1

A Convergence Proof of Projected Fast IterativeSoft-thresholding Algorithm for Parallel Magnetic

Resonance ImagingXinlin Zhang, Hengfa Lu, Di Guo, Lijun Bao, Feng Huang, Xiaobo Qu*

Abstract—The boom of non-uniform sampling and compressedsensing techniques dramatically alleviates the prolonged dataacquisition problem of magnetic resonance imaging. Sparsereconstruction, thanks to its fast computation and promisingperformance, has attracted researchers to put numerous effortson it and has been adopted in commercial scanners. Algorithmsfor solving the sparse reconstruction models play an essential rolein sparse reconstruction. Being a simple and efficient algorithmfor sparse reconstruction, pFISTA has been successfully extendedto parallel imaging, however, its convergence criterion is still anopen question, confusing users on the setting of the parameterwhich assures the convergence of the algorithm. In this work, weprove the convergence of the parallel imaging version pFISTA.Specifically, the convergences of two well-known parallel imagingreconstruction models, SENSE and SPIRiT, solved by pFISTA areproved. Experiments on brain images demonstrate the validityof the convergence criterion. The convergence criterion proofedin this work can help users quickly obtain the satisfy parameterthat admits faithful results and fast convergence speeds.

Index Terms—Parallel imaging, image reconstruction, pFISTA,convergence analysis

I. INTRODUCTION

MAGNETIC resonance imaging (MRI) is a non-invade,non-radioactive and versatile technique serving as a

widely adopted and indispensable tool in medical diagnose,however, the slow imaging speed impedes its development.The advent of sparse sampling and compressed sensing (CS)theory [1]–[3] meets the eager demand of fast scan throughsampling only a small amount of data points and recoveringthe missing data using well-developed reconstruction methods.

One principal assumption of CS lies in the transform domainsparsity of images. The sparse representation adopted to enablethe image to be sparse plays a crucial role when designing re-construction approach. Sparse representation approaches could

This work was supported in part by National Key R&D Program ofChina (2017YFC0108703), National Natural Science Foundation of China(61571380, 61871341, 61811530021, U1632274, 61672335 and 61671399),Natural Science Foundation of Fujian Province of China (2018J06018),Fundamental Research Funds for the Central Universities (20720180056),Science and Technology Program of Xiamen (3502Z20183053), and ChinaScholarship Council (201806315010). *Corresponding author: Xiaobo Qu.

Xinlin Zhang, Hengfa Lu, Lijun Bao, Xiaobo Qu are with Department ofElectronic Science, Fujian Provincial Key Laboratory of Plasma and MagneticResonance, School of Electronic Science and Engineering, National ModelMicroelectronics College, Xiamen University, Xiamen 361005, China.

Di Guo is with School of Computer and Information Engineering, FujianProvincial University Key Laboratory of Internet of Things ApplicationTechnology, Xiamen University of Technology, Xiamen 361024, China.

Feng Huang is with Neusoft Medical System, Shanghai 200241, China.

be categorized into two main genres: orthogonal systems [3]–[5] and redundant systems [6]–[12]. The orthogonal systemsfavor theoretical analysis, fast algorithm design, and also timeand memory-efficient computing. However, it exhibits insuffi-ciency in sparsely representing diverse images. In contrast,designed to capture image-specific features, the redundantsystems allow sparser image representation than the orthogonalsystems do, thus eventually suggest better noise removal andartifacts suppression in applications.

Unlike the orthogonal system characterized by orthogonalbasis, the redundant system is described by frame, mostlytight frame [13], [14], thus leading to two distinct kinds ofreconstruction models, the synthetic model and the analysismodel. Many researchers focus on designing efficient tightframes for MRI reconstruction and promising reconstructionsare achieved [7], [9], [15], but at that time, few efforts hadfocused on MRI reconstruction models. Analysis model andsynthesis model have different prior assumptions, analysismodel is based on the assumption that the coefficients intransform domain of an MRI image are sparse, while synthesismodel assumes that an MRI image can be formulated as alinear combination of sparse coefficients. Even with the sameMRI data, sampling pattern, and sparse transform, the analysismodel is observed to yield improved reconstruction resultscompared to the synthesis model [14], [16]. And it has beenshown that the transition from synthetic model to analysismodel comes the balanced model. In the context of MRIreconstruction, Liu et al. empirically explored the performanceof the balanced model and observed that balanced model hasa comparable reconstruction performance with the analysismodel [16].

