Array Compression for MRI With Large Coil Arrays

Martin Buehrer, Klaas P. Pruessmann, Peter Boesiger, and Sebastian Kozerke*

Arrays with large numbers of independent coil elements arebecoming increasingly available as they provide increased sig-nal-to-noise ratios (SNRs) and improved parallel imaging per-formance. Processing of data from a large set of independentreceive channels is, however, associated with an increasedmemory and computational load in reconstruction. This workaddresses this problem by introducing coil array compression.The method allows one to reduce the number of datasets fromindependent channels by combining all or partial sets in thetime domain prior to image reconstruction. It is demonstratedthat array compression can be very effective depending on thesize of the region of interest (ROI). Based on 2D in vivo dataobtained with a 32-element phased-array coil in the heart, it isshown that the number of channels can be compressed to asfew as four with only 0.3% SNR loss in an ROI encompassingthe heart. With twofold parallel imaging, only a 2% loss in SNRoccurred using the same compression factor. Magn ResonMed 57:1131–1139, 2007. © 2007 Wiley-Liss, Inc.

Key words: coil array; data compression; parallel imaging;SENSE; k-space

Coil arrays (1) have become essential for MRI, particularlywhen parallel imaging methods are applied (2–4). It hasbeen shown that the quality of parallel imaging resultsdepends on (among other parameters) the number andarrangement of the individual coil elements with respectto the object of interest (5–7). Until recently, the number ofindependent receiver channels supported by MR systemswas limited to typically six to eight. However, new spec-trometer hardware is becoming available that permits 32and more receiver channels to be used. This has triggeredresearch into applying large coil arrays with up to 128independent receive channels (8–11). The quest for largecoil arrays has been linked to improvements in parallelimaging performance, with increasing numbers of inde-pendent receive channels employed to further approachtheoretical limits (12,13). Also, as coil coverage is in-creased, positioning of the individual coil elements withrespect to the object of interest becomes less critical,thereby simplifying the setup and patient preparation pro-cess.

With the increasing number of independent receiverchannels, however, practical limitations, such as handlingand processing the large number of independent datastreams, have emerged. In particular, reconstruction speed

has decreased dramatically. Despite the steady evolutionof computer hardware, it is foreseen that the practicalgains in parallel imaging performance and SNR due to theincreasing number of receiver channels will be offset bythe computational load and memory limitations in imagereconstruction.

The idea of array compression is to reduce the number ofindependent data streams sent to the reconstruction hard-ware by combining raw data from multiple receiver chan-nels using a suitable transformation. By doing so, the com-putational load on the hardware and memory constraintscan be reduced significantly.

For the compression to be efficient, the raw data re-ceived by each physical coil element must be combined inthe time domain prior to transformation of the raw k-spacedata into image space. For that purpose a linear transfor-mation is applied on the data vector immediately afteracquisition. As a result, a new virtual set of coils is createdthat is composed of all the physically existing coils in theoriginal coil array.

The concept of array compression proposed herein isrelated to the MRI eigencoil described previously (14,15)in which, given a certain symmetry between the elementsof an array, a degeneracy of eigenmodes of the noise re-ceived in each channels may exist. This insight can beused to mix channels with degenerative eigenmodes in thetime domain, creating a reduced set of virtual coils. Sincethe coil elements are solely characterized by their noisestatistics, however, the method may not be optimal. Forexample, an individual coil element may be remote to animaging region and hence insensitive, and yet it is treatedas a unique channel since its noise characteristic does notreflect this relative insensitivity. Furthermore, by consid-ering noise characteristics only, properties of parallel im-age reconstruction cannot be taken into account, and thusthe virtual array created may not be optimal for parallelimaging.

In this work an optimal virtual coil array is chosen suchthat the signal-to-noise-ratio (SNR) in the reconstructedimage is maximized or, equivalently, the noise amplifica-tion is minimized while maintaining unit signal response.The optimization process may further be tailored withrespect to a region of interest (ROI) that can be smaller thanthe object covered and can be defined in a scout scan. Forexample, in cardiac imaging the heart as the object ofinterest occupies only a fraction of the total field of view(FOV) of the array.

