Visual Stabilization of Beating Heart Motion by Model-Based Transformation of Image Sequences

Transcription

1 Vsual Stablzaton of Beatng Heart Moton by Model-Based ransformaton of Image Sequences Evgenya Bogatyreno and Uwe D. Hanebec Abstract In order to assst a surgeon whle operatng on a beatng heart, vsual stablzaton maes the beatng heart appear stll by provdng the current heart vew as statonary and non-movng. In ths way, the surgeon s not dsturbed durng an operaton by the moton of the heart and has the mpresson of performng conventonal surgery. In contrast to exstng methods for vsual stablzaton, the proposed approach nvolves a model-based transformaton of mage sequences provded by a camera system. hs transformaton ncorporates the nowledge of physcal characterstcs of the heart n form of a mathematcal model of the heart surface. he man advantage of ths transformaton s that the uncertantes of the model and measurements are consdered. hs occurs by estmatng the parameters of the transformaton. Furthermore, the qualty of the vsual stablzaton s addtonally mproved by adaptng the parameters of the underlyng physcal model. he performance of the proposed approach s evaluated n an experment wth a pressure-regulated artfcal heart. In comparson to standard approaches, t provdes superor results llustratng the hgh qualty of the vsual stablzaton. I. INRODUCION Beatng heart surgery may allow hgh-rs patents to undergo bypass surgery. In comparson wth conventonal surgery, t s advantageous because of a sgnfcantly lower mortalty rate. However, beatng heart surgery s more challengng for a surgeon. For synchronzng surgcal nstruments wth the heart surface, the surgeon has to superpose the desred moton wth the beatng heart moton. hs demands a hgh concentraton, short reacton tme, and accurate handeye coordnaton from the surgeon. o overcome these dffcultes, a computer-asssted surgcal system wll facltate operatng on the beatng heart [1]. When the surgcal robot taes care of the synchronzaton, the provded vsual stablzaton wll enable the presentaton of the beatng heart surface as f t were statonary and non-movng. In ths way, the surgeon wll not be dsturbed durng an operaton by the moton of the heart and has the mpresson of performng conventonal surgery. When predcton of the heart moton for the synchronzaton of surgcal nstruments s nvestgated [1] [4], only few methods are proposed for the vsual stablzaton of the beatng heart. In [5], the use of electrocardogram-trggered strobed lght for mang the heart appear stll to a surgeon s explored. he presented expermental results show that ths method s not able to compensate the moton of the heart. he applcaton of strobed lght leads to ncreasng demands on the surgeon s concentraton and to fatgue. he hs wor was partally supported by the German Research Foundaton (DFG) wthn the Research ranng Group GRK 1126 Intellgent Surgery. E. Bogatyreno and U. D. Hanebec are wth the Intellgent Sensor-Actuator-Systems Laboratory (ISAS), Insttute for Anthropomatcs, Karlsruhe Insttute of echnology (KI), Karlsruhe, Germany. bogatyreno@t.edu, uwe.hanebec@eee.org approach to vsual stablzaton proposed n [6] smulates the stablzed vew of the heart surface by changng the extrnsc camera parameters. hs study shows that the approach performs well f a very small and smooth area of the heart surface s observed. However, t fals n the presence of local dstortons. In [7], the vsual stablzaton s based on mappng of the mages acqured at dfferent tme steps. For ths purpose, a lnear nterpolaton s appled. As can be easly recognzed, ths method demands a fne dscretzaton for achevng a hgh qualty of the stablzaton. Furthermore, the vsual stablzaton may become rough [8] snce no smoothness constrants regardng the heart surface moton are consdered. In summary, the qualty of the vsual stablzaton provded by all these approaches can deterorate n case of naccurate measurements. he man reason for ths s that uncertantes are gnored. Furthermore, for the processng of measurement data, these approaches do not nvolve models of both physcal phenomena and the measurement process, and neglect a pror nformaton. As a result, the measurement dsturbances, such as stochastc uncertantes caused by camera nose, envronmental dsturbances le smoe due to tssue cuttng, and naccuraces of feature extracton