JOURNAL OF ENGINEERING AND APPLIED SCIENCE, VOL. 55, NO. 5, OCT. 008, PP. 44-456 FACULTY OF ENGINEERING, CAIRO UNIVERSITY requred k-space. Sparse MRI s a fast magng method based on undersampled k-space samplng and non-lnear reconstructon []. Ths approach s nspred by theoretcal RAPID CARDIAC MRI USING RANDOM RADIAL TRAJECTORIES ABSTRACT S. M. EL-METWALLY, K. Z. ABD-ELMONIEM, A. M. YOUSSEF AND Y. M. KADAH Usng Sparse MRI wth random radal trajectores allows MRI mages reconstructon from a small number of acqured k-space data. Ths fulflls the demands of dynamc MRI. Dfferent random radal trajectores are generated by varyng the parameters of the added random perturbatons to the radal trajectores. Images reconstructon s performed usng the non-lnear L norm reconstructon. Entropy s computed for the reconstructed mages, as a quanttatve measure for the reconstructed mage qualty. Both phantom smulaton and real cardac mages are used n the experments of ths work. The results show that more sparsely sampled mages can be reconstructed wth hgher qualty compared to usng non-randomly sampled radal k- space trajectores. KEYWORDS: Cardac Magnetc Resonance Imagng (MRI), radal k-space samplng, dynamc magng.. INTRODUCTION Patent moton durng MRI cardac magng causes artfacts n the reconstructed mage that obscure anatomcal detals. The man sources of these artfacts are cardac and respratory moton. Dynamc MRI captures an object n moton by acqurng a seres of mages at a hgh frame rate, thereby reducng moton artfacts. Dynamc magng places conflctng demands requrng both hgh spatal resoluton to resolve anatomcal detal, and hgh temporal resoluton to montor rapd changes n sgnal. However, k-space samplng that obeys the Nyqust theorem usually precludes smultaneous achevement of both ams. k-space undersamplng speeds up the acquston by only samplng part of the Department of Systems and Bomedcal Engneerng, Caro Unversty, Caro, Egypt Department of Electrcal and Computer Engneerng, John Hopkns Unversty, USA results n sparse sgnal recovery [,3]. It has been shown that f the underlyng mage s compressble t can be recovered from randomly undersampled frequency data, an dea known as compressed sensng. It explots the fact that medcal mages often have a sparse representaton n some doman such as fnte dfferences, wavelets, Fourer, etc., where the number of coeffcents needed to descrbe the mage accurately s sgnfcantly smaller than the number of pxels n the mage. Unform undersamplng of the Fourer doman results n alasng. When the undersamplng s random, the alasng s ncoherent and acts as addtonal nose nterference n the mage representng ncoherent nterference of the sparse transform coeffcents. Sparsty s exploted by constranng the reconstructon to have a sparse representaton and to be consstent wth the measured k- space data []. The success of the reconstructon depends on the sparsty of the coeffcents and that the nterference s small, havng random statstcs. Ths approach has been used wth randomly perturbed undersampled sprals [4] and wth randomly undersampled 3D Fourer Transform 3DFT trajectores [5]. It has been shown that the used non-lnear L norm reconstructon outperformed conventonal lnear reconstructon, recoverng the mage even wth severe undersamplng [4,5]. Radal trajectores have many favorable ntrnsc propertes wth respect to the demands of dynamc MRI ncludng [6]: a) Moton-nduced artfacts result predomnantly n radal streaks wth only low ntensty near the source of moton and reduce moton-nduced ghostng. No ghosts dsplaced along phase-encodng drecton are present. b) The coverage of the k-space center n each radal lne avods contrast contnutes and preserves the contnuty of the process. Also, oversamplng of the low spatal frequences provdes ntrnsc averagng of the gross features of the subject. c) By applyng a magntude reconstructon, a reduced senstvty to statstcal phase errors may be acheved, although the ampltude of moton-nduced artfacts ncreases. 44
