Regularized Least Squares Estimating Sensitivity for Self-calibrating Parallel Imaging

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1 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY Reguaized Least Squaes Estimating Sensitivity fo Sef-caibating Paae Imaging LIU Xiao-fang Coege of Compute Science and Technoogy, Zhejiang Univesity, HangZhou, China Depatment of Biomedica Engineeing, China Jiiang Univesity, HangZhou, China Emai: YE Xiu-zi and ZHANG San-yuan Coege of Compute Science and Technoogy, Zhejiang Univesity, HangZhou, China LIU Feng Schoo of Infomation Technoogy & Eectica Engineeing,The Univesity of Queensand, Bisbane, Austaia Abstact Caibation of the spatia sensitivity functions of coi aays is a cucia eement in paae magnetic esonance imaging (pmri). The sef-caibating technique fo sensitivity extaction has compemented the common caibation technique that uses a sepaate pe-scan. In ode to impove the accuacy of sensitivity estimate fom sma numbe of sef-caibating data, which is extacted fom a fuy samped centa egion of a vaiabe-density k-space acquisition in sef-caibating paae images, a nove scheme fo estimating the sensitivity pofies is poposed in the pape. On consideation of tuncation eo and measuement eos in sef-caibating data, the issue of cacuating sensitivity woud be fomuated as a eguaized east squaes estimation pobem, which is soved by the peconditioned conjugate gadients agoithm. When appying the estimated coi sensitivity to econstuct fu fied-of-view(fov) image fom the unde-samping simuated and in vivo data, the nomaized signa-to-noise atio () of econstuction image is evidenty impoved, and meanwhie the nomaized mean squaed eo (NMSE) is emakaby educed, especiay when a athe age acceeate facto is used. Index Tems paae magnetic esonance imaging (pmri), sef-caibating technique, eguaized east squaes (RLS), peconditioned conjugate gadients (PCG), geneaized encoding matix(gem) econstuction I. INTRODUCTION Paae magnetic esonance imaging (pmri) is a apid acquisition technique and consideed to be one of the moden evoutions in the fied of MRI. The technique simutaneousy sampes the educed k-space data and uses the infomation fom mutipe eceives to econstuct fu Fied-Of-View (FOV) image. In paae imaging, since a cetain amount of the spatia encoding, taditionay achieved by the phase-encoding gadients, is substituted by evauating data fom sevea coi eements with spatiay diffeent coi sensitivity pofies, the choice of sensitivity caibation stategy is at east as impotant as the choice of econstuction stategy [1]. Unfotunatey, the existing techniques fo detemining sensitivity functions ae not satisfactoy. The most common technique has been to deive sensitivities diecty fom a set of efeence images obtained in a sepaate caibation scan befoe o afte the acceeated scans. This caibation step can poong tota examination time, patiay counteacting the benefits of deceased acquisition time associated with PMRI. whie, it aso intoduces a possibe souce of eo into the PMRI econstuction, as it is difficut to ensue that the patient and coi aay wi be in the same positions duing both the caibation scans and the acceeated data acquisitions. Adaptive sensitivity estimation is anothe technique fo detemining sensitivity functions, which is a majo concen in dynamic imaging appications. Based soey on the data fom acceeated scans, the method uses unaiasing by Fouie-encoding the oveaps using the tempoa dimension (UNFOLD) [2] to geneate owtempoa-esoution, aiasing-fee efeence images fo sensitivity estimation. Howeve, UNFOLD is imited to dynamic appications in which at east haf FOV emains static ove time. A moe genea method is the sef-caibating technique, which aso eiminates a sepaate caibation scan but acquies vaiabe-density k-space data duing the acceeated scan [3,4]. In addition to the down-samped ines at oute k-space, the vaiabe-density acquisition incudes a sma numbe of fuy samped ines at the cente of k-space, known as auto-caibation signa (ACS) ines in geneaized auto-caibating patiay paae acquisitions (GRAPPA) [5], namey sef-caibating ines. These sef-caibating ines afte Fouie tansfomation poduce ow-esoution efeence images esoution [ f( ) c ( )]ow as fo any given component coi, whee ow-esoution supescipt indicates that use of ony the centa k-space positions, that esuts in a ow- Poject paty suppoted by ZheJiang povince Natua Science Foundation(Y , Y , Y ) doi: /jcp

