An algorithm for sparse MRI reconstruction by Schatten p-norm minimization

Available online at www.sciencedirect.com

Magnetic Resonance Imaging 29 (2011) 408–417

An algorithm for sparse MRI reconstruction by Schattenp-norm minimization

Angshul Majumdar⁎, Rabab K. WardDepartment of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC, Canada

Received 15 July 2010; revised 3 September 2010; accepted 4 September 2010

Abstract

In recent years, there has been a concerted effort to reduce the MR scan time. Signal processing research aims at reducing the scan time byacquiring less K-space data. The image is reconstructed from the subsampled K-space data by employing compressed sensing (CS)-basedreconstruction techniques. In this article, we propose an alternative approach to CS-based reconstruction. The proposed approach exploits therank deficiency of the MR images to reconstruct the image. This requires minimizing the rank of the image matrix subject to data constraints,which is unfortunately a nondeterministic polynomial time (NP) hard problem. Therefore we propose to replace the NP hard rankminimization problem by its nonconvex surrogate — Schatten p-norm minimization. The same approach can be used for denoising MRimages as well.

Since there is no algorithm to solve the Schatten p-norm minimization problem, we derive an efficient first-order algorithm. Experimentson MR brain scans show that the reconstruction and denoising accuracy from our method is at par with that of CS-based methods. Ourproposed method is considerably faster than CS-based methods.© 2011 Elsevier Inc. All rights reserved.

Keywords: Schatten p-norm minimization; Compressed sensing-based methods; K-space

1. Introduction

Magnetic resonance imaging (MRI) is a comparativelyslow imaging modality. Speeding up the data acquisitiontime has always been of interest to the MRI researchcommunity. Until recently, most of the effort in decreasingthe MR scan (data acquisition) time had been dedicated toimproving the hardware of the scanner. Once the full K-space data are acquired, reconstructing the image is almosttrivial — applying 2D inverse Fourier transform. In recentyears, signal processing researchers have shown that it ispossible to reduce the scan time by acquiring less K-spacedata followed by a nonlinear reconstruction technique basedon the recent findings of compressed sensing [1–7].Compressed sensing (CS)-based methods reconstruct theimage by framing a nonlinear optimization problem that

⁎ Corresponding author. Department of Electrical and ComputerEngineering, University of British Columbia, Kaiser 20102332 MainMall, Vancouver, BC, Canada V6T1Z4.

E-mail address: [email protected] (A. Majumdar).

0730-725X/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.mri.2010.09.001

exploits the sparsity of the MR image in a transform domainsuch as wavelet, contourlet or total variation.

Our work has the same interest in mind— reconstructingthe MR image from subsampled K-space data. But instead ofapplying CS-type methods, we will show that similarreconstruction accuracy can be achieved by exploiting therank deficiency of the image. The way we formulate thereconstruction problem requires solving a nonlinear optimi-zation problem that minimizes the Schatten p-norm (0bp≤1)[8] of the image. This work does not compete against CS-based MR reconstruction techniques; rather, it proposes analternate approach. Our reconstruction accuracy is similar tostandard CS-based methods, but takes considerably less time.

Rank deficiency can also be exploited to denoise MRimages from legacy scanners. Current MR scanners samplethe full K-space and reconstruct the image by applying aninverse Fourier transform. Since the K-space data is itselfcorrupted by noise, the reconstructed image is noisy as well.Such images require an additional denoising step. The MRimage is assumed to be rank deficient (has a small number ofhigh singular values); however, the noise is not. The noise isfull rank but has very small singular values. Thus it is possible

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409A. Majumdar, R.K. Ward / Magnetic Resonance Imaging 29 (2011) 408–417

to denoise the image by properly thresholding the singularvalues of the noisy image. However, instead of applyingarbitrary thresholding to the singular values noisy image, wepropose to denoise it through Schatten p-norm minimization.

The rest of the article is organized into several sections.The following section discusses the CS-based MR imagereconstruction. Section 3 formulates the reconstructionproblem as one of Schatten p-norm minimization. In Section4, the formulation behind the denoising problem is provided.The algorithmic development for solving the minimizationproblem is described in Section 5. The experimentalevaluation is performed in Section 6. Finally, in Section 7,the conclusions of the work are discussed.