The present work introduces a coil array compressiontechnique that is suitable for conventional and parallelimaging applications using sensitivity encoding (SENSE)(1). Based on model and in vivo cardiac coil array data, itis demonstrated that significant coil compression factorscan be achieved with minor losses in SNR relative toconventional multicoil reconstruction. Conventional mul-

Institute for Biomedical Engineering, University of Zurich and Swiss FederalInstitute of Technology, Zurich, Switzerland.Grant sponsor: SEP Life Sciences/ETH Zurich; Grant number: TH7/02-2;Grant sponsors: Stifterverband der deutschen Wirtschaft; Philips MedicalSystems.*Correspondence to: Sebastian Kozerke, Ph.D., Institute for Biomedical En-gineering, University and ETH Zurich, Gloriastrasse 35, 8092 Zurich, Switzer-land. E-mail: kozerke@biomed.ee.ethz.chReceived 20 October 2006; revised 7 February 2007; accepted 21 February2007.DOI 10.1002/mrm.21237Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 57:1131–1139 (2007)

© 2007 Wiley-Liss, Inc. 1131

ticoil reconstruction refers to methods that use data fromeach individual physical coil element of an array.

MATERIALS AND METHODS

The purpose of coil array compression is to reduce thenumber of independent data streams from n coil inputchannels to m output channels by applying a suitablelinear combination in the time domain prior to imagereconstruction, as outlined in Fig. 1.

Let �� denote a vector containing the data from all n coilsfor k-space point �. Then the combination of coils can beexpressed as a linear transformation described by a matrixA with dimension m � n applied on the data vector �� toreduce its length from n to m elements:

��� � A�� [1]

This may also be viewed as creating a new virtual set ofm coils (m � n) on which the subsequent reconstructionprocess is performed. The transformation matrix A is cho-sen such that the root mean square (RMS) of the noise in anROI of the final SENSE-reconstructed image is minimizedwhile unit signal response is maintained. For a given un-dersampling factor R and undersampling directions, thefolded ROI, ROIfolded, consists of superimposed pixels thatreceive signal contributions from R locations of the object(Fig. 2). If � is defined as a pixel in ROIfolded, the imagenoise after SENSE reconstruction in the unfolded pixels of� can be expressed as the diagonal elements of the imagenoise matrix:

X� � �S�H�1S�

1 [2]

where S� denotes the complex coil sensitivities from allcoils in the superimposed pixel � (3). The receiver noisecovariance matrix is denoted as �. By applying a lineartransformation A, the sensitivity matrix S� and the noisecovariance matrix � are transformed to:

S�� � AS� [3]

�� � A�AH

Substituting Eq. [3] into Eq. [2] yields the noise matrix inthe unfolded pixels of � upon transformation A:

X�� � �S�HAH�A�AH1 AS�

1 [4]

Since image quality is optimized in a limited region, afilter F� is defined, selecting only the diagonal elements of��� corresponding to pixels inside the ROI. Minimization ofthe RMS noise in the ROI can now be expressed as mini-mizing the sum of traces of the transformed and filterednoise matrices ��� in all pixels � of the folded regionROIfolded:

��εROIfolded

Tr�F�X�� � ��εROIfolded

Tr F��S�HAH�A�AH1 AS�

1�

� min [5]

A�AH can be seen as the noise covariance matrix ob-tained from a virtual set of m coils after transformation A.Without restrictions to the solution, we can demand the mcoils to be decorrelated: A�AH � identity. This may beexpressed by defining a transformation T that transformsthe noise covariance matrix into identity: ��T�TH �identity. The sensitivities upon transformation T becomeS� � TS�, and A modifies to A 3 A. The claim for the mvirtual coil elements to be decorrelated translates into theconstraint for A to be unitary, i.e., AAH � identity. Accord-ingly, the expression in Eq. [5] modifies to:

��εROIfolded

Tr F��S�HAHAS�

1� � min [6]

s.t.AAH � identity

Minimization of Eq. [6] is computationally demanding,since the unknown A is placed inside an inversion and thematrices S� and A are generally not invertible by them-selves. Nevertheless, using numerical methods, Eq. [6]may be solved with a considerable computational load. Toavoid computationally intensive numerical methods, theproblem may be simplified using appropriate approxima-tions rather than solving for the optimum transformation Athat minimizes Eq. [6].

Principle Component Analysis (PCA)

A commonly used method for reducing the dimension-ality of a problem is the PCA. Applied to coil array com-

FIG. 1. Schematic drawing of the reconstruction process. The sampled signals from all physical coils (stored in vector ��) are combineddirectly after acquisition in the time domain using the linear combination A. Only a reduced virtual set, consisting of m coils, is passed onto the reconstructor. The FFT and the subsequent merging of coils (SENSE, sum of squares, etc.) are performed on the reduced data.