cannot be removed. Ignorng these dsturbances can reduce the qualty of the vsual stablzaton. he specalty of the method proposed n ths paper s the vsual stablzaton of the beatng heart moton usng a modelbased transformaton of mage sequences. hs method has the followng three partcular characterstcs: 1) ncorporatng nowledge of the underlyng physcal characterstcs of the beatng heart n form of a mathematcal model along wth a measurement model, 2) consderng the model uncertantes as well as accompanyng measurement nose drectly n the vsual stablzaton, and 3) mprovng the vsual stablzaton by parameter adaptaton of the underlyng mathematcal model. II. PROBLEM FORMULAION hs secton provdes a bref ntroducton to the problem of vsual stablzaton, whch s formulated as a transformaton of mage sequences. As common n computer-asssted surgery, we assume that an elastcally deformable object such as the surface of the beatng heart s observed by a multcamera system. Every camera provdes an mage sequence contanng nformaton about the deformaton of the object. We defne every mage as a set of quadratc pxels wth the assgned color values. Every pxel s dentfed by two ndces. For example, as llustrated n Fg. 1, the pxel p s dentfed by the ndces p x, and p y, at tme step t. he

2 reference mage at tme step t current mage at tme step t x, p mage transformaton : p p x, p pxel p y, p surface reconstructon reconstructon f l nterpolaton l s mage feature f pxel p y, p mage feature f mage reconstructon nterpolaton l s s projecton p moton propagaton l l surface pont s landmar l surface pont s landmar l Fg. 1. Vsual stablzaton of the elastcally deformable object ncorporatng model-based mage transformaton. hs transformaton s defned by a transformaton functon that provdes the mappng between the pxel ndces of the reference p and current p mages. hese mages represent the camera projectons of the movng object at two dfferent tme steps t and t. For better clarty, only a landmar l s placed on the surface of the object. ndces of all pxels are collected n the vector p := p x, p y. For vsual stablzaton, every mage of the mage sequence should be transformed to a certan reference mage wth the am to compensate the shft of the pxels,.e., shft of ther ndces. In other words, ths transformaton s descrbed by a mage transformaton functon that maps the pxels of the current mage to the pxels of the reference mage. he pxels of the reference mage are descrbed by the vector of ndces p := p x, p y. he defnton of ths transformaton functon deserves careful attenton. We are convnced that t should ncorporate not only geometrc but also physcal constrants on the heart moton for achevng hghly accurate vsual stablzaton. herefore, the parameters of the mage transformaton functon should depend on the physcal characterstcs of the heart. Furthermore, the uncertantes of the camera measurements and naccuraces of the transformaton functon should be consdered. Most exstng mage transformaton methods, such as warpng [8], [9] or morphng [1] gnore the uncertantes by assumng that the measurements and the transformaton functon are exactly nown. However, the real-world measurements are corrupted by dsturbances. Also the transformaton functon may not exactly reproduce the real transformaton and nvolves stochastc and systematc uncertantes. Ignorng these errors can lead to an naccurate vsual stablzaton. III. KEY IDEA he ey dea of the proposed approach s the desgn of a three-dmensonal model-based mage transformaton. As shown n Fg. 1, the current and the reference mages represent the camera projectons of the movng heart surface at two dfferent tme steps t and t. herefore, the shft of the ndces of the pxels p p can be descrbed by the dsplacements of the heart surface. For example, n ths fgure, the shft of the pxel p n the reference mage to the pxel p n the current mage s determned by the changng poston of the correspondng heart surface pont from s to s. Unfortunately, the measurements of the heart surface dsplacements are avalable only at certan measurement ponts, whch are represented by the landmars, whose threedmensonal coordnates are collected n the vector l := [ l x, l y, l z ]. (1) In ths paper, for achevng a hgh accuracy of measurements, artfcal landmars, such as crcular marers, are attached to the heart surface. For estmatng the dsplacements