906 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 An Adaptve-Focus Deformable Model Usng Statstcal and Geometrc Informaton Dnggang Shen and Chrstos Davatzkos AbstractÐAn actve contour (snake) model s presented, wth emphass on medcal magng applcatons. There are three man noveltes n the proposed model. Frst, an attrbute vector s used to characterze the geometrc structure around each pont of the snake model; the deformable model then deforms n a way that seeks regons wth smlar attrbute vectors. Ths s n contrast to most deformable models, whch deform to nearby edges wthout consderng geometrc structure, and t was motvated by the need to establsh pont-correspondences that have anatomcal meanng. Second, an adaptve-focus statstcal model has been suggested whch allows the deformaton of the actve contour n each stage to be nfluenced prmarly by the most relable matches. Thrd, a deformaton mechansm that s robust to local mnma s proposed by evaluatng the snake energy functon on segments of the snake at a tme, nstead of ndvdual ponts. Varous expermental results show the effectveness of the proposed model. Index TermsÐActve contour, snake, statstcal shape models, adaptve focus deformable model. INTRODUCTION æ DEFORMABLE models [] have been used extensvely n mage analyss, especally n medcal or bologcal magng applcatons [2], [3]. In cases where strong a pror knowledge about the object beng analyzed s avalable, t can be embedded nto the formulaton of the snake model [4]. For applcatons n whch a tranng set s avalable, statstcal models can be appled [5]. Cootes et al. [6] have developed a technque for buldng compact models of the shape and appearance of varable structures n 2D mages, based on the statstcs of labeled mages contanng examples of the objects. A herarchcal statstcal modelng framework s also developed for representaton, segmentaton, and trackng of 2D deformable structures n mage sequences [7]. Flexble Fourer contour and surface models [8], performng statstcs on Fourer parameters [9], were appled nto the segmentaton of 2D and 3D objects from MRI volume data. Besdes, researchers [0] also consdered ncorporatng a type of smoothness constrant nto the covarance matrx where neghborng ponts are correlated. An extensve revew of deformable models can be found n [2]. In ths paper, we nvestgate a herarchcal method usng a seres of affne transformatons to deform the ntal model onto the desred object. The man characterstcs of our model are the followng:. A model contour s frst constructed from a tranng set. Attached to each pont of the model are two knds of nformaton. Frst, an attrbute vector s used, whch reflects the shape characterstcs of the model around each pont, from a local and fner scale to a more global and coarse scale. The attrbute vectors are essental n our formulaton snce they dstngush dfferent parts of a boundary accordng to ther shape propertes. Second, our model uses statstcal nformaton about the expected shape. The authors are wth the Department of Radology, JHOC 3230, School of Medcne, John Hopkns Unversty, 60 N. Carolne St., Baltmore, MD 2287. E-mal: dgshen@cbmv.jhu.edu, hrstos@rad.jhu.edu. Manuscrpt receved 29 June 999 ; revsed 27 Dec. 999; accepted 5 May 2000. Recommended for acceptance by Y.-F. Wang. For nformaton on obtanng reprnts of ths artcle, please send e-mal to: tpam@computer.org, and reference IEEECS Log Number 045. varaton, whch s expressed n a way that allows the model to emphasze partcular aspects of the shape to be reconstructed. Ths formulaton overcomes the lmtaton of prevous statstcal shape models [6] n whch larger features of a shape domnate over relatvely smaller, yet mportant features merely because ther large sze nfluences the measures of shape varablty. The adaptvefocus model also allows for a herarchcal formulaton n whch the focus of the deformable model can be steered each tme toward matches of relatvely hgher confdence. 2. The degree of smlarty between a partcular snake confguraton and the model s not merely Eucldean dstance n the 2D or 3D space, but rather the dstance between ther correspondng attrbute vectors. 3. The snake deformaton s acheved by employng a herarchy of transformatons of a varyng degree of smoothness. Importantly, our model doesn't deform snake ponts ndvdually, but t deforms segments of the snake at a tme. It s shown that ths procedure helps the snake avod local mnma. Our approach s partcularly suted to bomedcal applcatons for three reasons. Frst, the snake does not deform freely to nearby edges. Instead, t looks for ponts n an edge map that have smlar geometrc structure. Ths mples that the ponts n the model deform toward anatomcally correspondng ponts n a bomedcal mage. Second, statstcal knowledge about the anatomy can be used to restrct the snake deformaton to the space of anatomcally meanngful shapes. Fnally, our model s robust to mssng or erroneous data, whch s often the case n hstologcal sectons. 