REVIEW Open Access

Compressed sensing MRI: a review fromsignal processing perspectiveJong Chul Ye

Abstract

Magnetic resonance imaging (MRI) is an inherently slow imaging modality, since it acquires multi-dimensional k-space data through 1-D free induction decay or echo signals. This often limits the use of MRI, especially for highresolution or dynamic imaging. Accordingly, many investigators has developed various acceleration techniques toallow fast MR imaging. For the last two decades, one of the most important breakthroughs in this direction is theintroduction of compressed sensing (CS) that allows accurate reconstruction from sparsely sampled k-space data.The recent FDA approval of compressed sensing products for clinical scans clearly reflect the maturity of thistechnology. Therefore, this paper reviews the basic idea of CS and how this technology have been evolved forvarious MR imaging problems.

Keywords: MRI; compressed sensing; k-space

BackgroundMagnetic resonance imaging (MRI) exploits the nuclearmagnetic resonance (NMR) phenomena to enable highcontrast imaging. Since the first observation of NMR ab-sorption in a molecular beam in 1938, the main researchthrust was to understand and utilize the NMR phenom-ena for spectroscopy applications. Then, Lauterbur in-troduced in 1973 the gradient field to encode the spatialorigin of the radio waves emitted from the nuclei of theobject. This breakthrough allowed for multi- dimen-sional imaging by NMR physics.The idea of Lauterbur can be easily understood using

k-space interpretation that describes MR sampling asFourier encoding in 2D or 3D spaces. Specifically, datacollected by an MRI scanner are samples of the spatialFourier transform of an object image. Hence, in order toobtain an image without aliasing artifacts, k-space sam-ples need to satisfy the Nyquist sampling criterion.Despite this close link to signal sampling theory, until

the first demonstration of the sensitivity encoding(SENSE) technique by Prussemann [1], MR imaging wasnot considered as an important research topic for signalprocessing. Specifically, Prussemann et al. [1] showedthat spatial diversity information from coil sensitivity

maps have additional information that can be exploitedfor fast signal acquisition. Furthermore, Sodickson et al.[2] proposed the simultaneous acquisition of spatial har-monics (SMASH). These works gave a birth of parallelimaging and iterative reconstruction methods, and hasresulted in a flurry of novel ideas and algorithms, includ-ing Generalized Autocalibrating Partially Parallel Acqui-sitions (GRAPPA) by Griswold [3] and k-t space methodfor cardiac imaging such as [4–9].The common theme in these approaches is that the

data redundancy can be exploited to reduce the requiredsampling rate. Because redundant data can be compactlyrepresented in some transform domains, it is also closelyrelated to the concept of “sparsity”. Originally investi-gated by Bresler and his students for on- the-fly Fourierimaging [10, 11], the sparsity regularization has becomethe main workhorse in modern accelerated MRI re-searches thanks to the introduction of the compressedsensing theory [12, 13]. Ever since the first demonstra-tion of compressed sensing MRI by Lustig et al. [14, 15],the compressed sensing MRI has become the essentialtools in modern MR imaging researches. In this paper,we will review these ideas in more detail.

Correspondence: jong.ye@kaist.ac.krDepartment of Bio and Brain Engineering, Korea Adv. Inst. of Science &Technology (KAIST), 291 Daehak-ro, Daejeon, Korea

BMC Biomedical Engineering

© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Ye BMC Biomedical Engineering (2019) 1:8 https://doi.org/10.1186/s42490-019-0006-z

Main textMR forward modelsBefore we introduce the concept of compressed sensingMR, we begin with the discussion of MR forward model.Mathematically, the forward model for k-space measure-ment can be described by.

biðkÞ ¼Z

γiðrÞe− j2πkT rdr; i ¼ 1;…;C; ð1Þ

where r ∈ R^d and k ∈ Rd, d = 2, 3 denote the image do-main and k-space coordinates, respectively, C is thenumber of the coil and the i-th coil image γi is given by

γi rð Þ ¼ ν rð Þsi rð Þ ð2ÞHere, ν(r) denotes the contrast weighted bulk

magnetization distribution, and si(r) is the correspondingcoil sensitivity of the i-th coil. Although the expression maygive an impression that the measurement is two- or three-dimensional, this is indeed originated from a 1-D measure-ment since the k-space trajectory is a function of time, i.e.k:= k(t), and we acquire one k-space sample at each timepoint t. This makes the MR imaging inherently slow, sincewe need to scan through a 3-D object via 1-D trajectories.Dynamic MRI is another important MR technique to

monitor dynamic processes such as brain hemodynamicsand cardiac motion. Among the various forms of dy-namic MR modeling, here we mainly focus on the k-tformulation. Specifically, consider a discrete imagingequation for cartesian trajectory for simplicity. Becausethe samples along the readout direction are fully sampled,most of the dynamic MR formulation is applied separablyafter taking the Fourier transform along the read- out dir-ection. More specifically, let γ(s, t) denote the unknownimage content (for example, proton density, T1/T2weighted image, etc.) on the spatial coordinate s along thephase encoding line at time instance t. Then, the k-t spacemeasurement b(k, t) at time t is given by

bðk; tÞ ¼Z

γðs; tÞe− j2πksds ð3Þ

where γ(s, t) is the spatio-temporal image which may beweighted by coil sensitivity map for the case of parallelimaging.Throughout the paper, for simplicity we often use the

operator notation for (1):

bi ¼ F ½Si�ν; i ¼ 1;⋯;C;

where [Si] is a diagonal operator comprised of thei-th coil sensitivity. Similarly, we use operator nota-tion for (3).

b ¼ Fγ ð4Þ

or

bi ¼ Fγ i; i ¼ 1;⋯;C ð5Þ

for the case of parallel imaging.

Compressed sensing theoryPerformance guaranteesCompressed sensing (CS) theory [12, 16, 17] addressesthe accurate recovery of unknown sparse signals fromunderdetermined linear measurements and has becomeone of the main research topics in the signal processingarea for the last two decades [18–23]. Most of the com-pressed sensing theories have been developed to addressthe so-called single measurement vector (SMV) prob-lems [12, 16, 17]. More specifically, let m and n be posi-tive integers such that m < n. Then, the SMVcompressive sensing problem is given by

P0ð Þ : minimize xk k0subject to ‖y � Ax‖ < ϵ;

ð6Þ

where y ∈ R^m, A ∈ R^m\×n, x ∈ R^n, and ϵ denotesthe noise level. (P0) implies that the solution favours thesparsest solution.Since (P0) requires a computationally expensive com-

binatorial optimization, greedy methods [24], reweightednorm algorithms [25, 26], or convex relaxation using thel1 norm [12, 27] have been widely investigated as alter-natives. In particular, the convex relaxation approach ad-dresses the following l1 minimization problem:

P1ð Þ : minimize xk k1subject to ‖y � Ax‖ < ϵ;

ð7Þ

One of the important theoretical tools of CS is theso-called restricted isometry property (RIP), which en-ables us to guarantee the robust recovery of certain in-put signals [17]. More specifically, a sensing matrixA ∈ Rm�n is said to have a k- restricted isometry prop-erty (RIP) if there is a constant 0 ≤ δk < 1 such that

1−δkð Þ‖x‖2≤‖Ax‖2≤ 1þ δkð Þ‖x‖2:

for all x ∈ R^n with ||x||0 ≤ k. It has been demon-strated that δ2k <

ffiffiffi2

p−1 is sufficient for l1/l0 equivalence

[12]. For many classes of random matrices, the RIP con-dition is satisfied with high probability if the samplingpattern is incoherent and the number of measurementssatisfies m ≥ ck log(n/k) for some constant c > 0 [17].