Developng an MR samplng pattern s centered on developng the gradent waveforms, whose ntegrals traces out a trajectory n the k-space. In current systems, gradents are lmted by maxmum ampltude and maxmum slew rate. Also, physology provdes a fundamental lmt to gradent system performance as hgh gradent ampltudes and rapd swtchng can produce perpheral nerve stmulaton [7]. Random samplng may not be feasble n MR as the k-space trajectores have to be smooth due to these hardware and physologc consderatons. Radal trajectores other than beng fast and tmeeffcent, they result n k-space samplng wth a varable densty ncreasng lnearly wth the nverse of the dstance from the k-space orgn. Therefore, they are good canddates for random samplng approxmaton as they are far from beng regular as n Cartesan grd samplng despte spannng k-space unformly []. In ths paper, we ntroduce the applcaton of Sparse MRI wth random radal trajectores to explot the ntrnsc advantages of these trajectores wth respect to the demands of dynamc MRI. Dfferent random radal trajectores are generated by varyng the parameters of the added random perturbatons to the radal trajectores. Images reconstructon s performed usng the non-lnear L norm reconstructon. Entropy s then computed for the reconstructed mages, as a quanttatve measure for the reconstructed mage qualty. The obtaned results show that more sparsely sampled mages can be reconstructed wth hgher qualty compared to those obtaned usng unformly sampled radal k-space trajectores. The paper s organzed as follows: Secton two focuses on the methods appled for practcal random trajectores generaton and mage reconstructon. Secton three descrbes the data used n mage reconstructon. The results and dscussons are ncluded n secton four. The conclusons are presented n secton fve.. METHODOLOGY The block dagram n Fg. summarzes the man appled steps n ths work. Ths secton descrbes these steps n detals.. Random Radal Perturbatons Fg.. Block dagram for the man steps employed. Radal lnes are perturbed by slght random devatons taken from Gaussan dstrbuton wth zero mean and varyng varances. Dfferent schemes for the random perturbatons are used n the smulatons. These nclude a) usng constant varance along radal lnes, and b) usng lnearly ncreasng varance wth the dstance from the k-space orgn..e., varance s ncreased n areas of low samplng densty and decreased n areas of hgh samplng densty. Due to the mpractcalty of pure random samplng of k-space [], a practcal ncoherent samplng scheme s amed to closely mmc the nterference propertes of pure random undersamplng. Therefore, the generated random radal trajectores are processed usng a numercal algorthm to keep the gradents ampltude and slew rate below the maxmum permssble lmts. Ths algorthm has been used to reshape D selectve pulses n such a way that one or the other of the appled gradents s always near ts maxmum allowable ampltude or slew rate, thereby mnmzng pulse duraton [8]. It nvolves makng dscrete steps along the radal trajectory and checkng the gradent g ( t ) = g ( t ), g ( t ) and the slew rate s ( t ) = s ( t ), s ( t ) along the way. The th x y Random radal perturbatons wth dfferent varances Processng of trajectores for practcal gradents lmtatons Image reconstructon usng L norm nonlnear algorthm Entropy computaton x y gradent vectors are expressed as 443 444
