Recent improvements in magnetic resonance image (MRI) reconstruction from partial data have been reported using spatial context modelling with Markov random field (MRF) priors. However, these algorithms have been developed only for magnitude images from single-coil measurements. In practice, most of the MRI images today are acquired using multi-coil data. In this paper, we extend our recent approach for MRI reconstruction with MRF priors to deal with multi-coil data i.e., to be applicable in parallel MRI (pMRI) settings. Instead of reconstructing images from different coils independently and subsequently combining them into the final image, we recover MRI image by processing jointly the undersampled measurements from all coils together with their estimated sensitivity maps. The proposed method incorporates a Bayesian formulation of the spatial context into the reconstruction problem. To solve the resulting problem, we derive an efficient algorithm based on the alternating direction method of multipliers (ADMM). Experimental results demonstrate the effectiveness of the proposed approach in comparison to some well-adopted methods for accelerated pMRI reconstruction from undersampled data

Aelterman

Afonso

Chambolle

Donoho

Lustig

Panić

Panic, Marko

Aelterman, Jan

Crnojevic, Vladimir

Pizurica, Aleksandra

Ghent University Academic Bibliography

Multi-coil Magnetic Resonance imaging reconstruction with aMarkov Random Field PriorMarko Panic´a, Jan Aeltermanb, Vladimir Crnojevic´a, and Aleksandra PizˇuricabaBioSense Institute, University of Novi Sad, SerbiabTELIN-IPI, Ghent University - imec, Ghent, BelgiumABSTRACTRecent improvements in reconstructing Magnetic Resonance Images (MRI) from partial data have been reportedusing spatial context modelling with Markov Random Field (MRF) priors. However, these algorithms havebeen developed only for magnitude images from single-coil measurements. In practice, most of the MRI imagestoday are acquired using multi-coil data. In this paper, we extend our recent approach for MRI reconstructionwith MRF priors to deal with multi-coil data i.e., to be applicable in parallel MRI (pMRI) settings. Instead ofreconstructing images from different coils independently and then obtaining final image using sum of squaresmethod, we consider recovery of MRI image using undersampled measurements from all coils jointly with theirestimated sensitivity maps. The proposed method is derived from a maximum a posteriori probability (MAP)formulation of the reconstruction problem on which is then applied an alternating direction method of multipliers(ADMM). Therefore, an algorithms steps of the proposed method can be divided in two blocks. The first onecontains few iterations (up to 10) of conjugate gradient (CG) method for obtaining a temporary image estimatefrom multi-coil measurements which although are undersampled significantly overlap at low frequencies. Thesecond block contains steps which refer to applications of MRF regularizations on temporary image estimateobtained from the first block. This procedure is repeated through iterations until some stopping criterion issatisfied (e.g. number of iterations). Experimental results demonstrate the effectiveness of the proposed approachin comparison to some well-adopted methods for accelerated pMRI reconstruction from undersampled data.Keywords: Parallel MRI (pMRI), Markov Random Field (MRF), Alternating direction method of multipliers(ADMM)1. INTRODUCTIONSlow MRI acquisition is still a barrier in everyday clinical usage. Parallel imaging in MRI (pMRI) and usinga compressed sensing theory in MRI (CS-MRI) are the two possible directions how this speed limitation canbe overcome. In this work we combine both these techniques for the purpose of multi-coil MRI reconstructionstarting from the method which already proved it potential in single coil reconstruction scenario. A matematicalmodel for a single coil MRI acquisition is given in the following equationy = Ax+ n (1)where image x ∈ CN has to be recovered from a k-space measurements data y ∈ CM with the present of whiteGaussian noise n ∈ CM .1,2 Matrix A ∈ CM×N , where M  N , denotes the undersampled Fourier operatorformed according to a predefined k-space undersampling scheme e.g. (radial, spiral or random). Estimation ofimage x is usually obtained as a solution of the following optimization problemminx∈CN12‖Ax− y‖22 + τφ(x) (2)Further author information:Marko Panic´: E-mail: panic@biosense.rsJan