Ye BMC Biomedical Engineering (2019) 1:8 Page 2 of 17

Optimization approachesIn practice, the following unconstrained form ofoptimization problem is often used:

minx

12

y−Axk k2 þ λ Ψxk k1 ð8Þ

where y is noisy measurement, λ is the regularizationparameter, and Ψ refers to an analysis transform suchthat Ψx becomes sparse. Note that for the special caseof Ψ = I, (8) is reduced to (P1).One of the technical issues in solving (8) is that the

cost function is not smooth at Ψx = 0, which makes thecorresponding gradient ill-defined. This leads to the de-velopment of various techniques, which are culminatedas the new type of convex optimizaton theory calledproximal optimization [28]. For example, the popularop- timization methods such as forward-backward split-ting (FBS) [29], split Bregman iteration [30], alternatingdirectional method of multiplier (ADMM) [31],Douglas- Rachford splitting algorithm [32], andprimal-dual algorithm [33] have been devel- oped tosolve compressed sensing problems. However, the com-prehensive coverages of these techniques deserves an-other review paper, so in this section we mainly reviewthe ADMM algorithms which have been extensivelyused for compressed sensing MRI ever since its firstintroduction to compressed sensing MRI [34].More specifically, we convert the problem (8) to the

following constraint problem:

minγ;u

12

y−Aγk k2 þ λ uk k1 ð9Þ

subject to

u ¼ Ψγ: ð10ÞThen, the associated ADMM is given by

γ kþ1ð Þ ¼ arg minx

12

y−Aγk k2 þ μ2

ζ kð Þ þΨγ−u kð Þ��� ���2

u kþ1ð Þ ¼ arg minu

λ uk k1 þμ2

ζ kð Þ þΨγ kþ1ð Þ−u��� ���2

ζ kþ1ð Þ ¼ ζ kð Þ þ x kþ1ð Þ−u kþ1ð Þ;

Now, each step of ADMM has a closed-form solution.Algorithm 1 summarises the resulting ADMM iteration,where shrink1 denotes the soft-tresholding:

shrink1 x; τð Þ ¼ sgn xð Þ max 0; jxj−τf g; ð11Þfor the threshold value τ > 0.

In addition to the analysis prior in (8), the total vari-ation (TV) penalty has been also extensively used forimaging applications, because the finite differenceoperator

can sparsify smooth images. Specifically, the TV mini-misation problem assumes the following form:

minx

12

y−Axk k2 þ λTV xð Þ; ð12Þ

where T V (x) is given by

TV xð Þ ¼ ∇xk k1;p; ð13Þ

where ‖∙‖1, p, p = 1, 2 deotes the l1/lp-mixed norm. Ind-dimensional space (e.g. d = 2 for images), the discre-tized implementation can be defined as

∇xk k1;p ¼Xn

i¼1∇x ið Þk kp

where ∇x(i) ∈ Rd denotes the gradient of x at the i-thcoordinate and n denotes the number of discretizatedsamples.In order to apply ADMM for (12), we need to focus

on the primal formulation of the total variation penalty:

TV xð Þ ¼ ∇xk k1;p ¼Xn

i¼1∇x ið Þk kp ð14Þ

Now, we define a splitting variable u(i) =∇x(i) ∈ Rd.Then, the constraint opti- mization formulation is given by

minx; u ið Þf gni¼1

12

y−Axk k2 þ λXn

i¼1u ið Þk kp ð15Þ

Subject to u ið Þ ¼ ∇x ið Þ; i ¼ 1;⋯; n ð16ÞThen, the associated ADMM is given by

x kþ1ð Þ ¼ arg minx

12

y−Axj jj j2 þ η2

Xiζ kð Þ ið Þ þ ∇x ið Þ−u kð Þ ið Þ

��� ���2

u kþ1ð Þ ið Þ ¼ arg minu ið Þ

λ u ið Þk kp þη2

ζ kð Þ ið Þ þ ∇x kþ1ð Þ ið Þ−u ið Þ��� ���2

Ye BMC Biomedical Engineering (2019) 1:8 Page 3 of 17

ζ kþ1ð Þ ið Þ ¼ ζ kð Þ ið Þ þ ∇x kþ1ð Þ ið Þ−u kþ1ð Þ ið ÞEach step has the closed form expression. Algo-

rithm 2 summarises the TV-ADMM algorithm, whereshrinkvec,2(x) for x ∈ Rd denotes the vector shrinkage[34]:

shrinkvec;2 x; τð Þ ¼ xxj j max 0; xj j−τf g; x∈Rd ð17Þ

for the threshold value τ > 0.

Basic MR ingredients for compressed sensingIn order to apply compressed sensing theory for specificimaging applications, the unknown signal should be sparsein some transform domain, and the sensing matrix shouldbe sufficiently incoherent. This section shows why theseconditions can be readily satisfied in MR imaging, whichis one of the main reasons that allows for successful appli-cations of CS theory to MR imaging.

