Fast parallel MRI reconstruction using B-spline approximation (PROBER)[6142-134]

Fast Parallel MRI Reconstruction Using B-splineApproximation (PROBER)

Jan Petra, Jan Kybica, Vaclav Hlavaca, Sven Mullerb and Michael Bockb

aCenter for Machine Perception, Dpt. of Cybernetics, Fac. of Electrical Engineering,Czech Technical University in Prague, Karlovo nam. 12, 121 35 Praha 2, Czech Republic

bDepartment of Medical Physics in Radiology (E020), German Cancer Research Center (dkfz)Im Neuenheimer Feld 280, D-69120 Heidelberg, Germany

ABSTRACT

Parallel MRI (pMRI) is a way to increase the speed of the MRI acquisition by combining data obtained simul-taneously from several receiver coils with distinct spatial sensitivities. The measured data contains additionalinformation about the position of the signal with respect to data obtained by a standard, uniform sensitivitycoil. The idea is to speed up the acquisition by sampling more sparsely in the k-space and to compensate thedata loss using the additional information obtained by a higher number of receiver coils.

Most parallel reconstruction methods work in the image domain and estimate the reconstruction transfor-mation independently in each pixel. We propose an algorithm that uses B-spline functions to approximate thereconstruction map which reduces the number of parameters to estimate and makes the reconstruction fasterand less sensitive to noise.

The proposed method is tested on both phantom and in vivo images. The results are compared with com-mercial implementation of GRAPPA∗ and SENSE† algorithms in terms of time complexity and quality of thereconstruction.

Keywords: Restoration, Parallel MRI, Magnetic resonance imaging, B-splines

1. INTRODUCTION

In parallel MRI (pMRI), multiple receiver coils with distinct spatial sensitivities are used in parallel to acquirethe MRI signal. pMRI is used to decrease the time needed to acquire an MRI image. The acquisition speed isincreased by sampling k-space more sparsely in the phase-encoding direction, which causes aliasing in the imagedomain. The task of the pMRI reconstruction process is to combine the set of the images acquired in parallel toremove this aliasing.3, 4

pMRI decreases the total time of patient examination which helps to avoid motion artifacts and allows tovisualize fast dynamic processes. However, with current reconstruction algorithms the quality of the reconstructedimages is lower as compared to a fully sampled image acquired using the same coil configuration1, 2 mainly dueto the k-space subsampling.

The pMRI reconstruction is a complex process and takes considerably more time than a standard MR imagereconstruction. Thus the retrieved image cannot be displayed immediately after the signal retrieval. The aimof our work is to design a reconstruction algorithm offering shorter reconstruction times while maintaining orimproving the image quality. One of the reasons for the long reconstruction times of methods working in theimage domain is that the reconstruction coefficients are estimated independently in each point. In the algorithmdescribed here, an implicit regularization is used by approximating the coefficients of the reconstruction map

Further author information: (Send correspondence to J.P.)J.P.: E-mail: [email protected],Telephone: +420 22435 5729S.M.: E-mail: [email protected], Telephone: +49 6221 424775

∗Generalized Autocalibrating Partially Parallel Acquisitions1

†Sensitivity Encoding for Fast MRI2

by a small number of smooth functions. This reduces the number of estimated parameters and therefore speedsup the estimation process. It also allows to control the smoothness of the reconstruction transformation andmakes it more robust to noise in the reference images. The described algorithm is an extension of previouslypublished work.5 The numerical computations were optimized to achieve higher reconstruction speed. Theestimation was extended by a statistical criterion that minimizes the propagation of noise from input images tothe reconstructed image and this way improves the reconstruction quality.

2. PROPOSED RECONSTRUCTION METHOD

In this section, a new reconstruction method Parallel MRI Reconstruction using B-spline Approximation(PROBER) is presented. Initially, set of reference images is used in an estimation step to estimate the re-construction map. Reference images are images without aliasing acquired using a coil-array. Reference imagesinclude array-coil images rSl and an ideal image rS. The ideal image is an image without aliasing or any sen-sitivity variation (i.e. like the body-coil image) that represents the ideal outcome of the reconstruction. Thetechnique to obtain the ideal image from the reference array-coil images rSl is covered in detail in Section 2.5.