2. CS-based reconstruction from subsampledK-space measurements

MR images are spatially redundant, i.e., a pixel value at aparticular location is highly dependent on the values ofneighboring locations except at discontinuities along theedges. Since the images are spatially redundant, they arecompressible, i.e., one does not require storing all the pixels,rather one can only store a few wavelet/DCT/singular valuecoefficients from which the image can be easily generated.

The spatial redundancy of MR images is well captured bydifferent types of transforms (wavelets/DCT). The transformcoefficients of an MR image are approximately sparse, i.e., ifthe image is of size N×N, there are only a few (much less thanN2) high-valued wavelet coefficients, while the rest are eitherzeroes or negligibly small. Therefore one only needs to storethe few high-valued coefficients. During reconstruction, theinverse transform is applied to the stored coefficients togenerate the image.

Long before the CS framework was established,transform domain coding had been employed for medicalimage compression [9–14]. What needs to be noticed fromthese studies is that the total information of the image canbe captured in a few high-valued coefficients in thetransform domain. This forms the basis for transformdomain compression. This is also the basis for CS-basedimage reconstruction.

The MR image reconstruction problem from subsampledK-space can be expressed as follows:

ym�1 = Fm�nxn�1 + gm�1; n = N 2 and mbn ð1Þ

where x is the image to be reconstructed, y is the K-spacedata, F is a subsampled Fourier operator and η is the noiseassumed to be normally distributed. The image is of sizeN×N and n is the total number of pixels in the image. Thenumber of K-space samples (m) is less than the total numberof pixels in the image. The problem is to estimate x.

Since (1) is an underdetermined system of equations andx is dense, it is impossible to find a reasonable solution in itscurrent form. One can incorporate the transform domain

information into (1). Most transforms employed in MRimage reconstruction are orthogonal, i.e.,

SST = I = STS; where I is the identity and S the

transform matrix ð2Þ

The analysis and synthesis equations connecting theimage to the transform coefficients are as follows:

Analysis: a = SxSynthesis: x = STa

ð3Þ

The transform domain representation α is approximatelys-sparse, i.e., it has s high-valued nonzero coefficientswhile the rest are all zeroes or near about zero. Theinformation content in the small coefficients is small and ifwe neglect these small coefficients (i.e., treat them as zeroes)the reconstruction is not hampered much.

Moreover, the small-valued coefficients are virtuallyindistinguishable from noise. Any attempt to reconstructthese small-valued coefficients yields a noisy estimate ofthe transform coefficients. Therefore it is pragmatic to findonly the s high valued transform coefficients in α and treatthe rest as noise. Once the transform coefficients (α) areobtained the image can be constructed by applying thesynthesis equation.

In order to frame the image reconstruction problem in theCS framework, Eq. (1) is represented in the transformdomain as follows,

y = FSTa + g ð4Þ

We omit the dimensionalities to keep the expressionuncluttered.

The interest is in recovering the s coefficients of α (out ofthe possible n). To specify an s-sparse vector, only 2s piecesof information are required — s positions of the coefficientsin α and the corresponding s values. Thus the n-lengthtransform coefficient has only 2s degrees of freedom inreality. To solve 2s unknowns, in principle one only needsthe same number of equations. Therefore if the number of K-space samples is m≥2s, then that would be enough to solve(4). Mathematically, one can write the reconstructionproblem as follows,

minOaO0 subject toOy − FSTaO22Ve ð5Þ

where the l0 norm is just the number of nonzeroes in thevector and ɛ is the noise variance.

Solving (5) is NP hard and there is no tractable algorithmto solve it in polynomial time [15]. Thus solving (5) is notpractical. One is interested in getting the MR image with aslittle K-space data as possible (faster scan time), but on theother hand it is very important to reconstruct the MR imageas fast as possible as implied earlier.