Aelterman: E-mail: Jan.Aelterman@ugent.beVladimir Crnojevic´: E-mail: crnojevic@uns.ac.rsAleksandra Pizˇurica: E-mail: Aleksandra.Pizurica@ugent.bewhere the 12‖Ax− y‖22 is a data fidelity term, φ : CN 7→ R ∪ {−∞,+∞} is a regularization function andparameter τ which denotes the amount of involved regularization. The pMRI reconstruction problem can beformulated in image domain (SENSE method from3) or in k-space domain (GRAPPA method from4). In thiswork we opt for the maximum likelihood (ML) formulation of the multi-coil reconstruction problem given asminx∈CNNc∑i=1‖yi − FCix‖22 (3)where matrix F represents Fourier transform and index i indicates the considered coil data yi and its coresspond-ing spatial sensitivity Ci. If we consider reconstruction from a single coil (i = 1) we can see that equation (3)simplifies to form similar to the data fidelity term in (2). The difference is in the presence of undersampling inFourier domain denoted with operator A instead of F. This means that involving CS in pMRI (pMRI-CS) re-construction through the sampling of coils data below the Nyquist rate demands some sort of regularization sincenow reconstruction problem becomes ill-posed. In this work we extend the single coil reconstruction methodsfrom5 with MRF regularization for the more general pMRI-CS problem formulation.2. PARALLEL IMAGING IN MRIIn the pMRI reconstruction the multiple k-space data yi obtained from different reciever antenna coils are utilizedin order to recover the MR image. The method SENSE3 combines the aliased coil images xi, reconstructed usinginverse discrete 2-dimensional Fourier transform (2D-IDFT) from each coil measurements yi (or density correctedadjoint non-uniform Fourier transform in case of non-Cartesian k-space trajectories), and creates a compositeMR image with the a priori knowledge of coils sensitivities. Although an important issue with SENSE method isestimation of the coil sensitivity profiles, it is to date the most widespread employed pMRI techinque offered bymany companies. The issue with estimation of sensitivity profiels can be circumvented with GRAPPA method4which doesn’t require knowledge of the coil sensitivities. It reconstructs the missing k-space data using itsadjacent neighbourhood in k-space from all coils. GRAPPA uses a kernel which defines how the missing k-spacefrom the ith coil data are interpolated using the acquired k-space data from all of the coils. It is learned fromthe low-frequency spectrum of the k-space data of every coil (so-called Auto Calibration Signal (ACS)).6 Afterinterpolation of the missing k-space data an 2D-IDFT is employed to obtained a xi coil images which are usuallywith the Sum of Squares (SoS) techique:x =√√√√ Nc∑i=1|xi|2 (4)combined to reconstruct the composite MR image x. An approach for pMRI reconstruction SPIRiT (iterative self-consistent parallel imaging reconstruction) is coil-by-coil method (reconstruct each coil image separately and thencombine them to obtained the composite MR image) based on GRAPPA method. It synthesis observed missingk-space point using acquired and reconstructed missing points in its neighborhood from all coils. It achievedbetter noise reduction and more accurate reconstruction compared to traditional GRAPPA-like approaches. Amethod which bridge the gap between SENSE and GRAPPA, called ESPIRiT uses eigenvalue decompositionin image space for computation of robust sensitivity maps. It combines advantages of SPIRiT and GRAPPAmethods and restrict solution to a subspace span by coil sensitivity maps.For the case when the sampling of coils data is below the Nyquist rate a problem of image reconstructionbecomes ill-posed and demands some sort of regularization. A method COMPASS from7 expresses a multi-coilreconstruction problem as Basis Pursuit (BP) optimization problem with `1 norm as regularization on imagesparse representation. Authors in8,9 proposed method which also combines parallel imaging with partial Fourieracquistion named LORAKS (LOw-RAnk modelling of local K-Space neighborhoods). It takes into account thefact that many MRI images have limited spatial support and smoothly varying phase through the constraints inoptimization framework. In the following we propose a new method for the pMRI-CS reconstruction problem,based on method from.5 We create joint framework for utilization of parallel imaging and compressed sensingin MRI and compare it with the state-of-the-art methods in the field.3. PROPOSED METHODLet us