k& ( t ) g = = ( γδt )[ k ( t ) k ( t ) ], γ where, steps evenly n tme,.e., t = ( ) δt, =,,3... N γ, the gyromagnetc rato, s set to 4.8 ms - G - (or 48 µ s - T - ) and δ t = T N s set to 0µ sec. () δ t s chosen n order to meet practcal samplng rates [8]. The dscrete slew rate s expressed as the second dervatve of k: ( ) ( ) ( ) ( ) s = γ δt k t k t + k t. The computed gradent and slew rate magntudes are compared to maxmum lmts of 0.04T/m and 50T/m/sec (or 4G/cm and 5G/cm/ms) respectvely [8]. The k-space coordnates are modfed n accordance to satsfy these lmtatons. The generated random radal trajectory s frst processed usng a local medan flter along each radal lne n order to smooth the added varatons before applyng the practcal lmts, to preserve the shape of the radally emanatng trajectores. Correctons are then made to the smoothed trajectory to satsfy the practcal lmtatons. Fgures to 4 show an example for the generated normalzed radal k-space trajectores. Fgure shows a randomly perturbed radal trajectory usng added varance of 3. The trajectory smoothed usng the medan flter s shown n Fg. 3. The resultng trajectory after correctons to satsfy the practcal lmtatons s shown n Fg. 4. () Fg. 3. Randomly-perturbed radal trajectores usng added varance of 3 after applyng medan flter. Fg. 4. Randomly-perturbed radal trajectores usng added varance of 3 after practcal correctons.. Grddng Reconstructon Fg.. Randomly-perturbed radal trajectores usng added varance of 3. In MRI, grddng has been used routnely wth respect to nonunform, non- Cartesan samplng of the k-space [9]. Conventonal grddng s appled here to compare wth non-lnear L norm reconstructon usng the randomly perturbed radal trajectores. The grddng algorthm s bascally performed n four steps: a) Precompensate the data wth nverse of the samplng densty to compensate for the varyng densty of samplng n k-space. The samplng space s parttoned nto several 445 446
cells, each cell representng the neghborhood assocated wth a sample pont. The area of each cell s used as the densty compensaton factor for the correspondng sample pont [0]. b) Convolve wth a Kaser-Bessel wndow and resample onto a Cartesan grd. A grddng wndow of wdth 3 and β parameter of 9 s used. c) Apply an nverse two-dmensonal Fourer transformaton d) Postcompensate to remove the apodzaton of the convoluton kernel by dvdng by the transform of the Kaser-Bessel wndow..3 Non-lnear L Norm Conjugate- Gradent Reconstructon Image reconstructon s performed by solvng the followng constraned optmzaton problem []: mn λ Ψ ( m ) + αtv ( m ), (3) s. t. NFFT ( m ) y < ε where m s the reconstructed mage, y s the measured k-space data, Ψ s the sparsfyng transform operator, TV or Total-varaton s the fnte-dfferences sparsfyng transform. Mnmzng the objectve functon promotes sparsty by both the specfc transform and fnte-dfferences at the same tme. NFFT stands for the Non-unform Fast Fourer Transform of the mage.ε controls the fdelty of the reconstructon to the measured data. The threshold parameter ε s usually set below the expected nose level. α trades Ψ sparsty wth fnte-dfferences sparsty. λ s a regularzaton parameter that determnes the trade-off between the data consstency and sparsty. Equaton (3) poses a constraned convex optmzaton problem. Ths s converted to the unconstraned problem: mn arg NFFT ( m ) y + λ Ψ ( m ) + λ TV ( m ), (4) m where λ and λ are the weghtngs of the sparsty and the total varaton terms. They represent regularzaton parameters that determne the trade-off between the data consstency, sparsty and total varaton. The values of λ and λ can be determned usng tral and error where Eq. (4) s solved for dfferent values, then the values are chosen such that NFFT ( m ) y ε. An teratve non-lnear conjugate gradent descent algorthm wth