2 858 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY 2011 esoution measuement of the poduct of the image of tansvese magnetization f( ) and the coi sensitivity c( ).To deive the sensitivities, these ow-esoution efeence images ae divided by thei sum-of-squaes(sos) combination[3,6,7]: ow esoution [ f( ) c ( )] cˆ ( ) ow esoution 2 [ f( ) c ( )] (1) In genea, the appoximation in Eq.(1) equies the ange of spatia fequencies cove in the sef-caibating ines is much boade than the spatia fequency band of the coi sensitivity functions[8]. Howeve, this incease the acquisition time associated with pmri and contadicts the goa of pmri. If a sma numbe of sef-caibating ines ae used, tuncation of high spatia fequency components of tansvese magnetization f( ) esuts in Gibbs inging atifacts in the extacted sensitivity efeence images. These inging eos become seious especiay at ocations whee the object tansvese magnetization has high-spatia-fequency components. Howeve, these Gibbs inging eos in efeence images cannot be canceed by the division of thei SoS combination as Eq.1, and can hady be educed by the commony used poynomia-fitting[9] o waveetdenoising [10] techniques fo sensitivities. Consequenty, the pmri econstuction intepets vaying degees of Gibbs inging as actua featues of the sensitivities, and coesponding sensitivity-mismatch atifacts can esut. Theefoe, to impove the sensitivity accuacy with a sma numbe of sef-caibating ines is cucia fo pmri to achieve a high acceeation. Towads quiete and faste imaging, the pape woud study the method fo impoving the accuacy of sensitivity estimate fom sma numbe of sef-caibating data, and the quaity of econstuction images woud be used to evauate the effectiveness of the poposed method. Using ony the sef-caibating data in a vaiabe-density acquisition, a nove method fo estimating the sensitivity pofies woud be poposed in the pape. On consideation of Gibbs inging atifacts and noise in sensitivity efeence images, which is geneated fom sefcaibating data, this method woud view the issue of estimating c ( ) fom these images as a inea estimation pobem, and eguaized east-squaes methods ae used to estimate the sensitivity pofies. In ode to obtain the stabe soution, the conjugate gadients agoithm, togethe with an acceeation scheme as cicuant peconditione, is used to sove this estimation pobem. In the pape, the suitabe numbe of sef-caibating centa ines fo geneating the sensitivity efeence images woud aso be discussed. In combination with the geneaized encoding matix (GEM) econstuction [11] method, the effectiveness of this sef-caibation method woud be demonstated via phantom and in vivo bain imaging study. II. THEORY AND METHODS A. Figues and Fomuation of sensitivity estimation in sef-caibating paae imaging To impove SNR and educe acquisition times, the use of mutipe eceive cois has become inceasingy popua in MRI. Let c ( ) denote the sensitivity of the th coi, fo =1,..., L, whee L denotes the numbe of cois. Let y ( ) denote the ecoded measuements associated with the th coi, then the genea fowad mode fo the MR measuement signa associated with the th coi is : () ()() ik y c e d 1,... L (2) ) denotes the object s tansvese Whee ρ( magnetization; k is the chosen k-space tajectoy indices (e.g., k (kx, ky)) epesenting a tota numbe of acquied data points measued in the pesence of vaious fequency- and phase-encoding gadients. The measuement eos ε ae modeed by additive, compex, zeo-mean white gaussian noise [12,13,14]. With ony a modest oss of geneaity[15], the intega in Eq. (2) may be appoximated with a discete sum: y k c ( ik ) ( ) ( )e ( ) ( ) Hee the vecto index in the sum indicates a summation ove a discete pixe positions in the image. Simpifies Eq.(3) and yieds the foowing matix-vecto equation: Figue 1. A sampe vaiabe-density k-space tajectoy made up of a eguay unde-samped oute potion and a fuy samped inne potion. The inne potion may be used as a ow-esoution sensitivity efeence fo PMRI econstuctions. (3) y =c ρ+ε (4) Fo coi sensitivity vaies sowy as a function of dominant spatia vaiations position, ow-esoution invivo images suffice to fom sensitivity efeences data [16]. In sef-caibating paae imaging, the vaiabedensity data acquisition schemes ae usuay adopted (see Fig.1). Hee, k-space is effectivey spit into two egions: a centa egion in which a phase-encode ines ae fuy samped, and an oute egion in which the ines ae unifomy unde-samped. The centa ines of k-space ae extacted and Fouie tansfomed to yied ow-esoution efeence images as the measuement data of sensitivity efeence y efeence. Due to the Gibbs inging atifacts in the efeence images, a sufficient numbe of fuy samped centa k-space ines shoud be acquied to extact the intena efeence images. The minimum numbe of centa ines is geneay 10~20.

