1 Medicl Imge Anlysis 13 (29) Contents lists ville t ScienceDirect Medicl Imge Anlysis journl homepge: Modelling of imge-ctheter motion for 3-D IVUS Misel Rosles, Peti Rdev, Oriol Rodriguez-Leor c, Deor Gil d, * Lortorio de Físic Aplicd, Fcultd de Ciencis, Deprtmento de Físic, Universidd de Los Andes Mérid, Venezuel Centre de Visió per Computdor, Universitt de Brcelon, Spin c Universitry Hospitl Germns Tries i Pujol, Bdlon, Spin d Centre de Visió per Computdor, Universitt Autonom de Brcelon, Spin rticle info strct Article history: Received 12 Jnury 27 Received in revised form 13 June 28 Accepted 19 June 28 Aville online 3 July 28 Keywords: Intrvsculr ultrsound (IVUS) Motion estimtion Motion decomposition Fourier Three-dimensionl intrvsculr ultrsound (IVUS) llows to visulize nd otin volumetric mesurements of coronry lesions through n explortion of the cross sections nd longitudinl views of rteries. However, the visuliztion nd susequent morpho-geometric mesurements in IVUS longitudinl cuts re suject to distortion cused y periodic imge/vessel motion round the IVUS ctheter. Usully, to overcome the imge motion rtifct ECG-gting nd imge-gted pproches re proposed, leding to slowing the pullck cquisition or disregrding prt of IVUS dt. In this pper, we rgue tht the imge motion is due to 3-D vessel geometry s well s crdic dynmics, nd propose dynmic model sed on the trcking of n ellipticl vessel pproximtion to recover the rigid trnsformtion nd lign IVUS imges without loosing ny IVUS dt. We report n extensive vlidtion with synthetic simulted dt nd in vivo IVUS sequences of 3 ptients chieving n verge reduction of the imge rtifct of 97% in synthetic dt nd 79% in rel-dt. Our study shows tht IVUS lignment improves longitudinl nlysis of the IVUS dt nd is necessry step towrds ccurte reconstruction nd volumetric mesurements of 3-D IVUS. Ó 28 Elsevier B.V. All rights reserved. 1. Introduction The introduction of intrvsculr ultrsound (IVUS) in the field of medicl imging (Yock et l., 1988; Grhm et l., 1989; Mintz et l., 21; Whle et l., 1999; Evns et l., 1996) s n explortory technique hs significntly chnged the understnding of rteril diseses of coronry rteries. Ech IVUS plne visulizes the cross-section (Fig. 1) of the rtery llowing, mong others the ssessment of different vessel plques (thromus of the rteriosclerotic plques nd clcium deposits) nd the determintion of morpho-geometric prmeters (Metz et l., 1992, 1998; Jumo nd Rimund, 1998). 3-D IVUS is otined y ssemling in stck 2-D IVUS imges cquired during n utomtic pullck of the ctheter. 3-D IVUS is of gret clinicl interest s it llows to evlute nd quntify the effect of therosclerotic plque on rteril distensiility (Ginnttsio et l., 21), to follow-up the plque nd lumen fter intervention (Nissen, 22), nd to guide the selection of n optiml interventionl procedure (Nissen nd Yock, 21). Crdic dynmics ( Mrio et l., 1995; Delchrtre et l., 1999; Roelndt et l., 1994; Thrush et l., 1997) nd 3-D shpe of the vessel (Whle et l., 1999) introduce n imge mislignment tht cn e pprecited in the IVUS sequence s rottion nd displcement of the vessel cross-section. This is min rtifct tht hinders * Corresponding uthor. E-mil ddress: (D. Gil). the longitudinl visuliztion of vessel morphology nd ny ccurte ssessment of morphologicl structures long the dt sequence. Their effect in 3-D IVUS imges hs een descried (Whle et l., 1999; Ndkrni et l., 25; Di Mrio et l., 1993, 1995; Berry et l., 2; de Korte, 1999; Bruining et l., 1998; Dijkstr et l., 1998) s sw-tooth-shped longitudinl ppernce of the vessel wll tht decreses precision in volumetric mesurements. Intrinsic rottion nd trnsltion of the imging ctheter introduce rtifcts in the longitudinl views mking structures to pper nd dispper s well s provoking tooth-shped dventiti nd clcified plque, etc. This motion minly ffects the computtion of the vessel dimeter in the L-views. Fig. 1 shows the sinusoidl wvy pttern of the vessel wll due to the hert dynmics nd lood pulstion tht troules the vessel wll detection nd nlysis of vessel structures long the sequence. The most disturing phenomenon induced y vessel periodic motion is tht the displcement invlidtes ny point correspondence long the sequence s, given fixed ngle in the short-xis IVUS imge, points from different temporl frmes do not correspond to the sme vessel wll segment. On the other hnd, if we project the grey-level pixels in the temporl direction (longitudinl vessel direction) of the IVUS sequences, we cn oserve tht slient fetures usully representing the clcium nd firous plque nd dventiti-medi orders descrie trjectory of semi-rottion of ring round stick (in this cse, the ctheter) (Fig. 1c). Tht is, in the short-xis /$ - see front mtter Ó 28 Elsevier B.V. All rights reserved. doi:1.116/j.medi
2 92 M. Rosles et l. / Medicl Imge Anlysis 13 (29) c Shde zone Plque Ctheter Medi Distole Systole Longitudinl cut line reference Distole Intim Ctheter Ctheter Lumen Adventiti Guidewire rtifct Distole Systole Systole Vessel wll evolution Fig. 1. Morphologicl rteril structures nd rtifcts in n IVUS imge (). Sinusoidl shpe in longitudinl cut () nd grey-level shift in cross sectionl view (c). Red nd lue ellipses show the loction of vessel wll t systole nd distole, respectively. view, the rottion nd trnsltion of the ctheter led to high mount of vessel motion. A usul strtegy used to minimize the impct of the ove descried rtifcts is to synchronize sequence cquisition with hert dynmics. This cn e chieved y either specil ECG-gted devices (Bruining et l., 1998) or y n imge-sed ECG-gted 3-D IVUS (De Winter et l., 24; O Mlley et l., 27, 28). In the first cse, specil ultrsonic device performing the ctheter pullck t the sme systolic pek is required nd the cquisition time significntly increses. In the second cse, one discrds ll frmes etween two consecutive systolic peks, thus, running the risk of loosing vlule informtion. Keeping ll frmes nd ligning them y suppressing their rottion nd displcement would llow to get relile rdil mesurements during the whole cycle relted to the estimtion of the elstic properties of the vessel. Alterntively, reconstructing the pth using iplne ngiogrphy (Evns et l., 1996; Whle et l., 1999) could llevite the geometric distortions due to the position nd ngulr rottion of the ctheter y trcking the Frenet tringle ccording to the vessel tortuosity, lthough in this cse, imge rottion due to hert dynmics is not considered. The gol of this work is to estimte nd remove the rigid motion effect y using only IVUS imges keeping t the sme time ll the dt. The complex movement of the imging ctheter inside the coronry vessel cuses motion rtifcts tht cn e explined y three phenomen: () rottion of the IVUS imging ctheter with respect to the vessel (i.e. round the xis tngentil to the ctheter), () trnsltion of the imging ctheter in the plne perpendiculr to the ctheter xes, nd (c) trnsltion of the ctheter long its xis forwrd nd ckwrd cusing swinging effect. In this pper, we ddress the first two motion rtifcts in order to help the short-xis s long-xis vessel disply, to visulize nd quntify different morphologicl vsculr structures like plque, stents, lumen dimeter, lesions, etc. We present hypothesis tht the vessel wll motion in IVUS imges is due to two min fctors: systemtic contriution cused y vessel wll geometry nd dynmicl periodic contriution due to the hert pulstile contrction influence. This decomposition of the vessel profiles serves to model the vessel dynmics nd determines roust procedure for vessel motion suppression. Aiming to correct rigid trnsformtion the vessel wll is pproximted y n ellipticl model tht is trcked in IVUS frmes to estimte its rottion nd trnsltion. The ellipticl pproximtion (Andrew et l., 1999) is computed on rough segmenttion of slient vessel structures using Neurl Network lgorithm (Ptrick, 1994). By pplying our stiliztion pproch, vessel ppernce in short-xes views llows etter nd esier perception of vessel structures distriution nd quntity. In longitudinl views we remove the tooth-shpe nd thus ssure tht chnges in vessel dimeter re due to chnges in morphology. In this wy, we ese the nlysis of plque progression long the vessel compred to the originl L-views. Note tht this step is crucil for further nlysis of vessel iomechnics (plpogrphy, mesuring elstic properties s strin or stress, etc.) since their computtion requires trcking mteril points of the vessel long the crdic cycle. Recently there hs een n incresing interest in compensting motion (Dnilouchkine et l., 26; Leung et l., 26, 25) of the whole stck of IVUS imges in order to improve tissue elstogrphic studies. The existing pproches for motion compenstion in IVUS rely on either registrtion (Leung et l., 26) or trcking (Dnilouchkine et l., 26) of imge intensities. In Leung et l. (25) the uthors report severl techniques s follows: (1) Glol rottion lock mtching (GRBM), (2) contour mpping (CMAP), (3) locl lock mtching (LBM) nd (4) ctheter rottion nd trnsltion (CRT). These methods re sed on similrity estimtion of locl imge fetures defining ROI on frmes of interest. However, when the IVUS segment sequence does not possess ny chrcterizing structure, the similrity etween locks cn ecome very sensitive. In the GRBM cse, the minimizing lgorithm is prone to get stuck t locl minim unless exhustive (computtionlly inefficient) serch is performed. In the cse of trcking lgorithms, lrge displcements cnnot e properly modelled. In ny cse, chnges in imge intensity from one frme to the next one sustntilly ffect the performnce of the lgorithms. Our method is sed on registering the glol representtion of vsculr structures defined y the geometry of the coronry vessel. The proposed pproch does not require ny minimiztion process, since prmeters re given y n explicit formul. It follows tht our computtion of motion prmeters is not ffected y ny intensity chnges etween consecutive frmes. Furthermore, since points on vessel structures re computed seprtely on ech imge, our methodology nturlly hndles lrge displcements. An dvntge of our strtegy is tht it does not need very precise segmenttion of vessel structures in order to correct vessel motion. The set of reported experiments includes vessel motion simultions nd vlidtion of vessel dynmics suppression in rel-dt of 3 ptients. Our results revel suppression of rigid trnsformtion up to 97% in synthetic dt nd up to 85% in rel-dt. The pper is orgnized s follows: The theoreticl grounds of the model re given in Section 2: ssumptions on vessel dynmics re presented in Section 2.1 nd the mthemticl formultion in Section 2.2. Our pproch to computtion of prmeters is given in Section 3. Experiments re reported in Section 4 vlidting the model on phntom dt in Section 4.1 nd results on experimentl dt in Section 4.3. The rticle finishes with Discussions nd Conclusions.
3 M. Rosles et l. / Medicl Imge Anlysis 13 (29) A dynmicl model of vessel wll ppernce in IVUS 2.1. Model ssumptions The dynmics of coronry rtery is minly governed y the left ventricle dynmic evolution, lood pressure nd intrinsic geometric vessel properties (Mzumdr, 1992; Young nd Heth, 2; Ndkrni et l., 23; Holzpfel et l., 22). The first order pproximtion to vessel dynmics is given y liner trnsformtion comining trnsltion, rottion nd scling (Wks et l., 1996). The min ssumptions in the model we propose re the following: 1. Vessel trnsltion nd rottion hve two min contriutions: dynmic periodic motion nd systemtic geometric one. Vessel displcement long the sequence is reflected y the vessel wll profile in longitudinl cuts (Gil et l., 1996). Visul inspection of lrge longitudinl cuts (see Fig. 2) shows tht there is periodic wvy profile (lue nd red sinusoidl lines) nd systemtic contriution (green se-line). We tke the temporl evolution of the lumen center long the sequence s the min descriptor of vessel displcement. The lumen centers were estimted s the centers of n ellipticl pproximtion to the vessel wll points. The sptil vrition of the lumen center, (Dx, Dy), is min geometric mesure tht provides with relevnt informtion on hert dynmics nd geometric contriutions to the IVUS displcement. Fig. 3 shows the sptil evolution of the lumen center horizontl coordinte, Dx. We note tht such profile decomposes into two min curves: crdic periodic oscilltion (lue line) nd se curve (red thick line) representing vessel wll position. The power spectrl density (Fig. 3) reflects these two min phenomen, s there re two predominnt frequency rnges. Low frequencies (in the rnge (1, 2) = (.2,.33) Hz) correspond to the geometric lumen evolution, while, those frequencies locted t (3,4) = (.78,.91) Hz re hert dynmics contriutions. By the ove considertions, we decompose lumen displcement into geometric, (Dx g,dy g ), nd dynmicl, (Dx d,dy d ), contriution terms: Dx g + Dx d nd Dy = Dy g + Dy d. The decoupled terms re given y the Fourier series: Dx g ðtþ ¼ Xn¼ 2 n¼ 1 ða n cosðnxtþþb n sinðnxtþþ; Using similr rgumenttions, we lso decouple the ngle of rottion, D, into geometric nd dynmicl terms: D g ðtþ Xn¼ 2 n¼ 1 ða n cosðnxtþþb n sinðnxtþþ; D d ðtþ Xn¼ 4 n¼ 3 ðc n cosðnxtþþd n sinðnxtþþ These Fourier coefficients give the hert dynmics nd geometric contriutions mplitudes to vessel motion nd ply centrl role in the simultion of vessel dynmic profiles (see experimentl Section 4). 