1132 Buehrer et al.

pression, PCA performs a singular value decomposition ofthe Hermitian matrix:

Q � ��εROIfolded

S�S�H [7]

such that Q � VFVH. An approximation of the matrix A canthen be found by selecting the first m rows of the unitarymatrix VH or, mathematically equivalently, by definingmatrix C � �id�0 with dimensions m � n and multipli-cation with VH. Thus

APCA � CVHT [8]

It can be shown that the resulting transformation APCA

maximizes the following sum (see Appendix):

��εROIfolded

Tr �S�HAPCA

H �APCA�APCAH 1APCAS�� [9]

which is the same term as in Eq. [5] without the inverseand the filter. Accordingly, PCA maximizes the totalamount of sensitivity in the reduced virtual set of coils.Geometrically speaking, the columns of the sensitivity ma-trix S� can be seen as a set of vectors that span an R-dimensional subspace in �n, where R is the total under-sampling factor. Array compression by the transformationA can now be seen as a unitary transformation of thesevectors followed by a projection onto an m-dimensionalCartesian subspace in �n. APCA is then the transformationthat maximizes the squared length of these projectionssummed over all pixels in the unfolded ROIfolded. Whilethis is generally a reasonable approach, it has one majordrawback: the sum in Eq. [7] includes a weighting accord-ing to the length and phase of the sensitivity vector in apixel. Accordingly, PCA prefers pixels with high sensitiv-ity over pixels with low sensitivity, and therefore resultsin inhomogeneous virtual sensitivity maps. For illustra-tion consider the following example: Let us chose a com-bination A such that one pixel in the combined virtual setof coils receives zero sensitivity. Due to the inversion inthe image noise formula (Eq. [6]) this results in a singularterm and therefore infinite noise in that pixel. For thatreason such a combination can never be optimal in termsof noise minimization. However, when using PCA the

equivalent term of that pixel in Eq. [9] is simply zero dueto the lack of the inverse, and the sum can still be maxi-mized if all the other pixels consist of high-sensitivityvalues. Thus the PCA tends to maximize the sensitivityvalues in some pixels while neglecting others, resulting ininhomogeneous virtual sensitivity maps.

Optimized Transformation

In order to address the problem of singular summands inEq. [6] as created by the PCA, orthonormalization of thesensitivity matrix S� is required. Mathematically, this re-quirement is equivalent to replacing the Hermitian in Eq.[7] by the pseudo inverse. The sum in Eq. [7] then reads:

P � ��εROIfolded

S�S�† [10]

and an optimized transformation A is obtained:

A � CUHT [11]

where the unitary matrix U is defined by the singular valuedecomposition of P, such that P � UFUH. Geometrically,orthonormalization creates an orthogonal basis in the sub-space of �n spanned by the column vectors of S�. Thetransformation A then maximizes the squared length of therotated and projected basis in the m-dimensional subspacesummed over all pixels in the unfolded region of ROIfolded.As such, transformation A approximates the requirementfor minimal total image noise (Eq. [5]). Orthonormalizationof the sensitivity matrix in Eq. [10] is the key differencebetween PCA and the optimized compression. Accord-ingly, the angular differences between the sensitivity vec-tors are taken into account, but their lengths are not.

FIG. 2. Illustration of the desired ROI with and without undersam-pling. Pixel � in the folded ROI (ROIfolded) is a superposition of pixelsp1, p2, and p3 in the unfolded ROI.

FIG. 3. Computer simulation of a coil array consisting of 32 inde-pendent coils around a sphere. The middle slice was chosen as theROI for subsequent simulations.

Coil Array Compression 1133

Therefore, all pixels are treated identically and a morehomogeneous sensitivity map with nonzero sensitivityvalues in the ROI is obtained.

Computer Simulations

An artificial coil array was created that consisted of 32independent circular coils around a spherical object witha diameter of 0.2 m (see Fig. 3). The radius of a single coilwas 3.5 cm. Coil sensitivity information was calculatedaccording to the method described in Ref. 16. Conductivityand permittivity of the spherical sample were chosen ac-cording to average values from brain tissue (16). The cen-tral slice of the sphere was chosen as the ROI. Virtual coilarrays with different numbers of output channels m werecomputed using the proposed optimized transformationand compared with the results of the PCA method. TheSNR across the central slice ranged from 5 to 150. Imagesreconstructed from virtual sets of coils were evaluatedagainst images produced from the original set of 32 coils.