A seminal work in CS [16] shows that it is possible torelax the NP hard l0 norm to its closest convex surrogate, the

410 A. Majumdar, R.K. Ward / Magnetic Resonance Imaging 29 (2011) 408–417

l1 norm, and still be guaranteed to obtain the correct sparsesolution. Therefore instead of (5) one can solve,

minOaO1 subject to Oy − FSTaO22Ve ð6Þ

where l1 norm is the sum of the absolute values in the vector.Solving (6) is a quadratic programming problem and over

the recent years a large number of fast solvers have beendeveloped to solve it. However, the number of samplesneeded to reconstruct the vector is increased by a logarithmicfactor, i.e., one now requires m≥Cslogn K-space samples.

The l0 norm is NP hard to solve but requires the leastnumber of K-space samples, while the l1 norm is convex andcan be solved efficiently but requires a large number ofsamples. Somewhere between the two extremes lies the lpnorm (0bpb1) minimization problem. The nonconvexsurrogate (lp norm) better approximated the NP hard l0norm and therefore yields better results, i.e., requires lessnumber of samples. Moreover, the lp norm can be solved asefficiently as its convex counterpart (l1 norm).

The nonconvex lp-norm minimization problem for MRimage reconstruction is as follows:

minOaOpp subject toOy − FSTaO2

2Ve; 0bpb1 ð7Þwhere OaOp

p =Piapi .

It has been shown in Ref. [17] that the number of samplesrequired by (7) to solve the reconstruction problem is,

mzC1s + pC2s logn ð8ÞOne can easily see that as the value of p diminishes, thesecond term gets negligible and the number of measurementsincreases almost linearly with the sparsity of the vector.

Minimizing the lp norm being a nonconvex problem isalways fraught with the danger of being stuck in a localminima. Even though this possibility exists, nonconvexalgorithms have been successfully used in the past for MRIreconstruction [18–20].

3. MR image reconstruction as a Schatten p-normminimization problem

From the discussion in Section 2, we understand thatwhen the number of K-space samples is considerably morethan the number of degrees of freedom of the solution, it isfeasible to solve the reconstruction problem. Transformdomain representation is not the only way the informationcontent of the image is compactly captured. In many cases,singular value decomposition (SVD) can also efficientlyrepresent the information content of the MR image.

Owing to the spatial redundancy, MR images do nothave full rank, i.e., the rank of an image of size N×N isrbN. For an image of rank r, the number of degrees offreedom is r(2N−r). Thus when r=N, the number of degreesof freedom is much less than the total number of pixels N2.This idea has been successfully used in recent studies to

propose SVD-based compression schemes for images [21]and [22]. CS exploits sparsity in the transform domain toreconstruct the MR image from subsampled K-space data;our approach will exploit the low-rank property of themedical images for reconstruction.

An early work [23] proposed an SVD-based method forMR image reconstruction from undersampled K-space data.SVD was used as an interpolation method for estimating themissing K-space samples from multiple coils. The imagewas obtained from the interpolated K-space data by inverseFourier transform. Our technique towards MR imagereconstruction is completely different from Ref. [23] eventhough both exploit the rank deficiency of the data (image/K-space samples).

For our approach, we represent (1) in a slightlydifferent fashion,

ym�1 = Fop XN �Nð Þ + gm�1 ð9Þ

where X is the image in matrix form, Fop is a mapping fromthe image domain to the Fourier domain (K-space) and η isthe noise.

Here Fop is the Fourier operator applied to the wholeimage X. Since the Fourier transform is separable, it can beequivalently expressed as,

y = FT1D � F1D

� �x + g; where xn�1 = vec XN �Nð Þand F

= FT1D � F1D ð10Þ

where ⊗ represents the Kronecker product.This expression (10) is the same as (1). The Fourier

operator mentioned in (1) is actually a Kronecker product ofthe 1D Fourier operators as shown in (10).