2. Wll shpe evolution cn e descried y mens of the rottion nd trnsltion of n ellipticl model. In order to correct rigid motion rtifcts due to the reltive displcement of the vessel versus the imging ctheter, we restrict the vessel wll evolution to rigid trnsformtion, tht is, trnsltion followed y rottion. There re two possile wys of computing such trnsformtions: registrtion of IVUS dt sed on imge gry level (Lester nd Arridge, 1999) or correspondence of vessel structures/vessel order (Rosles et l., 25). Using n intensity sed imge registrtion runs the risks of ligning imges tking into ccount the most externl prt of the imge, which contin no structurl informtion on vessel orders ut re minly shdow nd right rtifct res. In order to void such mislignment, we trck the most slient vessel structures locted round the interfce intim, medi nd dventiti. To such purpose, points conforming to the gry vlue chrcteristics defining the trnsition lumen-tissue re extrcted y mens of neurl network vessel wll detection (see Section 3). Insted of serching for points correspondence etween two consecutive frmes, for the ske of roust nd efficient pproch, we extrct nd trck glol model of the set of points extrcted. Since we focus on glol liner rigid trnsformtion n elliptic model suits our purposes. The motion-trcking lgorithm will succeed s fr s the computed elliptic model keeps stle long the sequence since, in this cse, ny rtifct due to n unsuccessful fitting does not ffect the ellipticl vessel wll pproximtion. Dy g ðtþ Xn¼ 2 ða n cosðnxtþþb n sinðnxtþþ n¼ 1 Dx d ðtþ ¼ Xn¼ 4 ðc n cosðnxtþþd n sinðnxtþþ; n¼ 3 Dy d ðtþ Xn¼ 4 ðc n cosðnxtþþd n sinðnxtþþ n¼ 3 ð1þ ð2þ 2.2. Model formultion The ove ssumptions on vessel dynmics nd the ellipticl shpe pproximtion of vessel cross sections simplify the formultion of vessel motion to descriing rigid trnsformtion of n ellipticl vessel wll shpe. The vessel wll c k (u)=(x k (u),y k (u)) t time t k cn e written s follows: Cut C1 Vessel wll A Ctheter S1 S2 S3 S4 S5 S6 S7 Vessel wll B Fig. 2. Longitudinl cut of IVUS dt cquired during the ctheter pullck. Blue nd red points show the upper nd down vessel wll oundry. The green line shows the systemtic contriution of ctheter motion.
4 94 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Δx in [mm] Amplitude in [u] Time in [sec] 2 1 Vessel Geometry contriutions Vessel Geometry contriutions Hert dynmic contriutions Hert dynmic contriutions Frequency in [Hz] Fig. 3. Temporl evolution of the lumen center Dx coordinte () nd its Fourier spectrum (). Red lines show vessel geometry contriution nd lue line illustrtes the hert dynmic contriution. x k ðuþ ¼ cosðhðuþþd k Þþcx k y k ðuþ ¼ sinðhðuþþd k Þþcy k where < h 6 2p determines the ngulr position of the corresponding point on the ellipse, u is the internl prmeter of the model curve, (,) re the minor nd mjor rdii of the ellipse, d k is its orienttion nd C k =(cx k,cy k ) is its center. We ssume tht the lumen center C k of c k cn e chnged from position (cx k,cy k ) t time t k to C k+1 with position (cx k+1,cy k+1 ) t time t k+1 due to the periodic hert movement. Therefore, the position of the center C k+1 of the new vessel shpe c k+1 t time t k+1 cn e written s cxðt kþ1 Þ¼cxðt k ÞþDx k ; cyðt kþ1 Þ¼cyðt k ÞþDy k where Dx nd Dy give the geometric nd dynmicl chnge induced y the vessel geometry nd hert pulstile contriution on the lumen center sptil nd temporl evolution, given y Eqs. (1) nd (2). When chnging to the new lumen center, new rotted vessel wll c k+1 =(x k+1,y k+1 ) is given, which coordintes hve chnged ccording to rottion of ngle k. Using the imge center s rottionl center, the new vessel wll coordintes cn e written s x kþ1 ðuþ cosð kþ1 Þ sinð kþ1 Þ xk ðuþ ¼ ð5þ y kþ1 ðuþ sinð kþ1 Þ cosð kþ1 Þ y k ðuþ Thus, if we know the rottion ngle k t ech time k, we re le to find the corresponding mteril point position in the next frme. In order to otin the rottion ngle we define s reference point, the point of the vessel model closest to the rottion center pproximted y the center of the imge. The ngle determined y the vectors of the reference point in ech pir of consecutive imges determine the rottion ngle for these frmes. Fig. 4 shows the reltion of geometric prmeters of our model. Without loss of generlity, to illustrte our pproch we use circulr vessel model. Fig. 4 is showing two vessel models c 1 nd c 2 with their corresponding centers C 1 nd C 2 nd rottion ngle D. Note tht k+1 cn e written s k+1 = k + D k, where k+1 = rctn(cy k /cx k ) nd D k ¼ rctnððcy k þ Dy k Þ=ðcx k þ Dx k ÞÞ: 3. A procedure for rigid motion estimtion nd suppression The procedure for rottion suppression splits into three min steps: ð3þ ð4þ ð6þ Ellipticl vessel pproximtion this step consists in vessel detection y Neurl Network trined to clssify slient structures of vessel order nd ellipticl pproximtion of the vessel wll. Motion prmeters computtion this step consists in estimtion of the rottion long the imge sequence using trcking procedure of reference point. Imge motion correction this step consists in correction of imge rottion nd displcement using the extrcted prmeters of the ellipticl pproximtion Ellipticl vessel pproximtion For the computtion of the vessel ellipticl representtion, structures on vessel wll long the sequence must e locted. The detection of the vessel wll is sed on physicl prmeters tht chrcterize the intim, medi nd dventiti lyers. Such descriptors re defined y glol nd locl imge grey-level properties otined from IVUS imges in polr form (Rosles et l., 24) with the origin t the ctheter center. In this coordinte system, the imge intensity, nmely I, depends on the pixel distnce (rdius r in polr coordintes) to the ultrsound trnsducer (Jensen, 21; Vogt et l., 1998). The decrese in the initil em intensity, I, is exponentilly proportionl to the sorption coefficient f, the frequency of the ultrsound, f, nd the prticle size or sctterer numer, N h, locted long the ultrsound em pth: IðrÞ ¼I expð fn h frþ: The sorption coefficient gives the rte of diminution with respect to the distnce long trnsmission pth (Jensen, 21) nd it is loclly otined from the regression line slope (Vogt et l., 1998) of the imge profile. Fig. 5 shows rdil grey-level profile of n IVUS imge, the lumen-vessel order is pproximtely represented y the steepest grey-level trnsition tht is used to trin the Neurl Network. The set of chrcteristics chosen to find slient points on the vessel wll re the sorption coefficient f, imge pixel grey-level I(r) nd rdil stndrd men l nd devition r of the dt. The sorption coefficient gives locl informtion out the lumenvessel trnsition cting s n edge detector. The sttistics considered (l nd r) contin regionl informtion on vessel structures: l gives the seline of glol grey-level intensity nd r gives simple descriptor of the regionl texture. In order to find potentil cndidte structures on the vessel wll, Perceptron Multilyer Neurl Network (6:5:6:3) ws trined using stndrd Bck
