Comprehensive Quantification of Signal-to-Noise Ratio and g-factor for Image-Based and k-space-based Parallel Imaging Reconstructions

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1 Magneti Resonane in Mediine 60: (2008) Comprehensive Quantifiation of Signal-to-Noise Ratio and g-fator for Image-Based and k-spae-based Parallel Imaging Reonstrutions Philip M. Robson, 1 * Aaron K. Grant, 1 Ananth J. Madhuranthakam, 2 Riardo Lattanzi, 1,3 Daniel K. Sodikson, 4 and Charles A. MKenzie 5 Parallel imaging reonstrutions result in spatially varying noise amplifiation haraterized by the g-fator, preluding onventional measurements of noise from the final image. A simple Monte Carlo based method is proposed for all linear image reonstrution algorithms, whih allows measurement of signal-to-noise ratio and g-fator and is demonstrated for SENSE and GRAPPA reonstrutions for aelerated aquisitions that have not previously been amenable to suh assessment. Only a simple presan measurement of noise amplitude and orrelation in the phased-array reeiver, and a single aelerated image aquisition are required, allowing robust assessment of signal-to-noise ratio and g-fator. The pseudo multiple replia method has been rigorously validated in phantoms and in vivo, showing exellent agreement with true multiple replia and analytial methods. This method is universally appliable to the parallel imaging reonstrution tehniques used in linial appliations and will allow pixel-by-pixel image noise measurements for all parallel imaging strategies, allowing quantitative omparison between arbitrary k-spae trajetories, image reonstrution, or noise onditioning tehniques. Magn Reson Med 60: , Wiley-Liss, In. Key words: signal-to-noise ratio; image noise; g-fator; parallel imaging; image reonstrution Parallel imaging approahes are widely used for aelerating MR image aquisitions (Simultaneous Aquisition of Spatial Harmonis (SMASH) (1), Sensitivity Enoding (SENSE) (2), Generalized Auto-Calibrating Partially Parallel Aquisition (GRAPPA) (3). Reeiving signals simultaneously in the independent elements of a radiofrequeny oil array allows aquisition of some of the phase-enoded signals to be omitted. The distint spatial sensitivity profiles of the elements ontain spatial information that may be used for the purpose of spatial enoding in the image that is normally provided by appliation of magneti field gradients. The penalty for aquiring fewer signals is a loss of Signal-to-Noise Ratio (SNR) in the final image by a fator 1 Department of Radiology, Beth Israel Deaoness Medial Center, Boston, Massahusetts. 2 Global Applied Siene Lab., GE Healthare, Boston, Massahusetts. 3 Harvard-MIT Division of Health Sienes and Tehnology, Boston, Massahusetts. 4 Department of Radiology, New York University Shool of Mediine, New York, New York. 5 Department of Medial Biophysis, The University of Western Ontario, London, Ontario, Canada. *Correspondene to: Philip M. Robson, Beth Israel Deaoness Medial Center, Department of Radiology, Room, Ansin 242, 330 Brookline Avenue, Boston, MA probson@bidm.harvard.edu Reeived 30 Otober 2007; revised 7 April 2008; aepted 15 May DOI /mrm Published online in Wiley InterSiene ( Wiley-Liss, In. 895 of the square root of the aeleration fator R due to redued signal averaging. In parallel imaging image-noise is further amplified by the ill-onditioning of the image reonstrution proess. In general, the noise amplifiation is spatially variant and depends on the speifi geometry of the radiofrequeny oil array used and is therefore haraterized by the (geometry) g-fator (2). An aurate and quantitative method of analyzing noise amplifiation is essential for objetive omparison between parallel imaging tehniques and between image reonstrution methods when developing new methods and designing linial imaging protools. Spatial variation of the image noise preludes the onventional (and simple) Region-of-Interest (ROI) approah for SNR estimation whih uses a region of signal within the objet and a region of noise outside of the objet (4,5), making SNR diffiult to deal with pratially in parallel imaging. Quantifiation of noise amplifiation in parallel imaging has been studied previously (2,6), produing methods for diret alulation of image noise, g- fator (2) and SNR (7) for some lasses of parallel imaging strategies. However, all existing tehniques are subjet to ertain regimes in whih it is impossible to alulate SNR analytially. Diret image noise matrix approahes (2) require memory for matries of size O(n 2 ), where n may be as large as N 2 for arbitrary k-spae trajetories and generalized SENSE image reonstrution (8), where N is the image matrix dimension. Typially N may be 256, whih, for omplex floating-point data, leads to a reonstrution matrix of approximately 30 Gb. For diret GRAPPA approahes n may be as large as N 3 (9) leading to a reonstrution matrix of approximately 2000 Tb. Furthermore, matrix inversion approahes require O(n 3 ) operations (8). Thus, diret omputation rapidly may beome intratable suh that urrently there is no universal approah for measuring SNR or g-fator. Statistial methods, for example Monte Carlo and bootstrapping methods (10), are often used in funtional parameter estimation from MRI (e.g., Jones and Steger et al.) (11,12). This work develops a simple Monte Carlo method for rigorously alulating image noise propagating through the image reonstrution itself (similarly to methods explored previously) (13,14). This method allows alulation of SNR and g-fator for all parallel imaging tehniques whih use a linear image reonstrution algorithm, irrespetive of whether diret alulation is available or omputationally tratable. In this work we demonstrate noise analysis for two lasses of parallel imaging methods (SENSE and GRAPPA) routinely used in linial imaging appliations.