Analysis models, though enable better reconstructions withsmaller errors, still has a compelling demand for fast algo-rithms that allows favorable convergence speed and fewerparameters. The alternating direction methods of multipliers(ADMM) [17], [18] can solve both synthesis and analysismodels, however, it is vulnerable to parameter selections andis memory-demanding owing to the introduced dual variables.The iterative shrinkage threshold algorithms (ISTA) [19] andits acceleration version - fast ISTA (FISTA) [20] are efficientand robust, nevertheless, they are limited to solve the synthesismodel. Our group designed a projected iterative soft-thresholdalgorithm (pISTA) and its acceleration version - pFISTA [14],by rewriting the analysis model into an equivalent synthesis-like one and calculating the proximal map of non-smoothterms in the objective function approximately. Being essen-

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tially a variation of ISTA, pFISTA has only one adjustableparameter, and its convergence criterion has been provided inthe paper [14]. Also, Liu et al. [14] theoretically proved thatthe pFISTA converges to a balanced model.

The pFISTA permits lower reconstruction error compared toFISTA, and faster convergence speed than the state-of-the-artmethods, such as smoothing-based FISTA [14]. The pFISTA,however, is limited to tackle single-coil image reconstructionproblem. Ting et al. independently proposed a computationallyefficient balanced sparse reconstruction method in the contextof parallel MRI under tight frame [21], named bFISTA,and applied bFISTA to two widely adopted parallel imagingmodels, sensitivity encoding (SENSE) method [22] and iter-ative self-consistent parallel imaging reconstruction (SPIRiT)[23]. However, they did not provide proof of convergence ofbFISTA, that is to say, in practice there is no guidance abouthow to choose the parameter, thus, and the algorithm usersmay encounter a problem of choosing a proper parameterto produce faithful results, and we will demonstrate thisproblem in Fig. 2 (Section III-B). Considering the importanceof parallel imaging, it is necessary to give a clear mathematicalproof of its convergence to assist setting a proper algorithmparameter.

In this work, we prove the sufficient conditions for theconvergence of parallel imaging version pFISTA, and presentconvergence criteria and performances of applying pFISTAon solving two exemplars of parallel imaging reconstructionmethods - SENSE and SPIRiT. In addition, we discuss theresults of applying pFISTA on parallel reconstruction modelsunder different tight frames. Last, for the unique parameter ofpFISTA, we offer a recommended value to permit the fastestconvergence speed as well as promising results.

The rest of the paper is organized as follows. In SectionII, we introduce the notations. In Section III, we introducesome related works, firstly the pFISTA, and then SENSE andSPIRiT. In section IV, we prove that the parallel imagingversion pFISTA converges under proper selection of algorithmparameter. And we offer the convergence criteria of pFISTAwhen applied to tackle SENSE and SPIRiT models. In SectionV, we demonstrate the usefulness of the criteria we providedwith multiple parallel imaging brain images. Finally, conclu-sions will be drawn in Section VI.

II. NOTATIONS

We first introduce notations used throughout this paper. Wedenote vectors by bold lowercase letters and matrices by bolduppercase letters. The transpose and conjugate transpose of amatrix are denoted by XT and XH . For any vector x, ‖x‖1and ‖x‖2 denote the `1 and `2 norm for vectors, respectively.For a matrix X, ‖X‖2 denotes the `2 norm for matrix, whichis the largest singular value of matrix X and also the squareroot of the largest eigenvalue of the matrix XHX.

Operators are denoted by calligraphic letters. Let DMdenotes block diagonalization operator which places any Mmatrices of the same size, X1, · · · ,XM , along the diagonal

entris of a matrix with zeros:

DM (X1, · · · ,XM ) =

X1 0. . .

0 XM

. (1)

III. RELATED WORK

A. pFISTA for single-coil MRI reconstruction

An analysis model for single-coil sparse MRI reconstructioncould be formulated as

minxs

λ‖Ψxs‖1 +1

2‖ys −UFxs‖22 , (2)

where xs ∈ CN denotes the single-coil MR image datarearranged into a column vector, ys ∈ CM the undersampledk-space data, U ∈ RM×N (M � N) the undersamplingmatrix, and F ∈ CN×N the discrete Fourier transform. Ψ is atight frame, and the constant λ is the regularization parameterto balance the sparsity and data consistency.

To solve the problem (2), pFISTA rewrites the above-mentioned formula as a synthetic model as

minα∈Range(Ψ)

λ‖α‖1 +1

2‖ys −UFΨ∗α‖22 , (3)

where Ψ∗ denotes the adjoint of Ψ, and specifically satisfiesΨΨ∗ = I. α contains the coefficients of an image under therepresentation of a tight frame Ψ∗.