5 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Vessel wll γ 2 t time t 2 y P 2 (x,y ) 1 y P 2 (x,y ) Δα P 1 (x,y) Δα Δα P 1 (x,y) Δα C 2 C 2 Δy C 1 α 1 C 1 Ctheter (x o,y o ) Δx x (x,y ) Δα (x o,y o ) x C 1 (xc 1,yc 1 ) nd C 2 (xc 2,yc 2 ) Lumen centers position t time t1 nd t2 Vessel wll γ 1 t time t 1 (x,y ) Fig. 4. Geometric prmeters of the model (). Estimtion of the rottion ngle ()..8 Grey level Liner fitting to estimte the sorption coefficient (ζ) Rdil coordinte (R) Fig. 5. Vessel structures loction. Rdil grey-level profile () nd extrction of positive (+) nd negtive ( ) ptterns () to trin the neuronl network. Propgtion Algorithm (Ptrick, 1994). Fig. 5 shows trining exmples of positive (+) corresponding to intim nd negtive ( ) pttern, corresponding to lood, dventiti, shdows, nd rtifct zones. Once pplied the neuronl network to clssify slient points on the vessel order, n ellipticl pproximtion to the vessel wll points is djusted using the men squre ellipse fitting method descried in Andrew et l. (1999) Motion prmeters computtion Once defined the ellipse center s new origin, it is necessry to estimte the ngulr rottion profile of the vessel wll structure. In rottion, it is sufficient to provide the temporl evolution of single point on the vessel wll structure (Kittel et l., 1991; Broucke, 1978), to mesure the ngulr difference etween two consecutive frmes. Since the ultrsound intensity grdully decreses in the rdil direction, it follows tht one of the est visile structure points on the vessel wll is the point nerest to the imge center (Courtney et l., 22; Rosles et l., 24). The sptil loction of this reference point is determined s the positions ðx ; k yþ in k frme k nd ðx kþ1 ; y kþ1þ in frme k + 1 on the vessel structure (see Fig. 4) tht hve the miniml Eucliden distnce to the imge centers (cx k,cy k ) nd (cx k+1,cy k+1 ). The distnce criterion cn e written s ðx k ; y k Þ¼rgmin uððx o x k ðuþþ 2 þðy o y k ðuþþ 2 Þ 1=2 where x k (u) nd y k (u) re the coordintes of the points of the ellipticl vessel model. written s ðt k Þ¼ þ D k where ¼ rctnðy =x Þ is the reference ngle corresponding to the initil frme, nd D k rctnðy k =x kþ is the rottion ngle of frme k Imge motion correction The suppression of the rottion is given y the following liner trnsformtion. The current imge frme I k (x,y) is trnslted y (cx k,cy k ) to center the ellipse on the imge center, following y rottion through n ngle k : x k y k ¼ cosð k Þ sinð k Þ sinð k Þ cosð k Þ x k y k cx k cy k where ðx ; k y kþ nd (x,y) re the new nd old crtesin imge coordintes, k is the rottion ngle nd (cx k,cy k ) is the ellipticl center. By itertively pplying this eqution, we chieve to lign ll frmes within the IVUS sequence. 4. Vlidtion nd results As mentioned ove, our work is to use the geometric model to explin nd remove the discontinuous longitudinl vessel ppernce ð7þ ð8þ
6 96 M. Rosles et l. / Medicl Imge Anlysis 13 (29) in IVUS dt; hence, we pln the following set of experiments to vlidte the motion compenstion procedure: 1. Roustness of the estimtion of motion prmeters we check the impct of inccurcies in the estimting of the ellipsoidl shpe s well s the roustness of the estimtion of motion prmeters y mens of dynmic vessel phntom. 2. A roust mesure of rottion suppression in experimentl dt since in rel cses it is difficult to know in dvnce the expected motion prmeters, it is required to define first quntity properly ssessing the ccurcy in motion correction. The synthetic phntom helps us to define roust mesure of rottion suppression in terms of the vessel wll profile in longitudinl cuts. 3. Assessment of motion suppression in rel sequences we check the efficiency of the proposed technique y study of short sequences of 3 imges ech one extrcted from 3 ptients. 4. Stilized Vessel Model in order to illustrte the improvements provided y IVUS imge lignment we show the results of vessel reconstruction efore nd fter correction Roustness of the estimtion of motion prmeters The vessel phntom for motion correction is generted y pplying motion profile to synthetic IVUS imges. The Fourier decomposition given in Section 2.1 serves to compute the rottion (Eqs. (6) nd (7)) nd the vessel displcement (Eqs. (1) nd (2)) profiles. The result of dding hert dynmics contriutions to vessel geometry is the sinusoidl lue line nd the resulting geometric profile due to the ctheter pullck in tortuous vessel displyed y the red solid line in Fig. 6. We generted the vessel position corresponding to the different temporl frmes nd otined the vessel profiles in the simulted longitudinl IVUS cuts. Two different phntom models hve een used: n ellipticl shpe nd vessel profile extrcted from rel-dt. The ellipticl model llows to detect rtifcts in the geometric computtion of the rottion ngle. The vessel shpes extrcted from mnul segmenttion of IVUS sequences, ssess the roustness of the method ginst the elliptic modelling of vessel orders. For the ellipticl model we used 73 ellipses with eccentricities uniformly smpled in the rnge [, 1]. For the genertion of the rel-dt sed phntom, we used, for ech frme k only 25% of the vessel wll dt points rndomly selected. In this wy we cn check the model roustness to lck of informtion in disesed sections of the vessel. The error mesure used to ssess the ccurcy of the estimted rottion profiles is the solute difference etween ngles computed from the true synthetic profile. The correltion coefficients (the slope, m, nd the independent term, ) etween the estimted ngle nd the theoretic one hve een computed. In the idel cse of perfect correltion m = 1 nd =. Fig. 7 shows the regression line otined for n ellipse eccentricity with n = 37.5%, with slope m = 1.3 nd =.23. Fig. 7 shows the ngulr error s function of the eccentricity n. When the ctheter is ner to the lumen center ( 1 6 n 6 1) the liner coefficient m hs singulrity due to the equidistnce etween the ctheter center nd the vessel wll points. In these cses, the reference point s the closest point to the ctheter center, used to estimte the rottion ngle mkes no sense. In the cse of rel-dt sed phntoms (see Fig. 8), the liner coefficients present stle ehvior long the sequence, with sttisticl vlues for m equl to l 1.2, r.1. This represents pproximtely 1% of error in the estimtion of the rottion ngle which, tking into ccount tht only 25% of the vessel wll dt point hs een used, is n cceptle experimentl result. The liner coefficient gives the se line ngle etween theoreticl