2 896 Robson et al. IMAGE NOISE ANALYSIS FOR LINEAR RECONSTRUCTION ALGORITHMS As desribed by Pruessmann et al. (2), for a linear image reonstrution the propagation of noise from the sampled values in k-spae into the image noise is desribed by noise matries. The image reonstrution is desribed by the operation of the reonstrution matrix F on the vetor of k-spae data s, to generate image pixel values in the vetor (Eq. [1]). Fs [1] The variane of the pixel values in the p th element of is given by the p th diagonal entry in the image noise matrix X. X F F [2] The sample noise matrix (2) is the Kroneker produt of the noise ovariane matrix and an identity matrix of dimension N aq equal to the number of aquired k-spae points (the length of vetor s divided by the number of oil elements) (Eq. [3]). RI Naq [3] The noise ovariane matrix desribes the level and orrelation of noise in the signals reeived in eah element of the oil array. The noise ovariane matrix may be expressed in terms of the individual noise reords n ik (noise samples 1 k N k for oil i) reeived in eah oil element in the absene of NMR signal (Eq. [4], where * represents omplex onjugation) (7). ij 1 N k n 2N ikn jk* [4] k Trivially, a onventional Fourier Transform image reonstrution is a linear operation that may be expressed in the matrix form of Equation [1]. Noise amplifiation may also be haraterized analytially using this formalism. The g-fator is simply the ratio of the SNR for an optimal unaelerated image and the SNR of the aelerated image with an additional fator of the aeleration fator R whih aounts for the SNR loss due to averaging fewer aquired signals (Eq. [5]). k 1 g SNR optimal,unaelerated SNR aelerated R SD aelerated SD optimal,unaelerated R Equivalently, assuming the image signal is the same in the aelerated and unaelerated optimal reonstrutions (whih is expliitly required by the formalism of the SENSE reonstrution in Pruessmann et al.) (2) the g-fator may be expressed in terms of the image noise standard deviations (SD) leading to the familiar expression of g- fator in terms of image noise matrix terms (Eq. [6]). g p R X pp [5] aelerated X pp [6] optimal,unaelerated The optimal SNR image is ahieved by onsidering the noise orrelations between elements of the oil array as desribed by Roemer et al. (15). Therefore, it follows that an unaelerated image reonstrution may be suboptimal and may by our urrent definition have a g-fator greater than unity. This analytial approah allows image reonstrutions to inlude division by the image noise to generate pixel values in units of SNR, whereby the normal image pixel is divided by the alulated image noise for that pixel (hereafter referred to as the diret SNR method ). Kellman and MVeigh (7) desribe image reonstrutions in SNR-units by the diret SNR method for fully gradient-enoded (unaelerated) aquisitions and aelerated aquisitions using a SENSE tehnique. The diret method provides an image with pixel-by-pixel SNR values whih faithfully represent the spatial variation in image noise resulting from parallel imaging aquisition to omplement the information in the onventional magnitude image. Most linial appliations of parallel imaging follow either the SENSE or the GRAPPA approahes. SENSE image reonstrutions may be generalized (2,8,16) as a pseudo-inverse of the enoding matrix E whih expresses the k-spae signals reeived by eah element of a oil array reeiver s as the Fourier transform of the objet multiplied by the omplex spatial sensitivity of the oil element (Eqs. [7] and [8]). s E [7] E 1 E 1 E 1 s [8] It an be seen in Equation [8] that all generalized SENSE image reonstrutions obey the linear formulation required in Equation [1] for the analytial image noise matrix formalism of Equation [2] to hold. GRAPPA tehniques reonstrut missing phase-enoded lines in k-spae by linearly ombining neighboring lines. The weighting fators for the ombination are determined from fitting lines to a small additional number of autoalibration signal (ACS) lines aquired around the enter of k-spae. This is a generalization of similar k-spae fitting approahes (1,17). In other tehniques, for example, AUTO-SMASH (17), the linear ombination of k-spae lines produes a single omposite k-spae from whih a final image is reonstruted by Fourier transform. As desribed by Griswold et al. (3), the GRAPPA approah reonstruts fully sampled k-spae data for eah element of the oil array before reonstruting an image for eah oil by Fourier transform and enjoys the SNR advantage of a root-sum-of-squares (RSS) ombination of the separate oil images. The RSS step, speifially intended to improve SNR (and avoid phase anellation problems), is nonlinear and preludes use of the image noise matrix formalism to alulate image noise and g-fator diretly. This has proven an obstale to detailed analysis of the harateristis of noise propagation in the GRAPPA method beyond appreiating the exellent appearane of various images reonstruted by the GRAPPA method. However, the RSS oilombination is a simplifiation of the full omplex oil

3 Comprehensive Quantifiation of SNR and g-fator 897 ombination desribed by Roemer et al. (15) for the unaelerated ase. Use of omplex ombinations allows the GRAPPA reonstrution to be written as a linear matrix operation (9). In this alternative formulation, eah of the steps of the GRAPPA reonstrution may be expressed as a matrix operation. First, missing phase-enoded lines in the undersampled k-spae are reonstruted by the operation of a transfer matrix T, whih performs linear ombinations of neighboring lines. Seond, a Fourier transform matrix H is applied to the now repleted k-spae data matrix from eah oil. Finally, a omplex oil-ombination matrix Ĉ (9) is applied to form the vetor of final image pixels (Eq. [9]). ĈHTs [9] Complex oil ombination allows the GRAPPA reonstrution to be expressed as a linear operation and, therefore, noise propagation may be analyzed using the image noise formalism of Eq. [2]. The oil sensitivities used in the matrix Ĉ may be alulated from a separate image or from the entral lines of the aquired k-spae data in a self-alibrating approah. The linear matrix representation omplex-grappa approah is disussed in Robson et al. (9). Limitations of Conventional Noise Analysis The formality of writing the reonstrution in a matrix operation form for the purpose of noise analysis arries the high omputational burden of handling many data at one. The size of the reonstrution matrix for SENSE and GRAPPA methods with 1-D aeleration is O(N y by N ky ), where N y and N ky are the number of data in the imageand k-spae y-dimension, and is the number of oil elements. The omputation for the generalized SENSE method is more demanding as this involves inversion of a matrix of this order of magnitude (8). In volumetri image aquisitions where aeleration is applied in two dimensions, whih is ommonly the ase in linial appliations, the matries beome larger, O(N y N z by N ky N kz ), where N z and N kz are equivalent quantities in the z-dimension; furthermore, in GRAPPA methods where the reonstrution kernel may span three dimensions (18), the matrix expands into a further