Our work hinges on the assumption that the MR image isrank deficient. The most logical step is to reconstruct theimage X such that it has the minimum rank. This hints at thefollowing optimization problem:

min rank Xð Þsubject to Oy − FxO2

2Ve; F = FT1D � F1D and x = vec Xð Þ

ð11Þ

Unfortunately, (11) is an NP hard problem. The sameproblem arises in CS where the problem is to minimize the l0norm of the sparse coefficient vector. In CS, the NP hardproblem is bypassed by replacing the l0 norm by its nearestconvex surrogate — the l1 norm. Similarly for (11), it isintuitive to replace the NP hard objective function of rankminimization by its tightest convex relaxation— the nuclearnorm. Therefore, instead of (11) we solve the following,

minOXO4

subject to Oy − FxO22Ve

ð12Þ

where ||X||⁎ is the nuclear norm of the image X. It is definedas the sum of its singular values.


The fact that the rank of the matrix can be replaced by itsnuclear norm for optimization purpose has been justifiedtheoretically as well. Recent theoretical studies by Rechtet al. [24–26] and others [27] show the equivalence of (11)and (12). We refrain from theoretical discussion on thissubject and ask the interested reader to peruse Refs. [24–27].

A problem closely related to ours is the matrix completionproblem, where the objective function is the same (rank ofthe matrix) but the constraint is a subsampling operatorinstead of a projection. The matrix completion problem isdealt with using the same way, i.e., the NP hard minimizationproblem is replaced by the convex nuclear norm minimiza-tion problem. Since matrix completion is not the topic of thisarticle, we do not discuss it further. The interested reader isencouraged to refer to Ref. [28].

Our work is motivated by the efficacy of the lp-normminimization in nonconvex CS. The actual problem in CS isto solve (5) which is the l0-norm minimization problem. Thisproblem being NP hard is replaced by its closest convexsurrogate, the l1 norm. The l1-norm minimization problem ismore popular in CS because its envelope is convex andtherefore easier to theoretically analyze. However, the lpnorm better approximates the original NP hard problem.Consequently, it gives better results [17,29] both theoreti-cally and practically.

In this work, our main target is to solve the NP hardproblem (11). But following previous studies in this area, wereplace the NP hard rank minimization problem by its closestconvex surrogate the nuclear norm. However, the nonconvexSchatten p-norm (13) will be a better (albeit nonconvex)approximation of the original problem (11).

OXOp4 =Xi

rpi ; r0is are singular values of X ð13Þ

Intuitively, we believe that better reconstruction resultscan be obtained if we replace the NP hard problem (11) byits nonconvex counterpart Schatten p-norm (0bpb1) insteadof the convex nuclear norm. This is because a fractionalvalue of p in the Schatten p-norm is a better approximationof the matrix rank than the nuclear norm. The samereasoning is used in CS to justify the use of nonconvex lpnorms instead of the convex l1 norm to approximate the NPhard l0 norm. Unfortunately, there are no theoretical resultsto corroborate our intuition. However, in a recent work [30],it is shown that minimizing the reweighted nuclear normleads to better results. This work was motivated by the workon re-weighted l1-norm minimization in CS [31]. The re-weighting scheme in some sense solves the nonconvex lp-norm minimization [31] or Schatten p-norm minimization[30] problem by successively solving a series of convexproblems (weighted l1 norm or nuclear norm minimization).We hope the results in our article will motivate others tojustify theoretically the benefits of using Schatten p-norm inmatrix recovery problems.

In summary, we propose to solve the MR imagereconstruction problem (1)/(11) by minimizing its Schattenp-norm directly. Mathematically, the optimization problemis as follows,

min OXOp4; 0bpV1subject to Oy − FxO2

2Veð14Þ

There is no existing algorithm to solve the problem (14).In Section 5, we derive an efficient first-order algorithm tosolve it.

4. Denoising via Schatten p-norm minimization

An MR image is in most cases corrupted by whiteGaussian noise. When the MR images are reconstructed byCS-based methods or by our proposed method, the image isdenoised implicitly during reconstruction. However, inlegacy MR scanners, the reconstruction algorithm doesnot incorporate any denoising step. They used to scan thefull K-space data and the reconstruction step consisted onlyof applying a 2D inverse Fourier transform. Since the K-space data is itself corrupted by noise, applying the inverseFourier transform reconstructs the noise as well. Thus thefinal MR image is noisy. To remove the noise an additionaldenoising step is required. Our proposed Schatten p-normminimization can be utilized for denoising images fromsuch legacy scanners.