of all surface ponts, ther three-dmensonal coordnates are assembled n the vector s := [ s x, s y, s z ]. (2) We explot a physcal heart surface model, whch s ntalzed based on the three-dmensonal reference postons of the landmars, descrbed by the vector l := [ l x, l y, l z]. (3) As llustrated n Fg. 1 wth the example of landmar l, the postons of the landmars are reconstructed from twodmensonal postons of the mage features, assembled n the vector f := f x, f y. (4) In ths fgure, only a sngle landmar s depcted to smplfy the explanaton. When the heart deforms, the model represents the current deformatons of the heart surface and thus, provdes estmates of current three-dmensonal postons of all surface ponts s := [ s x, s y, s z ]. (5) he postons of these ponts are determned by nterpolatng between the current postons of the landmars l s. Fnally, by projectng the deformed model to the current mage s p, the correspondences between the pxel ndces and also postons of the mage features n the reference and current mages are establshed. Gven ths

3 current mages pxel ndces p nown unform exctaton force reference mages pxel ndces p u mˆ surface reconstructon and state-space model consstng of system model derved from a physcal heart surface model measurement model landmars l parameters θ surface ponts s smultaneous state and parameter estmaton state augmentaton predcton step flterng step predcted state updated state mage transformaton mage reconstructon color transfer stablzed mages Fg. 2. Model-based transformaton of the current mages to the stablzed mages ncorporatng the state-space model, whch consders the physcal characterstcs of the heart surface. he smultaneous state and parameter estmaton allows for consderng measurement and model uncertantes and adaptng model parameters. mappng, stablzed mages can be obtaned by transformng the color nformaton of the current pxels to reference pxels. For mprovng the vsual stablzaton, the dfferences between the estmated postons of the mage features, collected n the vector f := f x, f y (6) and the correspondng measured postons x ˆf := [ˆf, ˆf y ] (7) are traced bac and used to adapt the underlyng physcal model and thus, the mage transformaton. IV. VISUAL SABILIZAION he purpose of ths secton s the detaled explanaton of the proposed approach for the vsual stablzaton, whch s depcted n Fg. 2. Frst, the mage transformaton functon s derved n Secton IV-A. hen, the underlyng statespace model ncorporatng the physcal heart surface model s ntroduced n Secton IV-B. Fnally, the parameters of the physcal model are estmated usng a smultaneous state and parameter estmaton, whch wll be descrbed n Secton IV- C. A. Image ransformaton Functon By recallng that the mage transformaton s defned by the projectng the observed object to the camera mages at the reference tme step and current tme step, ths functon can be wrtten as p x p x =r 12u x + r 12u y + r 12u z + r 14 r 31 u x + r 32u y + r 33u z + r 34 p y p =r 21u x + r 22u y + r 23u z + r 24 y r 31 u x + r 32u y + r 33u z + r, 34 where the r j, = 1,..., 3, j = 1,..., 4 represents the elements of the projecton matrx P provded by a calbraton of the cameras [11]. he vector u = [ u x, u y, u z ] collects the three-dmensonal dsplacements of the heart surface. By usng (8), we can establsh the correspondences between the pxels of the current mage and reference mage. Fnally, the color nformaton of the current mage can be wrtten to the reference mage., (8) However, snce the camera measurements are avalable only at dscrete ponts and corrupted by uncertantes, the dsplacements of the heart surface between these two tme steps should be estmated. B. Model-Based Descrpton For estmatng the heart surface dsplacement, a statespace model consstng of system and measurement equatons, also called system and measurement models, wll be derved n ths secton. he specalty of the proposed method s that here the system model ncorporates nowledge of the underlyng physcal characterstcs of the heart. he reason for ths s that the system model s derved from a mathematcal model that descrbes the heart surface as a physcal elastc body. It s mportant to note that only the nterventon area s consdered. Because of small sze of ths area, the assumpton of an sotropc materal structure can be made. Furthermore, ths area s stablzed