According to [14], the main iterations of pFISTA to solvethe problem in Eq. (3) are

x(k+1)s = Ψ∗Tγλ

(Ψ(x(k)s + γFHUT

(ys −UFx

(k)s

))),

t(k+1) =1 +

√1 + 4

(t(k))2

2,

x(k+1)s = x

(k+1)s +

tk − 1

tk+1

(x(k+1)s − x

(k)s

),

(4)

where Tγλ (·) is a point-wise soft-thresholding function de-fined as Tγλ (α) = max {|α| − γλ, 0} · α/|α|.

According to the Theorem 2 in the pFISTA paper [14],when the step size 0 < γ ≤ 1, the algorithm will converge.In addition, the larger γ is, the faster pFISTA converges.Therefore, γ = 1 is recommended in pFISTA to producepromising reconstruction with the fastest convergence speed.

B. pFISTA for multi-coil MRI reconstruction

According to [21], we can formula analysis models for theparallel MRI reconstruction problem into a unified form as

(pFISTA-parallel) mindλ‖Ψd‖1 + ‖y −Ad‖22 , (5)

where d represents the desired image to be recovered,y = [y1; y2; · · · ; yJ ] ∈ CMJ the undersampled multi-coilk-space data rearranged into a column vector, and yj ∈CM (j = 1, 2, · · · , J) is the undersampled k-space data vec-tor of jth coil, and A the undersampling matrix in parallelMRI containing the Fourier transform with multi-coil modu-lation and undersampling.

For parallel MRI reconstruction methods based on differentsignal properties, the explicit expressions of Eq. (5) would

3

vary. Two reconstruction algorithms based on SENSE andSPIRiT are discussed in [21], however, the convergence ofthese two algorithms has not been proven. Thus, in this work,we first prove the convergence of pFISTA of solving thegeneral parallel MRI reconstruction model, and then offer twoconcrete examples of multi-coils MRI analysis model, SENSEand SPIRiT, with convergence analysis. We first introduce howto tackle SENSE and SPIRiT using pFISTA.

Fig. 1. Parallel imaging reconstruction methods. (a) SENSE; (b) SPIRiT.Here � denotes Hadamard product.

1) pFISTA-SENSE: As shown in Fig. 1 (a), in SENSE [22],the image xj ∈ CN of the jth coil is represented as:

xj = Cjxc, j = 1, 2, ..., J, (6)

where xj and xc ∈ CN denote the jth coil image and thecomposite MRI image rearranged into a column vector, Cj ∈CN×N , (j = 1, 2, ..., J) is a diagonal matrix which containsthe sensitivity map of the jth coil.

The reconstruction problem based on SENSE can be for-mulated as:

(pFISTA-SENSE) minxc

λ‖Ψxc‖1 +1

2

∥∥∥y − UFCRxc

∥∥∥2

2, (7)

where U = DJ (U, · · · ,U), F = DJ (F, · · · ,F), C =DJ (C1, · · · ,CJ), and these three matrices are block diag-onal matrices. The matrix R = [I; · · · ; I] ∈ RNJ×N , hereI ∈ RN×N is an identity matrix. Here, the undersamplingmatrix A in Eq. (5) has its explicit expression as A = UFCR.

Using pFISTA, we can get the solution of Eq. (7) byiteratively solve the following problems:

x(k+1)c = Ψ∗Tγλ

(Ψ(x(k)c + γRTCH FHUT

(y − UFCRx

(k)c

))),

t(k+1) =1 +

√1 + 4

(t(k))2

2,

x(k+1)c = x

(k+1)c +

tk − 1

tk+1

(x(k+1)c − x

(k)c

).

(8)For simplicity, we call the pFISTA adopted to solve SENSE

model pFISTA-SENSE.2) pFISTA-SPIRiT: The SPIRiT [23] primarily bases on the

assumption that each k-space data point of a given coil is alinear combination of the multi-coil data of its neighboringk-space points, and the weights of linear combination areestimate from auto-calibration signal (ACS) (Fig. 1 (b)).Let x = [x1; x2; · · · ; xJ ] ∈ CNJ denote the multi-coilimage data rearranged into a column vector, where xj ∈CN , (j = 1, 2, · · · J) is the jth coil image vector, then thecalibration consistency in SPIRiT can be formulated as:

xj = [Gj,1,Gj,2, · · · ,Gj,J ] x, (9)

where Gj,i ∈ CN×N (i = 1, 2, · · · , J) is circulant matrixthat denotes the linear combination weights of ith coil for thedesired jth coil. Then, the l1-SPIRiT reconstruction can beformulated as:

(pFISTA-SPIRiT)

minxλ‖Ψx‖1 + 1

2

∥∥∥y − UFx∥∥∥22+ λ1

2

∥∥∥(G− I) Fx∥∥∥22,

(10)

where the matrix G ∈ CNJ×NJ is shown as below

G =

G1,1 G1,2 · · · G1,J

G2,1 G2,2 · · · G2,J

......