The noisy image can be modeled as follows,

Y = X + N

where X is the clean image (to be recovered), N is thenoise assumed to be normally distributed and Y is thenoisy image.

The traditional way to denoise the MR image is to exploitsparsity of the image in a transform domain, such as in Refs.[32] and [33]. The MR image is sparse in the transformdomain while the noise is not, i.e., the MR image yields fewhigh-valued coefficients in the transform domain while thenoise gives rise to dense low-valued coefficients. Therefore anonlinear thresholding operator is used iteratively to pruneaway the low-valued transform coefficients while keepingthe high-valued ones. When only the high-valued coeffi-cients remain, the inverse transform is applied to obtain thedenoised image.

In a purely CS framework, the denoising is performed bysolving the following optimization problem.

minOaOpp subject to OY − STaO2

FroVe ð15Þ

Instead of exploiting sparsity in the transform domain, wepropose to denoise the MR image by exploiting its rankdeficiency. The assumption being that the noiseless MRimage yields the high singular values, while the noise resultsin the small singular values. Instead of applying arbitrary


thresholding on the singular values, we propose to denoisethe image via the following optimization problem,

min OXOp4; 0bpV1subject to OY − XO2

FroVeð16Þ

where ‖.‖Fro is the Frobenius norm.The problem (16) is similar to (14). The only difference is

that the Fourier operator of (14) is replaced by an identity in(16) since the reconstructed image already exists. Thealgorithm for solving the Schatten p-norm minimizationproblem is developed in the next section. The samealgorithm will be able to solve both (14) and (16); only theinputs to the algorithm change.

5. Solving the Schatten p-norm minimization problem

There are quite a few efficient algorithms to solve thenuclear norm (Schatten 1-norm) minimization problem[34–36]. But there is no algorithm to solve our proposednonconvex Schatten p-norm minimization problem. In thissection, we derive a first-order algorithm to solve ourproposed optimization problem.

The problem is to solve (14). However, it is difficult tosolve the constrained problem directly. Therefore wepropose to solve the following unconstrained Lagrangianversion of (14) which is easier to solve,

J xð Þ = minOy − FxO22 + kOXOp4 ð17Þ

The parameters λ and ɛ are related, but in general thereexists no analytical relation between them for nonorthogonalF. In this work, we first develop an algorithm to solve (17).Later (Section 5.2), we will show how to solve theconstrained problem by solving a series of unconstrainedproblems with decreasing values of λ.

5.1. Optimization transfer

The problem (17) does not have a closed form solutionand needs to be solved iteratively. We follow the so-calledmajorization–minimization (MM) technique [37,38] to solveit. MM replaces the hard minimization problem J(x) by aniteration of easy minimization problems Gk(x). The iterationsproduce a series of estimates which converge to the desiredsolution, i.e., the minimum of (17).

MM AlgorithmInitialize: iteration counter k=0; initial estimate x0.Repeat the following steps until a suitable exit criterion

is met.

1. Choose Gk(x) such that:1.1 Gk(x)≥J(x), ∀x1.2 Gk(xk)=J(xk)

2. Set: xk+1=min Gk(x)3. Set: k=k+1 and return to Step 1.

At each iteration, we replace J(x) by the following,

Gk xð Þ = Oy − FxO22 + x−xkð ÞT I − FTF

� �x − xkð Þ

+ kOXOp4

where xk=xk−1+FT(y−Fxk−1).