2 GEOMETRIC DESCRIPTION OF A SHAPE Oneoftheelementsofourdeformablemodelsanattrbutevectorthat s attached to each pont along the contour. The attrbute vector holds parameters that descrbe the shape characterstcs of the model contour around each pont. Throughout ths paper, we have used a numberofaffne-nvarantstoformtheattrbutevector; eachattrbute s the area of a trangle shown n Fg.. Areas of larger trangles represent more global propertes of the contour. It s not hard to see that the attrbute vector correspondng to V s dfferent from attrbute vectors of other ponts. We use ths fact n the deformaton process to preserve the confguraton of the snake. For the detaled dervaton of the affne-nvarant attrbute, please refer to []. Let's defne a contour C by an ordered set of ponts, fv ˆ x ;y j ˆ ; 2;...;Ng. For the th pont V, ts correspondng attrbute vector s defned as the areas of R trangles, T F ˆ f ; f ;2...f ;R. Here, f;vs s the area of a trangle V vsš V V vsš and vsš ˆ vs NŠ%N. Notce that, f vs s close to, then f ;vs reflects the local shape nformaton. As vs ncreases, f ;vs gradually captures more global shape nformaton. Therefore, the attrbute vector F ntegrates dfferent levels of shape nformaton around the th pont. It can be made exactly affnenvarant by the normalzaton ^F ˆ F P N P R ˆ vsˆ ; j f ;vs j h where ^F ˆ ^f ; ^f;2... ^f T ;R. Thus, the shape nformaton of the curve C can be descrbed by a set of affne-nvarant attrbute vectors, ^F ;ˆ ; 2;...;N. 3 HIERARCHICAL SNAKE DEFORMATION In ths secton, a herarchcal snake deformaton mechansm s proposed. In ths mechansm, the segments of the model seek 062-8828/00/$0.00 ß 2000 IEEE
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 907 defnes the external energy, amng at deformng the snake towards a boundary n the mage. E data 3. Snake Energy Defnton We now defne the model energy term, E model, of () as the dfference between the snake's attrbute vector and the model's attrbute vector: Fg.. Schematc representaton of the concept of the ªattrbute vectorº on the th pont. The area of a trangle V vs ;V ;V vs s used as the vsth element of the th attrbute vector F. Here, vs R: mage boundares wth the smlar shape structure and are not smply nfluenced by nearby edges. Ths s acheved by an energy term, whch measures the degree of smlarty between a partcular snake confguraton and the model, as reflected by the Eucldean dstance between ther correspondng attrbute vectors. Ths mechansm s descrbed n Secton 3.. In Secton 3.2, a herarchy of global and local affne transformatons s desgned as a deformaton strategy for the snake. Ths deformaton s very robust to local mnma snce t deforms the segments of the snake, not ndvdual ponts. Fnally, n order to capture the fnest detals of the boundary of nterest, a technque based on a local-curve fttng s presented n Secton 3.3. In the followng, the defnton of our snake energy s gven. Let's defne a snake C snake n a way analogous to the defnton of the curve n Secton 2, as an ordered set of snaxels, fv ˆ x ;y j ˆ ; 2;...;Ng. The th snake segment s defned as the set of (2R ) ponts, V vsš j ; R vs R around the th snaxel V (See Fg. 2a, for example). As customary, the total snake energy s defned as the weghted summaton of several energy terms: E snake ˆ XN ˆ! E ˆ XN ˆ! E model E data ; where! s a weghtng parameter for the th snaxel. E s composed of two terms: E model and E data. The term E model defnes the degree of smlarty between the snake and the model. The term ^f Snk ;vs E model ˆ XR vs vsˆ ^fsnk Mdl 2; ;vs ^f ;vs 2 Mdl where and ^f ;vs are the normalzed attrbute elements (areas of trangles), respectvely, for the snake and the model. The model can ether be a representatve shape or the result of an averagng procedure. The parameter jvsj s the degree of mportance of the vsth attrbute element ^fsnk ;vs (or the pont V vsš ) n the segment under consderaton. R s the number of geometrc attrbutes and the length of snake segment. The data energy term s usually desgned for movng the snake toward a boundary of nterest n the mage. Accordngly, we requre that, n the poston of each snaxel V, the magntude of the mage gradent s hgh and the drecton of mage gradent s smlar to the normal vector of the snake. Snce we suggest deformng the whole snake segment around each snaxel V at a tme, the data energy term E data for the th snake segment s obtaned by summng ndvdual terms along the th snake segment: E data ˆ XR vsˆ R jvsj jri V vsš jj h V ~ vsš ~n V vsš j ; 3 where jri V j, valued between 0 and, s the normalzed magntude of the gradent on the snaxel V ; ~ h V s the drecton of the gradent; ~n V s the normal vector of the snake n a gven snaxel V, drected toward the snake nteror. 3.2 Snake Deformaton Mechansm We now descrbe a greedy deformaton algorthm [2] that fnds a soluton that mnmzes the energy of the snake segment. In ths procedure, a snake segment of 2R ponts s deformed by an affne transformaton at each deformaton stage. The reason for transformng snake segments by affne transformatons s that the value of the model energy term E model remans unchanged under Fg. 2. The defnton of the th snake segment and ts affne transformaton. The black arrows n (b), (c), (d) ndcate the affne transformatons of the current snake segments, whle the gray arrows represent the affne transformatons already completed. (a) the th snake segment, wth 2R snaxels; (b), (c), (d) the affne transformatons of dfferent snake segments to the gray object.