The methods were implemented in MATLAB (The Math-Works, Natick, MA, USA) on standard PC hardware.

Quality of Compression

Due to computational difficulties in solving Eq. [6] di-rectly, the accuracy of the approximate solutions as givenin Eqs. [8] and [11] with respect to the exact solution canonly be assessed indirectly. To this end, Monte Carlo sim-ulations with N � 1*106 were performed on the computermodel data. The relative noise enhancement obtained fromreconstructions using random unitary matrices A werecompared to results obtained with the PCA and the opti-mized transformation method.

In Vivo Measurements

Cine 2D short-axis views were acquired on a 1.5T PhilipsIntera System (Philips Medical Systems, Best, The Neth-erlands) using a 32-channel cardiac coil array covering

FIG. 4. Comparison between the PCA and the op-timized combination. The fully sampled data of all32 coils from the physical array were combinedinto one virtual coil. The resulting sensitivity mapsare shown for the (a) optimized combination and(b) PCA method. The reconstructed images of themiddle slice are given in c and d along with the (eand f) total image noise maps.

1134 Buehrer et al.

approximately 30 cm in both the feet–head and right–leftdirections. The coils were distributed in a hexagonal pat-tern and divided into two sets of 16 anterior and 16 pos-terior elements. A steady-state free precession (SSFP) se-quence was used with the following parameters: TR �3.4 ms, TE � 1.7 ms, flip angle � 60°, scan matrix � 192 �182, FOV � 320 � 320 mm2, slice thickness � 8 mm, 14lines/segment, and 30 cardiac phases.

Sensitivity maps were obtained from a SENSE referencescan (TR � 4.9 ms, TE � 1.1 ms, flip angle � 7°, scanmatrix � 64 � 64 � 20, FOV � 270 � 270 � 30 mm3, andeight signal averages). The region of the heart was roughlydefined on the reference scan images, and reduced virtualcoil arrays for that region were computed for a fully sam-pled dataset and twofold undersampled data with subse-quent SENSE reconstruction. Reconstruction results ob-tained with the virtual coil arrays were compared relativeto data reconstructed from all 32 coil elements in theoriginal array. Furthermore, the performance of array com-pression was investigated for different sizes of the ROI for

a compression factor of 8 (n � 32, m � 4). The ROI size,including the heart, was incrementally expanded using aregion-growing algorithm with the ROI coordinates savedat each incremental step. For each ROI setting, the noiseamplification in the image reconstructed from compressedarray data was compared relative to the noise in the imagereconstructed from all coils.

RESULTS

Computer Simulations

Figure 4 shows image results obtained with array compres-sion using the optimized combination relative to the PCAmethod. All 32 physical coils were compressed to onevirtual coil, and the image representing the mid-slice wasreconstructed from the single virtual coil dataset. It can beseen that the optimized combination results in relativehomogeneous virtual sensitivities (Fig. 4a) after array com-pression, while sensitivities obtained with the PCA

FIG. 5. The reconstructed phantom images andtotal image noise maps for the reduced virtual coilarrays compressed with the optimized combina-tion consisting of different output channels m with-out and with fourfold SENSE. The noticeable rect-angular shapes in the noise image with m � 4 andR � 4 are regions that exhibit fourfold folding.

Coil Array Compression 1135

method exhibit considerable gradients (Fig. 4b). In PCA,the high sensitivity is assembled in pixels at the border ofthe object according to the sensitivity profiles of the phys-ical coils. Images reconstructed from the virtual coil setobtained with the optimized combination show homoge-neous noise levels across the slice, while noise levelsincrease considerably toward the center of the slice whenthe virtual coil set constructed with the PCA method isused (Fig. 4b and c).

A comparison of images and noise maps reconstructedfrom virtual coil arrays compressed with the optimizedcombination with different numbers of output channels isshown in Fig. 5. for optimally reconstructed fully sampled(1) and fourfold undersampled data. The SNR dependenceon the number of channels m after compression is plottedin Fig. 6. From Fig. 5 and Fig. 6, it can be observed that thevirtual number of coils can be drastically reduced withvery little SNR loss. For an acceptable SNR loss of 5%, thearray can be compressed from 32 physical to six virtualcoils for R � 1, and to seven virtual coils for R � 4.