by a mechancal stablzer durng a heart operaton. he stablzer damps the heart moton so that the heart surface deformatons are small [12]. hs allows to assume a lnear elastc behavor of the heart surface. Moreover, the model maes use of the fact that the force generated by the myocardum can be represented as force volume relaton or pressure as stated n medcal expermental studes [13]. herefore, based on the lnear consttutve law, the heart moton s approxmately characterzed by a dynamc dstrbuted parameter system n form of partal dfferental equatons, accordng to [14]. Applyng the meshless collocaton method [15] and usng the mplct Euler method for temporal dscretzaton, the heart surface model s converted to a dscrete-tme system equaton n state-space form x +1 = A (ψ ) x + B (ψ ) ( ˆm u + w ), (9) where the vector ˆm u = [ ˆmu,x, ˆm u,y, ˆm u,z ] R 3M descrbes the nown model exctaton, whch s assumed to be unformly dstrbuted. he modelng errors denoted by w are assumed to be whte zero-mean Gaussan w N (, C w ) wth covarance matrx Cw R 6N 6N. he matrx A R 6N 6N represents the system matrx and the matrx B R 6N 3M denotes the nput matrx. hese matrces depend on the physcal parameters of the heart surface ψ = [ ρ, η 1, η 2 ], E, ν,

4 such as materal densty ρ, Raylegh dampng coeffcents η 1, η2, Young s modulus E, and Posson s rato ν. It should be noted that the state vector n equaton (9) x = [ c d, c ], (1) ncludes the fnte approxmaton coeffcents c = [ c x, c y, c z ] and ther dscrete dervatves (11) c d = (c c 1 )/ t, (12) where the t s the tme step and the vector c 1 denotes the approxmaton coeffcents from the prevous tme step. It s mportant to note that the approxmaton coeffcents c represent the parameters of the transformaton functon. hey are used for approxmatng the three-dmensonal veloctes and dsplacements of the heart surface. For example, the dsplacements of the heart surface are calculated by u Φ c, (13) where the approxmaton matrx Φ := dag { Φ, Φ, Φ } s assembled from the matrx ϕ (s 1 Φ, l1 ) ϕ (s 1, ln ) :=..... ϕ (s L, l 1 ) ϕ (s L, l N ) (14) wth elements defned by the movng least-squares meshless functon ϕ defned n [16]. It should be noted that the approxmaton ncorporates N landmars, whose threedmensonal postons l are averaged over the perod of the heart moton. he postons of these landmars are reconstructed accordng to [14]. he reconstructon s based on the correspondng mage features f extracted from all mages of the camera system. he reference postons of the L surface ponts s are obtaned by nterpolatng between the landmars l accordng to [17]. An mportant pont to emphasze s that n [14] the heart exctaton s assumed to be generated by a unformly dstrbuted pressure nsde the cardac chambers. However, the heart moton s ntated wth nternal forces generated n a dstrbuted fashon wthn the muscles. In our paper, the consdered heart surface model s extended so that we do not need an assumpton on the unformly dstrbuted heart exctaton. For ths purpose, an addtonal unnown nonunformly dstrbuted exctaton force m n s ntroduced n the system model (9) so that ths model taes the form x +1 = A (ψ ) x + B (ψ ) ( ˆm u + m n + w ). (15) Furthermore, the unnown model parameters are augmented n the parameter vector θ = [ψ, mn ], (16) where the vector ψ s ntalzed by a nonlnear least-squares method, descrbed n [14]. he measurement process model descrbes the relatonshp between the propagated state vector x and the measurements. o smplfy the explanaton, we focus on the dervaton of the measurement equaton when the measurements are provded by a sngle camera. By assumng that the postons of F mage features are currently measured, we defne the measurements by the vector ˆf R 2F. It should be noted that the number F of measured features can vary at every tme step, e.g., when these measurement ponts,.e., landmars, are occluded or features are not detected. Furthermore, n ths paper, the lens dstorton of the cameras s assumed neglgble. For gettng the measurement equaton, frst, the threedmensonal heart surface dsplacements are approxmated usng the state vector and movng least-squares meshless functons, accordng to (13). he approxmaton matrx Φ s then computed accordng to equaton (14) by substtutng the vector l nto s. Fnally, the current postons of the measurement ponts are projected to the