Sparsity of MR imagesAn MR image is rarely sparse in its own. However, oneof the important observations that led to the successfulapplications of CS to MR is that the sparsity is closelyrelated to signal redundancies. This is because redun-dant signals can be easily converted to sparse signalsusing some transforms.Basically, there are three major directions that have

been investigated in com- pressed sensing MRI: 1) thespatial domain redundancy, 2) the temporal domain re-dundancy, and 3) the coil domain redundancy. For ex-ample, as shown in Fig. 1(a), natural images can besparsely represented in finite difference or wavelet trans-form domain, although the image is not sparse in itsown. This observation is the main idea that allows fortotal variation (TV) and wavelet transform approachesfor image denoising, and reconstruction. Accordingly,TV and wavelets have been the main transforms thathave been extensively used in most of the CS MRI re-searches. On the other hand, dynamic MR images such

as cardiac cine, functional MRI, and MR parameter map-ping have significant redundancy along the temporal di-mension as shown in Fig. 1(b). For example, if the imageis perfectly periodic, then temporal Fourier transformmay be the optimal transform to sparsify the signal.However, in many dynamic MR problems, the temporalvariations are dependent on the MR physics as well asspecific motion of organs, so the analytic transform suchas Fourier transform may not be an optimal solution,but more data-driven approaches such as PCA or dic-tionary learning are better options. Indeed, these obser-vation leads to dictionary learning approaches that willbe discussed later.On the other hand, the coil redundancy is somewhat

distinct compared to the spatial- and temporal- domainredundancy. As shown in Fig. 1(c), the redundancy ofthe coil image stems from the underlying common im-ages, which results in cross- channel redundancies:

s j rð Þγ i rð Þ ¼ si rð Þγ j rð Þ; i≠ j; ∀r ð18Þwhere si denotes the i-th coil sensitivity map and γi isthe coil image. The rela- tionship in (18) is easy to showsince the coil image is given by (2). The main idea of theclassical parallel MRI is to exploit this relationship. Spe-cifically, the sensi- tivity encoding (SENSE) [1] exploitsthe image domain redundancy described in (18), whereasthe k-space domain approaches such as GRAPPA [3] ex-ploits its dual relationship in the k-space:

s j rð Þ � γ i rð Þ ¼ si rð Þ � γ j rð Þ; i≠ j; ∀r ð19Þ

where ∗ denotes the convolution, and s j , γ j enote theFourier transform of sjand γjrespectively. Later we willdescribe how these two expressions of the coil dimen-sional redundancies have been exploited in compressedsensing MRI.

Incoherent sampling patternIn compressed sensing MRI, downsampling pattern isvery important to impose the incoherence, but itsrealization is limited by MR physics. For example, in the2-D acquisition, the readout direction should be fullysampled, so there is only freedom along the phase en-coding direction in designing incoherence sampling pat-terns. Figure 2(a)-(c) shows the examples of realizablesampling patterns that have been exploited in the litera-ture: a) cartesian undersampling, b) radial trajectory, andc) spiral trajectory. In particular, the radial and spiral tra-jectories, which have been studied even before the ad-vanced of the compressed sensing [35], have moreincoherent radial sampling patterns compared to thecartesian undersampling.

Ye BMC Biomedical Engineering (2019) 1:8 Page 4 of 17

On the other hand, if we deal with 3-D imaging ordynamic imaging, there are more rooms for samplingpattern design, since there are two dimensional degreeof freedom. Figure 2(d) shows an example of 2-Drandom sampling pattern. For the case of wavelettransform-based sparsity imposing prior, Lustig et al.[14] showed that the incoherence can be optimized bygeneralizing the notion of PSF to Transform PointSpread Function (TPSF). Specifically, TPSF measureshow a single transform coefficient of the underlying ob-ject ends up influencing other transform coefficients ofthe measured undersampled object. To calculate theTPSF, a single point in the wavelet transform space atthe i-th location is transformed to the image space andthen to the Fourier space. Once the corresponding Fou-rier space data is subsampled by the given downsam-pling pattern, its influence in the wavelet transform

domain can be calculated by taking inverse transformfollowed by the inverse wavelet trans- form. To have thebest incoherence property, the side of TPSF should be assmall as possible, such that the side lobe contributioncan be removed by shrinkage op- eration. This was usedas the main criterion for sampling pattern design.

Historical milestonesThe earliest application of the compressed sensing forMRI was done by Lustig and his collaborators [14]. Theapplication of CS to dynamic MRI was pioneered byJung et al. [18] and Gamper et al. [36], which was latersignificantly improved by Jung et al. [20]. The positiveresults of these earlier works have resulted in flurry ofnew ideas in static and dynamic MRI.In terms of combining CS with parallel imaging, the

first application for static imaging was done by Block et

Fig. 1 Various types of sparsity in MRI. (a) Sparsity from spatial domain redundancy, (b) Sparsity from temporal redundancy, and (c) sparsity frommu.ti-channel redundancy

Fig. 2 Various under-sampling patterns: (a) Cartesian undersampling, (b) radial undersampling, and (c) spiral undersampling

Ye BMC Biomedical Engineering (2019) 1:8 Page 5 of 17

al. [37] by combining total variation penalty and parallelimaging; and by Liang et al. [38] in more general formusing SENSE. In dynamic imaging, the works by Jung etal. [18] and Feng et al. [39] are among the first that com-bine the SENSE type parallel imaging with the k-t do-main compressed sensing. Combination with CS andparallel imaging using GRAPPA type constraints waspioneered by Lustig et al. [40]. Coil compression tech-niques have been also investigated to reduce the numberof coils for CS reconstruction [41]. Feng et al. [42] latercombined compressed sensing, parallel imaging, andgolden-angle radial sampling for fast and flexible dynamicvolumetric MRI, which has been approved for clinical use.Aside from the standard l1 and TV penalty, several in-

novative sparsity inducing penalty have been used forcompressed sensing MRI. For example, Trzasko et al.[43, 44] proposed a direct l0 mimimization approaches,whereas Knoll et al. [45, 46] proposed a generalized totalvariation approaches, and Sung et al. [47] combined theapproximate message passing algorithm with parallel im-aging. This idea was then extended to the dictionarylearning [20, 48, 49] and motion compensation [20, 50].Then, the idea of low-rank regularization was soon in-troduced [49, 51–53] thanks to the theoretical advancesby Candes et al. [54]. The low-rank idea was further ex-tended to structured low-rank approach for parallel im-aging [55] and single coil imaging with the finite support[56]. The duality between the compressed sensing andlow-rank Hankel matrix approaches were discovered byJin and his colleagues [57–60] and Ongie et al. [61]. Inparticular, the unified framework for compressed sensingand parallel MRI in terms of low-rank Hankel matrix ap-proaches was presented by Jin et al. [57] and its theoret-ical performance guarantees was also given in [62].In the following, we provide more detailed reviews of

these historical milestones in compressed sensing MRI.