Once the estimation step is finished, input images iSl are acquired. Input images are array-coil images withaliasing acquired with a various acceleration factors. These images are used to reconstruct the image withoutaliasing iS using the already estimated reconstruction maps (the reconstruction step is described in Section 2.1).The reference and input images need not display the same object. However, it is important that both sets areacquired with the same coil configuration.

The problem of perfect reconstruction is analyzed according to several factors. The first factor is the squaredifference between the ideal image rS and the reconstructed image S which is covered by the perfect recon-struction condition (see Section 2.1). The second factor is propagation of noise from the input images iSl tothe reconstructed image S. The influence of the reconstruction on the noise level in the reconstructed imageand means to reduce this influence are discussed in Section 2.2. The reconstruction map is not minimized di-rectly. Instead, the map is approximated using a linear combination of B-splines (see Section 2.3). The B-splinecoefficients are estimated in order to minimize the reconstruction error (see Section 2.4).

2.1. Perfect reconstruction conditions

The image Sl acquired by the l-th receiver coil can be represented as a pointwise multiplication of the idealimage S and sensitivity of the coil Cl (see Figure 1)

Sl(x,y) = Cl(x,y) S(x,y), (1)

where S, Sl and Cl contain complex values. Note that image noise is neglected – the problem of noise in theinput images is discussed in Section 2.2.

In pMRI, input images are acquired with a reduced field-of-view (FOV) in the phase-encoding direction.This is done by sampling the k-space more sparsely while preserving the spatial resolution, which causes aliasing(see Figure 1). Relation between the image with aliasing SA (acceleration factor M – i.e. every M -th line isretrieved) and the fully-sampled image S is formulated in the following equation6

SA(x,y) =

M−1∑m=0

S(x,y+mκ), (2)

where Y is the original phase-encoding resolution and κ = Y/M .

Equations (1, 2) are both linear. This means the composite transformation from the ideal image S to thearray-coil image with aliasing SA

l is also linear. Invertibility of this composite transformation is assumed. Thisis generally the case with a reasonable coil configuration2, 7 and it is an implicit assumption of all reconstructionmethods. The inverse transformation is then also linear. Therefore, we search for a linear reconstruction map

(a) a body-coil image (b) an image of a single (c) 2 times FOV reduction of (b)in a coil array

Figure 1. An image of a vessel phantom is shown in (a), (b), (c). Images (a) and (b) are fully sampled. The samplingdensity in image (c) is reduced by a factor of two. This causes aliasing that is visible as a “fold-over” effect in image (c).The image in (a) was acquired using a body-coil with approximately constant sensitivity over the whole imaged object.The images in (b) and (c) were acquired by an array coil with a more localized spatial sensitivity.

α that combines the input images to the reconstructed image S, that should be as close as possible to the idealimage S for each pixel:

S(x,y+mκ) =L∑

l=1

αl(x,y+mκ) SAl (x,y), (3)

where αl(x,y) ∈ C is the reconstruction map for the l-th coil.

Substitution of the aliasing equation (2) in the reconstruction transformation (3) yields

S(x,y+mκ) =L∑

l=1

αl(x,y+mκ)

M−1∑m′=0

Sl(x,y+m′κ) =M−1∑m′=0

L∑l=1

αl(x,y+mκ) Sl(x,y+m′κ). (4)

The intensity value on position (x, y) is independent of values on any other spatial position, namely (x, y +mκ) for m �= 0. This observation gives rise to the perfect reconstruction condition that further constrain thereconstruction map α

L∑l=1

αl(x,y+mκ)Sl(x,y+m′κ) = δm,m′S(x,y+mκ). (5)

This can be illustrated on an example of estimation and reconstruction with the acceleration factor twofor pixels (y) and (y + mκ). The x-coordinate is not important in this example and therefore it is omitted.During the estimation step, both values rSl(y) and rSl(y+κ) in reference images are known. However, during thereconstruction step only the sum iSA

l (y) = iSl(y) + iSl(y+κ) is available in input images. The map α has to beestimated in order to fulfill the conditions (5) that a linear combination

∑l Sl(y) αl(y) gives the original value

S(y) (full lines in Figure 2) and similarly for∑

l Sl(y+κ) αl(y+κ), while the linear combinations∑

l Sl(y+κ) αl(y)

and∑

l Sl(y) αl(y+κ) are zero and do not affect the value of S(y) resp. S(y+κ) (dashed lines in Figure 2).

