International Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)

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1 Internatonal Assocaton of Scentfc Innovaton and Research (IASIR) (An Assocaton Unfyng the Scences, Engneerng, and Appled Research) Internatonal Journal of Emergng Technologes n Computatonal and Appled Scences (IJETCAS) ISSN (Prnt): ISSN (Onlne): Hybrd Samplng Technque for MRI Image Reconstructon Tanuj Kumar Jhamb 1, Vnth Rejathalal 2, V.K. Govndan 3 Department of Computer Scence and Engneerng, Natonal Insttute of Technology Calcut, Inda Abstract: The k-space data s obtaned from sgnals generated by Magnetc Resonance Imagng (MRI) scannng machne, and these sgnals get captured at the rado frequency cols. Accuracy of reconstructon of MRI mages nvolves many factors lke data acquston, data samplng and reconstructon algorthms. Ths paper nvestgates the effect of the type of samplng on the accuracy of the reconstructon. In ths context, varous samplng technques have been revewed, and the frequency encodng and the phase encodng for the k-space data have been explaned. The related works n mage reconstructon lke, Senstvty Encodng approach, SURE approach and sgnal acquston protocol have been revewed. The mechansm of k-space acquston has been dscussed, and a new approach for the k-space samplng has been proposed to mprove the samplng of the captured k-space, and hence to provde a better reconstructed mage. The performance of the proposed approach s evaluated usng mean absolute error, mean squared error and peak sgnal-to-nose rato. The results demonstrate that the proposed samplng approach, referred as hybrd samplng of k-space, can reconstruct better qualty mages when compared to the exstng conventonal samplng approaches. Keywords: Frequency Encodng, Phase Encodng, FEG, PEG, CS, Radal Samplng, Hybrd Samplng I. Introducton MRI s a rvetng magng technque that pctures human body or part of t. It s non-nvasve technque and does not cause any onzaton n the human body. MRI evolved from varous rudmentary nventons made durng 1980 s and t underwent enormous development snce the last three decades. Indeed, the objects maged by MRI have an mpressve qualty, even comparable to photographs taken by dgtal cameras. Now days, MRI s recognzed as the prme modalty for dagnosng many common dseases, ncludng cancer and stroke. It s not only used for magng hgh spatal resoluton mages, but also offerng good contrast for soft tssues. It has advantages over other magng technques lke SPECT (Sngle Photon Emsson Computed Tomography), Doppler Ultrasound, PET (Postron Emsson Tomography), and computed tomography (CT) whch cause onzaton n the human body. The examnaton procedure through MRI does not nvolve any rsk to the patent's health. The bascs of MRI magng processes s brefly ntroduced n the followng sub sectons. A. Frequency Encodng and Phase Encodng: A man magnetc feld B 0 s appled n the parallel drecton of an object whch algns the nuclear spns of the protons. All the spns have the same Larmor frequency [1] and they are ndependent of the spatal locaton of the scanned object. A secondary magnetc feld s appled n the form of gradent n Y and Z drecton. The nucle of an atom experences stronger magnetc feld n the postve X drecton, and thus undergo wth hgh Larmor frequency. Smlarly, nucle whch are n the opposte (negatve) X drecton experences low ntensty magnetc feld and thus undergo wth lower Larmor frequency. Ths s referred to as frequency encodng. The feld gradent s appled n Y drecton; t nduces spatal poston dependent phase-dfference. Ths s called phase encodng. Now, each spatal locaton of an object has a dstnctve Larmor frequency and phase shft. An mage can be reconstructed from these sgnals. B. k-space Data Acquston: The k-space data acquston n MRI takes place n followng steps: () An object s placed n the MRI scannng devce and a strong magnetc feld B 0 s appled, whch enables all protons of the object to undergo torque-nduced precesson at the Larmor frequency. () The RF pulse appled, whch tlts [2] the total magnetzaton present n the protons n a sngle drecton and enables the n-phase precesson for all protons n RF pulse. () The PEG (Phase Encodng Gradent) s appled for makng a phase dfference wth respect to the spatal poston of an object. (v) In ths step, the FEG (Frequency Encodng Gradent) s appled whch helps RF cols to capture the proton sgnal ntensty from the excted protons when they return to ther ground state. The FEG shfts the Larmor frequency of the protons whch s determned by the poston of the object and then IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 19