current camera vew at every tme step ˆf x f x =r 12u x + r 12u y + r 12u z + r 14 r 31 u x + r 32u y + r 33u z + r + v x, 34 ˆf y f y =r 21u x + r 22u y + r 23u z + r (17) 24 r 31 u x + r 32u y + r 33u z + r + v y. 34 he error terms v x and vy denote the measurement dsturbances v = [ v x, v y] due to mage nose and naccurate projecton. hese uncertantes are assumed to be whte zeromean Gaussan v N (, C v ) wth covarance Cv R 2F 2F. For processng the measurements provded by a camera system, the equaton (17) s constructed for every camera. C. Smultaneous State and Parameter Estmaton he proposed mage transformaton s characterzed by consderng transformaton uncertantes. hs s acheved by nvolvng a smultaneous state and parameter estmaton n the transformaton process. For that purpose, the system state (1) s augmented wth. the parameter vector (16) n the vector z = x, θ It can be easly shown that the substtuton of ths vector n the lnear system model (15) leads to a nonlnear estmaton problem that can be solved by dfferent Bayesan nonlnear estmators. In ths paper, the Gaussan flter, frst proposed n [18], s appled for smultaneous state and parameter estmaton. hs lnear regresson Kalman flter characterzes the state estmate by ts frst two moments,.e., mean and covarance matrx. hese moments are captured n an approxmate fashon by regresson ponts, whch are determned usng determnstc samplng approach. Propagatng these ponts through the nonlnear system and measurement equatons (15, 17) mplctly lnearzes these equatons. Assumng the process w and the measurement nose v as uncorrelated and by representng the ( augmented ) state as a Gaussan random vector z N µ e, Ce, the Gaussan flter recursvely updates the mean and covarance of the augmented state µ [ˆx e = e, ˆθ e ] {, C e := dag C x,e, Cθ,e n two steps: predcton step and flter step. he predcton step provdes the a pror densty of the predcted state }

5 Fg. 4. he measured posons of the small green marers are used for evaluatng the vsual stablzaton and therefore, they are not consdered by flterng. he measured postons of the large green marers are ncluded n the measurement vector ˆf. Fg. 3. Expermental setup for evaluaton of the mage stablzaton performance. he moton of the pressure-regulated artfcal heart s measured by a trnocular camera system. ( ) f p (z +1 ) N µ p, +1 Cp +1. Snce the system model s condtonally lnear, smlar to [19] and [2], the system equaton can be decomposed nto lnear and nonlnear substructures by means of Rao-Blacwellzaton. For effcency of the estmaton, only the nonlnear part s processed by the Gaussan flter n an approxmate fashon. Furthermore, the flter step provdes the ( a posteror ) densty of the updated state f e (z ) N µ e, Ce. Snce the measurement model (17) s nonlnear, the standard flter step provded by the Gaussan flter [18] s appled. In ths step, the devaton between the predcted poston of the mage features and correspondng measurements ˆf s used for correctng the state and thus adaptng the parameters of the transformaton functon and of the underlyng physcal model, as shown n Fg. 2. Usng the mean values of the updated state, the mean value of the three-dmensonal heart surface dsplacements û = [ûx, û y, û z ] s estmated by equaton û = H ˆx e, (18) where the matrx H := dag {, Φ } contans the approxmaton matrx (13), dependng on the postons of the landmars l and the postons of the measurement ponts f. As a result, by substtutng û nto u of (8), we obtan the mappng between the pxels of the reference mage and current mage. In contrast to the exstng approaches, ths model-based mage transformaton functon allows to transform the actual mage to the reference mage also when no measurement nformaton s avalable. In ths case, for reconstructng the heart surface dsplacements (18), the predcted state ˆx p s used. As a result, n spte of the loss of measurement nformaton loss, the correspondences between the pxels of the current mage and the reference mages are estmated and the color nformaton of the current mage can be wrtten to the reference mage. V. EVALUAION After a detaled descrpton of the expermental setup, the accuracy of the vsual stablzaton wll be analyzed n a seres of experments. A. Expermental Setup he performance of the proposed approach for vsual stablzaton s evaluated n an experment wth a pressureregulated