nd experimentl ngle. The sttisticl vlues of were l 5, r 4), hence, there is n cceptle se line shift etween theoretic nd rel ngle. The dvntges of incorporting vessel geometric informtion to the computtion of its rigid dynmics re reflected in the ccurcy chieved in synthetic phntoms. According to (Dnilouchkine et l., 26) only motion elow 3.5 cn e successfully modelled. For those rottions the men solute ngulr errors did not exceed 7.8. Our methodology supports lrge rottions in the rnge [ 25,25] degrees with n solute error of 5 ± 4. Errors sttisticl rnges etween theoretic nd estimted prmeters where extrcted from the regression coefficients (the slope m nd the independent term ) of the errors otined for ech vessel dynmic phntom. Since the verge slope is 1.2, the rnge for the independent term reflects the rnge for the solute error. Figs. 7 nd 8 show the regression lines for n elliptic (Fig. 7) nd rel vessel profile (Fig. 8) phntom models A roust mesure of rottion suppression Numericlly checking to wht extend the vessel wll rigid motion hs een correctly computed cn only e done in synthetic models, where the true displcement prmeters re known. In the cse of rel imges extrction of quntittive mesurement it is not possile nd one hs to se on motion correction in vessel wll visul ppernce. Vessel profiles in longitudinl cuts nd lignment of vessel slient structures re good cndidtes. After proper trnsltion nd rottion suppression (Pluim et l., 23), the vessel shpe ppernce of longitudinl cut is stright line, s oth sw-like periodic ptterns nd vessel curvture due to n origin different from lumen center hve een removed (Seemntini et l., 25; De Winter et l., 24). We propose using the Eucliden distnce, Dd, etween the vessel profile nd its liner fitting s ojective mesure of the mount of motion removed. Fig. 9 illustrtes the mechnism followed to compute Dd nd the difference etween the dynmic synthetic profile (Fig. 9) nd its corrected counterprt (Fig. 9). The sttisticl rnge (given y the men ± stndrd devition) of Dd long the segment mesures the reduction in vessel movement. The men vlue of Dd reflects the overll performnce of the proposed suppression nd the stndrd devition mesures its roustness. Fig. 1 displys Dd histogrm efore (Fig. 1) nd fter (Fig. 1) rottion suppression for the rel-dt sed α in [degrees] 1 5 Vessel geometry contriutions Hert dynmics contriutions Time in [sec] Fig. 6. Rottion profile for synthetic simultion of vessel dynmics. Blue line shows the hert dynmics contriution nd the red line the vessel geometry contriution.
7 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Estimted ngle (α) in [degrees] Liner coefficient m = 1.3 =.23 Error in [degress] Rel ngle (α) in [degrees] Ctheter eccentricity (ξ) in [%] Fig. 7. Rottion estimtion for elliptic phntom. Theoreticl vs. estimted rottion profile for n eccentricity n = 37.5%, (), nd error in ngle estimtion s function of the ellipse eccentricity, (). Fig. 8. Rottion estimtion for rel-dt phntoms. Rel (in lue) nd estimted (in red) rottion profile () induced to rel vessel wll points nd its liner correltion (). Vessel wll points Liner fitting Distnce (Δd) etween liner fitting nd vessel wll points Vessel wll points Liner fitting Distnce (Δd) etween liner fitting nd vessel wll points Fig. 9. Longitudinl cut efore () nd fter () rottion suppression. phntom. Before movement suppression, the distnce distriution, Dd A, is in the rnge (1.1 ±.5) mm, while fter correction the distnce, Dd B, is within (.4 ±.1) mm. This represents n verge movement reduction of 97% nd strighten of the sinusoidl vessel wll shpe induced y rottion nd trnsltion. The rtio, R, etween the distnce fter, Dd A, nd efore, Dd B, indictes the mount of movement suppressed. A vlue of zero is chieved when the vessel profile is stright line nd vlue ove 1 suggests tht the lgorithm hs fils to reduce motion. In experimentl dt, rdil diltion nd vessel curvture prevent the rtio R chieving its optiml lower ound even if rigid motion hs een properly removed. In fct, in short segments, rdil diltion is the min source of devition from stright pttern. According to clinicl studies (Willims et l., 1999; Dodge et l., 1992) lood pressure introduces n verge diltion of 14% of the vessel dimeter. In order to tke into ccount this seline elstic deformtion, we define rigid rtio s Rr ¼ jdd A :14 VesselDimj Dd B for VesselDim the men dimeter of the vessel segment under study. ð9þ
8 98 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Count Distnce Δd in [mm] Count Distnce Δd in [mm] Fig. 1. Histogrm of Dd efore () nd fter () rottion suppression Assessment of motion suppression in rel sequences In order to study the ehvior of the reported motion suppression strtegy we nlyzed set of sequences of 3 imges ech one for 3 ptients independent from the set used to trin the Neurl Network. The sequences were cquired using Boston Sci. equipment Boston Scientific Corportion, 1998 t 4 MHz t constnt pullck speed of.5 mm/sec. The processing time cn e divided in four steps (see Section 3): 1. Neuronl network trining is reltively fst, since the min chrcteristics difference etween vessel wll nd the rest is notly lrge, nerly 5 min (using PC Pentium t 2.4 GHz). 2. Vessel wll detection y Neuronl Network (.8 s/imge) 3. Vessel wll ellipticl fitting (.1 s/imge). 4. Correction of imge rottion nd displcement using the extrcted prmeters of the ellipticl pproximtion (.1 s/ imge). Longitudinl cuts efore nd fter rottion elimintion re shown in Fig. 11. One cn notice the smoothing effect of the rottion correcting on the ppernce of the lumen order (red line in Fig. 11) in the longitudinl view. Still complete smooth ppernce of the vessel in the longitudinl imges is not expected due to the different degree of rdil expnsion/contrction of the vessel (Fig. 11d nd f) ccording to its elstic properties (since we remove/correct only rigid vessel motion effects). We should lso tke into ccount tht y ligning vessel lumens, we hve n opticl rtifct due to the externl dventiti right tissue nd the ctheter which dopt wvy sw-shpe ppernce (see Fig. 11). The structures in longitudinl views tht reflect correct lignment of vessel IVUS plnes, re the ones ordered y the red lines tht show the trnsition etween lood nd tissue. Correction of center position strightens the vessel profile (see Fig. 11 c), removing most of the sw-shpe effect if present (see Fig. 11d f). Rottion correction produces n lignment of structures which is etter pprecited in the presence of slient plque such s clcium (ottom of Fig. 11). The periodic vessel rottion produces shdow-right nded profile t clcium sectors which disppers converting it into uniform right or drk nd (s it is the