backtrackng lne search s used, followng the work n []..4 Entropy Mnmzaton The entropy crteron, E, s defned as []: N B j B j E = ln, j = B ss B ss (5) where N s the number of mage pxels and B j s the modulus of the complex value of the jth mage pxel or the pxel brghtness. B ss s gven by the sum of squared brghtness. B ss N = B. (6) j = j When all the mage energy s located n a sngle pxel and the remanng pxels are black, the entropy E = 0. When a 8 x 8 mage has a unform brghtness, B j /B ss =/8 for all the pxels and the entropy E = 6. Therefore, entropy mnmzaton favors hgh contrast. Ths entropy crteron favors alteratons to the data that tend to ncrease the number of dark pxels. It has been used as a focus crteron to remove moton-nduced ghosts and blurrng from low ntensty regons of the mage that would otherwse be dark []. Entropy s used here as a measure of the reconstructed mage qualty. It s computed for the reconstructed mages usng the non-lnear L norm reconstructon at the dfferent varances added to the randomly perturbed radal trajectores. 3. EXPERIMENTAL VERIFICATION 3. Phantom Smulaton A D numercal SheppLogan phantom s used. The phantom mage s desgned as a lnear superposton of ellptcal objects, whose FTs are scaled jnc functons (jnc x = J (x)/(x), where J s a frst-order Bessel functon). k-space samples can thus be evaluated drectly, therefore the phantom smulates realstc k-space samplng and truncaton. Phantom mage reconstructon s done usng both non-lnear conjugate-gradent method and the conventonal grddng reconstructon. The reconstructed phantom 447 448
resoluton s 60 x 60. k-space undersamplng wth 8-fold s used. That s, k-space conssts of 0 radal lnes. Each lne conssts of 5 samples. Practcally, each radal lne s acqured durng a Repetton Tme TR nterval, whch s relatvely long. Therefore, the number of samples per lne can be ncreased, as much as permtted by TR, wthout any ncrease n the overall acquston tme. Image reconstructon s done n the sparse fnte dfferences doman usng λ = 0 andλ = 0.05. Ths s because the SheppLogan phantom represents a pece-wse constant object, and the TV term measures the fnte dfferences n the reconstructed mage. Therefore, mnmzng the objectve functon s equvalent to mnmzng the fnte dfferences n the pece-wse constant object. 3. MRI Data Cardac MRI magntude mages are Fourer transformed at varous generated random radal trajectores usng non-unform Fourer transform. Undersamplng factors of 8 and 0 are used. k-space conssts of 6 radal lnes wth 5 samples/lne n the case of 8-fold acceleraton, and conssts of 3 radal lnes wth 5 samples/lne n the case of 0-fold acceleraton. The reconstructed mage resoluton s 8 x 8. Random perturbatons taken from a Gaussan dstrbuton wth zero mean and varyng varances are added to the generated radal k-space trajectores. Snce medcal mages often have a sparse representaton n the wavelets doman, where the number of coeffcents needed to descrbe the mage accurately s sgnfcantly smaller than the number of pxels n the mage, the parameters used n mage reconstructon are λ = 0.00 and λ = 0.005. A tral and error strategy s followed for these parameters selecton such that NFFT ( m ) y ε. The TV term s ncluded n reconstructon n order to reduce the nose level n the reconstructed mages. 3.3 Nose Addton In order to nvestgate the performance of the non-lnear reconstructon n the presence of nose, nose s added to the noseless SheppLogan phantom at dfferent nose varances of 0.0, 0.03, 0.05, 0.08, and 0. correspondng to Sgnal-to-Nose Ratos SNR of 4, 5,, 8, and 6 db []. At each nose level, mages are reconstructed usng sparsty weghtng λ = 0 and varous TV weghtngs λ. Dfferent values of λ are tred, and the entropy s computed n each tme. The λ value that results n the lowest entropy can then be consdered to be the most approprate for reconstructon at that nose level. 