Basic formulation of compressed sensing MRIAlthough some MR images such as angiograms arealready sparse in the pixel representation, more compli-cated images are rarely sparse, but only have a sparserepresentation in some transform domain, for example,in terms of spatial finite- differences or their wavelet co-efficients. Based on this observation, Lustig et al. [14]proposed the first compressed sensing MRI using spatialdomain wavelet transform as a sparsifying transform.More specifically, the problem is formulated as

minx

Ψγk k1 ð20Þ

subject to y−DFγk k2 < ϵ

where ‖∙‖1 denotes the l1 norm, F is the Fourier trans-form, γ denotes the 2-D complex image, Ψ is a either

finite difference or spatial wavelet transform, D is thedownsampling pattern, and y is the downsampledk-space measurement. Eq. (20) was solved using a non-linear conjugate gradient method.The resulting optimization algorithms are, however,

computational expensive. For cartesian sampling trajec-tory, this problem can be overcome as follows. Specific-ally, the image update step for (20) in ADMMimplementation can be summarized as

F�D�DF þ μIð Þγ ¼ F�D�y−μΨ� ζ kð Þ−u kð Þ� �

;

which is computationally expensive due to the matrixinverse to obtain γ. Instead of using a direct matrix in-verse, by multiplying the Fourier transform F to bothsides, we have

D�Dþ μIð Þγb¼ D�yþ μFΨ� u kð Þ−ζ kð Þ� �

where γb¼ Fγ. Note that a diagonal matrix D*D consist-ing of ones and zeros. The ones are at those diagonal en-tries that corresponds to the sampled locations in thek-space. Let Ω denote the sampled location. Then,

γb kþ1ð Þ ¼ PΩ

D�yþ μFΨ� u kð Þ−ζ kð Þ� �

1þ μ

0@

1Aþ PΩc FΨ� u kð Þ−ζ kð Þ

� �� �;

ð21Þwhere PΩ and PΩc denotes the projection on the sam-pling location Ω and its com- plement Ωc, respectively.Then, the image update γ(k + 1)can be simply done by tak-ing the fast Fourier transform (FFT).The first experimental demonstration by Lustig et al.

[14] clearly confirmed the efficiency of the compressedsensing algorithm, which has led to many other CS ap-proaches using various sparsifying transform, andoptimization algorithms, etc. For example, to deal withseveral hyperparameters to trade off between sparsityand data fidelity terms. Many research efforts have beenmade to quantity such trade- off, provide different formsof reconstruction, even to propose hyperparameter freereconstruction method [63–70].

Advanced formulation for compressed sensing MRINon-Cartesian compressed sensing MRICompressed sensing with non Cartesian sampling hasbeen also extensively stud- ied, since they are really agreat combination given the sampling behavior of nonCartesian sampling schemes and the incoherence re-quirement in compressed sens- ing reconstruction.Aside from the radial and spiral sampling patterns dis-cussed before, the works in [71–83] have fo- cused ondesigning better sampling trajectories with good inco-herent properties. For example, Haldar et al. [82] and

Ye BMC Biomedical Engineering (2019) 1:8 Page 6 of 17

Puy et al. [83] proposed random phase encod- ingscheme to maximize the incoherency of the sensingmatrix. However, one of the main technical issues asso-ciated with non-cartesian compressed sensing MRI isthat the fast reconstruction trick shown in (21) cannotbe used, which increases the overall computational time.

Combination of parallel imaging with CSRecall that parallel MRI (pMRI) [1, 3] exploits the diver-sity in the receiver coil sen- sitivity maps that are multi-plied by an unknown image. This provides additionalspatial information for the unknown image, resulting inaccelerated MR data acquisition through k-space samplereduction. Because the aim of the parallel imaging andCS approaches is similar, extensive research efforts havebeen made to syner- gistically combine the two for fur-ther acceleration [18, 20, 38, 40, 84].One of the most simplest approaches can be a SENSE

type approach that explic- itly utilizes the estimated coilmaps to obtain an augmented compressed sensing prob-lem [18, 20, 38, 84]. More specifically, if the coil sensitiv-ity is known and given by the sensitivity[Si], i = 1, ⋯,C,then the SENSE type compressed sensing MRI prob-lem can be formulated as

minx

12

XC

i¼1bi−DF Si

� �x

�� ��2 þ λ Ψxk k1 ð22Þ

The optimization framework is the standardoptimization framework under sparsity constraint, soproximal optimization algorithms can be used to solvethis problem.On the other hand, l1-SPIRiT (l1-iTerative Self-consistent

Parallel Imaging Re- construction) [40] utilizes theGRAPPA type constraint as an additional constraint for acompressed sensing problem:

minx

‖Ψx‖1

subject to ‖bi−DF ½Si�x‖2 < ϵ; i ¼ 1;…;C

x ¼ Mxð23Þ

where M is an image domain GRAPPA operator. In bothapproaches, an accurate estimation of coil sensitivitymaps or GRAPPA kernel is essential to fully exploit thecoil sensitivity diversity. To address this problem, Ueckeret al. [85] developed a novel eigen space method to ex-tract the coil sensitivity maps directly from the k-spacedata, which is one of the most popular methods widelyused by MR researchers.

Blind compressed sensing MR using dictionary learningBlind compressed sensing approaches attempted to sim-ultaneously reconstruct the underlying image as well as

the sparsifying transform from highly undersampledmeasurements. Ravishankar and his colleagues pioneeredtwo distinct approaches - synthesis dictionary learning[48] and analysis transform learning [86] - when theunderlying sparsifying transform is unknown a priori.

Synthesis dictionary-based BCSMore specifically, let Pj, j = 1 …, N represents the oper-ator that extracts a m- dimensional patch as a vector Pjx∈ Cm from the image x, where N denotes the number ofpatches. Then, dictionary learning is to find the un-known dictionary D ∈ Rm�Q and the correspondingsparse coefficient matrix C ∈ Cm�n such that Y = DC,where Y = [P1x P2x…, PNx]The synthesis model allows each patch Pjx to be ap-

proximated by a linear combina- tion Dcj of a smallnumber of columns from a dictionary D ∈ Cn�K , wherecj ∈ CK is sparse. The columns of the learnt dictionary(represented by dk, 1 ≤ k ≤ K) in (P0) are additionallyconstrained to be of unit norm in order to avoid thescaling ambiguity. The dictionary, and the image patch,are assumed to be much smaller than the image. Thismodel can be used as a signal model, and Ravishankar etal. [48] proposed the following patch-based dictionarylearning regularizer:

R xð Þ ¼ minD;C

XN

j¼1P jx−DC j

�� ��2; s:t; dik k¼ 1; ∀i; C j

�� ��0≤k∀ j ð24Þ

where Cj and dj denotes the j-th column of C and D, re-spectively. Then, the associated BCS formulation is given by

P0ð Þ : minx;D;C

v Ax−bk k22

þXNj¼1

P jx−DC j

�� ��2; s:t; dik k

¼ 1∀i; C j

�� ��0≤s∀ j ð25Þ

where A: = DF denotes the downsampled Fouriertransform.To address the optimization problem (P0), Ravishankar

et al. [48] employed the following two-step alternatingminimization algorithm. First, the following mini- miza-tion problem is solved by fixing x:

minD;C

XNj¼1

P jx−DC j

�� ��2; s:t; dik k

¼ 1∀i; C j

�� ��0≤K∀ j ð26Þ

The K-SVD algorithm [87] was used to learn the dic-tionary. For a given dictionaryD, the image update can be done by

Ye BMC Biomedical Engineering (2019) 1:8 Page 7 of 17

minx

v Ax−bk k22 þXN

j¼1P jx−DC j

�� ��2n oð27Þ

These steps are alternated until convergence. The dic-tionary learning MRI have shown superior image recon-structions for MRI, as compared to non-adaptivecompressed sensing schemes.