dimension leading to a reonstrution matrix size O(N x N y N z by N kx N ky N kz ). Thus diret alulation of SNR and g-fator by means of image noise matries soon beomes omputationally intratable. It should be noted that 2D-Cartesian- SENSE (19) is omputationally tratable beause the regularly undersampled data permit ontration of the generalized matrix expression into small sub-bloks. For variable density k-spae sampling shemes oupled with 2D aeleration, the full matrix must be used. Non-Cartesian k-spae sampling trajetories, inluding radial and spiral tehniques, where it is not possible to form disrete sets of aliased pixels are another important lass of image aquisitions for whih an analytial and omputationally tratable matrix reonstrution may not be available due to the size of the reonstrution matrix. In all suh ases where the matrix approah of image reonstrution is not feasible an alternative approah is used. For SENSE tehniques, an iterative onjugate-gradient (CG) tehnique may be applied to reonstrut the image as desribed in Pruessmann et al. (8). For GRAPPA tehniques, many small onvolution operations are applied to the undersampled k- spae followed by a standard Fourier Transform algorithm (indeed this is the onventional implementation of the GRAPPA reonstrution for Cartesian sampling, and with additional regridding of the data for non-cartesian GRAPPA) (20). However, diret alulation of the image noise and g-fator is not possible without forming the reonstrution matrix; indeed iterative approahes for generalized SENSE do not yield expliitly the reonstrution weights whih form the reonstrution matrix. Diret alulation of image noise beomes intratable whenever the reonstrution matrix beomes too large to be omputationally feasible, inluding the following: (i) Cartesian SENSE with variable density k-spae traversal and 2D aeleration; (ii) Generalized SENSE with arbitrary k-spae trajetories (e.g., spirals); (iii) GRAPPA with 2D aeleration (inluding regular undersampling shemes); (iv) GRAPPA with 2D or 3D reonstrution kernels. A PSEUDO MULTIPLE REPLICA APPROACH As an alternative to the analytial matrix-based approah, image noise amplifiations may be determined from the variations in image pixel values aused by random noise flutuations in the input k-spae signals where the image reonstrution is onsidered to be a blak-box signal proessing blok. The gold-standard image noise measurement is the soalled atual multiple replia method where k-spae signals are aquired multiple times and reonstruted using the same blak-box into a stak of equivalent image replias whih differ only in their noise ontent. Image noise may then be found on a pixel-by-pixel basis by finding the standard deviation of pixel values through the stak of image replias. Aquisition of multiple image replias allows assessment of image noise and thus SNR on a pixel-by-pixel basis and is therefore useful for measuring the spatially-variant amplified noise in parallel imaging (4,5). The use of this gold standard is not feasible in pratie for real-world in vivo imaging due to exessive patient motion or physiologi noise between image aquisitions, suseptibility to instrument drift (21), and prolonged examination times. For a linear blak-box signal proessing blok the image noise is solely dependent on the noise in the signal and the reonstrution an be seen to operate on the image and the noise separately. We propose a simple Monte Carlo tehnique hereafter termed the pseudo multiple replia method for obtaining a robust measurement of image noise for all linear image reonstrutions whih may be used when a diret alulation is not possible. Corretly saled and orrelated syntheti random noise is added to the aquired k-spae before blak-box image reonstrution. This proess is repeated multiple times, eah time with different syntheti noise, to produe a stak of independent image replias from whih image noise may be alulated emulating the gold-standard atual multiple replia method. A previous study (13) has used this approah with syntheti Gaussian noise without onsider-

4 898 Robson et al. ing noise orrelations. Another study (14) applied a bootstrap method, reordering aquired noise, to estimate the noise propagation through the image reonstrution that inluded noise orrelations and saling without needing to generate fresh noise for eah replia. The pseudo multiple replia method is based on prior knowledge of the signal noise in k-spae. It does not require expliit knowledge of the reonstrution matrix and requires aquisition of k-spae only one to alulate SNR of the aelerated image. The iterative onjugate gradient method for SENSE methods is linear and thus may be used with the pseudo multiple replia method. For GRAPPA methods, the onventional image reonstrution whih applies many small onvolutions before RSS oil-ombination is nonlinear only in the final RSS step. Therefore, a linear omplex- GRAPPA may be implemented in both a full matrix form (transfer matrix used to reonstrut k-spae) and the onventional form of reonstrution. Thus a linear onventional GRAPPA tehnique may be used with the pseudo multiple replia method. ( RSS-GRAPPA will be speified expliitly hereafter, with onventional referring to k-spae reonstrution and omplex referring to omplex oil ombination.). THEORY Pseudo Multiple Replia Method On the assumption that repeatedly aquired k-spae data differs only in its noise ontent, the signal omponent being unhanged, syntheti omplex noise, orretly saled and orrelated between elements of the oil array, may be added to k-spae to emulate atual multiple replia aquisition before image reonstrution (see Fig. 1). Noise may be added uniformly to k-spae; the harateristi spatial variane of noise ours only in image spae after image reonstrution. The pseudo multiple replia method takes advantage of the linearity of the image reonstrution. Eah pseudo replia image has the orret representation of the image noise despite being noisier than the original aquired image by a fator of 2 after FIG. 1. a: The pseudo multiple replia method. Image k-spae and the noise presan data are aquired one (steps 1 3). The noise ovariane matrix is used to sale and orrelate unity SD Gaussian noise for every replia loop (steps 4 7). The reonstruted image is added to a stak of replias before repeating the noise-addition loop (step 8). For self-alibrated tehniques, the reonstrution matrix is alulated from the original data and is used for all replias. The stak of replias is then used to alulate SNR (step 9 10). In step 11, the multiple replia loop is repeated using an artifiial fully-sized zero k-spae to analyze noise from an un-aelerated aquisition and with an image reonstrution that is simply an FFT and an optimal omplex oil ombination. Finally, g-fator is alulated (step 12). b: The steps involved in forming the SNR and g-fator for both SENSE and GRAPPA image reonstrutions. Noise SD is found from the standard deviation of the real or imaginary parts of the omplex pixel values for eah pixel loation through the stak of replias (STEP 2). In STEP 5, the g-fator is found from the two noise SD maps divided by the square root of the aeleration fator R.