2.2. Noise amplification by reconstructionThe retrieved input images are not perfect. Both real and imaginary part of the input images are affected bynoise, which is assumed here to be white additive Gaussian noise. Noise in all receiver coils is characterized byvariance σ2

l . The proposed reconstruction (3) amplifies the noise and propagates it to the reconstructed images.2

Since noise in all channels is Gaussian with zero mean the resulting noise is also Gaussian with zero mean. Thevariance of the Gaussian noise in the final image σ2

R is described by the following equation

σ2R(x,y) =

∑l

αl(x,y)σ2l α∗

l (x,y). (6)

rSl(y)

rSl(y+κ) αl(y)

αl(y)αl(y)

αl(y+κ)

αl(y+κ)

αl(y+κ)

rS(y)

rS(y+κ)

iS(y)

iS(y+κ)

iSl(y)+iSl(y+κ)

Estimation Reconstruction

Figure 2. Visual example of the equation (5) for the acceleration factor 2 and two image points.

β0 β1 β2 β3

Figure 3. B-spline function of order 0 to order 3.

Minimizing the total variance∑

x,y σ2R(x,y) in each point is the crucial task in pMRI reconstruction apart

from removing aliasing (see Section 2.4).

2.3. B-spline approximation

It is assumed that coil sensitivities are smooth and slowly-changing in space. The same is assumed about thereconstruction map. Thus, the reconstruction weights α (3) need not to be calculated in each point independently.Instead, the solution is found in a restricted space of smooth functions represented by a B-spline basis.8

B-splines βp of degree p are piecewise polynomial functions of degree p with compact support. B-spline βp isdifferentiable up to p-th order. They are widely used for their properties in approximation of smooth functions.Zero-degree B-spline β0 is a box function (i.e. a piecewise constant). B-spline of degree p is obtained by pconvolutions of the zero-order B-spline with itself (see Figure 3). Cubic B-splines have been chosen for ourpurpose

β3(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

23 − |x|2 + |x|3

2 0 ≤ |x| < 1

(2−|x|)36 1 ≤ |x| < 2

0 2 ≤ |x|.

Several basis functions ϕi are used to represent the reconstruction coefficients gijl. ϕi are equally spaced andscaled B-splines. The scaling factor is chosen in a way so that values of three basis functions are non-zero ineach point of the reconstruction map in the original coordinate system

Hy = YI ,

ϕi(y) = β3( yHy

− i + 2), (7)

where I is the number of basis functions and Y is the image resolution.

Tensor product of shifted and scaled B-splines basis is used to represent the two-dimensional space of allreconstruction transformations

αl(x,y) =I∑

i=1

J∑j=1

gijl ϕi(y) ϕj(x), (8)

where J is the number of splines used in the x direction, I is the number of splines used in the y direction andgijl are the B-spline coefficients that represent αl for l-th coil. The choice of I and J differs with the resolutionof the reference images and coil configuration (five to eleven splines are used for each dimension).

It is possible to use any other linearly independent basis to represent the reconstruction map. The advantageof B-splines is that thanks to their good approximation properties even low number of splines is sufficient torepresent the reconstruction transformation. Also, only a limited number of B-spline coefficients interact at each(x, y) yielding a sparse block diagonal system.

In original SENSE2 algorithm, the reconstruction is estimated for each location separately, because storageand handling with all reconstruction coefficients at once requires too much time and memory. In the proposedapproach, the number of parameters is reduced significantly by using a B-spline approximation. Thus it ispossible to estimate all reconstruction coefficients at once by solving a system of linear equations.

2.4. Reconstruction errorThe main task of the method is to reconstruct the image with a minimal error. Fully sampled low-resolutionreference images without aliasing (rS and rSl) are typically used for the estimation. The reconstruction coefficientsgijl are estimated minimizing a reconstruction error criterion (defined below). For input images, the idealreconstruction iS is obviously not known. Therefore, it is assumed that the reconstruction transformation forthe reference and input images is the same, because the images are acquired with the same coil configuration.

An error criterion is specified using only reference images. We search for reconstruction coefficients gijl thatminimize this criterion.