4. RESULTS AND DISCUSSION The results of smulaton phantom reconstructon wth 8-fold acceleraton are shown n Fg. 5. It can be seen that the small ellpses, as ponted to by the arrows, become more resolved by ncreasng the varance of the added random devatons. Ths s also accompaned by an overall decrease n mage blurrng. Image reconstructed by conventonal grddng shows much lower qualty, compared to those obtaned usng nonlnear reconstructon. It reveals radal streaks arsng due to radal k-space trajectores undersamplng. However, these radal streaks dsappear gradually wth ncreasng the varance n the reconstructed mages usng non-lnear reconstructon. Table demonstrates the computed entropy of the reconstructed phantom at the dfferent varances used. It can be notced that entropy value decreases wth added varance ncrease. Lnearly-ncreasng varance shows entropy wth slghtly hgher values compared to constant varance. Fgures 6 and 7 dsplay cardac mage reconstructon usng 8-fold and 0-fold acceleratons, respectvely. It can be seen that the fne mage detals become clearer wth added random perturbatons compared to usng the standard non-random radal trajectory. Also, as varance ncreases, more contrast enhancement s notced. Even at very hgh undersamplng of 0-fold, mage reconstructon revealed more mprovement. Table demonstrates the computed entropy of the reconstructed mages usng non-lnear reconstructon wth random radal perturbatons at dfferent varances. It can be seen from the table that the entropy decreases wth the ncrease of varance. Images reconstructed usng conventonal grddng method show lower qualty compared to non-lnear reconstructon. 449 450
Fgures 8-0 show the results obtaned at selected SNRs of 4,, and 6 db. At each nose level, the reconstructed mages at dfferent TV weghtngs are shown. Table 3 dsplays the computed entropy for the reconstructed mages usng varous TV weghtngs λ at the dfferent nose levels. The lowest entropy obtaned s wrtten n bold letters. It can be seen that wth ncreasng nose levels, mage reconstructon s enhanced by ncreasng λ to some lmt where the computed entropy s lowest then the reconstructed mages become worse and entropy ncreases agan. Table. Entropy computed for phantom smulaton wth 8-fold acceleraton. Varance (normalzed squared Entropy spatal frequency m - /m - ) 0 88.057 0.5 79.56 78.68 69.86 3 64.9984 5 63.90 Lnearly ncreasng n the range 75.584 [0-] Lnearly ncreasng n the range 7.695 [0-3] Lnearly ncreasng n the range 67.03 [0-5] Table 3. Entropy computed for phantom mage reconstructon at dfferent nose levels usng varous TV weghts TV weghts ( λ ) 0.0 0.05 0.08 0. 4 66.648 68.6608 7.4463 74.0649 5 76.4939 7.0308 74.376 75.673 86.0366 79.503 77.9645 79.9497 8 99.8878 9.6 89.7 88.5640 6 07.98 99.3539 95.73 94.3866 SNR (db) Table. Entropy computed for cardac mage reconstructon wth 8-fold and 0-fold acceleratons. Varance Entropy (normalzed 8-fold 0-fold acceleraton squared spatal frequency m - /m - ) acceleraton 0 84.4865 88.06 0.5 79.0080 80.7773 78.487 80.663 77.778 79.9855 3 77.46 79.54 5 76.857 79.33 8 76.33 78.366 Fg. 5. Phantom mage reconstructon wth 8-fold acceleraton. (a) Orgnal mage. (b) Image reconstructed usng conventonal grddng. Non-lnear conjugate gradent reconstructon usng (c) non-random radal k- space, random radal k- space wth (d) varance=, (e) varance=3, (f) varance=5, (g) lnearly ncreasng varance n the range [0-], (h) lnearly ncreasng varance n the range [0-3], () lnearly ncreasng varance n the range [0-5]. 45 45