. . ....

GJ,1 GJ,2 · · · GJ,J

. (11)

Notice that strictly speaking, the Ψ in Eq. (10) should bewritten in the form of Ψ = DJ (Ψ, · · · ,Ψ) indicating thatthe Ψ is applied to each coil image. Here Ψ is still a tightframe which satisfies ΨΨ∗ = I, thus, we use only Ψ in therest of the paper for simplicity.

In order to express Eq. (10) in the unified form shown inEq. (5), we rewrite the Eq. (10) as:

(pFISTA-SPIRiT)

minxλ‖Ψx‖1 + 1

2

∥∥∥∥[ I0

]y −

[U

−√λ1 (G− I)

]Fx

∥∥∥∥22

,(12)

Here, the undersampling matrix A in Eq. (5) has its explicitexpression as A =

[U −

√λ1 (G− I)

]TF.

Using pFISTA, we can get the solution of Eq. (12) byiteratively solve the following problems:

x(k+1) = Ψ∗Tγλ

(Ψ(x(k) + γFH

(UH

(y − UFx(k)

)λ1(G− I)H(G− I)Fx

))),

t(k+1) =1 +

√1 + 4

(t(k))2

2,

x(k+1) = x(k+1) +t(k) − 1

t(k+1)

(x(k+1) − x(k)

).

(13)

4

It has been shown that pFISTA-parallel embraces fasterreconstruction speed than nonlinear conjugate gradient algo-rithm, rendering it having great potential in the clinic [21].However, the convergence analysis of pFISTA-parallel is stillan open problem. In other words, we don’t know explicitly inadvance which γ can guarantee the algorithm to converge. Liuet al. [14] have proved that under the condition γ = 1, pFISTAfor single-coil MRI reconstruction is guaranteed to converge.But if the same setting, γ = 1, is used in pFISTA-SENSEand pFISTA-SPIRiT, the algorithms may not converge (Fig.2). This is because the sensitivity map or convolution kernelwould affect the convergence property of pFISTA-parallel. Weobserved in experiments that a relatively large γ will leadto the divergence of pFISTA-parallel while a far smaller oneresults in the slow convergence of the algorithm (Fig. 2). Andthe range of γ allowing the algorithm to converge varies underdifferent tested data. Therefore, we aim to offer a explicit ruleabout how to choose a proper γ of pFISTA-parallel to hold afast convergence speed and promising results.

(b)(a)

Fig. 2. Empirical convergence of pFISTA-SENSE (a) and pFISTA-SPIRiT(b) with different γ. The reconstruction experiments were carried out on a32-coil brain image with 34% data acquired using a 1D Cartesian samplingpattern. The used data and the sampling pattern are presented in Fig. 3.

IV. CONVERGENCE ANALYSIS

In this section, we prove the convergence of pFISTA-parallel.

We present the analysis model of the parallel MRI recon-struction in a unified formula shown in Eq. (5) in which the un-dersampling matrix A has its explicit form A = UFCR if themodel is SENSE-based, and A =

[U −

√λ1 (G− I)

]TF

if the model is SPIRiT-based. According to [14], [20], let{d(k)

}be generated by pFISTA-parallel, and if the step size

satisfiesγ ≤ 1

L (γ), (14)

and Ψ is a tight frame, the sequence{α(k)

}={Ψd(k)

}converges to a solution of

minαλ‖α‖1+

1

2‖y −AΨ∗α‖22+

1

2γ‖(I−ΨΨ∗)α‖22 , (15)

with the speed

F(α(k)

)− F (α) ≤ 2

γ(k + 1)2

∥∥∥α(k) − α∥∥∥2, (16)

where α is a solution of (15) and F (·) is the objective functionin (15) and L is the Lipschitz constant for the gradient term.

Let B = ΨAHAΨ∗ − 1/γΨΨ∗, we have

L (γ) = maxi

(∣∣∣ei (B) + 1γ

∣∣∣) = maxi

{1γ,∣∣ei (AHA

)∣∣} , (17)

where ei (·) denotes the ith eigenvalue of matrix.Now we just have to analyze the largest eigenvalue of matrix

AHA in different reconstruction problems. In the following,we will explicitly discuss the convergence of pFISTA-SENSEand pFISTA-SPIRiT.

A. Convergence of pFISTA-SENSE

In this section, we provide the sufficient conditions for theconvergence of pFISTA-SENSE in the form of a theorem.