5 Comprehensive Quantifiation of SNR and g-fator 899 addition of the seond independent set of pseudo noise data. The atual image noise, whih annot be removed from the aquired k-spae, remains in every image replia and appears as part of the true image when the image noise is alulated from the stak of pseudo image replias. The noise is found from the standard deviation of either the real or imaginary omponents of the omplex image pixel values through the stak of replias beause the linear reonstrution transforms noise in an equivalent manner in either omponent of the omplex signal. As many replias as desired for alulating standard deviation may be formed. Linearity of an image reonstrution also ensures that Gaussian distributed white noise in the reeived NMR signal is transformed into Gaussian noise in the image domain. Thus, the standard deviation of image pixel values takes an intuitive interpretation defining the range in whih the true pixel value may be found, or, equivalently, the amount by whih the pixel value may vary due to random noise flutuations alone. Furthermore, linear reonstrutions do not suffer from noise biasing whih results from magnitude operations (7,22). Saling and orrelation of the noise is determined from the signal aquired during a noise presan when the reeiver is opened with no RF pulses and no normal MR signal present, as desribed by Kellman and MVeigh (7) (see Fig. 1a). The oils must be loaded as for imaging so that the noise reeived is that originating from the objet; and the oil must be in plae as for imaging so that the noise orrelations due to oupling of omponent oils are orretly measured. The bandwidth of the reeiver and the reeiver gains are set as for imaging to ensure the noise is saled orretly. Further details of the noise presan may be found in Kellman and MVeigh (7). The noise ovariane matrix ij measuring the noise level (diagonal elements) and orrelation (off-diagonal elements) between oils i and j (where i and j run from 1 to the number of oils, ) in the oil array may be formed (7,23) from the measured noise presan data aording to Equation [10] with the noise reords n ik of N omplex points sampled at time intervals indexed by k from the i th omponent oil. ij 1 N 2N n ik n* jk [10] k 1 Saled and orrelated noise reords for eah hannel i of orr the phased-array n ik are formulated from unorrelated Gaussian-distributed white noise n G ik with unity standard deviation by matrix multipliation with the prinipal matrix square root of the measured noise ovariane matrix (Eq. [11]) suh that the noise ovariane matrix is reovered upon multipliation of orrelated noise reords (Eq. [12]), given that the prinipal square root of the noise ovariane matrix is equal to its Hermitian onjugate. n ik orr j 1 1/2 G ij n jk [11] N 1 2N k 1 1 n orr ik n orr jk * 1 2N p 1 1 2N p 1 N 2N k 1 1/2 ip jp q 1 1/2 G ip n pk p 1 1/2 ip jq q 1 N 1/2 * k 1 1/2 *2N pq p 1 * 1/2 jq *n G qk q 1 n G pk n G qk * 1/2 ip 1/2 pj ij [12] 1/ The matrix square root 2 ij is given by the matrix of eigenvetors V ij and the diagonal matrix of square roots of eigenvalues S pq ij 1/2 1 V ip S pq V qj p 1 q 1 [13] from the eigen-deomposition of the noise ovariane matrix ij 1 ij V ip D pq V qj p 1 q 1 [14] where D pq is the diagonal matrix of eigenvalues. The prinipal matrix square root is given by taking all positive square roots of the eigenvalues. For the Hermitian-positive definite noise ovariane matrix, all eigenvalues are real and positive, and the eigenvetors form a unitary basis, so 1/ that it an be seen that 2 ij is equal to its Hermitian onjugate as required in Equation [12]. The prinipal matrix square root of the noise ovariane matrix was hosen arbitrarily as the orrelating matrix. The Cholesky deomposition of the noise ovariane matrix into a lower triangular matrix L ij and its Hermitian onjugate is an equivalent hoie for the orrelating matrix as previously used by Pruessmann et al. (8). To onfirm the freedom of hoie Gaussian noise was orrelated with the prinipal matrix square root, another of the matrix square roots from the eigen-deomposition, and the lower triangular matrix from the Cholesky deomposition. The noise SD of eah of the orrelated hannels was found to be the same for eah method used. Thus, any appropriate hoie of the orrelating matrix produes equivalent results when used with the pseudo multiple replia method. SNR Maps From the Pseudo Multiple Replia Method Maps of SNR are formed from the pseudo multiple replia method by first forming an image noise map. For eah pixel the standard deviation in either the real or imaginary omponent of the omplex image is found from the stak of image replias. The SNR is then given by the magnitude of the image in the original reonstruted image, that is, that without additional noise added, divided by the value of the noise standard deviation for that pixel. We use the real omponent of the image beause the image is ompletely real after reonstrution so as to avoid magnituderelated noise bias in low-snr regions. Note that the image

6 900 Robson et al. noise is always orretly measured, without noise bias, from the omplex data. The SNR map will always have itself the SNR of the aquired image from whih the signal omponent of SNR is taken whereas the SNR of the image noise map may be made as high as desired by inreasing the number of pseudo replias reonstruted and is limited only by total available omputation time. g-fator Maps From the Pseudo Multiple Replia Method g-fator maps are generally reated diretly from the reonstrution matrix as speified in Pruessmann et al. (2). To ensure the general appliability of the pseudo multiple replia method, a proedure is required for determining the g-fator that does not require aquisition of a fully sampled referene image to provide the image noise standard deviation for an unaelerated aquisition in the expression for the g-fator in Equation [5] (where R is the aeleration fator). Again taking advantage of the linearity of the reonstrution, the pseudo multiple replia