Fg. 6. Enlarged vew for the regon surrounded by the dashed rectangle showng orgnal and reconstructed cardac mages at 8-fold acceleraton. The heart muscles as ponted to by the arrows become more apparent at hgher varances. Fg. 8. Phantom mage reconstructon at SNR = 4 db. (a) Orgnal mage. (b) Nosy mage. Non-lnear conjugate gradent reconstructon usng (c) λ = 0.0, (d) λ = 0.05, (e) λ = 0.08, (f) λ = 0. Fg. 7. Enlarged vew for the regon surrounded by the dashed rectangle showng orgnal and reconstructed cardac mages at 0-fold acceleraton. Fg. 9. Phantom mage reconstructon at SNR = db. (a) Orgnal mage. (b) Nosy mage. Non-lnear conjugate gradent reconstructon usng (c) λ = 0.0, (d) λ = 0.05, (e) λ = 0.08, (f) λ = 0. 453 454
Fg. 0. Phantom mage reconstructon at SNR = 6 db. (a) Orgnal mage. (b) Nosy mage. Non-lnear conjugate gradent reconstructon usng (c) λ = 0.0, (d) λ = 0.05, (e) λ = 0.08, (f) λ = 0. 5. CONCLUSIONS In ths paper, the concept of compressed sensng s appled n MRI mage reconstructon usng randomly perturbed radal k-space trajectores. The effect of ncreasng the varance of the added random components has been nvestgated for mage reconstructon at dfferent undersamplng factors or acceleratons. The obtaned results have shown that usng randomly perturbed radal k-space enables more sparsely sampled mage reconstructon wth hgher qualty compared to usng non-randomly sampled radal k-space trajectores. Also, mage reconstructon at hgh undersamplng rates s enhanced by ncreasng the varance of added random perturbatons. The future research should nclude the nvestgaton of usng sparsty transform n non-lnear reconstructon of radally sampled mages. Also, a study of varyng the non-lnear reconstructon parameters may be done for dfferent mage models such as pece-wse varyng and smoothly varyng models. REFERENCES. Lustg, M., Donoho, D., and Pauly, J., Sparse MRI: The Applcaton of Compressed Sensng for rapd MR Imagng, Magnetc Resonance n Medcne, Vol. 58, Issue 6, pp. 8-95, 007.. Candès, E., Romberg, J. and Tao, T., Robust Uncertanty Prncples: Exact Sgnal Reconstructon from Hghly Incomplete Frequency Informaton, IEEE Transactons on Informaton Theory, Vol. 5, No., pp. 489-509, 006. 3. Donoho, D., Compressed Sensng, IEEE Transactons on Informaton Theory, Vol. 5, No. 4, pp. 89-306, 006. 4. Lustg, M., Lee, J., Donoho, D. and Pauly, J., Faster Imagng wth Randomly Perturbed Undersampled Sprals and L Reconstructon, Proceedngs of the ISMRM, Mam Beach, Florda, USA, 005. 5. Lustg, M., Donoho, D. and Pauly, J., Rapd MR Imagng wth Compressed Sensng and Randomly Undersampled 3DFT Trajectores, Proceedngs of the ISMRM, Seattle, Washngton, USA, 006. 6. Rasche, V., De Boer, R., Holz, D. and Proksa, R., Contnuous Radal Data Acquston for Dynamc MRI, Magnetc Resonance n Medcne, Vol. 34, pp. 754-76, 995. 7. Wrght G., Magnetc resonance magng, IEEE Sgnal Processng Magazne, Vol. 4, No., pp. 56 66, 997. 8. Hardy, C. and Clne, H., Broadband Nuclear Magnetc Resonance Pulses wth Twodmensonal Spatal Selectvty, Journal of Appled Physcs, Vol. 66, No. 4, pp. 53-56, 989. 9. Jackson, J., Meyer, C., Nshmura, D. and Macovsk, A., Selecton of a Convoluton Functon for Fourer Inverson Usng Grddng, IEEE Transactons on Medcal Imagng, Vol. 0, No. 3, pp. 473-478, 99. 0. Takahash, A. M., Consderaton for usng the Vorono areas as a k-space weghtng functon, Proceedngs of the ISMRM, 7th Annual Meetng, 999.. Atknson, D., Hll, D., Stoyle, P., Summers, P. and Keevl, S., Automatc Correcton of Moton Artfacts n Magnetc Resonance Images Usng an Entropy Focus Crteron, IEEE Transactons on Medcal magng, Vol. 6, No., pp. 903-90, 997.. Gamper, U., Boesgner, P., and Kozerke, S., Compressed Sensng n Dynamc MRI, Magnetc Resonance n Medcne, Vol. 59, pp. 365-373, 008... 455 456