (Table II, “Intensity only” column).Tables II and III show that all reconstruction methods benefit

from the pseudo-radial sampling, whether they are based onlyon sparsity or both low-rank and sparsity prior. This can be ex-pected because it can be seen that a direct inversion (zero-filledinverse Fourier transform, ZF-IDFT) already gives an improvedreconstruction performance when a pseudo-radial sampling pat-tern is employed over the Cartesian sampling pattern.2) Cardiac MRI: The second experiment is conducted on

cardiac in vivo MRI data, specifically a free-breathing cardio-vascular dataset of dimensions froma 3T MRI scanner. Apart from motion such as heartbeats andlarge breathing movements, this dataset has complex anatom-ical features that makes it more challenging to reconstruct thanthe numerical phantom. An acceleration factor of approximately8 is chosen which corresponds to about 12.5% of acquired sam-ples. It is necessary to add noise to in vivo data because the orig-

TABLE ICHARACTERISTICS OF THE DIFFERENT NOISELESS PHANTOMS

TABLE IIRECONSTRUCTION RESULTS USING METRIC (15) (EXPRESSED IN DECIBELS)FOR NUMERICAL PHANTOMS WITH CARTESIAN SAMPLING. NUMBERS IN

BRACKETS REFER TO REGULARIZATION PARAMETERS

TABLE IIIRECONSTRUCTION RESULTS USING METRIC (15) (EXPRESSED IN DECIBELS)FOR NUMERICAL PHANTOMS WITH PSEUDO-RADIAL SAMPLING. NUMBERS IN

BRACKETS REFER TO REGULARIZATION PARAMETERS

inal noise in the magnitude image becomes part of the apparentsignal when retrospectively undersampled.Time frames extracted from the different reconstruction

methods are shown in Fig. 8, although it is visually difficult toclaim objectively which method is the best. Quantitative resultsare given in Table IV for the different sampling strategies.Additionally, NMSE at each time frame are shown in Fig. 9 forboth Cartesian and pseudo-radial sampling.Based on these results, all methods performed similarly using

the polynomial variable density sampling, and a slight advan-tage can be seen for both - SLR and - RPCA when pseudo-radial sampling is used. In - RPCA, the selected reconstruc-tions were chosen with , which means that the selectedreconstruction has favoured the low-rank part rather than thesparse part in this context. - SLR did not successfully manageto obtain much better reconstruction results over other existingmethods as it previously did on the first experiment. One of thepossible reasons can be attributed to the fact that the cardiacMRI dataset did not have a particularly sparse gradient in spaceand time compared to the numerical phantoms.This last experiment demonstrates - RPCA as a competi-

tive and distinct dynamic MRI reconstruction method from par-tial measurements.

C. Exploiting the Separation

In the following section, the utility of the intrinsic separationof the reconstructed data into low-rank and sparse componentsis demonstrated in the context of motion estimation in dynamiccontrast enhanced (DCE) MRI.In DCEMRI, acquisition of multiple 2-DMR images is taken

before, during, and after the administration of a contrast agent(CA). The uptake and washout of the CA concentration over

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Fig. 8. Visual comparison of reconstruction methods for cardiac MRI data showing one time frame magnitude image (frame number ). First row corre-sponds to Cartesian sampling, second row to pseudo-radial sampling.

TABLE IVRECONSTRUCTION RESULTS USING METRIC (15) (EXPRESSED IN DECIBELS)FOR CARDIAC MRI DATA WITH CARTESIAN AND PSEUDO-RADIAL SAMPLING

Fig. 9. NMSE at each time frame for cardiacMRI data with Cartesian sampling(top) and pseudo-radial sampling (bottom).

time in the body corresponds to local changes of intensity in theMR images. Pharmacokinetic analysis can then be used to relateto tissue characteristics [42]. However, patient motion duringacquisition (such as heartbeats, breathing, or involuntary move-ments) produces inter-frame misalignment and complicates the

Fig. 10. Different types of separation into low-rank and sparse componentsusing - RPCA with different decomposition parameters . It can be observedthat this parameter acts as a trade-off between the two components. Undersam-pling rate is 0.25. (a) Low-rank time frames. (b) Sparse time frames. (c) -temporal profiles of low-rank component. (d) - temporal profiles of sparsecomponent.

estimation of the rate of the uptake by the tissue. Image regis-tration can be used to solve this problem, but the presence of thecontrast enhanced (CE) images interferes with the registrationprocedure because conventional algorithms can interpret localintensity changes as motion.Due to the embedded separation, the proposed - RPCA ap-

proach is expected to separate slow time-varying elements frommore abrupt changes. Hence, it is possible to separate to somedegree the local changes of intensity in the sparse componentwhen using an appropriate parameter. Fig. 10 demonstratesthe different decompositions obtained with different . Regis-tering low-rank images that include most of the motion and lesslocal intensity changes provoked by the CA is likely to provide a

1698 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014

Fig. 11. - temporal profiles used in the registration procedure. For the registration of - RPCA, only the low-rank part is used which mostly contains imageswithout contrast enhancement thanks to the separation process.