3 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY Accoding to Eq.(4), the foowing equation woud be poposed in the pape: efence ow esoution y () c() e (5) Hee, e denotes any eo, such as tuncation eo and measuement eos. Aiming at estimating sensitivity pofies, ρ ow-esoution woud be cacuated by the sum-ofsquaes (SoS) combination of these efeence images, which can be the ow-esoution econstuction image and noted as G in the pape. At ast, the foowing mode woud be used to cacuate the sensitivity pofies: y =Gc +e (6) As shown in Eq.(6), a cetain appoaches might be taken to emove the magnetization distibution and isoate the encoding effects of pue coi sensitivities. B. Refeences Reguaized east-squaes estimation of coi sensitivity maps Because c appeas inea scaing in Eq.(6), the pobem of estimating c woud be a inea estimation pobem, which can be esoved by minimizing the foowing eguaized east-squaes cost function: cˆ ag min ( c ) c 1 efeence 2 ( c) y Gc R( c) 2 2 (7) Whee ĉ denotes the estimation vaues of c, and y efeence denotes the sensitivity efeence images; R(c ) is eguaizing oughness penaty functions that encouages piecewise-smooth estimates, and β is eguaized paametes that conto the smoothness of the estimates. Thus, ou goa is to compute an estimate ĉ of c fom by finding the minimum of the objective function Ф(c ). In Eq.(7), an open pobem is how to best choose the eguaize R(c ). If this tem is not incuded, then the sensitivity estimate ĉ wi suffe fom noise and Gibbs inging atifacts because this invese pobem is iconditioned. The simpest choice is Tikhonov eguaization, anothe choice is a quadatic oughness penaty function, which penaizes diffeences between neighboing pixe vaues. The foowing eguaizing penaty function woud be used in this pape: K Rc ( ) k([ Dc] k) k 1 (8) Whee φ is a convex potentia function, and K is the numbe of pais of neighboing pixes; D is a K n d matix, and each ow of D contains one "+1" and one "-1" enty so that [Dc ] k coesponds to the diffeence between two neighboing pixe vaues. Fo coi sensitivity is assumed smooth, the quadatic potentia function φ(t)=t 2 /2 woud be adopted and second-ode diffeences ae used fo a esuts in ou study. The quadatic potentia function is convenient fo minimization [17], but it has the dawback of smoothing image edges, paticuay if the eguaization paamete β in Eq. (7) is too age. C. Minimization using peconditioned conjugate gadients agoithms Unde the above assumptions, we can detemine the unique minimize ĉ of the cost function Ф(c ) by finding the zeo of its gadient. Fo diffeentiabe eguaizes, the gadient of Ф(c ) is expessed as: ' ( c ) b Hc Whee H is the Hessian matix of Ф(c ); H=G G+βD D; efeence b G y is essentiay a back pojection of y efeence ; and ae espectivey the conjugate and gadient opeatos. Fo the quadatic penaty, the zeo of the gadient is simpy the soution of the inea system of equations H ĉ =b. Howeve, due to the size of H fo eaistic image sizes, we cannot compute diecty the anaytica soution. Thus, the iteative methods as peconditioned conjugate gadients (PCG) agoithm woud be used to compute ĉ. Fo quadatic objectives, peconditioning the CG agoithm is equivaent to soving the tansfomed inea system of equations M 1/2 HM 1/2 M -1/2 ĉ = M 1/2 b, whee M is the peconditione matix. The idea peconditione woud be M=H -1 because the convegence ate of the CG agoithm geneay impoves as the condition numbe of MH deceases towad unity. Since we cannot compute H- 1 fo age nd pixes, the cicuant peconditione is put fowad to appoximate H -1, which defined as M T Q Ω 1 (β)qt[18], whee the matix T is the identity matix; Q is the othonoma vesion of the 2-D discete Fouie tansfom (DFT) opeato, and Ω(β) is a diagona matix fo any β such that: H T Q Ω (β)qt. In genea, Ω consists of the 2-D DFT coefficients of the coumn of H coesponding to the pixe at the cente of the image. D. Post-pocessing As mentioned eaie, system of Eq. 7 is undedetemined and ony the ow-esoution of SoS econstuction image is used to cacuate the sensitivity pofie, so that the econstuction image is not necessaiy identica to a typica SoS econstuction image, especiay when the sensitivity pofies estimated by Eq. (7) with diffeent eguaization paametes ae used. This diffeence manifests itsef as sight changes in intensity distibution. By mutipying the econstuction image with the oot of the sum of squaes of the estimated coi sensitivities and dividing the coi sensitivities by the same quantity, this diffeence can be emoved in a simpe step. Athough not sticty necessay, this post-pocessing is usefu when cacuating the nomaized mean squaed eo (NMSE) of the econstuction images with the SoS of fu images as the god standad. Nevetheess, it is not necessay fo this post-pocessing when the nomaized signa-to-atio(snr) of econstuction images is used to quantitativey evauate the poposed method. III. DATA ACQUISITION AND ANALYSIS (9)