With rearrangement of terms, Gk(x) can be expressed as,

Gk xð Þ = Ox − xkO22 − xTk xk + yTy + xTk − 1 I − FTF

� �xk −1

+ kOXOp4

As all the terms apart from the first term are independentof x, we can thus minimize the following instead,

Gk V xð Þ = Ox − xkO22 + kOXOp4 ð18Þ

Now, x and xk are vectorized forms of matrices. Thefollowing property of singular value decomposition holdsin general,

Ovn2�1O22 = OVn�nO

2F = O

Xr� r

O2F = OSr�1O

22

where v is the vectorized version of the matrix V, Σ is thesingular value matrix of V and s is a vector formed by thediagonal elements of Σ (singular values of V). Now both xand xk have the same right and left singular vectors. With theuse of this property, minimizing (18) is the same asminimizing the following,

GkW sð Þ = Os − skO22 + kOsOp

p ð19Þ

where s and sk are the singular values of the matricescorresponding to x and xk, respectively.

The expression (19) is decoupled as:

GkW sð Þ =Xi

s ið Þ−s ið Þk� �2

+ ks ið Þp ð20Þ

It is now easy to minimize (20) by differentiating it term-wise. Skipping the mathematical manipulations it can beshown that (20) is minimized by:

s = signum skð Þmax 0; jsk j − k2p jsk jp−1

� �ð21Þ

This is a modified version of the famous soft thresholdingoperator and can be expressed as,

s = soft sk ;k2p jsk jp−1

� �

Based on this derivation, we propose the followingshrinkage algorithm for minimizing the unconstrainedoptimization problem (17).

Shrinkage Algorithm

1. xk=xk−1+FT(y−Fxk−1)

2. Form the matrix Xk by reshaping xk.3. SVD: Xk=UΣVT.

Table 1NMSE for noise free data with varying number of radial sampling lines

Slice name Technique 50 lines 70 lines 90 lines 110 lines

BrainWeb new l1 norm 0.19387 0.14855 0.11762 0.09765lp norm 0.18379 0.14721 0.11705 0.09766Proposed 0.19573 0.15926 0.13452 0.11390

BrainWeb old l1 norm 0.13199 0.09500 0.07082 0.05461lp norm 0.12587 0.09001 0.07080 0.05400Proposed 0.12066 0.08042 0.06012 0.04407

National Instituteof Health

l1 norm 0.23270 0.18807 0.15649 0.12988lp norm 0.23351 0.18991 0.15900 0.13072Proposed 0.23475 0.19456 0.16738 0.14236

The table shows that, for the BrainWeb new and the NIH data, thereconstruction error from our proposed method is slightly higher (1–2%than that of CS-based methods and, for the BrainWeb old data, our error isslightly lower (around 1%). However, these slight variations are practicallyindistinguishable in the reconstructed images. The reconstructed imagesfrom the proposed approach and the convex CS-based method [1] are shownin Fig. 2 for visual quality assessment. Since the results from the nonconvexmethod are quite similar to the convex one, we provide results only for theconvex one.


4. Soft threshold the singular values:R = soft diag Rð Þ; k

2 p jsk j p−1� �

5. Xk + 1 = UVT . Form xk+1 by vectorizing Xk+1.6. Update: k=k+1 and return to Step 1.

5.2. Constrained optimization via cooling

Our target is to solve the constrained problem (14). So far,we have discussed how to solve the unconstrained version(17). Although the parameters λ and ɛ are related, therelation is not analytical. In this article, we solve theconstrained problem by adopting a cooling technique asemployed by Lin and Herrmann [39].

Cooling algorithm

Initialize: x0=0; λ bmax(FTx)Choose a decrease factor (DecFac) for cooling λOuter Loop: While1 ||y−Fx||2Nɛ

Inner Loop: While2 Jk − Jk + 1Jk + Jk + 1

zTol

∘ Compute objective function for current iterate:Jk=||y−Fxk||2

2+λ||Xk||p⁎∘ Minimize Jk by the shrinkage algorithmdescribed earlier.

∘ Compute objective function for next iterate:Jk+1=||y−Fxk+1||22+λ||Xk+1||p⁎

End While2 (inner loop ends)λ=λ×DecFac

End While1 (outer loop ends)

The cooling algorithm consists of two loops. The innerloop is for minimizing the unconstrained problem (17) for aparticular value of λ. This loop either runs for a fixed numberof iterations or exits when the relative change in the objectivefunction (Jk−Jk+1)/(Jk+Jk+1) is less than the tolerance level.The outer loops reduce the value of λ. The outer loopcontinues as long as the value of λ does not reach a specifiedminimum value or as long as ||y−Mx||2Nɛ.