Theorem 1. Let{x(k)

}be generated by pFISTA-SENSE, and

if the step size satisfies

γ ≤ 1

c, c =

J∑j=1

‖Cj‖22, (18)

and Ψ is a tight frame, the sequence{α(k)

}={Ψd(k)

}converges to a solution of

minαλ‖α‖1 +

1

2

∥∥∥y − UFCRΨ∗α∥∥∥22

+1

2γ‖(I−ΨΨ∗)α‖22 .

(19)

Proof. In pFISTA-SENSE, we have A = UFCR, thus,

AHA = RTCHFHUT UFCR. (20)

Notice that U and F are block diagonal matrix, Eq. (20)can be rewritten as:

AHA = RTCH FHUT UFCR

=[

I · · · I]

CH1 FHUTUFC1 0

. . .0 CH

J FHUTUFCJ

I

...I

=

J∑j=1

CHj FHUTUFCj .

(21)

Let Q = FHUTUF,J∑j=1

CHj QCj is a Hermitian matrix.

For a Hermitian matrix, the largest eigenvalue is equal to the `2norm. In addition, notice that matrix `2 norm satisfies triangleinequality and consistency property [24], we can find the upper

bound of the largest eigenvalue of the matrixJ∑j=1

CHj QCj :

maxici

J∑j=1

CHj QCj

=

∥∥∥∥∥∥J∑j=1

CHj QCj

∥∥∥∥∥∥2

≤J∑j=1

∥∥CHj

∥∥2‖Q‖2‖Cj‖2.

(22)

Here, the matrix F is a unitary matrix, according to the unitaryinvariant of `2 norm, we have

‖Q‖2 =∥∥FHUTUF

∥∥2

=∥∥UTU

∥∥2, (23)

5

And UTU is a diagonal matrix with the diagonal elements 0or 1, indicating that

‖Q‖2 = 1. (24)

With Eq. (24), we can further simplify Eq. (22)

maxiei

J∑j=1

CHj QCj

≤ J∑j=1

∥∥CHj

∥∥2‖Q‖2‖Cj‖2

=

J∑j=1

∥∥CHj

∥∥2‖Cj‖2

=

J∑j=1

‖Cj‖22.

(25)

Let c =J∑j=1

‖Cj‖22, we have

L (γ) = maxi

{1γ,

∣∣∣∣∣ei(

J∑j=1

CHj QCj

)∣∣∣∣∣}

= 1γ, 0 < γ ≤ 1

c,

L (γ) = maxi

{1γ,

∣∣∣∣∣ei(

J∑j=1

CHj QCj

)∣∣∣∣∣}

= c, γ > 1c.

(26)

The Eq. (26) means that, when 0 < γ ≤ 1/c, one hasL (γ) = 1/γ, which satisfies the convergence condition ofpFISTA in Eq. (14); whereas when γ > 1/c, then L (γ) =c > 1/γ, which does not satisfy the convergence condition ofpFISTA. In summary, when 0 < γ ≤ 1/c, the pFISTA-SENSEis guaranteed to converge.

B. Convergence of pFISTA-SPIRiT

In this section, we provide the sufficient conditions for theconvergence of pFISTA-SPIRiT in the form of a theorem.

Theorem 2. Let{x(k)

}be generated by pFISTA-SPIRiT, and

if the step size satisfies

γ ≤ 1c ,

c = 1 + λ1

(J∑

m=1

J∑n=1‖Gm,n‖2 + 1

)2

,(27)

and Ψ is a tight frame, the sequence{α(k)

}={Ψd(k)

}converges to a solution of

minαλ‖α‖1 +

1

2γ‖(I−ΨΨ∗)α‖22

+1

2

∥∥∥∥y − [ U−√λ1 (G− I)

]FΨ∗α

∥∥∥∥22

.

(28)

Proof. In pFISTA-SPIRiT, we have

AHA = FH[

UT −√λ1(G− I)H

] [ U−√λ1 (G− I)

]F

= FH(UT U + λ1(G− I)H (G− I)

)F.

(29)

Since the F is a unitary matrix, then the matrix(UT U + λ1(G− I)

H(G− I)

)and the matrix AHA is

unitary similar. And(UT U + λ1(G− I)

H(G− I)

)is a

Hermitian matrix, thus, the eigenvalue of AHA and

(c)

(a) (b)

(d)

Fig. 3. Experimental dataset. (a-c) Three different brain images; (d) theCartesian sampling pattern of sampling rate 0.34.

(UT U + λ1(G− I)

H(G− I)

)are the same. Now, we anal-

ysis the eigenvalue of matrix UT U + λ1(G− I)H

(G− I).