Quality of Compression

Relative noise amplification from Monte Carlo simulationswas found to be always greater than noise amplification inimages reconstructed from data compressed with the op-timized transformation method (Fig. 7). In contrast, thePCA method resulted in inferior noise performance, withsome random transformation outperforming the resultfrom PCA.

In Vivo Measurements

Reconstructions from the in vivo cine dataset are shown inFig. 8. Images and noise maps obtained from fully sampleddata using all 32 physical coils are compared to imagesreconstructed from virtual coil sets consisting of four and

one output channels. Furthermore, images and noise mapsfrom twofold SENSE undersampled in the right–left direc-tion for m � 32, 4, and 2 are shown. For the fully sampleddataset the average noise in the ROI was amplified by 0.3%when the number of virtual coils was reduced to four, andby 13% for one virtual coil. When twofold undersamplingwas performed, the noise amplification in the ROI was 2%for a reduction to four virtual coils and 10% for two virtualcoils. The maximum noise amplifications for the fullysampled data were 2% (m � 4) and 15% (m � 1). Thesevalues increased to 9% (m � 4) and 74% (m � 2) for theundersampled data. Using array compression, the raw datastorage size of the fully sampled dataset could be reducedfrom 160MB to 20MB for four virtual coils and 5MB forone virtual coil.

Accordingly, the time required for SENSE unfolding(R � 1) or optimal coil combination (R � 1) was reducedby 15% for m � 4 and 30% for m � 1. It should be notedthat the overall gain in reconstruction speed will approachthe compression factor since processing steps such as dataapodization, filtering, and Fourier transform (FT) are per-formed before SENSE unfolding and are therefore appliedper coil.

The noise amplification dependence on the ROI size canbe observed in Fig. 9. For a region including only the heart,noise was amplified by 0.3% for the optimized combina-tion and 0.6% for the PCA. For an ROI including the wholethorax, this value increased to 8% for the optimized com-bination and to 35% for the PCA-based method.

The determination of the optimum combination matrixA is as fast as the computation of the sum in Eq. [10].Therefore, the computational effort for calculating A de-pends on the number of pixels in ROIfolded (N) and the size

FIG. 6. The SNR dependence on the number of virtual coils m forthe computer model without and with fourfold SENSE. Shown is thenormalized averaged SNR over the ROI. For comparison the SNR inimages reconstructed with the PCA method is plotted (dashed line).

FIG. 7. Solution distribution obtained from a Monte Carlo simula-tion. One million randomly chosen matrices were used for arraycompression, and noise amplifications relative to the reconstructionusing all 32 coils were compared to the optimized combination andthe PCA-based method. The optimized transformation method pro-vides lower noise amplification than any randomly chosen transfor-mation, while noise amplification from the PCA-based method isconsiderably greater.

1136 Buehrer et al.

of the sensitivity matrix S�. While the size of the sensitivitymatrix is given by the number of physical coils n and thereduction factor R, the number of pixels in ROIfolded de-pends again on R and the size of the unfolded ROI. Sim-ulations using nonoptimized MATLAB code on standardPC hardware (Pentium 4, 2.4GHz, 1GB of RAM) yieldedthe following empirical formula for the computational ef-fort of Eq. [10]:

t�n,R,N � �0.012 � n � 0.107 � R � N [12]

For the datasets presented above, this adds up to approx-imately 6 s for the simulation data and 0.2 s for the in vivodataset.

DISCUSSION

In this work the principle of coil array compression hasbeen proposed. Coil array compression has been shown to

enable significant reductions of the number of indepen-dent coil channels to be processed by creating virtual setsof coils with fewer elements from all physical coils prior toactual image reconstruction. Accordingly, coil array com-pression allows the memory and computational constrainsassociated with processing data obtained from large coilarrays to be reduced very efficiently.

Given appropriate analog combiner hardware (14), itmay also enable coil arrays to be operated on systems withfewer receivers than the number of independent coil chan-nels.

The fundamental difference between the MRI eigencoilconcept described in Refs. 14 and 15 and the one proposedherein relates to the fact that in the present work thecompression matrix is determined based on coil sensitiv-ities (and noise statistics), whereas only noise statistics areused in the MRI eigencoil method. Principal eigenmodescharacterize the noise behavior only, and as such do notnecessarily reflect the SNR modes. SNR modes can only be

FIG. 8. Images of an exemplary time frame fromthe cardiac cine acquisition as well as total imagenoise maps for the reduced virtual coil arrays con-sisting of different output channels m without andwith twofold SENSE. The undersampling was per-formed in the right–left direction. The ROI ismarked by the dotted line. Note that very littlenoise amplification occurs in the ROI after com-pression.