Sparsifying transform-based BCSHowever, the BCS Problem (P0) is both non-convex andNP-hard. Approximate iterative algorithms for (P0) typ-ically solve the synthesis sparse coding problem re- peat-edly, which makes them computationally expensive. Inorder to overcome some of the aforementioned drawbacksof synthesis dictionary-based BCS, Ravishankar et al. [86]proposed to use the sparsifying transform model in thiswork. Sparsify- ing transform learning has been shown tobe effective and efficient in applications, while also enjoy-ing good convergence guarantees. Specifically, they usedthe follow- ing transform learning regularizer:

R xð Þ ¼ minW ;C

XN

j¼1WPjx−C j

�� ��2þ λQ Wð Þ; s:t: Bk k0≤s ð28Þ

where W ∈ C^{m\× m} denotes the unknown transform,the function Q(W) is a regualizer for the transform givenby

Q Wð Þ ¼ − log detWj j þ 0:5 Wk k2FThe − log |detW| penalty eliminates degenerate solu-

tions such as those with repeated rows. The ‖W‖Fpenaltyhelps remove a scale ambiguity in the solution.Then, with additional constraint ‖x‖ ≤ E, the overall

optimization problem is given by

P1ð Þ : minx;W ;C

v Ax−bk k22 þXNj¼1

WPjx−C j

�� ��2þ λQ Wð Þ; s:t: Ck k0≤s; xk k≤E; ð29Þ

where A = DF again denotes the downsampled Fouriertransform. One of the important advantages of (P1) isthat there exists a closed-form update for W, so thecomputationally expensive dictionary learning step canbe avoided.

K-t methods for compressed sensing dynamic MRIDynamic MRI is a technique to acquire temporally vary-ing MR sequences such as cardiac cine, perfusion,time-resolved angiography, functional MRI, etc. In dy-namic MRI, there exists significant redundancies alongthe temporal directions, which can be extensively stud-ied in various compressed sensing approaches.

k-t SPARSEThe k-t SPARSE by Lustig et al. [88] is an earliest ver-sion of compressed sensing dynamic MRI. Recall thatthe k-space measurement b(k, t) at time t is given by

b k; tð Þ ¼Z

γ s; tð Þe− j2πksds ð30Þ

Another application of the Fourier transform along thetemporal direction

γ s; tð Þ ¼Z

p s; fð Þe− j2πftdf ; ð31Þ

results in the following 2-D Fourier relationship:

b k; tð Þ ¼ ∬p s; fð Þe− j2π klþftð Þdsdf ; ð32Þ

where ρ(s, f ) denotes the temporal Fourier transform ofγ(s, t).Note that p(s, f ) is usually sparse because the periodic

motions from heart or slow varying motions from fMRIcan be easily sparsified using the temporal Fourier trans-form. Accordingly, the k-t data can be represented asmapping from the spatial- temporal image:

b ¼ Aρ ð33Þ

where A: = DF and D is a k-t downsampling pattern,and F now becomes a 2D Fourier transform. Inaddition to the temporal Fourier transform, the au-thors in [88] used the wavelet transform in the spatialdimension to exploit the spatial redundancy. Then,k-t SPARSE is formulated based on the followingoptimization problem:

minγ

b−DFγk k22 þ λ Wγk k1 ð34Þ

where W denotes the spatial wavelet transform,respectively.

K-t FOCUSSPreliminary CS dynamic MRI approaches [36, 88]were seemingly different from the classical k-t ap-proach such as k-t BLAST/SENSE [5]. One of themost impor- tant contributions of the k-t FOCUSS byJung et al. [18, 20] was to reveal that the compressedsensing dynamic MRI is not very different from theclassical k-t approaches, but rather it can be obtainedby a very simple modification of the classical k-tBLAST/SENSE to ensure significant performanceimprovement.

Ye BMC Biomedical Engineering (2019) 1:8 Page 8 of 17

More specifically, rather than using wavelet transform,k-t FOCUSS exploited the x-f domain sparsity. Then, astandard compressed sensing formulation would be:

minρ

b−DFρk k22 þ λ ρk k1 ð35Þ

to enforce the sparseness in x-f image ρ. However,there exists two additional novel- ties in the k-tFOCUSS. First, rather than directly enforcing the sparse-ness of the x-f image, the k-t FOCUSS further sparsifiesthe x-f image using the initial estimate.Specifically, let ρ0 be the predictable initial estimate of

ρ. Then, the residual

x ¼ ρ−ρ0 ð36Þshould be much more sparse than the original

spatio-temporal image. Second, rather than directlyusing the l1 minimization, k-t FOCUSS employed thereweighted norm approaches. This results in the follow-ing equivalent minimization problem:

minx;v

b−Aρ0−Ax�� ��2

2 þ λ12

Xn

i¼1

xi2

viþ vi

� ; ð37Þ

Then, the normal equations with respect to x and vare given by

−xi2

vi2þ 1 ¼ 0

−FH b−Aρ0−Ax �þ λV −1x ¼ 0

where V is a diagonal matrix whose i-th diagonal elem-ent is vi. This result in the following FOCUSS iteration:

ρ nþ1ð Þ ¼ ρ0þW nð ÞAH AHW nð ÞAþ λI� �−1

v−Aρ0 � ð38Þ

where the weighting matrix is given by

W nþ1ð Þ ¼ diag x nþ1ð Þ�� ��12� �:

One of the most powerful observations in [18, 20] wasthat the first iteration of (38) has very similar form tothe classical k-t BLAST/SENSE algorithm, except thepower factor of weighting matrix. This observation ledto an innovative idea to con- vert k-t BLAST/SENSE tocompressed sensing approach. More specifically, byusing incoherence sampling patterns, multiple iterationsand correct weighting factor for the diagonal matrix, theauthors of k-t FOCUSS [18, 20] clearly demonstratedthe performance improvement. This observation

suggested that the improvement by the classical k-t algo-rithms such as k-t BLAST/SENSE [5] was not from theBayesian perspective as the original authors of [5] hadclaimed, but indeed is originated from exploiting thesparsity int the spatio-temporal domain [18, 20]. Further-more, by simply modifying the weight factor and samplingpatterns, several additional iter- ation can significantly im-prove the performance of k-t BLAST/SENSE.Another powerful aspect of k-t FOCUSS was that the

idea can be easily extended to exploit the sparsity inother transform domains. For example, the residual stepin (36) can be interpreted as sparsity promoting step bysubtracting the temporal mean images. Thus, Jung et al.[20] proposes motion estimated and compensated modi-fication of k-t FOCUSS to make the residual signalmuch sparser. More specifi- cally, rather than subtract-ing the temporal mean values, they subtracted the mo-tion estimated frame. Note that motion estimation andcompensation (ME/MC) is an essential step in videocoding that uses motion vectors to exploit the temporalre- dundancies between frames [89, 90]. In order to em-ploy ME/MC within dynamic MRI, there are severaltechnical issues to address. First, at least one referenceframe is required. This issue can be easily resolved if weacquire fully sampled data in one frame as often done indynamic MRI. The main technical difficulty, however,comes from the existence of the low quality currentframe. Fortunately, this issue can be addressed using anadditional reconstruction step before the ME/MC [20].In addition, the spatio-temporal signals can be further

sparsified using data-driven transform:

ρ ¼ Dc;

where D denotes the learned temporal dictionary basedfrom the images, whereas c denotes the coefficients.Then, the imaging problem can be formulated as

minc;v;D

b−Aρ0−ADC

�� ��22 þ λ

12

Xn

i¼1

Ci2

viþ vi

� ; ð39Þ

For example, in order to find the temporal basis thatcan sparsify ρ, Jung et al. [18, 20] performed the the sin-gular value decomposition (SVD) after the image recon-struction, then the new dictionary is used to estimatethe new coefficients. This procedure is closely related tothe partial separable function (PSF) [91] and k-t SLR (k-tsparse and low-rank decomposition) [49], which will bereviewed soon.

Partially separable function (PSF) approachIn the PSF model by Liang et al. [91], the k-t sam-ples are assumed to be decomposed in the followingform

Ye BMC Biomedical Engineering (2019) 1:8 Page 9 of 17

b k; tð Þ ¼XLl¼1

Ψl kð Þ∅l tð Þ

for some data dependent spectral and temporal basis

function fΨlðkÞgLl¼1 and f∅lðtÞgLl¼1 . Thanks to the par-tially separable assumption, the so-called CasoratiMatrix B for the fully sampled k-t data given by

B ¼b k1; t1ð Þ ⋯ b k1; tnð Þ

⋮ ⋱ ⋮b km; t1ð Þ ⋯ b km; tnð Þ

24

35

has at most rank L.In dynamic CS MRI, many of the k-t samples are miss-

ing and we are interested in finding the missing compo-nents. Hence, by utilizing the low-rankness of theCa-sorati matrix, the missing k-t samples can be esti-mated using a low rank matrix completion algorithm. Inparticular, the authors in [92, 93] proposed the followingmatrix factorization approach:

U ; V � ¼ arg min

U;VA UVH �

−B�� ��2 ð40Þ

where A denotes the k-t sampling operator that indicatesthe missing samples by 0, and U ∈ Cm�L , V ∈ Cm�L isused for the low rank matrix factorization B =UV H,and the optimization is performed for U and V alternat-ingly by fixing the other matrix using the previous esti-mate. Because the exact rank of B is not known, theauthors proposed the incremented powerFactorization(IRFP) algorithm [92], where (40) starts with L = 1 by in-creasing order with the initialization of the powerfactorization of L + 1 from that of L. To avoid an overfit-ting, the algorithm is terminated as soon as the data fi-delity is below some threshold values.

K-t SLR: K-t sparse and low rank approachk-t Sparse and Low Rank Approach (k-t SLR) model byLingala et al. [49] is a more systematic way of learningboth basis and sparse coefficients. In this approach, thespatio-temporal signal ρ(x, t) is first rearranged in amatrix form

τ ¼ρ x1; t1ð Þ ⋯ ρ x1; tnð Þ

⋮ ⋱ ⋮ρ xm; t1ð Þ ⋯ ρ xm; tnð Þ

24

35 ð41Þ

Then, using the low rank prior, the optimisation prob-lem can be formulated as

minΓ

kb−AðΓÞk2 þ λφðΓÞ,

where A: = DF is now a downsampled Fourier transform.Here, the rank prior is approximated using the generalclass of Schatten p-functionals, specified by

φ Γð Þ ¼ Γk kpp ¼Xi

σpi ;

where {σi} denotes the singular values of Γ.In dynamic imaging applications, the images in the

time series may have sparse wavelet coefficients orsparse gradients. In addition, if the intensity profiles ofthe voxels are periodic (e.g., cardiac cine), they may besparse in the Fourier domain. Based on this observation,Lingala et al. [49] proposed additional sparsity inducingpenalty in specified basis sets along with the low-rankproperty to further improve the recovery rate. Specific-ally, they chose the 2-D wavelet transform to sparsifyeach of the images in the time series, while can be a 1-DFourier transform to exploit the pseudo-periodic natureof motion. Then, the resulting composite minimizationproblem can be formulated as

minΓ

b−A Γð Þk k2 þ λ1 Γk kpp þ λ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXq−1i¼0

ΦHi ΓΨi

�� ��2vuut

������������1

ð42Þ

K-space structured low-rank approachesBasic theoryCompared to the standard compressed sensing ap-proaches, k-space structured low- rank approaches suchas SAKE [55], LORAKS [56], ALOHA [57, 59, 60] andGIRAF [61] are relatively new, but has significant poten-tials in MRI imaging. These ap- proaches are all derivedby the k-space convolution relationship and can be usedfor both static and dynamic imaging. So we first discussa matrix representation of the convolution. For simpli-city, we will consider the 1-D notation.Specifically, consider a fully sampled k-space measure-

ment from the multi-channel coils:

yi ¼ Fγ i; i ¼ 1;⋯;C

where γi denotes the unknown i-th coil images and yi

corresponds to its k-space data. Note that the matrixrepresentation of a k-space convolution of yi with ad-tap filter h is given by

zi ¼ C yi �

h ð43Þ

where CðyiÞ denotes the convolution matrix constructedby the vector yi:

Ye BMC Biomedical Engineering (2019) 1:8 Page 10 of 17

C yi � ¼

⋮ ⋮ ⋱ ⋮yi −1½ � yi 0½ � ⋯ yi d−2½ �yi 0½ � yi 1½ � ⋯ yi d−1½ �yi 1½ � yi 2½ � ⋯ yi d½ �⋮ ⋮ ⋱ ⋮

yi n−d½ � yi n−d þ 1½ � ⋯ yi n−1½ �yi n−d þ 1½ � yi n−d þ 2½ � ⋯ yi n½ �

⋮ ⋮ ⋱ ⋮

266666666664

377777777775ð44Þ

and h¯ denotes a vector that reverses the order of theelements. If we extract the n − d-rows of the convolu-tion matrix with n − d > d, we can obtain the followingHankel structured matrix:

Hd yi � ¼

yi 0½ � yi 1½ � ⋯ yi d−1½ �yi 1½ � yi 2½ � ⋯ yi d½ �⋮ ⋮ ⋱ ⋮

yi n−d½ � yi n−d þ 1½ � ⋯ yi n−1½ �

2664

3775ð45Þ

By defining Y = [y1,⋯, yC], we can further defined theextended Hankel matrix

HdjC Yð Þh i;kð Þ ¼ Hd y1 �

⋯Hd yC �� � ð46Þ

In the following, we will explain how these Hankelstructured matrix has been utilised for accelerated MRI.