method is applied to noise-only data. The unaelerated image noise may be found by reonstruting a syntheti noise-only fully sampled k-spae (i.e., the data has the full matrix size of the final image) instead of an aquired unaelerated image k-spae. Repeated reonstrution of multiple replias of noise-only k-spae data produes a stak of noise-only image replias whose pixel noise standard deviation gives the proper value for an unaelerated aquisition. Alternatively, a diret image-noise alulation for R 1 may be used (7). Care must be taken to find the optimal-snr image for the unaelerated ase to orretly define the g-fator. For both SENSE and GRAPPA tehniques, this is given by an FFT, orretly saled for unity gain of noise, followed by an optimal omplex oil-ombination desribed by Roemer et al. (15). The Roemer-optimum image is formed by the inlusion of the noise ovariane matrix into the optimal oil-ombination Ĉ whih appears in the GRAPPA matrixreonstrution (Eq. [9]) as desribed in Robson et al. (9). g-fator maps are then found from the ratio of the noise standard deviation maps for an aelerated and an unaelerated aquisition (Eq. [5]). The proedure for measuring the noise ovariane matrix and for performing the pseudo multiple replia measurement of the SNR and g-fator is desribed in Figure 1. Pseudo Multiple Replia Method With Self-alibrated Approahes FIG. 2. Plot of the standard deviation (losed irles) of repeated (five) estimates of the noise-sd (open irles) of separate unityvariane Gaussian-distributed noise data of varying number of samples. For a single estimate of the noise-sd to have a standard deviation less than 1% of the true value (here 0.01), the noise data must omprise greater than approximately 10 4 samples. The pseudo multiple replia tehnique relies on the image reonstrution being idential for eah replia in order that flutuations in the NMR signal are orretly translated into variane in the image domain. For externally-alibrated tehniques, this ondition is met. Conventionally, SENSE tehniques use oil sensitivities obtained from a separate alibration image and are therefore externally alibrated. However, GRAPPA tehniques are most often selfalibrating, finding the reonstrution weights from the aquired k-spae itself (i.e. from the Auto-Calibrating-Signal lines). It is possible to implement an externally alibrated GRAPPA-reonstrution by using weights alulated from a separate equivalent k-spae aquisition. In the ase of self-alibrated tehniques, it is important to use the same reonstrution for eah replia in the pseudo multiple replia method. For SENSE tehniques this requires the use of the same oil sensitivities for eah replia. For GRAPPA tehniques the same transfer matrix (or set of reonstrution weights) must be used as alulated for reonstruting the original aelerated image without pseudo-noise added (rather than using new weights alulated from syntheti noise-enhaned ACS in eah replia). METHODS Pre-San Noise Measurement and Saling The required number of noise samples whih must be taken to orretly measure the noise ovariane matrix has been determined by alulating the SD of Gaussian distributed random noise reords of various lengths N, whih eah have unit variane. This was repeated five times eah for different values of N between 64 and The SD of the five estimates for eah N is plotted in Figure 2 showing that for a measurement of noise SD to be aurate to within 2% N must be greater than approximately 2000 and within 1% greater than approximately 8000 (following a 1/ N pattern). These values an be ahieved by using image matries of and , respetively. It was onfirmed that digitization error was not important by observing a Gaussian harater in a histogram plot of both the real and imaginary omponents of the noise reeived in eah of the elements of the oil array with no NMR signal present. Twenty-one bins were used over the observed range of the noise. It was assumed throughout that the noise is white in harater, having equal power at all frequenies. Kellman and MVeigh desribe the presan noise measurement in greater detail (7). Experiments The pseudo multiple replia SNR measurement was validated with images of the manufaturer s standard spheri-

7 Comprehensive Quantifiation of SNR and g-fator 901 al phantom. Pseudo multiple replia SNR and g-fator measurements were ompared with gold-standard atual multiple replia SNR and g-fator measurements from 128 separately aquired 2D slie images and also ompared with the diret SNR method (7,9). Both SENSE and GRAPPA methods were investigated for various aeleration fators with undersampling in one dimension. 2D aeleration with variable density k-spae traversal was then investigated in phantoms where it is possible to validate the pseudo multiple replia method against the gold standard atual multiple replia method but not against a diret alulation. The pseudo multiple replia method was then applied in vivo after obtaining written informed onsent with approval of our institutional review board. First, in the brain, 1-D aeleration was used allowing the pseudo multiple replia method to be verified by a diret image-noise alulation, noting that no atual replia method is feasible in vivo. Finally, a 3D volumetri image of the abdomen was obtained using 2D aeleration, a variable density k-spae trajetory and self-alibration. A linial presription was hosen whih requires parallel imaging aeleration to aquire data in a single 22-s breath-hold demonstrating the pseudo multiple replia tehnique in a ase for whih no other method exists for alulating SNR and g-fator. Pseudo multiple replia alulations used 128 image replias throughout. The estimate of SNR and g-fator beomes inreasingly aurate as the number of replias is inreased, thus the optimal hoie of the number of replias is somewhat arbitrary and depends on the desired auray and total omputation time available. Using approximately 100 replias gives an adequate auray of 10% aording to a 1/ N saling relationship. All data were aquired on a 1.5 Tesla (T) Exite-HDx whole-body