2 Fgure 1 Generaton of Raw Data Matrx (k-space) [1] the sgnal represents spectrum of frequences. The phase also dffers from the spatal poston of the protons. Generally, each exctaton and the "fall back" of the proton energy to ground state s read-out from RF cols and collected n the raw data matrx n row major order as shown n Fgure 1. Bascally, the entres n the raw data matrx are the frequency components of the proton densty mage. C. Image Reconstructon Algorthm Image Reconstructon s the process of convertng k-space data to an mage that s vsble and pctorally meanngful. An mage reconstructon algorthm s requred to generate the spatal doman mages from the sampled k-space data obtaned from MRI scan. There exst many algorthms [3] to reconstruct mages from k- space data lke Zero Paddng Method, Phase Correcton usng Conjugate Symmetry method, Homodyne Reconstructon Algorthm, and Projectons onto Convex set (POCS) method. Formally, the k-space obtaned from the scannng machne can be represented n the form of contnuous doman [5] that relates the densty of protons m(r) to the sgnal receved p(k). The m(r) s related to p(k) by the followng equaton: p(k)= m( r) e 2k. r FOV where k represents the frequency doman pont and the r represents the spatal doman pont II. Related Work An enormous amount of work has been done n the feld of reconstructon of mages from the sgnals acqured by scannng devces. There has been staggerng mprovement n the qualty of reconstructon snce 1970 [4]. A few of the notable works are revewed n the followng subsectons: A. Senstvty Encodng One of the classcal works n mage reconstructon s that by Cheng and Dewey [6]. The authors descrbed the concept of reconstructon of mage from projectons or raw data as an nverse problem, and mplemented n cases where a large number of mages to be reconstructed. Expectaton maxmzaton s employed for determnng the class values that are not avalable. MRF (Markov Random Feld) model of dynamc objects s used to acheve local regularty and ncreased smoothness durng reconstructon. Pruessmann et al. [7] proposed a new approach for collectng the k-space data referred to as SENSE (Senstvty Encodng). In classcal technque, Fourer Imagng collects the data on RF col at a very slow rate, but wth the help of SENSE, the capablty of Fourer Imagng can be enhanced at the double rate. Ths also reduces the tme for scannng. In the normal Fourer encodng, the k-space data s sampled and transferred to the RF col. If the Fourer Imagng s used wth SENSE, t reduces the samplng densty and collects the k-space more effcently. Authors observed that SENSE enhanced gradent encodng and reduced scan tme as compared to Fourer magng. B. Sten's Unbased Rsk Estmator (SURE) Alfred et al. [8] revewed the theoretcal results from sparse solutons of systems that are lnear. Harmonc analyss of wavelets and applcatons of sparsty to sgnal n mage processng are dscussed. The sparsty based soluton of lnear equatons n the applcaton of mage compresson and mage reconstructon s also dscussed. The other concepts dscussed n the paper are LU Decomposton, and Least Square method. Rach and Alfred [9] proposed sparse mage reconstructon n rado astronomy. They proposed three sparse mage reconstructon methods; the frst uses Sten's unbased rsk estmator (SURE) for selectng the hyper-parameter for L1 estmator, and the other two are based on sparse pror; they used emprcal Bayes denosng. These three methods are smulated and found to produce better results than sparse Bayesan learnng (SBL) [10]. Sparse pror s the magnetzaton ntensty fallng on the body. Usng sparse pror wth SURE method enhanced the qualty of sgnals leadng to better reconstructed mages. C. Sgnal Acquston Protocol Candes et al. [11] presented how to reconstruct an mage from numercal optmzaton from the full-length sgnal/mage usng the raw data (k-space) that s also called projectons. They also descrbed a very smple and dr 1 IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 20