SAKEA calibrationless parallel imaging reconstruction method,termed simultaneous au- tocalibrating and k-space estima-tion (SAKE), is a data-driven, coil-by-coil recon- structionmethod that does not require a separate calibration stepfor estimating coil sensitivity information [55]. SAKE isbased on the following observation in GRAPPA:

XC

k≠iyk � w k;ið Þ ¼ yi ð47Þ

which implies that the i-th k-space measurement canbe represented as the linear combination of the filteredk-space data from other coils. In matrix form, this recur-sive relationship implies the existence of the null spaceof the Hankel matrix.

HdjC Yð Þh i;kð Þ ¼ Hd y1 �

⋯Hd yC �� �

h i;kð Þ ¼ 0 ð48ÞIn other word, Hd|C(Y) is low-ranked. Therefore,

SAKE solves the following low- rank matrix completionproblem to interpolate the missing k-space data:

minM

rank H Mð Þð Þ

subject to PΩ mi � ¼ PΩ yi

�; i ¼ 1;⋯;C;

where PΩ(·) denotes the projection on the measuredk-space samples on the index set Ω. The problem wassolved using the iterative singular value shrinkagemethod [94].

LORAKSThe low-rank modeling of local-space neighborhoods(LORAKS) [56] was inspired by the finite support condi-tion. More specifically, if the object γ has finite support,we can easily find the function w such that

γw ¼ 0;

This results in a convolution relationship in k-space

y � h ¼ 0⇒H yð Þh ¼ 0;

where y and h denote the Fourier spectrum of γ and w,respectively. Therefore, this gives a single channel ver-sion of the low-rank condition, which results in the fol-lowing rank minimization problem:

minm

rank H mð Þð Þ

subject to PΩ mð Þ ¼ PΩ yð Þ;

ALOHA and GIRAFAnnihilating filter-based low rank Hankel matrix(ALOHA) approach [57–60, 62] and GIRAF (Generic It-erative Reweighted Annihilating Filter) [61] can be con-sidered as the full generalization of SAKE and LORAKSfor general class of signals with the finite rate of innova-tions (FRI) for MR measurements. Moreover, the ap-proaches have unified the parallel imaging andcompressed sensing as a k- space interpolation with per-formance guarantees [62], and can be used for artifactcorrection [58]. This section describes the fundamentaldual relationship between transform domain sparsityand low rankness in reciprocal domain, which is the keyingredient. For better readability, we provide here a highlevel description by assuming 1-D signals.The Fourier CS problem of our interest is to recover

the unknown signal x(t) from the Fourier measurement:

x ωð Þ ¼ F x tð Þf g ¼Z

x tð Þe−iωtdt: ð49Þ

In classical Nyquist sampling, to avoid aliasing arte-facts, the grid size should be at most:

Ye BMC Biomedical Engineering (2019) 1:8 Page 11 of 17

Δ ¼ 2π=τ

when the support of the time domain signal x(t) is τ.Then, discrete Fourier data at the Nyquist rate is defined by:

x k½ � ¼ x ωð Þjω¼2πkτ: ð50Þ

We also define a length (r + 1)-annihilating filter hˆ[k]for xˆ[k] that satisfies

h � x� �

k½ � ¼Xr

p¼0h p½ �x k−p½ � ¼ 0; ∀k: ð51Þ

The existence of the minimum length finite length an-nihilating filter has been ex- tensively studied for FRIsignals [95–97]. Let r + 1 denotes the minimum size ofannihilating filters that annihilates discrete Fourier datax½k� . Then, a d-tap anni- hilating filter h with d > r + 1can be easily obtained by convolving an appropriate sizeFIR filter with the minimum length annihilating filter. Inmatrix form, this is equivalent to

Hd xð Þh ¼ 0 ð52Þwhere the Hankel structure matrix HdðxÞ is constructedas

Hd xð Þ ¼x 0½ � x 1½ � ⋯ x d−1½ �x 1½ � x 2½ � ⋯ x d½ �⋮ ⋮ ⋱ ⋮

x n−d½ � x n−d þ 1½ � ⋯ x n−1½ �

2664

3775 ð53Þ

Assume that min(n − d + 1, d) > r. Then, we can showthe following low rank property [62]:

RANKH xð Þ ¼ r; ð54ÞThanks to the low-rankness of the associated Hankel

matrix, the missing k-space data can be easily interpo-lated using the following low-rank matrix completion[62]:

minimizem∈ℂ n

RANK H mð Þsubject to PΩ mð Þ ¼ PΩ xð Þ; ð55Þ

where PΩ is the projection operator on the samplinglocation Ω. Moreover, as shown in [62], thelow-rank matrix completion approach (55) does notcompromise any optimality compared to the stand-ard Fourier CS.Note that signals may not be sparse in the image do-

main, but can be sparsified in a transform domain. Infact, this was the main idea of the compressed sensing.

Specifically, the signal x of our interest is a non-uniformspline that can be represented by:

Lx ¼ w ð56Þwhere L denotes a constant coefficient linear differentialequation that is often called the continuous domainwhitening operator in [98, 99]:

L≔aK∂K þ aK−1∂K−1 þ…þ a1∂þ a0 ð57Þand w is a continuous sparse innovation:

w tð Þ ¼Xr−1

j¼0c jδ t−t j

�: ð58Þ

For example, if the underlying signal is piecewise con-stant, we can set L as the first differentiation. In this case,x corresponds to the total variation signal model. Then,by taking the Fourier transform of (56), we have

z ωð Þ≔l ωð Þx ωð Þ ¼Xr−1

j¼0aje

−iωx j ð59Þ

where

l ωð Þ ¼ aK iωð ÞK þ aK−1 iωð ÞK−1 þ⋯þ a1 iωð Þ þ a0

ð60ÞAccordingly, the Hankel matrix HðzÞ from the weighted

spectrum z (ω) satisfies the following rank condition:

RANKH zð Þ ¼ r:

Thanks to the low-rankness, the missing Fourier datacan be interpolated using the following matrix comple-tion problem:

ðPwÞ minm∈Cn‖HðmÞ‖�subjecttoPΩðmÞ ¼ PΩðl⊙yÞ;ð61Þ

or, for noisy Fourier measurements y,

ðP′wÞ minm∈Cn‖HðmÞ‖�subjectto‖PΩðmÞ−PΩðl⊙yÞ‖≤δ

ð62Þ

where ⊙ denotes the Hadamard product, l and x denotes

the vectors composed of full samples of l (ω) and x(ω),respectively. After solving (Pw), the missing spec- traldata x (ω) can be obtained by dividing by the weight, i.e.

xðωÞ ¼ mðωÞ=lðωÞ assuming that lðωÞ≠0.