cse of Fig. 11) menwhile the rottion is suppressed. As for the geometric systemtic rottion induced y the vessel geometry, it results in continuous vrition of slient structures which do not suddenly dispper ut smoothly chnge long the longitudinl cut (see Fig. 11e). Our sttisticl nlysis focused on checking the roustness in prmeter estimtion s well s reduction in movement. Roustness in the estimtion procedure is ssessed y computing prmeters of originl nd corrected imges. Intuitively, we cn ssume tht the less vrition in prmeters extrcted from utomticlly computed still imges, the more relile the reported geometric motion estimtion is. The stndrd devition, r, long ech sequence indictes such vriility for ech prmeter. The stndrd devition in the rdil position of the lumen centers efore trnsformtion correction presents significtive decrement for ll cses from Dr (2.3 ± 3) mm to Dr (.5 ±.3) mm fter imge lignment. In the cse of rottion estimtion,wehvereductionfromd (69 ± 2) efore to d (11 ± 8) degrees fter rottion suppression tht supposes 85% of rottion suppression. Fig. 12 shows the stndrd devition of motion prmeters for the 3 ptients nlyzed, lumen center position in Fig. 12 nd rottion ngle in Fig. 12. The quntittive ssessment of motion reduction is given y the rigid rtio Rr defined y (9) nd using tht the verge vessel dimeter in our dt is pproximtely 2.5 mm. Fig. 13 shows the rigid rtio Rr nd its histogrm in percentges (Fig. 13) for the 3 cses. The rnge for Rr in percentges is 79.3 ± nd in 76% of the cses is over 72% Stilized vessel model In order to otin relistic model of the vessel shpe, we must seprte the geometric vessel properties from hert dynmics contriutions. Let c(t) =(x(t), y(t), z(t)) e the vessel order s function of time t written s n ellipticl pproximtion: xðtþ ¼ðtÞ cosðh þ dðtþþ þ cxðtþ; yðtþ ¼ðtÞ sinðh þ dðtþþ þ cyðtþ; zðtþ ¼czðtÞ ð1þ where ( < h 6 2p), ((t),(t)) re the minor nd mjor rdii of the ellipse, d(t) is its orienttion nd C =(cx(t), cy(t), cz(t)) its center. In 2.5-D IVUS reconstruction cz(t) is unknown, ut in our model we cn tke it s the ctheter position in longitudinl direction, cz = vt, where v is the ctheter velocity. Tking into ccount the decoupling into geometricl nd hert dynmics contriutions given in Section 2.1, the vessel wll prmeters cn e written s ðtþ ¼ g ðtþþ d ðtþ; ðtþ ¼ g ðtþþ d ðtþdðtþ ¼d g ðtþþd d ðtþ cxðtþ ¼cx g ðtþþcx d ðtþ; cyðtþ ¼cy g ðtþþcy d ðtþ where the suscript g stnds for the geometric component nd the d for the periodic hert movement. In order to reconstruct the vessel wll from ctheter point of view only the Fourier coefficient tht corresponds to geometricl contriution must e tken into ccount, so tht Eq. (1) reduces to
9 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Fig. 11. Longitudinl cut efore (( f) up) nd fter (( f) down) rottion correction Ellipses center efore Before rottion suppression std in [mm] Ellipses center fter std in [degrees] After rottion suppression Ptient numer Ptient numer Fig. 12. Stndrd devition of motion prmeters efore (lue) nd fter (red) rottion suppression: ellipses centers () nd rottion ngle (). xðtþ ¼ g ðtþ cosðh þ d g ðtþþ þ c xg ðtþ; yðtþ ¼ g ðtþ sinðh þ d g ðtþþ þ c yg ðtþ; zðtþ ¼vt Fig. 14 shows sttic vessel reconstruction of n IVUS pullck of 19 imges corresponding to 25 mm long vessel segment. Fig. 14 shows the 2.5-D grphicl representtion of the segment efore
10 1 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Rr rtio Ptient numer Fig. 13. Movement reduction in experimentl dt: reduction rtio for the 3 ptients () nd histogrm of reduction percentge (). Fig. 14. Sttic vessel model: two views of 2.5-D vessel wll reconstruction efore () nd fter () dynmic suppression nd the corresponding longitudinl IVUS cuts efore (c) nd fter (d) motion suppression. () nd fter () dynmics suppression. Fig. 14c nd d illustrte the corresponding longitudinl IVUS cuts efore nd fter motion compenstion. 5. Discussions 5.1. Ellipticl fitting of vessel orders When considering the prolem of imge registrtion nd lignment, nturl question is which generl strtegy to use: imgesed (where ll pixels of the imges contriute to the estimte of the imge trnsformtion) or lndmrk-sed (only some, usully hving specil mening, points of the imges re tken into ccount in order to lign the imges). Our pproch follows the second strtegy ligning the elliptic pproximtions to set of points corresponding to the lumen-tissue trnsition. Although n ellipsoidl shpe cn not exctly pproximte the vessel wll shpe, such inccurcies do not ffect the performnce of the proposed pproch since its purpose is to estimte the rigid trnsformtion of the vessel. By using n ellipticl pproximtion of the vessel represents glol pproch to detect motion nd rottion of the vessel; t the sme time the ellipticl pproximtion is le to etter cope with smll errors in the clssifiction of the contour points y the neurl network (compred to flexile snke model or other locl deformle models). Empiriclly, we found tht in 88% of segmented imges the ellipticl shpe model could pproximte the vessel eing the rest of imges minly cses of ifurctions to min vessels nd lrge NURD rtifcts. In order to evlute the qulity of the vessel pproximtion, we estimted distnce mp of the found contour points y the neurl network nd pproximted the vessel points y the ellipticl model. We consider tht the ellipticl model pproximtion is correct if the vlue of its points on the distnce mp is less thn given epsilon. In our cse, epsilon ws set empiriclly to.2 mm. Fig. 16 shows the men error nd stndrd devition of estimted distnces s well s their histogrms. In cse the distnce surpsses this threshold, hypothesis for segmenttion filure is generted nd the frme corresponding to this prt of the vessel is disregrded from nlysis nd motion estimtion is
11 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Fig. 15. Segmented vessel wll y neurl network (in yellow) nd fitted ellipse (in lue). Fig. 16. Men error ( k ) () its stndrd devition () etween segmented points y the neuronl network nd the djusted ellipse. The corresponding histogrm re given in (c) nd (d). computed ligning the next vlid frme. Finlly, dynmic prmeters of the filed frme re computed y interpoltion of the neighor vlid frmes. Ellipticl fitting qulity cn e visully checked in the imges of Fig. 15, where we show the points extrcted from the neurl network (in yellow) nd the fitted ellipse (in lue) for six consecutive frmes. From our tests on the 3 ptients IVUS videos we conclude the following: () Our methodology is not ffected y clcified plque presence s it presents n rupt trnsition from lumen to tissue which is esily identified y the neurl network (see Fig. 17).