Fig. 12. Displacement fields (zoom-in) over source images used for registration. Table V provides the associated quantitative results. (a) Ground truth noiselessphantom. (b) Noisy phantom with local intensity changes. (c) - FOCUSS. (d) - SLR. (e) - RPCA, low-rank part. It can be seen that the displacement fieldis better estimated in the region with local changes of intensity in - RPCA.

displacement field that is closer to the ground truth images (i.e.,the same signal without CE images). This displacement field canthen be employed for more accurate motion correction [43].As a proof of concept, the numerical phantom of the previous

section with a combination of intensity and motion is employedfor the purpose of demonstration with an acceleration factor of4 using the pseudo-radial sampling. - FOCUSS and - SLRare reconstructed as previously, using the best regularization pa-rameters. However, - RPCA reconstruction is now explicitlyselected with to end up with mostly the local intensitychanges in the sparse part, and motion in the low-rank part. Re-construction errors for - FOCUSS, - SLR, and - RPCAwere, respectively, 25.3 dB, 31.1 dB, and 26.3 dB.A sequential registration of each frame of - FOCUSS, -

SLR and the low-rank part of - RPCA was performed withNiftyReg2 [44], an efficient C++ implementation of a parallelformulation of the free-form deformation (FFD) algorithm [45]based on cubic B-splines. The time profiles of the differentreconstruction methods along with the ground truth are shownin Fig. 11. Note the ground truth was obtained with the noise-less phantom created without intensity changes but includingmotion. The reference images taken for registration are the lasttime frame images in the respective dynamic reconstructedsequences.Displacement fields (zoom-in with source images used for

registration) are shown in Fig. 12. To quantify these results, asimilar metric to (15) is used with the Jacobian of the 2-D dis-placement fields taken in the region of interest with local in-

2Local normalized cross correlation is used as measure of similarity (standarddeviation of the Gaussian kernel set to five pixels for all time points) and acontrol point spacing of two pixels in all directions.

TABLE VDISPLACEMENT FIELDS ERRORS USING THE JACOBIAN (TAKENIN THE REGION OF INTEREST WITH LOCAL INTENSITY CHANGES)

IN THE METRIC (15) (EXPRESSED IN DECIBELS)

tensity changes. Results are reported in Table V which shows aslight improvement for - RPCA over other methods.

V. DISCUSSION

A. Prior Assumptions and Regularization Parameters

Experiments suggest that from a reconstruction point of view,the prior assumption made in - RPCA is strong enough to re-main competitive with state of the art methods. The prior as-sumption in - FOCUSS is that the -space is a sparsesignal, which is appropriate with dynamic datasets that exhibitperiodicity in time. - SLR performed well over other methodsfor phantom simulations, but was not so successful for freebreathing cardiac MRI data.Generally, regularization parameters affect directly the

reconstruction results, and selecting the right one is alwayschallenging in the general topic of inverse problems. In -FOCUSS and - SLR, the best reconstructions were selectedby testing a range of different regularization parameters. Twostrategies for the choice of in - RPCA have been adopted.If interested in obtaining the best reconstruction, the choice ofshould be selected such that the best reconstruction is returned.However, can also be forced to a specific value to obtain a

TRÉMOULHÉAC et al.: DYNAMIC MR IMAGE RECONSTRUCTION–SEPARATION FROM UNDER-SAMPLED ( )-SPACE 1699

Fig. 13. Influence of regularization parameters on the reconstruction error (in dB) for - SLR and - RPCA. Numerical phantom simulations with 10-foldacceleration and Cartesian sampling. For - SLR, and refers to (3) and for - RPCA, and refers to (7).