First, an operator R is defined using (3, 8)

∀x = 1, . . . , X ∀y = 1, . . . , Y/M

∀m, m′ = 0, . . . , M − 1

R [Sl]m,m′ (x,y) =I∑

i=1

J∑j=1

L∑l=1

gijl ϕi(y+mκ) ϕj(x) Sl(x,y+m′κ). (9)

Parameters of this linear operator R are the B-spline coefficients gijl. The reconstruction from input images(3) is redefined using this operator

S(x,y+mκ) = R[SA

l

]m,0

.

The perfect reconstruction condition (5) is expressed by substituting the reference images in the operator

R [Sl]m,m′ (x,y) = S(x,y+mκ) δm,m′ .

The total reconstruction error is defined as a sum of the deterministic and stochastic part. The deterministicpart is the sum of square differences between the reconstructed image and the ideal reference image (the perfectreconstruction condition). The stochastic part minimizes the variance of noise in the reconstructed image (6) bysuppressing noise amplification during the reconstruction (see Section 2.2). The reconstruction error e is

e =M−1∑

m,m′=0

‖R[Sl]m,m′ − δm,m′ S‖2x,y + λ‖

L∑l=1

αlσ2l α∗

l ‖x,y, (10)

where ‖ ‖x,y is an l2 norm over all x = 0, . . . , X − 1 and y = 0, . . . , κ − 1. The aim is to find parameters gijl

that minimize the error e (10)gijl = argmingijl

e.

Partial derivatives of error criterion are computed for each i, j and l. This yields I · J · L linear equationswith the same number of variables (see Appendix A)

AG = B.

There are several ways how to solve this linear system of equations concerning using Gauss-Newton eliminationmethod or Singular value decomposition to invert the matrix A.9 The matrix A is Hermitian and positivedefinite (for proof see Appendix B). This means that it is possible to use Cholesky decomposition9 to solve thelinear system. Cholesky decomposition is faster and more stable than the methods mentioned above. Choleskydecomposition is used to obtain the decomposition A = QTQ, where Q is an upper triangular matrix. First, thelinear system QTG0 = B is solved (solving system with a triangular matrix QT is trivial). Vector G is solutionof the equation G0 = QG.

2.5. Choice of the reference images

The problem with using reconstruction coefficients calculated from a set of reference images is that the coil arrayconfiguration might change and the coefficients are no longer optimal.

Fortunately, the sensitivity maps are changing slowly in space. Therefore, it is not necessary to have full-resolution images for the estimation of the reconstruction coefficients. Instead, low-resolution images withoutaliasing are acquired together with the accelerated input images. For this purpose variable-density (VD) imagesare retrieved. VD images are fully sampled in the center of k-space and subsampled in the outer part. ForPROBER method, images with resolution 24 pixels in the phase-encoding direction are used for the estimation.Acquiring these additional pixels does not prolong the scan significantly. In VD scan, the input and referenceimages are acquired with the exact same coil configuration (this removes the artifacts that arise when the objector coil array moved between acquisition of reference and input images).1, 10, 11

Usage of low-resolution reference images is easily incorporated in PROBER algorithm. The basis functions ϕare sampled at lower resolution during the estimation. Reconstruction map α is evaluated from the coefficientsgijl using full-resolution basis functions (8).

The center of k-space usually carries most of the information about the image. The extra lines acquired inthe center of k-space can be used to improve the reconstruction quality. The easiest method to use the additionallines is to reconstruct the image the usual way (8) and then replace the center lines in the reconstructed imageby the originally retrieved lines, which increases the quality of the reconstructed image.

A problem arises from the ideal reference image. When using self-calibration by VD images, there are onlyarray-coil images Sl available (5). However, there is a need for an image with constant sensitivity over the wholeimaged object to serve as the ideal reference image S. Sum-of-squares (SoS) image is an ideal combinationof array-coil images with unknown sensitivities.12 However, SoS image lacks phase information. Therefore, aweighted sum over the array-coil images is used as the ideal image

S(x,y) = (∑

l

Sl(x,y))

√∑l S

2l (x,y)

|∑l Sl(x,y)| .

The second possibility is to consider array-coil image to be the ideal reference image1 (1)

Sl(x,y) = S(x,y)Cl(x,y) = Cl(x,y)C−1l′ (x,y)Sl′ (x,y) = Cl,l′ (x,y)Sl′ (x,y).