12 12 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Fig. 17. Performnce in the presence of rtifcts: clcifiction (), guide rtifct () nd ifurctions (c). () Imging rtifcts, such s sensor guide, constitute sprse phenomen leding to locl responses in the short-xes imge view tht do not ffect the ellipse pproximtion (see Fig. 17). (c) In the cse of NURDS, such sequences re of limited clinicl interest y their low qulity (impossiility to quntify vessel structures) nd re out of the scope of our nlysis, neither recommended to e quntified in clinicl studies. Still, the procedure for ellipticl fitting is vlid provided tht the NURD rtifct does not ffect more thn 3% of the vessel Motion rtifct removl The complex motion of the imging ctheter inside the coronry vessel cuses motion rtifcts tht cn e pproximted y three phenomen: () rottion of the IVUS imging ctheter with respect to the vessel (i.e., round the xis tngentil to the ctheter), () trnsltion of the imging ctheter in the plne perpendiculr to the ctheter xes, nd (c) trnsltion of the ctheter long its xis forwrd nd ckwrd cusing swinging effect. In this pper, we ddress the first two motion rtifcts in order to help the short-xis s long-xis vessel disply, to visulize nd quntify different morphologicl vsculr structures like plque, stents, lumen dimeter, lesions, etc. The intrinsic rottion nd trnsltion of the imging ctheter (relted to type () nd ()) introduce rtifcts in the longitudinl views mking structures to pper nd dispper s well s provoking tooth-shped dventiti nd clcified plque, etc. This motion minly ffects the computtion of the vessel dimeter in the L-views. In the short-xis view, the rottion nd trnsltion of the ctheter led to high mount of vessel motion. By pplying our stiliztion pproch, vessel ppernce in short-xes views llows etter nd esier perception of vessel structures distriution nd quntity. In longitudinl views we remove the tooth-shpe nd thus ssure tht chnges in vessel dimeter re due to chnges in morphology. In this wy, we improve the nlysis of plque progression long the vessel compred to the originl L-views. Note tht this step is crucil for further nlysis of vessel iomechnics (plpogrphy, mesuring elstic properties s strin or stress, etc.) since their computtion requires trcking mteril points of the vessel long the crdic cycle. Relile length mesurements long the vessel must either ccount (i.e. correct) for ctheter swinging or simulte imge-sed ECG-gting. We note tht the crdic components of the rottionl ngle cn serve to simulte n ECG-gting y smpling the sequence t the right (crdic) rte. Therefore, lthough we do not explicitly ddress it, our estimtion of dynmics enles relile volumetric mesurements s well s exploring vessel (continuous) elstic properties y keeping the originl informtion (complete set of dt O Mlley et l., 26) Our pproch nd IVUS gting In this pper we del with the prolem of rottion suppression of IVUS imges tht cn e oserved in non ECG-gted s well s ECG-gted sequences. Thus, our pproch is not depending nd completely pplicle to ECG-gted procedures (sed on hrdwre or imge-gting). ECG-gting hs the dvntge to llow more precise longitudinl mesurements; the price to e pid consists of incresing the cquisition time in up to 7 times with respect non-gted motorized pullcks t.5 mm/seg. (von Birgelen et l., 1997), which results in n increse of the intervention time of the ptient. Besides the specil cquisition devices re not commonly ville t ll medicl centers. The ove resons hve recently motivted the development of imge-sed ECGgted IVUS (Seemntini et l., 25; De Winter et l., 24). Such techniques discrd prt of the informtion (frmes) ville in n IVUS sequence. Since in our pproch to lign imges we do not put ny constrint on the pir of imges, our pproch llows using the whole stck of IVUS imges provided y the cquisition device s well s the ECG-gted suset of imges in order to overcome the mislignment rtifcts introduced y vessel rottion nd trnsltion. Current ECG-gting s well imge-gting pproches still cn suffer from the imge rottion rtifcts mking difficult to ddress prolems s plpogrphy estimtion, plque following long the vessel, volumetric mesurements, etc. Moreover, lthough we do not explicitly ddress it, we note tht the crdic components of the rottionl ngle cn serve to simulte n ECG-gting y smpling the sequence t the right (crdic) rte. Therefore, our estimtion of dynmics enles relile volumetric mesurements s well s exploring vessel (continuous) elstic properties y keeping the originl informtion (complete set of dto Mlley et l., 26) Ctheter oscillting oliquity Crdic dynmics introduce two min rtifcts ffecting the ctheter trjectory. Firstly, due to the chnge in vessel curvture during the crdic cycle, the ctheter moves long the xis of the vessel (longitudinl displcement). This forwrd nd ckwrd displcement implies tht, in mechnicl pullck, IVUS imges re not uniformly distriuted long the sequence z-xis. Secondly, tilting of the ctheter introduces n nisotropic scling of IVUS cross sections if the ctheter is not coxil within the vessel. Such phenomenon produces more ellipticl shped vessel, which might result in n overestimtion of the re nd misinterprettion of the vessel shpe (Mintz et l., 21).