The parameter λ in (17) actually balances the relativeimportance of the two terms ||y−Fx||22 and ||X||p⁎. Initially,the value of λ is kept high in order to get a very low ranksolution. As its value is gradually decreased, the rank of the

Fig. 1. Ground truth images: (left to right) BrainWeb new, BrainWeb old and NIH.

)

recovered matrix increases and the mismatch ||y−Fx||22 isreduced. The value of λ is reduced until the mismatch fallsbelow the given error tolerance.

6. Experimental results

We carried out the experiments on three slices: two arefrom the BrainWeb and one from the National Institute ofHealth database (see Fig. 1). All the images are of size 256by 256 pixels. Radial sampling was employed to collect theK-space data. The reason to simulate radial sampling is that itis by far the fastest K-space sampling method [40]. However,our work is generalized and can be employed with any othersampling scheme. Nonuniform FFT (NUFFT) [41] is used asthe mapping (F) between the real image space and thecomplex K-space. The NUFFT is not orthogonal; it has anadjoint but not a unique inverse.

Our proposed reconstruction technique is comparedagainst two standard CS-based reconstruction techniques,

Table 2Reconstruction times for noise free data with varying number of radiasampling lines




NationalInstitute ofHealth

l1 norm 92.767 92.0046 206.72 99.7556lp norm 170.810 251.029 421.008 140.957Proposed 13.3627 12.7485 45.1203 11.7448

The table shows that our proposed approach is considerably faster than CS-based techniques. However, we are not in a position to generalize this claimCS is a matured area of research, and choosing a different solver mayaccelerate CS-based reconstruction. Our algorithm for solving the Schattenp-norm minimization problem is a naïve one and its speed can be increasedas well. The core computational demand of our algorithm is in computingthe singular value decomposition in each iteration. Computing the SVDnaïvely is time consuming since we already know that the matrix is rankdeficient. Our current algorithm can be made even faster by employingPROPACK [45] for computing the SVD.

Fig. 2. Noise-free reconstruction results (220 lines). (Top row) Convex CS-right) BrainWeb new, BrainWeb old and NIH.


l

.

viz., the convex formulation (l1 minimization) [1] and thenonconvex formulation (lp minimization) [42]. There areseveral solvers for the l1-minimization problem, but theproperties of NUFFT are not suitable for most of them. The

based re

spectral projected gradient L1 (SPGL1) [43] is by far themost suitable solver for the current problem. For the lp-normminimization, the iterated reweighted least squares (IRLS)algorithm [44] is used.

Haar wavelet was used as the sparsifying transform forCSMRI. We tested other wavelets like Daubechies andsymlets, but found no improvement in reconstruction.Besides, they are considerably slower than the Haar wavelet.

Our proposed reconstruction approach, based on Schattenp-norm minimization, gave the best results when the value ofp is 0.9. Therefore for all the experiments we fixed p at thesaid value. For the lp-norm minimization [42], the bestresults were obtained for p=0.8.

We will provide both quantitative and qualitativereconstruction results. For quantitative comparison, thenormalized mean squared error (NMSE) between thereconstructed image and the ground truth is computed.However, the NMSE is not the best metric for imagereconstruction; therefore we will also provide the recon-structed images for qualitative assessment.

6.1. Reconstruction from noise-free K-space samples

In the first set of experiments, it is assumed that the K-space data is noise free. The number of radial sampling lines

construction; (bottom row) reconstruction from proposed approach. (Left to

Table 3NMSE for noisy data with varying number of radial sampling lines




National Institute of Health l1 norm 0.23053 0.18863 0.16082 0.13823lp norm 0.26805 0.22371 0.19182 0.16074Proposed 0.23614 0.19807 0.17273 0.14792

The nonconvex CS-based method gives the worst results. This is because the IRLS algorithm is not very robust to noise. The NMSE values between our proposedmethod and the convex CS-based method show only slight variations (1–2%) in reconstruction results. The distinction between our proposed method and theconvex CS-based method is practically indiscernible as can be verified from Fig. 3.


was varied from 50 to 110 in steps of 20. In Table 1, theNMSE values are shown, and in Table 2 and Fig. 2, thereconstruction times are tabulated.