Indeed, once the kernel has been estimated using ACS, thematrix G is determined, so that the largest eigenvalue can becalculated. However, the matrices U and G are too large tocalculate the eigenvalue conveniently, for example, as for a4-coil 256 × 256 image, the size of corresponding U and Greaches the size of 262144× 262144. Therefore, we relax thebound so as we could efficiently calculate it.

Notice that matrix UT U + λ1(G− I)H

(G− I) is Hermi-tian matrix and with the linearity and triangle inequality ofmatrix norm [24], so that

ei

(UT U + λ1(G− I)

H(G− I)

)=∥∥∥UT U + λ1(G− I)

H(G− I)

∥∥∥2

≤∥∥∥UT U

∥∥∥2

+ λ1

∥∥∥(G− I)H

(G− I)∥∥∥2

=∥∥∥UT U

∥∥∥2

+ λ1 ‖(G− I)‖22

≤∥∥∥UT U

∥∥∥2

+ λ1(‖G‖2 + ‖I‖2)2.

(30)

As mentioned, the matrix G is a block circulant matrix asshown below

G =

G1,1 G1,2 · · · G1,J

G2,1 G2,2 · · · G2,J

......

. . ....

GJ,1 GJ,2 · · · GJ,J

,

where Gm,n (m,n = 1, 2, · · · , J) is circulant matrix. Thus,now, the eigenvalue can easily be obtained by linear combi-nation weights.

6

(d)

Iteration=50 Iteration=300 Iteration=800Iteration=100

0.1/c

γ=

0.01/c

γ=

1/c

γ=

(a) (b) (c)

Fig. 4. Reconstructions of three different brain images by pFISTA-SENSE with different step size γ. The 8, 12 and 32 -coil data shown in Fig. 3 were usedin (a), (b) and (c), respectively. (d) is the reconstructions of pFISTA-SENSE at different iteration in 12-coil data. All experiments used the same samplingpattern depicted in Fig. 3.

Therefore,

ei

(UT U + λ1(G− I)

H(G− I)

)≤∥∥∥UT U

∥∥∥2

+ λ1(‖G‖2 + ‖I‖2)2

≤∥∥∥UT U

∥∥∥2

+ λ1

(J∑

m=1

J∑n=1

‖Gm,n‖2 + ‖I‖2

)2

= 1 + λ1

(J∑

m=1

J∑n=1

‖Gm,n‖2 + 1

)2

.

(31)

Let c = 1 + λ1

(J∑

m=1

J∑n=1‖Gm,n‖2 + 1

)2

we have

L (γ) = maxi

{1

γ,∣∣∣ei (UT U + λ1(G− I)H (G− I)

)∣∣∣}=

1

γ, 0 < γ ≤

1

c,

L (γ) = maxi

{1

γ,∣∣∣ei (UT U + λ1(G− I)H (G− I)

)∣∣∣}= c, γ >

1

c.

(32)

The Eq. (32) means that, when 0 < γ ≤ 1/c, one hasL (γ) = 1/γ, which satisfies the convergence condition ofpFISTA; whereas when γ > 1/c, then L (γ) = c > 1/γ, whichdoes not satisfy the convergence condition of pFISTA. Insummary, when 0 < γ ≤ 1/c, pFISTA-SPIRiT is guaranteedto converge.

V. EXPERIMENTAL RESULTS

In this section, we first conducted experiments on multi-coils MRI brain images to assess the validity of the con-vergence criteria we derived. And then we compared thereconstructions of pFISTA-parallel and the widely adoptedalgorithm - ADMM [14]. The ADMM softwares to solveSENSE and SPIRiT analysis reconstruction models were im-plemented by ourselves. Last, we discussed the convergenceand results under other tight frames with different γ.

Three datasets are used in experiments. The first braindataset shown in Fig. 3 (a) was acquired from a healthyvolunteer on a 1.5T Philips MRI scanner equipped with an8-coil head coil using the 2D T1-weighted fast-field-echosequence (matrix size = 256 × 256, TR/TE = 1700 ms/390

7

(d)

Iteration=200 Iteration=400 Iteration=800 Iteration=1600 Iteration=2600

0.1/c

γ=

0.01/c

γ=

1/c

γ=

(a) (b) (c)

Fig. 5. Reconstructions of three different brain images by pFISTA-SPIRiT with different step size γ. The 8, 12 and 32 -coil data shown in Fig. 3 were usedin (a), (b) and (c), respectively. (d) is the reconstructions of pFISTA-SPIRiT at different iteration in 8-coil data. All experiments used the same samplingpattern depicted in Fig. 3.