4 860 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY 2011 A. Data Acquisition Unacceeated anatomica images of a standad esoution phantom wee obtained fom a 3T (Siemens Medica Soutions) human using an 8-channe head aay coi. Imaging paametes: echo time (TE)=3.45ms, epetition time (TR)=2530ms, T1=1100 ms, Fip ange=7deg, sice thickness=1.33mm, FOV=256*256 mm. The B1 coi maps wee cacuated using Biot-Savat's aw, and the fuy samped k-space data wee obtained by invese Fouie tansfoming the acquied unacceeated images. A in-vivo fuy samped bain dataset was obtained fom PULSAR(a matab toobox fo paae MRI) [19], which was acquied using MR systems with eightchanne head aay and muti-channe eceive fom a heathy mae vountee with fast spoied gadient-echo sequence, TR/TE =300/10 ms, matix size = , tip ange=15 and FOV = 22*22 cm. To simuate the unde-samped datasets in the manne of taditiona vaiabe-density data acquisition, the acquied fu-samped k-space data was decimated using eduction factos, namey acceeated facto R=2, 3, 4 and 5 fo simuated and in vivo dataset. whie, the centa k-space data wee fuy samped to geneate the sensitivity efeence images. The numbe of centa ines aong phase-encode was chosen between 6 and 32. B. Anaysis The accuacy estimation of coi sensitivity is a majo deteminant of the quaity of paae magnetic esonance image econstuctions. Sef-caibation paae imaging techniques can eiminate the need fo an extena sensitivity efeence, making econstuction image ess susceptibe to miscaibation and image degadation. Howeve, in sef-caibation paae imaging, the sensitivity pofies cacuated by Eq.(1) might have the infomation of tansvese magnetization distibution, Gibbs inging and noise fom the data acquisition, and esut in the degadation of econstuction image. In ode to impove the accuacy of coi sensitivity estimation, we popose the eguaized east-squaes method to estimate the coi sensitivity pofies fom the efeence images, and image econstuctions ae impemented by GEM econstuction method in the MATLAB pogamming anguage to test ou method. Fo the eguaized east squaes method, choosing the eguaization function was a key pobem. Accoding to the chaacteistics of sensitivity function, we woud adopt a quadatic diffeentiabe penaty function. Fo eguaized east squaes estimating sensitivity pofies is a age inea estimation pobem, diect soution methods ike the Choesky factoization ae not pactica. In this case, an iteative technique known as the conjugate gadient (CG) method, togethe with an acceeation scheme cicuato peconditioning, povides us a vey efficient means of soving this inea system [18,20]. In ode to compaative evauation, two sets of coi sensitivity pofies woud be used in ou study. One set of coi sensitivity pofies is caed ough sensitivity map, which ae diecty estimated fom the intena efeence images by Eq.(1); The othe set is caed RLS sensitivity maps, which ae estimated by PCG agoithm to esove Eq.(7). They woud be used to econstuct fu FOV images, and the quaity of econstuction image woud be taget fo evauating the accuacy of estimated sensitivity pofies. In ode to quantitative anayze the quaity of econstuction image, signa-to-noise atio (SNR) and the nomaized mean squaed eo (NMSE) of image woud be cacuated. Howeve, we woud neve expicity cacuate the pue sensitivities in the sef-caibating econstuction. Fo the puposes of compaative evauation, we ae inteested in detemining SNR nomaized eative to an optima image fom the coesponding un-acceeated acquisition, so the so-caed pixe-to-pixe nomaized SNR woud be used as Eq.10 in the pape. SNR nomaised 1 () N ) ( ) ( ( ) ) ( ) N fu 1/2 efeence 2 1/ /2 acquied f c k, k (10) Hee, N fu, N acquied epesent the un-acceeated and acceeated numbe of phase-encoded ines espectivey. y efeence denotes a ow-esoution efeence image in a given component coi, which is Fouie tansfomation of the centa k-space ines. ĉ epesents the coi sensitivity pofies cacuated as intoduced befoe. It efeence ) ( f ( ) ) ( c ) 2 1/ /2 k, k coud be seen that is so-caed the geometica noise mutipication facto, o g-facto. NMSE is defined as the nomaized diffeence squae between