Each array-coil image is reconstructed separately from the input images and the final image is computed as asum-of-squares. The reconstruction speed and the quality of the reconstruction is discussed in Section 3.2.

3. RESULTS

Several experiments have been done to compare two versions of the proposed algorithms (PROBER and FastPROBER) with other available methods. Fast PROBER uses a single ideal reference image and PROBERreconstructs every array-coil image separately – see Section 2.5. The reconstruction quality of PROBER is higher

64 128 256 384 512 640 768 102410

6

107

108

109

1010

Resolution

Ope

ratio

ns

GRAPPAmSENSEPROBERFast PROBER

64 128 256 384 512 640 768 102410

6

107

108

109

1010

Resolution

Ope

ratio

ns

GRAPPAmSENSEPROBERFast PROBER

8 coils, AC factor 2 8 coils, AC factor 4

Figure 4. Number of operations needed for SENSE, PROBER, Fast PROBER and GRAPPA. The theoretical timecomplexity has been computed for different number of input coils and acceleration factors for resolutions from 64 to 1024.Note that number of operations is shown on a logarithmic scale.

than the quality of Fast PROBER however, Fast PROBER reconstruction is faster. Matlab implementation ofPROBER and commercial C++ implementations of GRAPPA1 and SENSE2 are used for the experiments.

The raw data were acquired by a 1.5T clinical MR system (Siemens Symphony). The spatial resolution of theimages is 2562 (Sets 1, 4, 7, 10, 13, 14, 17, 18) or 5122 pixels. The acceleration factor is two (Sets 1, 2, 4, 5, 7, 8,10, 11, 13, 15, 17, 19) or four. In both cases, variable-density scans with 12 additional lines in the k-space centerwere acquired, so that a total of 24 lines in the phase-encoding direction were available for the estimation of coilsensitivities. An eight-channel head coil was used to acquire the images of a distortion phantom (Sets 1-12) andthe brain of a volunteer images (Sets 13-16). The vessel-phantom images were acquired using a six-channel spinearray coil (Sets 17-20). The experiments below compare the speed and quality of the reconstruction for the fourabove mentioned methods.

3.1. Reconstruction speed

Theoretical time complexities of GRAPPA, SENSE, PROBER and Fast PROBER reconstruction have beencompared regarding the number of basic mathematical operations that have to be evaluated. All numbers referto a reconstruction with variable-density images with 24 reference lines. The number of operations was computedfor different acceleration factors and numbers of coils (see Figure 4).

The graphs show that the reconstruction speed of SENSE is comparable with that of PROBER. GRAPPAis faster then both techniques. Reconstruction speed of Fast PROBER is slower than GRAPPA for imageresolutions up to 128 or 192 pixels. However, Fast PROBER is getting faster than GRAPPA for higher resolutionsand is faster than the other methods for any resolution exceeding 192 pixels. This applies for the accelerationfactors 2-4 and the number of receiver coils from 4 to 16.

Unfortunately, a C++ implementation of PROBER algorithm is not available yet and therefore it is notpossible to compare the real reconstruction times.

3.2. Quality of reconstruction

The SNR of the reconstructed images was computed as

20 ∗ log10

‖S‖2

‖S − S‖2, (11)

where S is the reconstructed image and S is the ideal reconstruction. The ideal image was obtained as asum-of-squares of array-coil images averaged over ten acquisition to suppress noise in the images.

set/method 1 2 3 4 5 6 7 8 9 10

PROBER 49.4 33.1 24.5 56.5 40.8 27.5 60.7 45 29.8 64.2Fast PROBER 47.5 31 25.7 54.3 38.8 29.2 58.1 44.1 31.2 61.3

GRAPPA 46.4 34.3 32.5 51 39.8 36.3 53.9 43.5 38.6 57SENSE 48.5 32.1 15.8 55.5 40.4 19.4 59.8 45.9 22.9 63

set/method 11 12 13 14 15 16 17 18 19 20

PROBER 51.7 32 44.7 40.5 31.1 26.7 58.3 48.1 50.3 35.3Fast PROBER 49.5 32.4 42.7 40.3 28.3 27.6 55.4 37.2 47.4 27.7

GRAPPA 46.8 40.5 43.8 38 31.2 27.5 57.3 48.2 50.6 38SENSE 51.9 27.3 44 30.8 29.8 20.2 56.1 35.4 49.6 29.9

Table 1. SNR of the four methods tested on a set of 20 images (in dB).