6.2. Reconstruction from noisy K-space samples

Practical MR data is corrupted by noise. Our reconstruc-tion method can address the noisy reconstruction problem(Fig. 3). Table 3 shows the results from noisy reconstruction.The number of radial sampling lines was varied from 50 to

Fig. 3. Reconstruction results for data corrupted by 5% noise (220 lines). (Topproposed approach. (Left to right) BrainWeb new, BrainWeb old and NIH.

110 in steps of 20. The K-space data is corrupted by 5%additive white Gaussian noise.

6.3. Denoising

Finally, we will show that our proposed approach can alsobe employed for MR image denoising. The same algorithmdeveloped in Section 6 is used for denoising. But instead ofthe Fourier operator, an identity operator is used.

row) Convex CS-based reconstruction; (bottom row) reconstruction from

Table 4Denoising results

Slice name Technique No noise 2% noise 5% noise 10% noise

BrainWeb new CS 0.03969 0.03990 0.04359 0.06573Proposed 0.03093 0.03295 0.04338 0.07118

BrainWeb old CS 0.04104 0.03612 0.03931 0.06132Proposed 0.03764 0.03380 0.03853 0.06641

National Instituteof Health

CS 0.04248 0.04192 0.04422 0.07067Proposed 0.0278 0.03057 0.04329 0.07609

As seen earlier, the NMSE values show slight differences in denoisingresults. But these differences are indistinguishable in the actual denoisedimages as seen from Fig. 4.

Fig. 4. Denoising results for data corrupted by 10% noise. (Top row) CS denew, BrainWeb old and NIH.


We propose to denoise the image by solving (16),

minOXOp4; 0bpV1subject toOY − XO2

FroVe

Our denoising approach is compared against the followingCS-based denoising method (15),

min OaOpp subject toOY = STaO2

FroVe

Once the above problem is solved, the image is obtainedfrom the wavelet coefficients (α) by the synthesis equation.

noising

For this experiment, we have changed the amount of noisefrom 0 to 10%. Our proposed approach (16) was comparedagainst a CS-based denoising method (15). The results areshown in Table 4 and Fig. 4.

7. Conclusion

Reducing the data acquisition time forMR scans has alwaysbeen a challenge. In this article, we address the problem from asignal processing perspective. A new approach to reconstructMR images from subsampledK-space measurements has beenpresented here. It exploits the rank deficiency of the MRimages to frame the reconstruction problem.

In recent years, CS-based MR image reconstructiontechniques have been successful in reconstructing imagefrom subsampled K-space measurements. Our proposedapproach aims at the same goal, but from an entirely newperspective. The reconstruction results show that theproposed approach and the CS-based techniques providevirtually indistinguishable results. The same is true for thedenoising experiments. Our proposed method gives practi-cally the same results as CS-based denoising. Experimentalresults in this article indicate that our proposed method isconsiderably faster than the CS-based methods. However, the

; (bottom row) denoising from proposed approach. (Left to right) BrainWeb


speed of optimization is heavily dependent on the algorithmused to solve it, and one may get different results by choosinga different set of algorithms than employed in this article.

It is possible to speed up the Schatten p-norm solver evenfurther. In each iteration of our proposed optimizationalgorithm, the SVD of the matrix needs to be computed. Weemploy the SVD computation algorithm in-built into Matlab.For low rank matrices, computing the SVD in this fashion isslow. The PROPACK [45] is a much faster algorithm tocompute SVD for rank deficient matrices. In the future, wewill increase the speed of our Schatten p-norm solver byusing PROPACK to solve for the SVD in each step of theoptimization algorithm.

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An algorithm for sparse MRI reconstruction by Schatten p-norm minimization