908 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 Fg. 3. Constrants mposed on the deformaton of a snake segment. Gray smooth curves (dotted or not) represent the segment under transformaton, from V to V t. See text for detaled meanngs. affne transformaton of the th snake segment. Accordngly, the new confguraton of a partcular snake segment can be determned drectly by mnmzng an energy term E data that s found by ntegratng the ndvdual energy terms along the snake segment. The affne-transformaton of a snake segment can be completed by the affne transformaton of the trangle V RŠ V V RŠ. Let a tentatve poston of V n the greedy algorthm be V t. Then, ths tentatve selecton of V wll lead to the shape change of the related trangle from the orgnal trangle V RŠ V V RŠ to the new tranglev RŠ V t V RŠ (Fg. 2b). The related deformaton procedure can be vewed as an affne transformaton A. Suppose that all the snaxels (V RŠ V RŠ ) on the th snake segment move wth the three vertces (V RŠ, V, and V RŠ ) accordng to the same affne transformaton A. Then, the value of the energy term E model wll reman unchanged. In the greedy algorthm, we only need to examne the match between the currently affne-deformed snaxels and the object boundares by mnmzng E Data. Fg. 2b shows the deformaton of the th snake segment, whle deformatons for the upper-left and the lower snake segments can be found, respectvely, n Fg. 2c and Fg. 2d. In practce, we have to confne the moton doman of the th snaxel. For example, n Fg. 3, f the th snaxel V moves from the current poston to any poston n the gray area, ths moton wll reverse the correspondng segment. Snce ths transformaton, whch s determned from three ponts s appled to the whole snake segment, reversal lke the one descrbed above sometmes causes extreme and unrealstc shape deformatons. Moreover, f V moves nto the doman below the thck black lnes and above the Fg. 4. Local-curve fttng. See the text for the meanng of these ponts and curves. gray area, the th segment wll ntersect other segments. To avod such cases of the curve beng reversed, the sgn of the determnant of A should be postve. Also, to avod the cases of the curve beng should be far from the thck black lnes. Our deformaton mechansm s mplemented herarchcally and t can be summarzed next: self-ntersected, the new poston of the th snaxel V t. Use a large value of R to determne the best affnetransformed confguraton of the snake segment around V. At ths pont, the number of snake segments that are consdered s small and the search area s large. 2. Reduce the value of R to update the affne-transformed confguraton of the snake segment around V. The number of snake segments that are consdered becomes larger, whle the search area becomes smaller. 3. Fnally, set R equal to. Update the affne-transformed snake segment around V. At ths level, all snaxels are consdered. 3.3 Fne Deformaton by Local-Curve Fttng The deformaton mechansm n Secton 3.2 s very robust, but often at the expense of smoothng out the very fne detals of the boundary. Actually, varous local features such as curvature can be used together for gudng the snake to exactly localze the desred object boundary [3]. In order to acheve better conformty to the shape of an object, we employ a curve fttng procedure that s descrbed next. Ths procedure consttutes the fnal fne-tunng step of our algorthm. Fg. 5. An llustraton of the dfferental weghtng strategy n our adaptve-focus statstcal model. The left sde of the fgure shows the case of equal weghts on the whole face, whle the rght sde of the fgure shows the case of larger weghts assgned to the eyes. For both cases, the face mode corresponds to the egenvector wth the th largest egenvalue and the face shapes n the th row show the effect of varyng parameter of the face mode n turn between 3 s.d.. It can be observed from the left sde of the fgure that the varaton of eyes s not represented by the frst three egenvectors. However, on the rght sde of the fgure, the frst face mode clearly ncorporates varaton of the eyes after emphaszng the landmarks of the eyes.