sanner, (GE Healthare, Waukesha, WI) using an 8-hannel head oil array exept the abdomen image whih used an 8-hannel body array. In the phantom and the brain images k-spae data were aquired fully sampled and later deimated by removing some aquired phaseenoded lines from the data set to mimi aelerated aquisitions of various net redution fators and k-spae sampling patterns. Noise data were aquired from an equivalent separate single image with the amplitude of RF pulses set to zero and without hanging transeiver settings, giving greater than 4096 data points, whih was suffiient to alulate the noise ovariane matrix. An axial 2D slie of the phantom was aquired using a fast gradient realled eho pulse sequene; imaging parameters inluded: matrix size of 64 64, field of view of 20 m, slie thikness of 1 mm, aquisition bandwidth of khz, flip angle of replias of the same slie were aquired onseutively in a single experiment. k-spae data from the first replia aquisition were used with the diret SNR method and the pseudo multiple replia method. The hosen imaging parameters for the phantom experiment resulted in an image SNR of approximately 40 for a fully sampled image, whih is suffiiently low that noise dominates shot-to-shot variation in the sanner in the determination of SNR from the atual multiple replias (6). For a seletion of pixels, pixel intensity in eah replia was plotted against replia number (not shown) to onfirm that no temporal drift was present in the atual multiple replia data. An axial 2D slie of the brain was aquired using a T 1 -weighted spin-eho pulse sequene; imaging parameters inluded: matrix size of , field of view of 22 m, slie thikness of 5 mm, aquisition bandwidth of khz, TR/TE of 500/20 ms. A oronal 3D volume of the abdomen was aquired with at 1 -weighted fast gradient realled eho pulse sequene; imaging parameters inluded: in-plane matrix size of , with 80 3-mm slies, field of view of 36 m, frequeny-enode diretion superior-inferior, aquisition bandwidth of 62.5 khz, TR/TE of 3.9/1.7 ms, and flip angle 15. In-house implementations of an iterative onjugate-gradient generalized SENSE image reonstrution and a omplex-ombined GRAPPA method with onventional onvolution-like k-spae reonstrution (9) were used for the multiple replia images. In-house implementations were also used for the diret matrix-inverse generalized SENSE reonstrution and the diret omplex-grappa matrixreonstrution (9). All image reonstrution and image replia analysis was performed using Matlab (The Math- Works, Natik, MA) and run on a standard PC with a Pentium IV 2.8 GHz CPU and 512 MB of RAM. The omputation time required to generate SNR and g-fator maps is dominated by the reonstrution of the aelerated image replias (the generation of pseudo noise, the formation of unaelerated image replias, and finding the final image noise standard deviation are all relatively rapid). Computation times varied aording to the size of the image matries being handled and reonstrution type; they were generally rather long, ranging from a few minutes for 128 onventional GRAPPA replias with a matrix size of to approximately 24 hr for 128 replias of a CG-generalized SENSE reonstrution with a matrix size of However, the Matlab ode used in this work had not been optimized for speed, and signifiant redutions in omputation time are likely to be possible. g-fator maps for the gold-standard atual multiple replia method (in the phantom) were formed by dividing the image-noise SD-maps alulated from the stak of atual aelerated image replias by the image-noise SD-map alulated from the stak of atual optimal-snr unaelerated image replias. It was onfirmed that the image-noise SD-map alulated from the unaelerated atual multiple replias and the SD-map alulated using pseudo fully sampled noise and the pseudo multiple replia method were equivalent (not shown). Unaelerated image-noise maps for the pseudo replia method used pseudo fully sampled noise throughout. In all ases, for both SENSE and GRAPPA methods oil sensitivities were alulated from the fully sampled enter of k-spae whih is filtered by a Kaiser-Bessel window with 2. In all ases exept in the phantom for an outer redution fator of 3 the undersampled k-spae had a variable density sampling pattern. RESULTS Figure 3 shows the SNR-saled images (intensity units of SNR) and g-fator maps for the 2D slie through the

8 902 Robson et al. FIG. 3. SNR-saled images and g- fator maps (three leftmost olumns, SNR top, g-fator bottom) for various aeleration fators (ORF shown at left) using generalized SENSE reonstrution. There is exellent agreement between all methods: (i) atual multiple replia, (ii) pseudo multiple replia, and (iii) diret alulation. Iterative CG reonstrution is used for the multiple replia methods and matrix-inversion for the diret SNR method. Differene maps (two rightmost olumns) for eah aeleration between methods show negligible residual struture between all methods. (Phase-enoding/aeleration diretion is left right.) phantom reonstruted using a SENSE image reonstrution. Measurements have been made using three methods: (i) atual multiple replia method, (ii) pseudo multiple replia method, and (iii) diret SNR method. SNR maps are shown for: (a) fully sampled image, (b) threefold regularly undersampled, and () variable-density undersampling with an outer redution fator (ORF) of 4 and dense fully sampled enter of 16 lines giving total aeleration fator R 2.3. Differene maps between the pseudo multiple replia measurement and gold standard methods are shown for both SNR and g-fator maps. Differene maps are normalized to the gold-standard measurement. Figure 4 shows equivalent images using a GRAPPA image reonstrution. Interestingly, in the g-fator maps for the GRAPPA image reonstrution shown in Figure 4 there are loalized regions in whih the g-fator is less than one. This is a surprising feature whih is disussed further below. There is exellent agreement between the SNR and g- fator maps alulated with all methods and for all aelerations both in spatial distribution and in the overall saling of SNR and g-fator values. The differene maps show there is no residual differential struture in the SNR or g-fator maps between methods. Computations of the mean value in the differene maps inside a mask of the phantom show that systemati error in the SNR and g- fator maps from the pseudo multiple replia method are always within 2% of the gold standard measurement. These small global errors are likely due to small inauray in the measurement of the noise ovariane matrix