The methods were tested on a set of 20 images. The images were acquired using a fast gradient echosequence.13

Each dataset was reconstructed using all four methods (GRAPPA, SENSE, PROBER and Fast PROBER).SNR was measured over region excluding background. In the images acquired using the 8 channel head coil,there was insufficient signal in the center of the head. All methods failed to reconstruct the image in the centralpart of the object for the acceleration factor four. Therefore, for the acceleration factor of four the error wasmeasured only in the region remainder of the image center. The SNR of GRAPPA, SENSE, PROBER and FastPROBER are given in Table 1.

The experiments show that the SNR of PROBER reconstruction is higher than the SNR of the other methodsin most cases for the acceleration factor two using the eight-channel head coil. The SENSE reconstruction is betterthan PROBER for dataset 8 resp. 11 (distortion phantom with resolution 5122 and slice thickness 7 mm resp. 11mm). In the remaining ten datasets, the PROBER reconstruction is better than SENSE reconstruction in termsof SNR. The SNR of the GRAPPA reconstruction is lower than the SNR of the PROBER reconstruction exceptin dataset 2. However, the GRAPPA reconstruction of dataset 2 contains severe aliasing artifacts. All threemethods offer aliasing free reconstruction for the acceleration factor two using the spine-array. The PROBERreconstruction is the best of the three methods in terms of SNR for resolution 2562 (dataset 17) and is betterthan SENSE reconstruction for the dataset 19 with resolution 5122 (see Figure 5 and 6).

For an acceleration factor of four all methods show significant deviations from the ideal image. The SENSEreconstruction is aliasing free, but there is a high level of noise. In the GRAPPA reconstruction, severe artifactsare present. PROBER reconstruction maintains a low level of noise and artifacts. No method is clearly betterthan the others. Several examples of reconstructed images from all methods are shown in Figure 5.

4. CONCLUSION

A reconstruction algorithm (PROBER) that uses a smooth model instead of computing the reconstructioncoefficients independently and for each pixel was proposed. This reduces the number of unknown parameters inthe estimation and speeds up the reconstruction. The algorithm can estimate coefficients in flat areas thanks tothe smoothing effect of the B-spline representation.

The earlier reconstruction method5 was improved in terms of both speed and quality. The error criterion wasmodified in order to reduce the noise level in the reconstructed images and thus to improve the reconstructionquality. The numerical evaluations were made more efficient using the separability in computations and by usingfaster and more stable methods to solve the linear system.

The PROBER method offers the same reconstruction quality as GRAPPA and SENSE for most coil con-figurations. For some configurations it even outperforms them. The Fast PROBER method has the advantageof being faster especially for image sizes above 192 pixels with the reduced reconstruction quality. This has alarge number of applications and is advantageous especially for high-resolution scans or 3D scans where it couldreplace the currently used reconstruction methods.

full FOV GRAPPA PROBER Fast PROBER SENSESNR=32.5dB SNR=24.5dB SNR=25.7dB SNR=15.8dB

full FOV GRAPPA PROBER Fast PROBER SENSESNR=39.8dB SNR=40.8dB SNR=38.8dB SNR=40.4dB

Figure 5. Examples of the reconstructions of an image of a distortion phantom (datasets 3 and 5). The full-FOV imageis an average over ten measurements and serves as a reference. Note that the images are enlarged and cropped to focuson the interesting parts of the image.

APPENDIX A. COMPUTING PARTIAL DERIVATIVES OF THERECONSTRUCTION ERROR

The technique how to differentiate the error criterion to find the optimality conditions for the reconstructioncoefficients is indicated in this section. The image values of S and S in (10) are complex. Thus the norm ‖z‖2

k

is rewritten as∑

k z(k)z(k)∗. The error criterion after this modification yields equation (10)

e =M−1∑

m,m′=0

X−1Y −1∑

x,y=0

(R[Sl]m,m′(x,y) − S(x,y)δm,m′) (R∗[Sl]m,m′(x,y) − S∗(x,y)δm,m′)+

+ λ

X−1Y −1∑x,y=0

L∑l

IJ∑

ij,i′j′gijlϕi(y)ϕj(x)σ2

l g∗i′j′lϕi′ (y)ϕj′ (x), (12)

where R∗ is the complex conjugate of the reconstruction operator.