3 effcent sgnal acquston protocol that samples the sgnal whch s ndependent of the rate of change n the projectons. The fnal set obtaned after the samplng allows mprovng the reconstructon power of sparse data, and permts better reconstructon of mages. Inspte of the large amount of exstng development works n MRI mage reconstructon, there stll has ssues lke complexty n k-space acquston process and requrements of large scan tme for generatng good qualty mages. Thus, there are further scopes for mprovements n the qualty of reconstructed mages by ntroducng mprovements n k-space acquston process and reducng the scan tme by enhancng gradent encodng. Ths paper proposes a hybrd samplng technque to mprove the k-space acquston process. III. Proposed k-space Samplng Approach Generally, the samplng of k-space can be done n two ways: () Cartesan Samplng, and () Radal Samplng. The order n whch the RF sgnals are detected plays an mportant role n MRI mage reconstructon. Durng the scannng of an object from the MR camera, the k-space data s collected at dfferent tmes, whch have dfferent spatal ntenstes (state) of an object. As dscuss n subsectons gven below, the central regon of k-space s havng more structural nformaton about the scanned part and the farthest regon of k-space (perpheral regon) contans less nformaton about the structural part of the object. In vew of the above fact, to take care of ths nonunformty of the k-space, we proposed a combned approach for samplng n whch the central regon s sampled by radal samplng and the perpheral regon s sampled by cartesan samplng. The bascs of conventonal samplng approaches and the proposed combned approach are presented n the followng subsectons: A. Cartesan Samplng In Cartesan samplng, the samplng s done by scannng each spatal locaton of k-space data n a rectlnear fashon row by row as shown n Fgure 2. All the sampled ponts are equdstant from each other. Hence, the central regons are not much focussed for the reconstructon, whch results n a poor qualty reconstructed mage. The samples obtan by Cartesan approach are all equdstant from each other whch treat k-space unformly. Hence, there should be more samples from the central regon as compared to perpheral regons, but here t s not so. So, ths samplng approach s not that acceptable for samplng k-space. Fgure 2 Cartesan samplng of k- space data [12] Fgure 3 Radal samplng of k-space data [12] Fgure 4 Co-ordnate of k-space n 2 Dmenson [13] Y (0, k) Fgure 5 Dagrammatc representaton of hybrd k-space samplng (0, k/4) (-k/4, 0) (k/4, 0) (k, 0) X (0, -k/4) IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 21

4 B. Radal Samplng In radal samplng, the k-space data s processed from the low frequency regon (orgn) and contnue n angular fashon as shown n Fgure 3. The data near to the orgn n k-space have the most nformaton about the structure of an mage lke contrast, and the SNR, whereas the data at farther regon helps n reconstructng smaller structures n an mage [12]. Wth the help of radal k-space samplng, the low frequency regons (central regons) of an mage get processed at every scan. Hence, usng the radal samplng of k-space can generate good reconstructed mages, but the perpheral regons are not gettng processed much whch sometmes result n lower qualty reconstructon of small structures of an mage. C. Hybrd Samplng It has been dscussed that the reconstructed mages from the radal samplng of k-space are lackng nformaton of small structural regons of the scanned object. Also, the drawback of cartesan samplng s that t assumes, the nformaton n the k-space, as unformly dstrbuted. We can elmnate the drawbacks of the above two approaches by combnng both the radal and Cartesan samplng technques. In ths approach, the central regon of k-space data (Fgure 4) s sampled usng radal approach and the perpheral regon s sampled usng cartesan approach. As t can be seen from the Fgure 5, the central regon of k-space wll get processed frequently and the dstance between the samples s also very less. Here, the perpheral regons are also gettng processed better when compared wth a radal samplng approach. IV. Algorthm for Hybrd approach It has been proved [13] that the most of the nformaton n k-space les up n regon -k/4 <= x <= k/4 and -k/4 <= y <= k/4 as ndcated n Fgure 4. There s an assumpton that the centrod of k-space data s the orgn of 2D coordnates. Algorthm for hybrd samplng s presented as follows: Algorthm 1: Hybrd Samplng Algorthm: Gven: S 1 : Cartesan sampled raw data, S 2 : Radal Sampled raw data, S: k-space data whch s used to represent pctoral vew of k-space, RI: Reconstructed mage after applyng hybrd samplng reconstructon algorthm Input: S 1, S 2 Output: RI, S 1. Fx the extreme ponts for radal samplng as -k/4 and +k/4 at X and Y axes as shown n Fgure 4. The rest of k k space wll be sampled n rectangular Cartesan form. 2. Start samplng from (-k, k) to (k, -k) n row by row order: For I = 0 to k-1 For J = 0 to k-1 f (-k/4 <= I <= k/4) and (-k/4 <= J <= k/4): Reconstruct usng radal samplng (S 2, I, J, S) else: Reconstruct usng cartesan samplng (S 1, I, J, S) 3. Image and k-space data can be dsplayed usng RI and S. () Reconstructon usng radal samplng: [14] Input: S 2, I, J, S Output: Updated RI and Updated S Compute the mage reconstructed 1) Each