artfcal heart. he expermental setup s shown n Fg. 3. he moton of the stablzed beatng heart, accordng to the medcal study performed n [12], s.59 mm n the lateral plane (x-y-axs) and 2.1 mm out-of-plane (zaxs). he obtaned ampltude of the artfcal heart moton s 1.3 mm and 1.2 mm n x- and y-drecton, respectvely, and 2.5 mm out-of-plane. It s generated by a pressure sgnal wth ampltude of 1 hpa and a frequency of 1.2 Hz. herefore, the moton of the artfcal heart s assumed to be comparable to the real heart deformatons. For measurng the heart surface moton, a trnocular camera system s nstalled at a dstance of 5 cm above the montored operaton area. he three cameras PIKE F-21C wth resoluton 192 pxel 18 pxel provde an accuracy of.2 mm/pxel n y-drecton and.11 mm/pxel n x-drecton of the mage coordnate system. he camera baselnes are about 57 cm, ther focal length s about 35 mm, and the feld of vew s 12.8 cm 17.2 cm. he mage sze of every camera s reduced by cuttng out the defned regons of nterest. For evaluatng the performance of the vsual stablzaton, addtonal evaluaton ponts are ntroduced. hese ponts are presented n Fg. 4 by small green marers. her measured poston s not consdered by the estmator n the flter step. For the evaluaton, sx mage sequences are collected, each of these conssts of 3 mages. he frame rate s about 22 frames/ sec. hese sequences are fnally transformed to the reference mage. For every mage sequence, the dstances between the stablzed and the reference poston averaged over the number of the evaluaton ponts s defned as a transformaton error at every tme step. B. Expermental Results For demonstratng the hgh qualty of the vsual stablzaton, the proposed model-based mage transformaton s compared wth the geometrc mage transformaton based on the drect processng of the measurement data. hs transformaton s ntroduced n [9]. In Fg. 5(a), the transformaton error of both approaches averaged over 6 mage sequences s shown over tme. In order to emphasze the challenges of the vsual stablzaton, the transformaton error of the non-stablzed moton of the evaluaton ponts s depcted n Fg. 5(b). Accordng to the presented results, the maxmum error of the geometrc mage transformaton s 3.2 pxel, where the maxmum error of the model-based mage transformaton s 2.6 pxel. herefore, the proposed physcs-based mage transformaton provdes about 19% hgher accuracy

6 transformaton error n pxel model based geometrc tme n sec (a) he transformaton errors of the geometrc and model-based mage transformaton methods. transformaton error n pxel unstablzed tme n sec (b) he transformaton error of the non-stablzed mage sequence for representng the challenges of the mage transformaton. Fg. 5. Comparson of the results provded by the geometrc and model-based vsual stablzaton. he transformaton error s averaged over sx mage sequences contanng 3 mages. Accordng to the maxmum transformaton error, the model-based mage transformaton provdes about 19% hgher accuracy then the geometrc mage transformaton. then the geometrc approach. It s especally effcent n compensatng strong dsplacements of the evaluaton ponts, whch range up to 18 pxel. VI. CONCLUSION he man dea of the vsual stablzaton s assstng a surgeon durng an operaton on a beatng heart by presentng hm a beatng heart as non-movng. In ths way, the surgeon can operate wthout beng dsturbed by the heart moton. hs enhances the surgeons capabltes. he proposed approach for vsual stablzaton of the beatng heart s characterzed by three new deas. Frst, n contrast to exstng methods based on drect processng of measurement data, t explots nformaton from a stochastc physcal heart surface model. hs allows vsual stablzaton also when no measurement nformaton s avalable. Second, ths approach copes wth uncertantes of the measurements and the mage transformaton by usng nonlnear estmaton. hrd, the qualty of vsual stablzaton s contnuously mproved durng a beatng heart operaton by adaptng the parameters of the transformaton functon. In future wor, the extenson of the proposed approach to three-dmensonal vsual stablzaton s planned. hs s possble by projectng two stablzed stereo vews to a headmounted dsplay of the surgeon or transformaton of the stablzaton n three-dmensonal space. REFERENCES [1] Y. Naamura, K. Ksh, and H. 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