The partial derivatives of the error function e(gijl, g∗ijl) (10, 12) are computed with respect to g∗ijl for each

full FOV GRAPPA, SNR= 43.8dB PROBER, SNR=44.7dB

Fast PROBER, SNR=42.7dB SENSE, SNR=44.0dB

Figure 6. Examples of the reconstruction of datasets 13 – brain images of volunteer. Note that the images are enlargedand cropped to focus on the interesting parts of the image. Also note that full-FOV image is averaged over ten acquisitiontherefore, the quality is higher as compared to the other images.

i, j and l and are set to zero in order to find the minimum of the error function e(gijl)

0 =M−1∑

m,m′=0

X−1κ−1∑

x,y=0

(R[Sl]m,m′(x,y) − S(x,y+mκ))ϕi(y+mκ) ϕj (x) S∗l (x,y+m′κ)

+ λ

X−1Y −1∑x,y

IJ∑

i′,j′ϕi(y)ϕj(x)σ2

l gi′j′lϕi′ (y)ϕj′ (x),

where κ = YM . The equation can be further rearranged as

X−1Y −1∑x,y=0

ϕi(y) ϕj (x) S∗l (x,y) S(x,y) =

∑i′,j′,l′

gi′j′l′

M−1X−1κ−1∑m,m′x,y=0

ϕi′ (y+mκ) ϕj′ (x) ϕi(y+mκ) ϕj(x)

(Sl′ (x,y+m′κ) S∗

l (x,y+m′κ) +1M

δl,l′λσ2l

),

which is a system of linear equations. This can be rewritten in a matrix notation

AG = B, (13)

where the matrix A and the vector B are defined as

Aijl,i′j′l′ =

M−1X−1∆−1∑m,m′x,y=0

ϕi′ (y+mκ) ϕj′ (x) ϕi(y+mκ) ϕj(x)(Sl′ (x,y+m′κ) S∗

l (x,y+m′κ) + λσ2l,l′

),

Bijl =

X−1Y −1∑x,y=0

ϕi(y) ϕj(x) S∗l (x,y) S(x,y) (14)

and G is the vector of the reconstruction coefficients gijl.

APPENDIX B. CHOLESKY DECOMPOSITION – PROOF OF APPLICABILITY

The conditions for the use of Cholesky decomposition are that the matrix is Hermitian and positive definite.First, the symmetry of the matrix A (14) is checked. The basis functions ϕ are real-valued, thus it is clear thatthe matrix is symmetric with respect to indices i and j: Aijl,i′,j′,l′ = Ai′j′l,ijl′ . The (ijl, i′j′l′) and (i′j′l′, ijl)differs only in the Sl elements. The equation Sl′ S∗

l = (S∗l′ Sl)∗ shows that the Sl part of the matrix A is

complex-conjugate of its symmetric counterpart. Therefore, Aijl,i′j′l′ = A∗i′j′l′,ijl which is the definition of a

Hermitian matrix.

Secondly we prove that the matrix A is positive definite. (i.e. qHAq > 0‡ for any vector q ∈ C). For anyr ∈ C is

0 ≤∑l,l′

r∗l rl′

0 ≤∑l,l′

q∗l S∗l ql′Sl′

The equality arises only when ∀l rl = 0 – i.e. ∀l qlSl(x, y) = 0. That will not happen with real images Sl. Thebasis function ϕ are positive on the domain. It is derived that the matrix A is positive-definite

0 <∑

i,i′,l,l′q∗il ϕi(y) qi′l′ ϕi′ (y)(S∗

l (x,y) Sl′ (x,y) +1M

δl,l′σ2l )

0 <∑

ijl,i′ ,j′,l′

∑m,m′x,y

q∗ijl qi′j′l′ϕi′ (y+mκ) ϕj′ (x) ϕi(y+mκ) ϕj(x)(Sl′ (x,y+m′κ) S∗l (x,y+m′κ) +

1M

δl,l′σ2l )

0 < qHAq =∑

ijl,i′,j′,l′q∗ijlAijl,i′,j′,l′qi′j′l′ ,

ACKNOWLEDGMENTS

The authors were supported by The Grant of the Czech Academy of Sciences under project 1ET101050403.

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‡qH is complex conjugation of the transposed vector q: qH = (qT )∗

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