desirable decomposition. Fig. 13 presents reconstruction errors(in dB) using different regularization parameters for - SLRand - RPCA. This figure and previous experiments using thephantom suggest that the low-rank a priori information in -SLR is not playing an important role in this case. When ,it is observed that a good reconstruction can be obtained usingonly the sparsity a priori information. This can be attributed tothe fact that piecewise constant signal such as the phantom isvery sparse when the norm of the gradient is computed. For- RPCA, the regularization parameter is a trade-off betweendata consistency and the low-rank plus sparse decomposition.As shown in Fig. 13, the solution is not regularized whenand an appropriate value must be selected to obtain a goodregularized solution. The decomposition parameter can bevaried, although it also affects the reconstruction results.In this paper, the selection of regularization parameters was

optimized but this is an unrealistic strategy since the groundtruth is not available in a practical scenario. However, methodscan be adapted to find ideal regularization parameters such asthe discrepancy principle if noise properties are known, and forexample L-curve or generalized cross-validation methods if not[46].

B. Decomposition

While a separation into low-rank and sparse components iseasy to see mathematically, it may be difficult to interpret phys-iologically what it represents in a dynamic MRI context. Oneof the reasons is that the decomposition depends on the typeof dynamic data. Further work in this direction should be ad-dressed to understand more deeply how it can be interpretedphysiologically.Generally, the decomposition provides a separation into two

components that have different characteristics. The low-rankcomponent will tend to have slow time-varying elements whilethe sparse component will capture more abrupt changes, butcan be modified to balance between the two parts.In the simulations, a specific example has been shown where

partial isolation of local changes of intensity in the sparse com-ponent from the general motion leads to a better estimationof the displacement field. Departing from this example, it is

likely that this combined reconstruction-separation approach of-fers further applications that could be investigated. For example,motion-related applications where sparse and localized motionelements interfere with the general background signal, or arte-facts removal where the outlier component would cause unde-sired alterations of data. In the latter case, an artefact correctionalgorithm for RF spike noise has been proposed based on RPCAas a postprocessing technique [47].

C. Noise

This study has included complex-valued noise to simulatemore realistic experiments. However, this study has not evalu-ated the reconstruction (and separation) performance of the pro-posed approach as a function of added noise. Generally in -RPCA, increased noise is inclined to interfere with the sparsecomponent. Both - FOCUSS and - SLR may be relativelymore robust to noise regarding reconstructed data, since in -FOCUSS the noise in -space will generally not be rep-resented by highly sparse coefficients, and the spatio-temporaltotal variation norm will tend to smooth a noisy solution in -SLR.

D. Acquisition and Sampling

Two strategies to undersample the ( )-space, a Cartesianand pseudo-radial sampling patterns, have been used in thispaper. There were respectively based on polynomial variabledensity and random rotations across each acquisition frame toproduce incoherent sampling artefacts. From the experimentsection, in particular Tables II, III, and IV, it is suggested thatthe pseudo-radial undersampling strategy provides better recon-struction results for all methods either based on only sparsity as- FOCUSS, or based on low-rank and sparse prior informa-tion ( - SLR, - RPCA). However, these sampling strategieswere based on retrospectively undersampling the ( )-spaceand they would ideally need to be validated using prospectivelyundersampled data from an MRI scanner.

E. Computational Times

The purpose of this paper was not to focus on computationalaspects of the different reconstruction methods. The different

1700 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 33, NO. 8, AUGUST 2014

algorithms used were implemented in MATLAB and not opti-mized. However, it should be noted that during our simulations,- FOCUSS reconstructions could be obtained in less than aminute, whereas - SLR and - RPCA could require severalminutes of computation in contrast ( min).

VI. CONCLUSION

Dynamic MR imaging is a widely used technique inmedicine. Numerous methods have been developed to reduceacquisition time, but it can still benefit from higher accelerationrates and efficient reconstruction algorithms.This paper has presented a method termed - RPCA that

jointly reconstructs and separates dynamic MR data from par-tial measurements by employing a low-rank plus sparse regu-larization prior. While providing a competitive reconstructionmethod for accelerated dynamic MRI as comparisons with stateof the art methods have shown, this technique also provides aseparation into two components having different characteristicsthat can have potential when tailored to the right application. Inthis paper, the decomposition was used to partially separate thecontrast enhanced region in a simulated DCE sequence and helpin motion estimation.Overall, this study supports the use of low-rank and sparsity

prior information in dynamic MR image reconstruction tech-niques from highly undersampled data. Interestingly, the pro-posed reconstruction—separation approach suggests a methodthat is not only about finding the closest representation of thetrue object, but is also a step towards methods that would di-rectly infer relevant characteristics from limited measurements.

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