ms, FOV = 230 mm×230 mm, slice thickness = 5 mm). Thesecond brain dataset depicted in Fig. 3 (b) were acquired froma 3T GE MRI scanner equipped with a 12-coil head coilusing the 2D T1- weighted SPGR sequences (matrix size =256 × 256, TR/TE = 400 ms/9 ms, FOV = 240 mm×240mm, slice thickness = 6 mm). The third brain dataset depictedin Fig. 3 (c) was acquired from a healthy volunteer usingthe 2D T2-weighted turbo spin echo sequence (matrix size =256 × 256, TR/TE = 6100 ms/99 ms, field of view = 220mm×220 mm, slice thickness = 3mm). It was obtained froma 3T SIEMENS Trio whole-body scanner equipped with a 32-coil head coil.

Relative `2 norm error (RLNE) is adopted as objectivecriteria to quantify the reconstruction performance. The RLNEis defined as

RLNE =‖xref − xrec‖2‖xref‖2

, (33)

where xref denotes the vectorized reference image that is asquare root of sum of squares (SSOS) of the fully sampledimage and xrec the vectorized reconstructed image that is

the SSOS image of pFISTA-SPIRiT reconstructed image andmodular image of pFISTA-SENSE reconstructed image. Weshould point out that a lower RLNE, a higher consistencybetween the reference image and the reconstructed image.

For SENSE, the fully sampled 256×64 area of the k-spacecenter is used to calculate sensitivity map, and for SPIRiT,the fully sampled 256×22 area is used to estimate linearcombination weights. The shift-invariant discrete waveletstransform (SIDWT) [7], [25], [26], if not mentioned otherwise,is adopted as the tight frame in experiments. In all experimentsinvolving SIDWT, Daubechies wavelets with 4 decompositionlevels are utilized. For pFISTA-SENSE, λ = 10−3 is set andfor pFISTA-SPIRiT, we set λ = 10−3 and λ1 = 1, and 5× 5SPIRiT kernel is used. All computation procedures run on aCentOS 7 computation server with two Intel Xeon CPUs of3.5 GHz and 112 GB RAM.

A. Main Results

As mentioned above, once if the parameter meets the con-dition 0 < γ ≤ 1/c, both the pFISTA-SENSE and pFISATA-SPIRiT converge. Thus, here we perform reconstructions by

8

pFISTA-SENSE and pFISTA-SPIRiT with various γ in therecommended range, respectively, to verify if the recom-mended γ could enable the convergence of the algorithm.

As shown in Fig. 4, for three tested brain images, pFISTA-SENSE converges when using the γ ranged from 0.01/c to1/c. And all γ used eventually allow the RLNEs decreasing toa comparatively low level. More importantly, the larger the γ,the faster the algorithm converges, this observation is consis-tent with the Eq. (16). The intermediate reconstructed imagesmanifest the convergence speeds of pFISTA-SENSE withvarious γ. The undersampling artifacts were quickly removedwithin 100 iterations when with parameter γ = 1/c and thealgorithm produced a nice image (Fig. 4 (d)). As γ decreased,the algorithm took more time to converge to the final stageyielding satisfying results, for instance, when γ = 0.01/c,the program cost about 800 iterations to eventually eliminatethe undersampling artifacts (Fig. 4 (d)). All recommended γenable promising results but with different convergence rate.Importantly, the algorithm reaches the fastest convergencespeed when γ = 1/c, thus we recommend γ = 1/c whencarrying out pFISTA-SENSE experiments. It is also worth topoint out that the number of coils of parallel imaging makes noinfluence on the convergence of pFISTA-SENSE with offeredrange 0 < γ ≤ 1/c. In a word, the convergence criteriawe provided can escort pFISTA-SENSE to achieve satisfyingresults of different coils parallel imaging experiments.

In addition, we observe similar phenomenon on pFISTA-SPIRiT experiments (Fig. 5). With 0 < γ ≤ 1/c , pFISTA-SPIRiT empirically convergences to a level of promising lowRLNE. The larger the γ, the faster the algorithm converges,and the fastest convergence speed is achieved when γ = 1/c,thus we also recommend γ = 1/c for pFISTA-SPIRiT ex-periments. The intermediate results of pFISTA-SPIRiT withmonotonically decreasing γ also reveal the increasing conver-gence rate as γ rises (Fig. 5 (d)). The pFISTA-SPIRiT couldbe applied in multi-coils imaging experiments with guaranteedconvergence if 0 < γ ≤ 1/c.