the econstucted image and the standad image f standad, which was the SoS combination of unacceeated images: 2 NMSE f econstucted ( ) fst andad ( ) 2 f ( ) st andad (11) This definition is equivaent to the atifact powe (AP), and a highe vaue of NMSE (o AP) epesents educed image quaity, which suggests both inceased image atifacts and noise. IV. RESULTS A. Simuated esuts Fig.2 shows the B1 maps, the efeence images, ough sensitivity maps and RLS sensitivity pofies of 8 eceive cois in simuated study. In Fig.2, the B1 maps ae cacuated by Biot-Savat's aw; the efeence images ae extacted fom 16 sef-caibating ines aong phase encoding. Fom the efeence images, the ough sensitivity maps ae cacuated by Eq.(1), and RLS sensitivity pofies ae estimated by Eq.(7). In ode to sove Eq.(7), iteative method peconditioned CG agoithms ae used to estimating the sensitivity pofies,

5 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY whee eguaizing penaty function R(c ) adopts a convex diffeentiabe function fo quadatic penaizing second finite diffeences of c. RLS sensitivity pofies in Fig.2 ae obtained afte 8 odes iteative, and the initia guess fo iteative pocedue is the edge of efeence images detected by Canny method. Figue2. Refeence images and sensitivity pofies of individua eceive coi. Hee, the efeence images ae extacted fom centa k-space data and numbe of centa ines was 16. Fom efeence images, ough sensitivity pofies and RLS sensitivity pofies ae estimated by Eq.(2) and Eq.(7) espectivey. Fo RLS sensitivity pofies, iteative ode=8, eguaized paamete β=4, the quadatic potentia function φ(t)=t 2 /2, diffeent ode=2. Using the coi sensitivity maps shown as Fig.2, fu- FOV images ae econstucted fom the unifom undesamping data by GEM econstuction method. Fig.3 shows the econstuction images and thei pixe-to-pixe nomaized SNR maps when acceeation facto 4 is used. Tabe I Nomaized SNR and NMSE of econstuction images in phantom study R B1 COIL MAPS ROUGH SENSITIVITY RLS SENSITIVITY of NMSE of NMSE of NMSE e e R: Unde-samping ate, namey acceeate facto; : pixe-to-pixe nomaized SNR; NMSE: nomaized mean squaed eo. Fom the fuy-samped data in 16 k-space centa ines, Rough sensitivity wee estimated by Eq.(1), and RLS sensitivity estimated by Eq.(7), which soved by PCG agoithm, the paametes: iteative ode=8, eguaized paamete β=3, the quadatic potentia function φ(t)=t 2 /2. intena sensitivity efeence images is aso studied in the pape. In Fig. 4a and b, the mean of nomaized SNR and NMSE of econstuction images fom the diffeent undesamping data ae potted as a function of the numbe of sef-caibating data ines. Fig.4a shows mean of nomaized SNR of the econstuction images which use the ough sensitivity and RLS sensitivity pofie, whie Fig.4b shows NMSE of econstuction images. As seen in Fig.4, mean of nomaized SNR of the econstuction Figue 3. The fist ow shows the econstuction images fom a set of eight-channe unde-samping data using (a) simuated b1 maps (b) ough sensitivity pofies (c) RLS sensitivity pofies. Hee, the eduction facto R was 4.Thei pixe-to-pixe nomaized SNR maps ae aso showed on the second ow. With the SoS of fu images as the god standad, Tabe.1 ists mean of nomaized SNR and NMSE of econstuction images espectivey using B1 maps, ough sensitivity pofies and RLS sensitivity maps shown in Fig.2. In Tabe 1, ough sensitivity maps and RLS sensitivity maps ae estimated fom 16 sef-caibating ines data. Using these RLS sensitivity maps, the standad deviation of nomaized SNR of econstuction images fom unde-samping data of eduction facto R= 2, 3, 4 and 5 ae , , and espectivey. As a sef-caibating technique, the suitabe numbe of centa k-space phase-encoding ines fo geneating the Figue 4. (a) nomaized SNR and (b)nmse of econstuction images. Hee, ough denotes ough sensitivity; RLS denotes RLS sensitivity pofies. They ae extacted fom 6 to 32 sef-caibating ines espectivey, and econstuction images fom unde-samping data when acceeated facto R is 2, 3, 4. images is impoved, and NMSE of econstuction images is emakaby educed. B. In vivo study esuts Fig.5 shows the efeence images, ough sensitivity maps and RLS sensitivity pofies of eight eceive cois espectivey fo in vivo study. In Fig.5, the efeence images ae extacted fom the fuy-samped data in 16 centa k-space ines aong phase encoding diection. Fom the efeence images, the ough sensitivity maps ae cacuated by Eq.(1), and RLS sensitivity pofies ae estimated by Eq.(7), which is esoved by iteative method PCG agoithms as above simuated study.