formulate the pMRI-CS problem as follows:minx∈CNφ(Px) subject toNc∑i=1‖yi −ACix‖22 ≤  (5)where P stands for sparsyfying transform and  ≥ 0 denotes a parameter which is usually related to the noisevariance. Since the measurements from the coils yi are undersampled with operator A, some regularization isnecessary in order to find a suitable solution. Problem formulation in (5) has very similar form to the problemformulation from.5 The difference is that we have measurements gathered from more than one coil, whichusually have overlaping in the region of low frequency components. This means that regularization step inalgorithm for solving (5), has to be applied on image x which is initially (temporary) reconstructed using all coilmeasurements. If we introduce notation for augmented measurement vector ya = [yT1 ,yT2 ...yTNc]T and augmentedvectorized image xa = Tx, where T is formed by stacking Nc times row-wise the indentity matrix IN×N , thenwe can reformulate problem from (5) asminx∈CNφ(Px) subject to ‖ya −ABCBxa‖22 ≤  (6)where AB consists of repeated A along the diagonal while CB is formed by stacking coil sensitivity maps Cialong diagonal. A constrained optimization problem in (6) can be transformed to its unconstrained form, with ausage of an indicator function,5 which is equivialent to maximum a posteriori probability (MAP) formulation forestimation of original image x. Obtained unconstrained form of the problem in (6) is then solved by applicationof alternating direction method of multipliers (ADMM). Derived ADMM algorithm steps can be divided in twoblocks the first one which refers to obtaining temporary image estimate from all coils measurements using fewiterations (up to 10) of conjugate gradient (CG) and the second block which contains steps for application ofMRF regularizations.4. EXPERIMENTAL EVALUATIONIn the first experiment we compared our proposed method with the COMPASS7 method on a saggital MRI sliceacquired using measurements from 4 coils with 12% of the Nyquist sampling rate on a k-space spiral trajectory.This leads to an ill-posed problem since the overall number of measurements is less than the Nyquist samplingrate. The results, shown in Fig. 1, demonstrate the advantage of our regularization based on MRF prior comparedto the `1 norm regularization in the multi-coil reconstruction scenario. Although both methods used the samecoil sensitivity maps, the proposed approach yields an improvement in the peak signal to noise ration (PSNR)of more than 6dB. The reconstructed image is much sharper and has preserved much more original details, as italso evident from the diference image.The second experiment was evaluated on data publicly available on http://www.eecs.berkeley.edu/~mlustigwhich is used in the experimental evaluation of methods ESPIRIT10 and SPIRIT.11 The lower sampling rate isobtained by decimating the data with acceleration factor greater than 1. We use measurements undersampledwith acceleration factor 3 from 4 virtual coils obtained from 8 coils using SVD coil compression technique. Thisexperimental setup is the same as the one used in.10,11 In the proposed method the coil sensitivity maps areestimated from undersampled measurements using approach from.12 From the presented results in Fig. 2 we seethat proposed method outperforms all compared methods. An image artifacts caused by undersampling trajec-tory are better suppresed in image reconstructed by proposed method than in images recovered by comparedmethods.In the last experiment we compare our method with the P-LORAKS and SENSE-LORAKS methods.8,9We use four-channel T1 weighted brain image from9 with provided coil sensitivity profiles. Reconstructionis conducted using 14% of measurements from each coil sampled using two different trajectories random anduniform. A golden standard image is created using all measurements from four coils with the SoS techique.Achieved reconstruction performances are given in Table. 