We observed in our experiments that pFISTA-parallel allowsas fast convergence speed as the fastest ADMM offers and alsoclose reconstruction error of ADMM (Fig. 6). This indicatesthat the relaxation of the convergence criterion of pFISTA-SPIRiT is reasonable as it enables comparable results as theADMM provides with the best hand-crafted optimal penaltyparameter β. As shown in Fig. 6, the reconstruction of the8-coil T1-weighted brain image, the convergence of ADMMis sensitive to the parameter β selection, relatively larger orsmaller β would result in noticeable discrepancy (Figs. 6 (a-b)). And β = 0.01 yields the fastest convergence speed ofADMM. Importantly, pFISTA-parallel with the recommendedparameter γ = 1/c very close reconstruction error as well asreconstructed images (Figs. 6 (c-j)).

B. Discussion on other tight frames

Tight frame is crucial for sparse MRI reconstruction. Inthis section, we conduct experiments using pFISTA-SENSEand pFISTA-SPIRiT with four other tight frames, contourlet[15], [27], shearlet [28], patch based directional wavelets

(c) (d)

(g) (h)

(e) (f)

(i) (j)

(a) (b)

Fig. 6. Reconstruction results of ADMM and pFISTA. (a) is the conver-gences of ADMM and pFISTA under SENSE-based reconstruction; (b) is theconvergences of ADMM and pFISTA under SPIRiT-based reconstruction; (c-d) are images of reconstruction results by ADMM (β = 0.01) and pFISTAunder SENSE-based reconstruction; (e-f) are images of reconstruction resultsby ADMM (β = 0.01) and pFISTA under SPIRiT-based reconstruction; (h-j)are the reconstruction error distribution (10x) corresponding to reconstructedimage above them. Note: 8-coil image in Fig. 3 (a) and sampling pattern inFig. 3 (d) are adopted in all experiments.

(PBDW) [9], and PBDW in SIDWT domain (PBDWS) [29].By exploring the image geometry, contourlet provides a sparseexpansion for images that have smooth contours [27], andapplied to preserve smooth edges on MRI reconstruction [15].Shearlet combines ability to capture the geometric featuresof data with the power of multiscale methods, and thereforeprovides improvements compared to the contourlet transform[28]. PBDW trains the geometric directions on the pixels ofimage patches and provides an adaptively sparse representationfor image [9]. PBDWS extend the PBDW into SIDWT domainto enhance the ability of sparsifying [29]. Here, the filters usedin contourlet are ladder structure filters and the decompositionlevels are [5,4,4,3], the filters used in shearlet are Meyer filters[30] and the decomposition level used in shearlet is 4, and thefilters used in PBDW and PBDWS are the Haar wavelets andthe decomposition level is 3.

The results shown in Fig. 7 reveal that the RLNEs ofpFISTA-SENSE with PBDW and PBDWs tight frames aresmaller than that under SIDWT and shearlet tight frames. Theresult is reasonable since PBDW and PBDWS are tight framestrained using pre-reconstructed image, admitting better sparserepresentation of the image. Notably, parameter γ = 1/cenables the fastest convergence speed though the tight framevaries. Those results indicates that the tight frames usedwill not affect the convergence of both pFISTA-SENSE andpFISTA-SPIRiT.

9

(f) (h)

(a) (b) (c) (d)

(e) (g)

Fig. 7. Empirical convergence using four other tight frames. (a-d) are the RLNEs of pFISTA-SENSE with different γ using contourlet, shearlet, PBDW, andPBDWS, respectively; and (e-h) are the RLNEs of pFISTA-SPIRiT with different γ using contourlet, shearlet, PBDW, and PBDWS, respectively. Note: 8-coilimage in Fig. 3 (a) and the Cartesian sampling pattern with sampling rate of 0.25 are adopted in all experiments.

VI. CONCLUSION

As a simple and fast algorithm to solve sparse reconstructionmodel, pFISTA has been successfully extended to solve paral-lel imaging problems, but its convergence criterion needs to beproved to help quickly and conveniently choose a satisfyingparameter. In this work, we prove the sufficient conditionsfor the convergence of parallel imaging version pFISTA forsolving spare reconstruction models. More explicitly, we offera bound served as guidance about how to choose the pFISTAparameter for solving both SENSE and SPIRiT. Experimentalresults evince the validity and effectiveness of the convergencecriterion. This work is useful to help user quickly choose aproper parameter to obtain faithful results and fast convergencespeed, and to promote the application of sparse reconstruction.

ACKNOWLEDGMENTS

Authors appreciate the help of Yunsong Liu for revisingthe manuscript. Xiaobo Qu is grateful to Prof. Chun Yuan forhosting his visit at University of Washington.

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