sample pont s havng area 'dk, whch can be represented n contnuous doman as: where mˆ ( r) p(k)= k space FOV p( k) e m( r) e 2k. r 2k. r Here, k represents the frequency doman pont and the r = (I, J) represents the spatal doman pont and, m( ) s the proton densty and p(k) s the sgnal receved. In the dscrete doman, the above can be wrtten, wth a weghng functon to account for the non-unform samplng, as: 2k. r 4 mˆ ( r) W( k ) p( k e dk dr I, J I, J ) where W(k I, J ) s the weghtng functon Weghng functon (refer [3]) s used for determnng the non-unform samplng and t s used to compensate the data ponts from radal lnes nto grds. Wthout the weghng, the radal sampled data cannot be processed further. The unform dstrbuton of spatal frequency n k-space data can be obtaned by multplyng weghng functon to the partal k-space n the equaton above. 2) Update RI(I, J) wth m ˆ ( r), whch s the contrbuton of k-space p(k I,J ) to the mage, RI(I,J). 2 3 ˆ r IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 22

5 3) Update S wth the grndng procedure [5] appled to p(k I, J ) () Reconstructon usng cartesan samplng: [15] Input: S 2, I, J, S Output: Updated RI and Updated S 1) All the sample ponts S 1 are equdstant from each other. They all are n the form of Cartesan ordnate space; the proton densty (mage) can be gven as follows: 2k. r 5 mˆ ( r) p( k I, J ) e where parameters are same as equaton -4 2) Update RI(I, J) wth mˆ ( r) whch s the contrbuton of k-space p(k I,J ) to the mage, RI(,j). 3) Update S wth p(k I,J ) V. Experment The experment s performed usng Matlab, (R2010a), 32-bt (glnxn86) runnng under Lnux Operatng System (ubuntu 32-bt, kernel generc). The cartesan sampled raw data can be downloaded from and the radal sampled raw data from MRI machne can be downloaded from AsterodDemo/blob/master/Matlab%20Code/rawdata.mat. The smulaton of Cartesan samplng, radal samplng, and hybrd samplng (the proposed approach) are performed usng the downloaded raw data (both cartesan and radal sampled data). The expermental results of reconstructon wth varous samplng approaches consdered are gven n Fgure 6, Fgure 7, and Fgure 8, respectvely, for cartesan samplng, radal samplng and hybrd samplng. The correspondng mages for all the k-space data are also gven. It can be observed from the pctoral representatons of k-space that the central regon vares n all the samplng approaches. Fgure 6 Cartesan samplng mage reconstructon (a) Gves the pctoral vew of k-space obtaned after cartesan samplng, (b) Red color porton represents the hgh ntensty, black dots ndcate the unform dstrbuton of samples and yellow area represents the ntensty of sample ponts. It can be seen that the all the regon near to center and away from center has unform dstrbuted samples, (c) Represents the fnal reconstructed mage from ths k-space. (a) (b) (c) Fgure 7 Radal samplng mage reconstructon (a) Gves the pctoral vew of k-space obtaned after radal samplng, (b) Red color porton represents the hgh ntensty, black dots ndcate the unform dstrbuton of samples and yellow area represents the ntensty of samples ponts. It can be seen that the center regon s denser than perpheral regons, (c) Represents the fnal reconstructed mage from ths k-space. (a) (b) (c) Fgure 8 Hybrd samplng mage reconstructon IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 23

6 (a) Gves the pctoral vew of k-space obtaned after hybrd samplng, (b) Red color porton represents the hgh ntensty, black dots ndcate the unform dstrbuton of samples and yellow area represents the ntensty of sample ponts. It can be seen that the samples are denser at central regon due to radal samplng and there s also better samplng at perpheral regon due to cartesan samplng, (c) Represents the fnal reconstructed mage from ths k-space. (a) (b) (c) From the pctoral vew of the results, t can be observed that the mage reconstructed from k-space obtaned usng hybrd samplng provdes better qualty mages. The objectve evaluaton and comparatve analyss of hybrd samplng wth radal samplng, and cartesan samplng are presented n the next secton. V. Performance Evaluaton and Comparatve analyss There are varous measures for evaluatng the qualty of reconstructed mages. Ths paper uses the Root Mean Square (RMS), Mean Absolute Error (MAE), and Peak Sound to Nose rato (PSNR) for evaluatng the qualty of the reconstructed mage. The varous performance measures obtaned for Cartesan samplng, radal samplng, and hybrd samplng are presented n Table 1. The performance of dfferent approaches s graphcally represented as n Fgures 9. It s observed that the proposed radal approach s superor n terms of MSE, MAE and