1. Reconstruction obtained with the proposed methodoutperformed compared methods. Visual comparison is presented in Fig. 3 and in Fig. 4 for the uniform andFigure 1. Reconstruction of sagittal slice from 4 coils with 12% of measurements per coil sampled with spiral trajectory.First row: Reconstructed images with COMPAS7 (27.75 dB) and proposed method (35.42 dB) method respectively.Second row: Coresponding reconstruction errors.random sampling trajectory respectively. We can see in reconstructed image with the proposed methods a muchsmaller amount of noise present compared to the reconstructions obtained by P-LORAKS and SENSE-LORAKS.Table 1. Reconstruction of image from9 using undersampled measurements obtained from 4 coils with uniform / randomtrajectory. A comparison of P-LORACS and SENSE-LORAKS methods with the proposed one is given in terms ofstructure similarity measure (SSIM) and normalized root-mean-squared-error (NRMSE).Methods SSIM NRMSEP-LORAKS 0.88 / 0.92 0.07 / 0.06SENSE-LORAKS 0.83 / 0.87 0.08 / 0.07Proposed method 0.95 / 0.96 0.04 / 0.035. CONCLUSIONIn this work we extend our recent approach for MRF based MRI reconstruction to deal with more general multi-coil acquisitions. The extension is based on (MAP) signal estimation from undersampled multi-coil measurementswhich is conducted using ADMM. The derived algorithm contains block of steps regarding temporary MRIimage estimate and block of steps which refer to MRF regularization which alternate one after the other throughFigure 2. Reconstruction of phantom test image using measurements sampled with acceleration factor 3 from 4 virtualcoils. First column: Reconstructed images with SPIRIT (32.22 dB), ESPIRIT (31.32 dB) and proposed method (32.64dB) respectively. Second column: Coresponding reconstruction errors.iterations until some stoping criterion is satisfied. From the experemental results we see that proposed methodoutperforms some well-adopted method in multi-coil reconstruction scenario.Figure 3. Reconstruction of T1 weighted brain image from9 form uniformly sampled measurements from 4 coils. First col-umn: Reconstructed images with P-LORAKS, SENSE-LORAKS and proposed method respectively. Second column:Coresponding reconstruction errors.Figure 4. Reconstruction of T1 weighted brain image from9 form randomly sampled measurements from 4 coils. First col-umn: Reconstructed images with P-LORAKS, SENSE-LORAKS and proposed method respectively. Second column:Coresponding reconstruction errors.6. ACKNOWLEDGMENTSThis research has been supported by the H2020 project ANTARES No664387. We are grateful to Prof. Dr.Karel Deblaere from the Radiology Department of UZ Gent hospital for providing the data.REFERENCES[1] Lustig, M., Donoho, D., and Pauly, J. M., “Sparse MRI: The application of compressed sensing for rapidMR imaging,” Magnetic Resonance in Medicine 58(6), 1182–1195 (2007).[2] Lustig, M., Donoho, D. L., Santos, J. M., and Pauly, J. M., “Compressed sensing MRI,” IEEE SignalProcess. 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Q., Goossens, B., Pizˇurica, A., and Philips, W., “Compass: a joint framework forparallel imaging and compressive sensing in mri,” in [Image Processing (ICIP), 2010 17th IEEE InternationalConference on ], 1653–1656, IEEE (2010).[8] Haldar, J. P. and Zhuo, J., “P-loraks: Low-rank modeling of local k-space neighborhoods with parallelimaging data,” Magnetic resonance in medicine 75(4), 1499–1514 (2016).[9] Kim, T. H., Setsompop, K., and Haldar, J. P., “Loraks makes better sense: Phase-constrained partial fouriersense reconstruction without phase calibration,” Magnetic resonance in medicine 77(3), 1021–1035 (2017).[10] Uecker, M., Lai, P., Murphy, M. J., Virtue, P., Elad, M., Pauly, J. M., Vasanawala, S. S., and Lustig,M., “Espirit—an eigenvalue approach to autocalibrating parallel mri: where sense meets grappa,” Magneticresonance in medicine 71(3), 990–1001 (2014).[11] Lustig, M. and Pauly, J. M., “Spirit: iterative self-consistent parallel imaging reconstruction from arbitraryk-space,” Magnetic resonance in medicine 64(2), 457–471 (2010).[12] Aelterman, J., Naeyaert, M., Gutierrez, S., Luong, H., Goossens, B., Pizˇurica, A., and Philips, W., “Au-tomatic high-bandwidth calibration and reconstruction of arbitrarily sampled parallel mri,” PloS one 9(6),e98937 (2014).

Multi-coil magnetic resonance imaging reconstruction with a Markov Random Field prior

Marko Panić

Jan Aelterman

Vladimir Crnojević

Aleksandra Pižurica

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

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Multi-coil magnetic resonance imaging reconstruction with a Markov random field prior

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Multi-coil magnetic resonance imaging reconstruction with a Markov Random Field prior

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Ghent University Academic Bibliography

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

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