6 862 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY 2011 function of the numbe of sef-caibating data ines. As seen in Fig.7, when RLS sensitivity pofies ae used, mean of nomaized SNR of the econstuction images is impoved, and NMSE of econstuction images is emakaby educed compaed with ough sensitivity pofies used. Figue 5. Sensitivity pofies of eight eceive coi. (a) The sensitivity efeence images extacted fom 16 sef-caibating ines. Fom these efeence images, (b) Rough sensitivity pofies estimated by Eq.(1); (c) RLS sensitivity pofies estimated by Eq.(7), which is soved by the same method as RLS sensitivity in Fig.2. Using the ough sensitivity pofies and RLS sensitivity shown in Fig.5, the fu-fov MR images econstucted fom the unde-samped in vivo data by GEM econstuction method ae shown in Fig.6, when eduction facto R=2,3,4 is used. Figue7. (a) nomaized SNR and (b)nmse of econstuction images. Hee, ough denotes ough sensitivity; RLS denotes RLS sensitivity pofies. They ae extacted espectivey fom 6 to 32 sef-caibating ines, and econstuction images fom undesamping data when acceeated facto R is 2, 3, 4. V. DISCUSSION Figue 6. The fist ow shows in vivo econstuction images using RLS sensitivity pofies in Fig.4, which fom the unde-samping data when acceeated facto R is 2, 3, 4 espectivey. The second ow shows thei pixe-to-pixe nomaized SNR maps as (a) (b) (c). Using ough sensitivity and RLS sensitivity pofies as shown in Fig.5, Tabe.2 ist mean of the nomaized SNR and NMSE of econstuction images espectivey fom the unde-samping data of acceeate facto R= 2, 3, 4 and 5. Tabe II and NMSE of econstuction images in in-vivo study R ROUGH SENSITIVITY RLS SENSITIVITY of Std of NMSE of Std of NMSE R, NMSE, Rough sensitivity, RLS sensitivity, : definitions as Tabe.1; Std: standad deviation Same as simuated study, the suitabe numbe of sefcaibating ines fo geneating the sensitivity efeence images is aso studied via in vivo study. Fig.7 shows the mean of nomaized SNR and NMSE of econstuction images fom the diffeent acceeate data ae potted as a The accuacy of coi sensitivity estimates is a majo deteminant of the quaity of paae magnetic esonance image econstuctions. Sef-caibating the coi sensitivity pofies can eiminate the need fo an extena sensitivity efeence, and theeby educes tota examination time. The pape poposed a nove sef-caibating sensitivity method, which viewed the issue of estimating the sensitivity pofies fom sef-caibating data as a inea estimation pobem [21,22]. On consideation of measuement eos and tuncation eo, the eguaized east-squaes method was used to estimate the sensitivities pofies. When the estimated sensitivity pofies wee used to econstuct fu FOV image fom unde-samping data, as seen in Tabe1 and Tabe 2, the quaity of econstuction images was emakaby impoved. Though the methods poposed and the expeiment pesented, the foowing aspects coud be taken into account: (1) In sef-caibation paae imaging, the numbe of fuy-samped centa ines fo geneating the intena efeence images must be caefuy chosen. In ode to be faste imaging, it shoudn t be necessay to acquie much sef-caibating data. Howeve, when a sma numbe of centa ines data ae used fo sensitivity estimation, these data ae tuncated fom the oigina fu data with a sma tuncation window, which causes the estimated sensitivities to have seious Gibbs inging. In ou study, the numbe of sef-caibating ines is chosen between 6 and 32. As seen in Fig.4, when sef-caibating ines is smae than 10, because of Gibbs ing atifacts, NMSE of

7 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY econstuction image woud be inceased and mean of nomaized SNR be deceased, and these Gibbs inging atifacts coudn t be emoved even by ou method. On the othe hand, as the numbe of sef-caibating ines inceases, NMSE woud decease and nomaized SNR woud incease. Howeve, moe sef-caibating ines impy sowe imaging. Fo a fixed acceeation facto R, acquiing moe fuy samped ines at the cente of k- space educes the Gibbs inging, but at the pice of educing the tue acceeation facto, caed as net acceeation facto, which cacuated by the foowing expession fo a vaiabe-density acquisition: Net acceeation facto= fu N cente fu cente N ( N N )/ R (12) Whee N cente denotes the numbe of fuy samped centa ines in k-space; N fu denotes the numbe of phase encoding ines. Fo exampe, when the acceeation facto is 4 and N fu is 128, the net acceeation facto cacuated by Eq. 12 is 3.5, 3.36, 3.24, 3.12, 2.9, 2.7, 2.56, 2.41, 2.28 espectivey fo 6, 8, 10, 12, 16, 20, 24, 28, 32 ines sef-caibating data. Consequenty, we must be faced with the question that is how to baance the numbe of