9 Comprehensive Quantifiation of SNR and g-fator 903 FIG. 4. SNR-saled images and g- fator maps (three leftmost olumns, SNR top, g-fator bottom) for various aeleration fators (ORF shown at left) using GRAPPA reonstrution. There is exellent agreement between all methods: (i) atual multiple replia (ii) pseudo multiple replia, and (iii) diret alulation. Conventional onvolution-like reonstrution is used for the multiple replia methods and matrix-inversion for the diret SNR method; both use SNR-optimal omplex oil-ombination. Differene maps (two rightmost olumns) for eah aeleration between methods show negligible residual struture between all methods. (Phase-enoding/aeleration diretion is left right.) when omparing to atual multiple replia noise, or due to some small saling differene that exists between the diret image reonstrutions and the reonstrution methods used for the multiple replia measurements. As disussed by Kellman and MVeigh (7) the noise in the diret SNR method is determined from more samples (64 64) of the noise in the noise ovariane matrix than the number of replia images (128); therefore, the SNR and g-fator maps for the multiple replia methods appear noisier than those from the diret SNR method. The light bands at the edges of the FOV evident in the gold-standard multiple replia differene maps in Figures 3 and 4 are a onsequene of a sharp roll-off to the top-hat frequeny response of the digital reeiver whih is not repliated in the pseudo noise; this does not affet the noise in the bandwidth inluding the image. Figure 5 shows SNR and g-fator maps for both a SENSE and a GRAPPA image reonstrution method. Data were undersampled with an ORF of 2 in twodimensions and inluded a dense enter of k-spae of 8 lines in both dimensions giving a total redution fator R 3.1. Exellent agreement is observed between the pseudo multiple replia method and the gold standard atual multiple replia method. Mean values of the differene maps within the mask of the phantom show systemati differenes in the SNR and g-fator maps between the pseudo and atual multiple replia methods of less than 2%. It is interesting to note that despite being applied to exatly the same data, the SENSE and GRAPPA reonstrutions give very different patterns of SNR and g-fator, as also observed by Thunberg and Zetterberg (13).

10 904 Robson et al. FIG. 5. SNR-saled images and g-fator maps (SNR middle-top, g-fator middle-bottom) for 2D aelerated generalized SENSE and GRAPPA image reonstrutions. A fully sampled and aelerated image is shown (top-left/ top-right for eah reonstrution method); an ORF of 2 in both dimensions was used with a dense enter of 8 lines in eah dimension giving R 3.1 for the full matrix size of CG-Generalized SENSE and onvolution-like GRAPPA with omplex-ombination reonstrutions were used. There is exellent agreement between pseudo multiple replia and atual multiple replia methods. Differene maps (bottom) between methods show negligible residual struture. Figure 6 shows SNR and g-fator maps for the axial 2D brain image reonstruted using a SENSE and a GRAPPA tehnique undersampled with an ORF of 4 and inluding a dense fully sampled enter of 32 lines giving a total aeleration R 2.9. The SNR-saled images and g-fator maps alulated using the pseudo multiple replia method are ompared with those alulated using the diret SNR method and show exellent agreement (note that atual multiple replia measurement is not feasible due to subjet motion limitations). Mean values of the differene maps within the mask of the brain show systemati differenes in the SNR and g-fator maps between the pseudo multiple replia and diret SNR methods of less than 1%. The g-fator maps in the brain are familiar from those in the phantom where the overall geometry and the oil sensitivity distributions are similar. Figure 7 shows SNR and g-fator maps for a oronal and axial-reformatted slie from the 3D volumetri image of the abdomen. Aquired data was undersampled with an ORF of 2 in both phase-enoded diretions with a densely sampled enter of 20 lines in eah diretion and a total aeleration fator R Image reonstrution used a generalized SENSE approah. No diret alulation of SNR and g-fator is omputationally feasible with this k-spae trajetory, beause the reonstrution matrix was approximately 6 Gb (for a matrix size of being undersampled by R 2.63 and with 8 oils, see Limitations of Conventional Noise Analysis in the Introdution setion); nor is an atual multiple replia method feasible in vivo. DISCUSSION g-fator Less Than Unity for GRAPPA It was noted in the Results setion that a g-fator less than 1 is found when using a GRAPPA reonstrution. It is well established that a g-fator less than one is not possible for SENSE reonstrutions that use a matrix inverse of the entire under-sampled k-spae data (2). Any image reon-