Ye BMC Biomedical Engineering (2019) 1:8 Page 12 of 17

The idea can be generalized for any transform domainsparse signals as long as the transform can be repre-sented using shift-invariant filters. Wavelet domainsparse signal belongs to this class. In this case, theweight kernel in the Fourier domain is obtained as thespectrum of the subband filters [57–60, 62]. For ex-ample, Fig. 3 showed the construction of the Hankelmatrix for the MR parameter mapping, where the

k-space weighting from wavelet weighting is applied onlyalong the phase encoding direction, whereas no weigh-ing is applied along the t-domain since the correspondtemporal spectrum is already sparse [59].The idea can be also easily generalised to the par-

allel imaging by exploiting (19). Specifically, (19) canbe equivalently represented using the matrixrepresentation:

Fig. 3 Construction of Hankel matrix for MR parameter mapping [59]

Fig. 4 Construction of Hankel matrix for multi-channel filter data for the case of MR parameter mapping [59]

Ye BMC Biomedical Engineering (2019) 1:8 Page 13 of 17

Hd γ i �

s j ¼ Hd γ j �si;∀i≠ j; ð63Þ

This implies that an exten^ded Hankel matrix HdjCð½γ1⋯γC �Þ in (46) is low ranked.For example, the multi-channel construction of Hankel

matrix for MR parameter mapping [59] is shown in Fig. 4.One of the most important advantages of the Hankel

matrix formulation is that the coil diversity can be read-ily exploited in addition to the image domain redun-dancy in a unified framework. This makes the separatecoil sensitivity estimation unnecessary.

Clinical applicationsSince the compressed sensing MRI allows significant accel-eration of MR acquisition, it has been extensively appliedfor various clinical applications such as fast cardiac MRI,whole heart MRI, dynamic contrast enhanced (DCE)-MRI,diffusion MRI, spectroscopic, etc., that usually require sig-nificant acquisition time using standard methods.For example, Otazo et al. [84] applied the CS method

to the first pass cardiac perfusion MRI and demon-strated feasibility of 8-fold acceleration in vivo imagingusing standard coil arrays. They showed that CS methodresults in similar temporal fidelity and image quality toGRAPPA with 2-fold acceleration [84]. Hsiao et al. [100]applied combined parallel imaging and compressed sens-ing to achieve 4D phase contrast for the quantificationof cardiac flow and ventricular volumes pediatric pa-tients during congenital heart MRI examinations.Vincent et al. [101] employed CS to evaluate LV functionand volumes and found that CS strategy with the singlebreath hold provided similar results to multi breathholdimaging protocols.For free-breathing contrast-enhanced multiphase liver

MRI, Chandarana et al. [102] showed that a combinationof compressed sensing, parallel imaging, and radialk-space sampling demonstrated the feasibility ofbreath-hold cartesian T1 weighted imaging. Espagnet etal. [103] employed golden-angle radial sparse paralleltechnique for DCE-MRI to evaluate the permeabilitycharacteristics of the pituitary gland.For diffusion MRI, Landman et al. [104] showed that

CS reconstruction using standard data can resolve cross-ing fibers similar to a standard q-ball approach usingmuch richer data with longer acquisition time. Kuhnt etal. [105] showed that High Angular Resolution DiffusionImaging (HARDI) + CS is a promising approach for fibertractography in clinical practice.Finally, for the spectroscopic imaging, Larson et al.

[106] developed a CS method for acquiring hyperpolar-ized 13C data using multiband excitation pulses andachieved 2 s temporal resolution with full volumetriccoverage of a mouse. Geethanath et al. [107] demonstrated

a potential reduction in acquisition time by up to 80% ormore for hydrogen 1 MR spectroscopic imaging using CS,with negligible loss of clinical information.With the commercially available CS reconstruction

methods, we expect to see more clinical applications ofCS in the near future.

ConclusionsNowdays, compressed sensing has become an maturetechnology, as reflected by recent approval by FDA.Major vendors have started to sell the compressed sens-ing reconstruction softwares, and many clinical re-searchers have been evaluating its clinical usefulness.Despite of this maturity, some of the main technical is-

sues of the compressed sensing are 1) the computationalcomplexity of the algorithm is relative high, and at high ac-celeration, image quality degradation is still reported. Al-though the recent state-of-the art CS techniques such asstructured Hankel matrix approach can address the qualitydegradation problems, it also increases the computationalcomplexity, which may interfere the clinical workflow.Fortunately, for the last two years, the MR image re-

construction field have been rapidly changed thanks tothe successful demonstration of of the deep learning-based MR reconstruction technologies [108–113]. Thesudden popularity of deep learning approaches can beattributed to the real-time reconstruction in spite of thesignificant improvement of the image quality. Thus,when originally presented, these techniques wereregarded as totally different technology that is nothingto do with the compressed sensing. However, recent the-oretical analysis [114] showed that the deep convolu-tional neural network is closely related to the Hankelmatrix decomposition. Therefore, we can still argue thatthe compressed sensing MRI has renewed interests inthe form of deep learning, and it will be interesting tosee how this exciting and rapidly evolving field will de-velop for the coming decades.

AbbreviationsALOHA: Annihliating filter-based low-rank Hankel matrix approach;CS: Compressed sensing; FOCUSS: FOCal Underdetermined System Solver;GRAPPA: Generalized Autocalibrating Partially Parallel Acquisitions; l1-SPiRIT: l1-iTerative Self-consistent Parallel Imaging Reconstruction;LORAKS: Low-rank modeling of local-space neighborhoods; MRI: Magneticresonance imaging; SAKE: Simultaneous autocalibrating and k-space estima-tion; SENSE: Sensitivity encoding

AcknowledgementsThe author would like to thank his students, Yoseob Han and Dongwook Lee,for providing figures. The author also like to thank two anonymous reviewerswho comments have significantly improve the quality of the paper.

FundingThis work is supported by National Research Foundation of Korea, Grantnumber NRF-2016R1A2B3008104.

Ye BMC Biomedical Engineering (2019) 1:8 Page 14 of 17

Availability of data and materialsData sharing not applicable to this article as no datasets were generated oranalysed during the current study.

Author’s contributionsThe author has read and approved this article.

Ethics approvalNot applicable.

Consent for publicationNot applicable

Competing interestsThe author declares that he has no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Received: 3 November 2018 Accepted: 4 February 2019

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