centa ines and the acceeation facto R in sef-caibating paae imaging. In ou study, the numbe of ines at the cente of k-space woud be fisty taken into account, which shoud be enough so that the coi sensitivities can be fuy chaacteized. In pactice, the pecise owe imit on Ncente that ae necessay wi depend on both the coi aay being used and the geomety of the imaged pane. Afte the numbe of centa ines is fixed on, impoving acceeation facto R can incease the net acceeation facto so as to achieve the desied net acceeation facto. As seen in Fig.6, Tabe 1 and Tabe 2, acceeation facto R is the main eement of degading the quaity of econstuction images, but the exact coi sensitivity coud paty aeviate this contadiction. In sef-caibating paae imaging, the accuacy of coi sensitivity estimates depends on both the intena efeence images and the effective method which can eave infomation about the spatia fequency content of the coi sensitivities neay distubed fom any eo in efeence images. (2)On consideation of measuement eos and tuncation eo in sensitivity efeence images, the eguaized east squaes method is used to estimate the accuacy sensitivity pofies. Fo this method, choosing both the eguaizing penaty function and the eguaization paamete β is a key issue. They conto the smoothness of the estimates can ensue that the iteative agoithm conveges to a stabe esoution. Fo the sensitivity pofies vaies sowy as a function of spatia position and is assumed smooth, the quadatic eguaization is pefeabe. Moeove, quadatic penaty function has the advantage of being diffeentiabe and easy to anayze, especiay with Gaussian noise. Seection of the eguaization paamete β is anothe pactica chaenge with eguaized methods. When β is vey sma, fiteing of the noise is inadequate. On the othe hand, age β coud fite out the noise and Gibbs atifacts, but the object own decoding key is aso fiteed and ead the eduction of nomaized SNR. Lcuve method is the common method fo choosing β. Howeve, this method is expensive because it equies evauating ĉ fo sevea vaues of β. Fo quadatic eguaization, thee is a we-deveoped theoy fo choosing β in tems of the desied spatia esoution popeties of the coi sensitivity maps. Based on this theoy, Fesse et a poposed that simpe measues ike fu width at haf maximum (FWHM) might be easonabe esoution metics. In ou study, eguaization paamete β was simpy chosen between 0.25 and 32. Though tia in tems of the quaity of econstuction image, eguaization paamete 3 was confimed suitabe. (3)The soution to the inea estimation pobem Eq. 7 is anothe issue in ou study. In ode to sove Eq. 7, the foowing symmetic positive definite(spd) inea system is used in this pape: (S+D D)c=b (13) Whee D is penaty deivatives vecto defined above, and D D=R; D denotes conjugate opeation of D; S=G G; b=g yefeence. Fo S in Eq. 13 is SPD, the peconditioned CG agoithm is used to efficienty esove it. Compaed with CG agoithm, the iteative ode is educed fom 256 to 8, and then the stabe soution coud be obtained. Howeve, the PCG agoithm equies a peconditioning step, whie CG does not. (4)The initia guess is impotant fo iteative estimating the sensitivity pofies by CG agoithm. In ou study, the initia guess is the edge of individua caibation images, which acquied by Canny method. Compaed with the zeos and unifom initia guess, this initia guess evidenty impoves the accuacy of sensitivity pofies estimates and esut in highe quaity of econstuction images. Though ou study, it is found that the edge acquied by diffeent method fo initia vaues coud affect the accuacy of sensitivity pofies estimate. Consequenty, the accuacy edge of caibation images woud be main cause fo estimating sensitivity by CG agoithm. (5)In ou study, ony the sef-caibating data is used to eguaized east squaes estimate the sensitivity pofies. Howeve, a detemination of coi sensitivities fom ony the cente of k-space does not take advantage of a avaiabe infomation. Athough the infomation about a smooth coi pofie is mosty ocaized in the k- space cente, the measued data epesents the convoution of the coi pofies with the object function which shift infomation fom the cente of k-space to its oute pats. An optima method fo estimating the sensitivity shoud expoit a avaiabe k-space data athe than ony a sma pat in its cente. Ying et a and Uecke et a ecenty poposed that image econstuction and sensitivity estimation shoud be joint by iteativey optimizing both the coi sensitivities and the image content unti a joint soution was found. (6)The study indicates that atifact and noise suppession in the econstucted image coud be achieved by impoving the eiabiity of the coi sensitivity maps.