11 Comprehensive Quantifiation of SNR and g-fator 905 FIG. 6. SNR-saled images and g- fator maps (SNR middle-top, g- fator middle-bottom) in vivo for generalized SENSE and GRAPPA image reonstrutions. A fully sampled and aelerated image is shown (top-left/top-right for eah reonstrution method); an ORF of 4 was used with a dense enter of 32 lines giving R 2.9 for the full matrix size of CG-Generalized SENSE and onvolutionlike GRAPPA with omplex-ombination reonstrutions were used with the pseudo multiple replia method; matrix-inverse generalized SENSE and full matrix GRAPPA image reonstrution was used for the diret method. There is exellent agreement between pseudo multiple replia and diret alulation methods. Differene maps (bottom) between methods show negligible residual struture. strution (inluding modifiations to SENSE tehniques) may inlude expliitly some degree of numerial regularization to improve the apparent SNR and in suh ases a g-fator less than unity may be found (24). We infer from the observation of g 1 that our implementation of the GRAPPA tehnique inludes some impliit onditioning of the noise. This has been observed and disussed in Robson et al. (9). The agreement of a diret alulation of g-fator and an atual multiple replia method indiates that this observation of g 1 is a genuine finding. Suh inherent onditioning is likely to stem from alulation of the weighting fators for fitting missing lines in k-spae on a least-squares basis. Indeed, in the related SMASH reonstrution tehnique (1), the fitting proess is expliitly approximate. Pseudo Multiple Replia The pseudo multiple replia method has been shown to be an aurate method of quantifying the image noise for both SENSE and GRAPPA tehniques when ompared with diret alulation of image noise and g-fator from the image noise matries and to the gold-standard atual multiple replia values. Validation of the pseudo multiple replia method has been further demonstrated against the gold-standard atual replia method for a variable-density 2D-aelerated parallel imaging aquisition for whih it is not feasible to alulate image noise diretly due to the extremely large reonstrution matrix required for the diret alulation of image noise. While omputation time for the pseudo multiple replia method is also long it requires repetitions of smaller omputations and is therefore a preferred approah when diret approahes beome unfeasible. The method has been used to produe maps of g-fator without requiring aquisition of an unaelerated image permitting the method to be used in vivo when the need to aquire a lengthy unaelerated image would be prohibitive (e.g., for volumetri image aquisition). Generation of g-fator maps in addition to SNR maps is benefi-

12 906 Robson et al. FIG. 7. SNR and g-fator maps alulated using the pseudo multiple replia method for slies from a 3D volumetri image aquired in vivo in a 22-s breathhold. A CG-generalized SENSE image reonstrution of data undersampled in two dimensions with ORF 2 2, a dense enter of lines and total aeleration R 2.63 was used. The graysale maximum of 30/40 for SNR maps orresponds to oronal/axial orientations, respetively. ial for assessing noise amplifiation beause SNR maps inlude spatial variations due to the magnetization density as well as noise amplifiation. Furthermore, SNR maps inlude artifats from the image reonstrution whih should not be onfused with loalized regions of noise amplifiation. Additionally, the pseudo multiple replia tehnique is immune to the influene of instrumental drift and is, therefore, likely to out-perform aquisition of atual multiple replias (4,5,21). The noise presan is both rapid and simple. For a typial reeiver bandwidth of 60 khz, the noise pre-san ould take as little as one sixth of a seond given 10 4 samples required to measure the noise standard deviation to within 1%. Data aquisition simply requires opening the reeiver without RF exitation, whih easily may be inorporated into any imaging pulse sequene program. In this work, we have made possible the omparison between the noise harateristis of different image reonstrution methods. In our implementations, it is apparent that GRAPPA tehniques intrinsially ontrol noise to a greater extent than SENSE tehniques and demonstrate a more diffuse spatial variation of image noise as seen in their g-fator maps (13). In future studies, the pseudo multiple replia method will allow omprehensive, quantitative omparison between reonstrution tehniques for whih diret image noise alulations may not be available or may be omputationally intratable. For example, in Figure 3, it is interesting to note that a lower total aeleration of R 2.3 displays higher g-fator values and lower SNR due to the variable density of its k-spae trajetory with ORF of 4 when ompared with equivalent values for the regularly undersampled threefold aeleration; similar investigation of variable density trajetories where diret alulation of image noise is not omputationally feasible will be possible. Furthermore, beause the pseudo replia method brakets the entire image aquisition and reonstrution proess inserting random noise at the beginning and omputing image noise in the final step, it is possible with this method to analyze the effet of image proessing steps, for example, an expliit noise onditioning step, or the stopping ondition hosen in iterative approahes. Computation time is the prinipal drawbak to the pseudo multiple replia method whih may prelude routine implementation of SNR analysis. The lengthy image postproessing time may be drastially redued by implementing the analysis with optimized C-programs run on a faster dediated omputer. Furthermore, reonstrution of separate image replias may be trivially parallelized on luster or multi-ore arhitetures. However, an important appliation of the SNR and g-fator analysis proposed here will be in developing new parallel imaging tehniques and designing linial imaging protools ases in whih omputation time may not be a primary onstraint. CONCLUSION The pseudo multiple replia image noise measurement outlined here is a simple, robust and aurate method for the quantifiation of SNR and g-fator for all parallel imaging tehniques whih use a linear image reonstrution algorithm, enompassing the majority of linial parallel imaging appliations. This allows noise analysis for reonstrutions that do not permit diret alulation of image noise. This approah will provide a useful tool for objetive omparison between the in vivo-performane of parallel imaging methods being universally appliable with any signal aquisition shemes, k-spae sampling shemes or trajetories, image reonstrution methods, image proessing, or regularization tehniques, hopefully guiding