13 M. Rosles et l. / Medicl Imge Anlysis 13 (29) Correction of ctheter longitudinl movement would require dynmic model of the ctheter 3-D trjectory nd is out of the scope of the presented work. Tilting of the ctheter cross-sectionl plne cn not e properly corrected unless complete 3-D reconstruction nd modelling of the vessel nd the ctheter inside the vessel re ville. Since we im t improving 3-D IVUS without ffecting vessel mesurements such s length or re, ctheter oliquity will e disregrded Rdil deformtion of the coronry vessel Rdil deformtion of crdic rteries is minly due to lood pressure nd vessel wll elstic properties. By Hooke s lw (Mzumdr, 1992; Ndkrni et l., 23; Holzpfel nd Weizsäcker, 1998; Humphrey, 1995), the rdil increment, rr, is proportionl to the grdient of lood pressure, rp vi the reltion (Mzumdr, 1992; Ndkrni et l., 23; Holzpfel nd Weizsäcker, 1998; Humphrey, 1995): rr ¼ðjDP=pÞ 1=2 ð11þ where j is the elsticity coefficient. At vessel sections presenting non homogenous plque, the elsticity coefficient vries for ech pixel. In order to estimte the rdil deformtion, it suffices to use constnt elstic coefficient (Hook s lw for uniform medi) descriing the verge deformtion rnges of the vessel wll. According to clinicl studies (Willims et l., 1999), j is in the rnge j = (.1 ±.2) mm 2 /mmhg nd DP 4 mmhg. By Hook s lw (11), it follows tht rr.35 mm. Tking into ccount tht the rdii of coronry segments (Dodge et l., 1992) is in the rnge r = 2.64 ±.3 mm, we hve tht the reltive rdil deformtion induced y lood pressure scling is dr =(rr/r) 1 13%. Applying scling of the vessel in the IVUS imges might remove the deformtion due to the lood pumping tht is directly relted to the elsticity of the vessel. Besides the existing rdil scling does not ffect the computtion of the trnsltion nd rottion prmeters. Therefore, rdil scling is not included into the model formultion Clinicl pplicility The methodology proposed nd its implementtion improve the visuliztion in short-view IVUS videos s well s in the longitudinl views with success of correctly ligned sequences. Currently, the methodology is to e implemented in pckge for IVUS nlysis nd plque chrcteriztion to e instlled in clinicl conditions. On the other hnd, the proposed methodology for rottion suppression hs different possile clinicl pplictions: 1. Generl mesures extrction from stilized IVUS sequence. We consider tht this ppliction is first nd necessry step towrds: () Estimtion of geometric prmeters such s vessel length using longitudinl cuts views sed on methodology pplying the ECG-gting to ddress the swinging rtifcts of the IVUS ctheter. () Vessel wll strin estimtion y plpogrphy. In order to otin vlid strin estimtion, we consider tht plpogrphy extrction needs: first, lignment of imges nd second, determining the frmes corresponding to end-systole nd end-distole peks (eing estimted y ECG-signl or imge nlysis). 2. Hert dynmic estimtion. Being possile to seprte the geometric contriution from dynmics contriution, we developed the sis to 2.5-D vessel reconstruction only using IVUS dt. This method cn give promising evidences for vessel wll dynmics estimtion such s the introduction of n lterntive technique to estimte locl hert dynmics. In this wy, we provide new possiility of studying the vessel dynmics nd geometry s well s estlishing new dignostic tools. For exmple, n extensive study of IVUS rottion profile versus pumping hert efficiency otined experimentlly using distolic nd systolic ngiogrphy hert views, cn hint positive correltion etween hert rottion nd pumping hert efficiency. 6. Conclusions In this rticle we developed geometric nd kinemtic model in order to study the evolution of coronry rtery wll. The model is sed on the ssumption tht the evolution of the rteril wll cn e modeled ssuming two principl contriutions tht come from different physicl resons. The first one, systemtic contriution cused y geometric intrinsic rteril properties, nd the second one, n oscillting contriution tht comes from ventricle dynmics. These contriutions govern in mjor degree the profiles ppernce of rteril wll in longitudinl views. Using this ssumption, we propose methodologicl strtegy in order to estimte nd suppress rigid IVUS dynmicl distortions. IVUS imge lignment is very importnt in prolems tht study vessel deformtion long the crdic cycle, such s plpogrphy which compres deformtions t distole nd systole. Exploring vessel wll kinemtics needs vessel order points trcking in order to extrct vessel deformtion nd to judge out coronry elsticity. Definitely, imge rottion oserved in current IVUS sequences hinders the trcking of the corresponding points/zones of the vessel order. On the other hnd, the ccurte rdil deformtion of the vessel on the imge not only depends on the mutul imge/ vessel rottion, ut lso on the rtifct produced y the oscillting longitudinl oliquity tht is induced y the ventricle pulstile dynmics. Usully, oscillting longitudinl ctheter motion is overcome y ECG-gted or imge-gted sequence construction tht improves significntly the longitudinl view of the vessel, ut cquires picture of the vessel shpe only in distole (De Winter et l., 24). Looking for lterntive technicl pproches to solve the prolem of imge sptil loction covering the whole crdic cycle is nother reserch opportunity to dvnce in the more ccurte longitudinl visuliztion, vessel rdil deformtion nlysis, vessel/plque elsticity estimte, prediction of therosclerotic ccumultion nd possile plque ruptures from IVUS dt. Seprting the geometric contriution from dynmicl contriution cn e the fundmentl sis to 2.5-D vessel reconstruction only using IVUS dt nd could e n importnt dvnce in vessel wll dynmics estimtion tht introduces n lterntive technique to estimte locl hert dynmics. In this wy, we provide new possiility of studying roustly the vessel dynmics, creting sttisticl models of vessel dynmics nd estlishing new dignostic tools. Acknowledgement We would like to thnk Crlos Rodríguez from the Mthemticl Dep. of the Universitt Autonom de Brcelon for his support nd dvise. This work ws supported in prt y reserch grnt from projects TIN C2, FIS-PI6129, PI7454 nd PI71188 nd CONSOLIDER-INGENIO 21 (CSD27-18). The lst uthor hs een supported y The Rmón y Cjl Progrm. References Andrew, F., Murizio, P., Roert, B., Direct lest squre fitting of ellipses. IEEE Trns. Pttern Anl. Mch. Intell. 21 (5), Berry, E., Kelly, S., Hutton, J., Lindsy, H., et l., 2. Intrvsculr ultrsoundguided interventions in coronry rtery disese, NHS RD HTA Progrmme.
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
A07 Uncertinties in Locl Anisotropy Estimtion from Multi-offset VSP Dt M. Asghrzdeh* (Curtin University), A. Bon (Curtin University), R. Pevzner (Curtin University), M. Urosevic (Curtin University) & B.
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, email@example.com
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
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Intensity trnsformtions Stefno Ferrri Università degli Studi di Milno firstname.lastname@example.org Methods for Imge Processing cdemic yer 2017 2018 Sptil domin The sptil domin of n imge is the plne tht contins
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1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
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Visul motion Mn slides dpted from S. Seitz, R. Szeliski, M. Pollefes Outline Applictions of segmenttion to video Motion nd perceptul orgniztion Motion field Opticl flow Motion segmenttion with lers Video
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On the Uncertinty of Sensors Bsed on Mgnetic Effects E. ristoforou, E. Kyfs, A. Kten, DM Kepptsoglou Ntionl Technicl University of Athens, Zogrfou Cmpus, Athens 1578, Greece Tel: +3177178, Fx: +3177119,
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
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How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex
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3SC Fctors ffecting the phontion threshold pressure nd frequency Zhoyn Zhng School of Medicine, University of Cliforni Los Angeles, CA, USA My, 9 57 th ASA Meeting, Portlnd, Oregon Acknowledgment: Reserch
Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University
7th World Conference on Nondestructive Testing, 25-28 Oct 2008, Shnghi, Chin Chrcteriztion of Impct Test Response of PCCP with System Identifiction Approches Astrct Zheng LIU, Alex WANG, nd Dennis KRYS
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. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order
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Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
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Technicl Appendix: Childhood Fmily Structure nd Schooling Outcomes: Evidence for Germny Mrco Frncesconi* Stephen P. Jenkins Thoms Siedler Universy of Essex nd Universy of Essex Universy of Essex Instute
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MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
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U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
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