PSNR. Mean Squared Error (MSE): MSE s an average of the square of the absolute error. n 2 ( y y ) where y MSE 1 s the actual value, y s the expermental value or predcted value at the th observaton. n Mean Absolute Error (MAE): MAE s an average of the absolute error. MAE N 1 p q N where p s the actual value, value at the th observaton. q s the expermental value or predcted Peak Sgnal to Nose rato (PSNR): Peak sgnal-to-nose rato (n db) s the rato of maxmum ntensty (power) of pxel (sgnal) at a sample pont to the ntensty (power) of corruptng nosed mage. MAX PSNR 10log 10 MSE 2 where MSE s the mean squared error, MAX s the maxmum pxel ntensty. Table -1: Performance Evaluaton Samplng MSE MAE PSNR (n db) Cartesan Radal Hybrd IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 24

7 Fgure 9 Comparatve Analyss of hybrd samplng wth radal and cartesan samplng n terms of MSE, MAE and PSNR From the bar dagram of MSE, MAE and PSNR, t can be seen that the MSE and MAE of our proposed samplng approach provdes low error values when compared to Cartesan and Radal samplng approaches. Also, the PSNR for the proposed samplng approach s hgher than that for Cartesan and radal samplng technques. VII. Concluson Ths paper proposes a hybrd samplng technque for mprovng the qualty of MRI reconstructed mages. The raw data acquston of k-space has been brefly dscussed. The varous features of k-space acquston lke phase encodng, phase shft, frequency encodng have been analyzed. The drawback of cartesan samplng s that t assumes unformly dstrbuton of k-space. The radal samplng of k-space gves better reconstructed mages, but has few dsadvantages lke the small structures of objects are not properly reconstructed. The dsadvantages of cartesan and radal samplng are elmnated by ntroducng the proposed hybrd samplng technque. The algorthm for hybrd samplng technque s mplemented and the performances are compared wth cartesan and radal samplng. The results demonstrate that the hybrd technque provdes better performance fgures n terms of PSNR, mean squared error and mean absolute error when compared to Cartesan and radal approaches. References [1] Patrk Brynolfsson "Usng Radal k-space Samplng and Temporal Flters n MRI to Improve Temporal Resoluton, edcal hyscs epartment of adaton cences me a nversty me a, Sweden 2010 [2] James Keeler "Understandng NMR spectroscopy", a book, Department of Chemstry, Unversty of Cambrdge, Publshed by John Wley & Sons, 2010 [3] Karla Mller "Partal k-space Reconstructon", Lecture Notes, FMRIB Centre, Unversty of Oxford, Nuffeld Department of Clncal Neuroscences, Unversty of Oxford, UK [4] Lauterbur PC (1973). "Image Formaton by Induced Local Interactons: Examples of Employng Nuclear Magnetc Resonance". Nature 242 (5394): Bbcode:1973Natur L. do: /242190a0. [5] Mara Magnusson, Olof Dahlqvst Lenhard, Patrk Brynolfsson, Per Thyr and Peter Lundberg2 "3D Magnetc Resonance Imagng of the Human Bran Novel Radal Samplng, Flterng and Reconstructon", publshed n Center for Medcal Image Scence and Vsualzaton (CMIV), Lnköpng Unversty, Sweden [7] Klaas P. Pruessmann, Markus Weger, Markus B. Schedegger, and Peter Boesger "SENSE: Senstvty Encodng for Fast MRI", Wley-Lss, Inc publsh n Magnetc Resonance n Medcne 42: [8] Alfred M., Davd L., Donoho Brucksten and Mchael Elad. From sparse solutons of systems of equatons to sparse modelng of sgnals and mages related databases Publsher: Socety for Industral and Appled Mathematcs, DOI: / (3):471 81, July [9] avv ach, Alfred chael Tng parse mage reconstructon usng sparse prors Electrcal Engneerng and Computer Scence, Unversty of Mchgan,Ann Arbor, MI , USA. [10].. Wpf and B.. ao, parse Bayesan learnng for bass selecton, IEEE Trans. gnal rocessng, vol. 52, no. 8, pp , [11] Candes Emmanuel J. and B. Wakn Mchael. An ntroducton to compressve samplng IEEE Sgnal Processng Magazne, March 2008, Pages-10, ISBN /08/$ IEE,. [12] zur Erlangung "Advanced Methods for Radal Data Samplng n Magnetc Resonance Imagng", mathematschnaturwssenschaftlchen oktorgrades der eorg-august- nverst at ottngen vorgelegt von a Tobas Block aus an ottngen 2008 [13] Jamng Chen "MRI Reconstructon From 2D Partal k-space Usng POCS Algorthm", 2009,1-4 E-ISBN : , Bejng DOI: /ICBBE Publsher: IEEE [14] Douglas C. Noll and Bradley P. Sutton "Grddng Procedures for Non-Cartesan K-space Trajectores", Dept. of Bomedcal Engneerng, Unversty of Mchgan, Ann Arbor, MI, USA [15] Rcardo Otazo, "k-space Samplng and Fourer Image Reconstructon", Practcal Magnetc Resonance Imagng II Sackler Insttute of Bomedcal Scences New York Unversty School of Medcne IJETCAS ; 2015, IJETCAS All Rghts Reserved Page 25