8 864 JOURNAL OF COMPUTERS, VOL. 6, NO. 5, MAY 2011 Howeve, in ode to emakaby impove the quaity of econstuction image, the constained econstuction was geneay shown to be an effective method [23, 24, 25]. VI. CONCLUSIONS In the pesent wok, a nove scheme fo estimating sensitivity pofies fom sef-caibating data was poposed and examined. Accoding to mean and NMSE of econstuction images, the sensitivity pofies estimated by this method coud evidenty impove the quaity of econstuction image, especiay when a athe age acceeate facto was used. REFERENCES [1] D. K.Sodickson, C. A.McKenzie, M. A.Ohige, et a. Recent advances in image econstuction, coi sensitivity caibation, and coi aay design fo SMASH and geneaized paae MRI. Magnetic Resonance Mateias in Physics, Bioogy and Medicine, 2002,13(3): doi: /BF [2] B.Madoe. UNFOLD-SENSE: a paae MRI method with sef-caibation and atifact suppession. Magnetic Resonance in Medicine, : doi: /mm [3] C. A. McKenzie, E. N.Yeh, M. A.Ohige, et a. Sef- Caibating Paae Imaging With Automatic Coi Sensitivity Extaction. Magnetic Resonance in Medicine, 2002,47: doi: /mm [4] S.O.Schoenbeg, O.Dietich, M. F.Reise. Paae Imaging in Cinica MR Appications. Spinge-Veag Bein Heidebeg, New Yok. 2007, p [5] M.A.Giswod, P.M.Jakob, R.M.Heidemann, et a. Geneaized autocaibating patiay paae acquisitions (GRAPPA). Magnetic Resonance in Medicine, 2002, 47: doi /mm [6] F.H.Lin, K.K.Kwong, J.W.Beiveau, et a. Paae imaging econstuction using automatic eguaization. Magnetic Resonance in Medicine, 2004, 51: doi: /mm [7] L.Ying, J.H.Sheng. Joint Image Reconstuction and Sensitivity Estimation in SENSE (JSENSE). Magnetic Resonance in Medicine, 2007, 57: doi: /mm [8] L.Yuan, L.Ying, D.Xu, et a. Tuncation effects in SENSE econstuction. Magn Reson Imaging, 2006, 24: doi: /j.mi [9] K.P.Puessmann, M.Weige, M.B.Scheidegge, et a. SENSE: Sensitivity encoding fo fast MRI. Magnetic Resonance in medicine. 1999, 42(5): doi: /(SICI) [10] F.H.Lin, Y.J.Chen, J.Beiveau, et a. A waveet-based appoximation of suface coi sensitivity pofies fo coection of image intensity inhomogeneity and paae imaging econstuction. Human Bain Mapping. 2003, 19: doi: /hbm [11] Puessmann, K.P.. Encoding and Reconstuction in Paae MRI. Magnetic Resonance in Medicine, 2006,19: doi: /nbm.1042 [12] A.Macovski. Noise in MRI. Magnetic Resonance in Medicine, 1996, 36(3): doi: /mm [13] K.F.Amanda, A.F.Jeffey, T.B.Y.Desmond, et a. Reguaized Fied Map Estimation in MRI. IEEE tansactions on medica imaging, 2008, 27(10): doi: /TMI [14] J. A.Fesse. Mode-based image econstuction fo MRI. IEEE Signa Pocessing Magazine, 2010, 81: doi: /msp [15] D.K.Sodickson, C.A.McKenzie. A geneaized appoach to paae magnetic esonance imaging. Medica Physics, 2001, 28(8): doi: / [16] D. K. Sodickson. Taioed SMASH image econstuctions fo obust in vivo paae MR imaging. Magnetic Resonance in Medicine, 2000, 44(2): doi: / [17] B.P.Sutton, D.C.No, T.A.Fesse. Fast, iteative image econstuction fo MRI in the pesence of fied inhomogeneities. IEEE Tans. Med. Imag. 2003, 22(2): doi: /TMI [18] J.A.Fesse, S.D.Booth. Conjugate-Gadient Peconditioning Methods fo Shift-Vaiant PET Image Reconstuction. IEEE tansactions on image pocessing, 1999, 8(5): [19] J.X.Ji, J. B.Son, S. D.Rane. PULSAR: A MATLAB Toobox fo Paae Magnetic Resonance Imaging Using Aay Cois and Mutipe Channe Receives.Concepts in Magnetic Resonance Pat B, 2007, 31B: doi /cm.b [20] C.R.Voge. Computationa methods fo invese pobems. SIAM Phiadephia. 2002, p F.Baue, S.Kannengiesse. An atenative appoach to the image econstuction fo paae data acquisition in MRI. Math. Meth.App.Sci, 2007, 30: doi: /mma.848 [21] M.Uecke, T.Hohage, K.T.Bock,et a. Image Reconstuction by Reguaized Noninea Invesion Joint Estimation of Coi Sensitivities and Image Content. Magnetic Resonance in Medicine, 2008, 60: doi /mm [22] Z.P. Liang, J.D. Hada, Henando. Constained Reconstuction. ISMRM2009(Intenationa Society fo Magnetic Resonance in Medicine 2009), Api, Hawai'i, [23] A.Raj, Y.Wang, R.Zabih. A Maximum Likeihood Appoach to Paae Imaging With Coi Sensitivity Noise. IEEE Tansactions on medica imaging, 26: (2007). doi: /TMI [24] A.Ribés, F.Schmitt. Linea invese pobems in imaging. IEEE Signa Pocessing Magazine, 7: 84-99(2008). doi: /msp Liu XiaoFang was bon in XanYang of SanXi povince, China, in She eceived the B.S., M.S. degees fom Depatment of Biomedica Engineeing of ZheJing Univesity, China in 1992 and 2002 espectivey. She is cuenty a POSTGRADUATE STUDENT of Coege of Compute Science and Technoogy in ZheJing Univesity, whee she investigates the technique of Magnetic Resonance imaging and medica images econstuction. She is a LECTURER in Depatment of Biomedica Engineeing of China JiLiang Univesity, china. He eseach inteests incude images econstuction, compex medica signa anaysis and its appication.