Coil Array Compression 1137

derived from noise statistics if the array fulfills symmetryconstraints with respect to the object, which is difficult toensure in the general setting (e.g., when using flexible coilarrays).

Furthermore, when only noise statistics are used, coilcompression can become very inefficient. For example, acoil element that is remote to the imaging region will notnecessarily have degenerate noise behavior relative to anelement close to that region. And yet such a coil elementwill contribute fully to the virtual array even though it isvery insensitive in the imaging region.

The efficiency of array compression as proposed heredepends on the degree of redundancy in sensitivitiesamong the physical coil elements with respect to the ROI.Compression is particularly efficient if the ROI is smallerthan the coverage of the coil array, a situation typicallyfound in a number of relevant imaging applications. In apractical setting, the achievable compression factor ismainly determined by the physical coil array configura-tion, ROI size, and ROI position. With respect to the ROIsize, it is insightful to consider the extreme situation inwhich the number of pixels in the ROI does not exceed thenumber of coils in the virtual set. In this case the solutionis exact and no SNR loss is imposed. This may be viewedas lossless compression since sensitivity information in allpixels of interest can be transformed into an m-dimen-sional subspace. The position of the ROI has an influenceon the weighting of the single coils in the combinationmatrix A. Coils close to the ROI have stronger influenceafter compression. Accordingly, the influence of the ROIposition on the compression quality strongly depends on

the sensitivity distribution and therefore on the design ofthe original coil array.

The quality of array compression can be quantified byassessing the loss in SNR relative to standard multicoilimage reconstruction. This in turn defines the maximumcompression factor given a certain acceptable SNR degra-dation in the ROI as defined by the user. Such an imple-mentation is referred to as “SNR-controlled compression.”

Alternatively, hardware constraints, such as limited re-construction memory or memory speed, may dictate theminimum compression factors in order to accommodatesufficient data or to fulfill minimum response require-ments. Accordingly, this may be referred to as “hardware-controlled compression,” which can be activated automat-ically as soon as hardware limits are exceeded.

Given the computational effort required to find the exactsolution of Eq. [6], an indirect validation of the approxi-mate solutions described by the PCA and the optimizedtransformation methods was chosen. Using Monte Carlosimulations, it was demonstrated that the optimized trans-formation method results in consistently lower noise am-plification than any randomly chosen transformation. Itremains, however, subject to future work to rigorouslyquantify the differences between the approximate solutionas described by the optimized transformation method andthe exact solution of Eq. [6].

CONCLUSIONS

Array compression allows a significant reduction in thenumber of channels to be processed in image reconstruc-tion, thereby alleviating the speed and memory constraintsthat emerge when using coil arrays with a large number ofindependent channels. The algorithm proposed facilitatesfast and efficient computation of a linear transformationmatrix, which reduces the number of physical coil chan-nels to a reduced virtual set. Given the steady increase incoil array channels, the proposed array compressionmethod is believed to be essential for time-critical anddata-intensive applications in which reconstruction la-tency or memory constraints currently limit the use of coilarrays with a larger number of independent channels.

APPENDIX

Inserting Eq. [8] into Eq. [9] yields:

��εROIfolded

Tr S�HTHVCH�CVHT�THVCH1CVHTS��

� Tr CVH ��εROIfolded

�S�S�HVCH� � Tr CVHQVCH�

� Tr CVHVFVHVCH� � Tr CFCH� [13]

With the use of Eq. [7] and the relationships: S� � TS�,T�TH � id and Q � VFVH.

Matrix F is diagonal and contains the sorted eigenvaluesof Q. C is a projection on a m-dimensional subspace.Therefore the trace in Eq. [13] is the sum of the m largesteigenvalues of Q and is at its maximum since V is a unitarymatrix.

FIG. 9. The performance of array compression as a function of thenumber of pixels in the ROI. The heart was manually segmented andused as the initial ROI, which was incrementally expanded using aregion-growing algorithm. The ROI range spanned from the initialROI to an ROI that included the entire thorax. The exemplary ROIsshown as dotted lines in the image include 190 and 530 pixels,respectively. The 32 physical coils of the cardiac array were com-bined into four virtual coils for the different-sized ROIs. Noise am-plification in the reconstructed image was evaluated inside the ROIrelative to the reconstruction using all physical coils.

1138 Buehrer et al.

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