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 909 Fg. 6. Some algned samples that are used n our model. The gray contour denotes the standard shape, whle the black one denotes the algned shape. There are n total 8 samples n the tranng set of ventrcles. Fg. 7. Qualtatve comparsons of our algorthm wth the standard snake and ASM. Contours n (a, a2, a3) are the manual ntalzatons, where crosses ª+º are labels used to track ndvdual ponts; (b, b2, b3) the result of the standard snake; (c, c2, c3) the results of ASM; (d, d2, d3) the results of our method. In partcular, we select the poston V fne of the snaxel V from the nonzero Canny edge ponts n the neghborhood of the snaxel V. Ths poston V fne s the one for whch the snake segment V Š V fne V Š and the locally-connected mage edge segment are n best agreement. The maxmum-compresson process employed by the Canny edge detector results n edge maps wth relatvely few nonzero ponts. Therefore, the search for edges n the neghborhood of a snaxel can be performed fast. In Fg. 4, the neghborhood of the snaxel V s drawn as a ƒƒƒƒƒƒƒƒ! gray dotted crcle. The snake segment V Š V V Š s shown as a thck gray curve, connectng three consecutve snaxels V Š, V and V Š. The snake segment V Š V fne V Š resultng from a tentatve placement of the th snaxel at V fne s shown as a thn black curve. The locally-connected mage edge segment, wth the fne pont V fne on t, s shown as a thck black curve. Notce here, for dfferent selected poston of V fne, the locally connected mage edge segments mght be dfferent. In Fg. 4, the value D s used to represent the sze of the neghborhood of the local snake curve V Š V fne V Š, whch s enclosed by the dotted lnes. The degree of smlarty between the extracted edge segment Seg V fne and the local snake V Š V fne V Š s defned as the total length of the extracted edge segment contaned n the neghborhood of V Š V fne V Š : Length Seg V fne ;V Š V fne : V Š The procedure of curve fttng, determnng the fnal poston of the snaxel V, s summarzed next:. Suppose V fne s the poston of a selected nonzero Canny edge pont n the neghborhood of the snaxel V. 2. Regard ths Canny edge pont as the seed pont and then track the connected edge segment, Seg V fne pont from Canny edge map. Dfferent V fne extracts dfferent mage edge segment. 3. Calculate the degree of smlarty,, of ths seed probably Length Seg V fne ;V Š V fne V Š ; between the extracted edge segment Seg V fne and the snake curve V Š V fne V Š. 4. The fnal fne poston of V s determned by the best fne poston V fne, whch maxmze the fttng degree
90 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 TABLE Quanttatve Comparsons of Our Algorthm wth the Standard Snake and ASM, Usng the Results Gven n Fg. 7 The unt of the value gven n the table s at pxel sze. Fg. 8. (a)-(h) More segmentaton results of MRI bran mages by our algorthm. Length Seg V fne ;V Š V fne V Š n the whole neghborhood of the snaxel V. 4 ADAPTIVE-FOCUS STATISTICAL MODEL In ths secton, an adaptve-focus statstcal model s suggested. Ths formulaton overcomes the lmtaton of other statstcal shape models. In the followng, we gve an overvew of our formulaton by ndcatng the lmtatons of prevous methods. The bref descrpton of our formulaton s gven n Sectons 4. and 4.2. In ASM [6], all landmarks are treated equally. Consequently, components of the shape that are ether more varable or are larger and therefore are comprsed of a larger number of landmarks wll domnate over ether smaller or less varable components. Such components, however, can stll be very mportant and, therefore, should be captured accurately. Moreover, our confdence of a certan component mght be relatvely hgher than the rest of the shape. Therefore, we mght want to focus, at frst, on those relable parts of the shape and, subsequently, to shft the focus to other parts as they become closer to ther respectve targets and therefore more relable. As an example, we use face reconstructon. In ASM, the top three egenvectors do not represent the varatons of the eyes, as can be observed on the left sde of Fg. 5. The reason s that the sze of the eyes s smaller than the szes of other components of the face, whch makes the varaton of eyes lower than other components. Moreover, the varaton of the eyes s almost ndependent of the varatons of other components. Accordngly, the egenvectors correspondng to the largest egenvalues, whch are typcally used to represent shape varatons, do not reflect varaton of the eyes. However, for some applcatons [4], t s very useful to exactly reconstruct the moton of the eyes. Therefore, the statstcal model should be able to capture the varaton of all the mportant components of a shape and, possbly, to adaptvely emphasze partcular aspects of the shape to be reconstructed. One way to acheve ths s va assgnng dfferent weghts to dfferent components. In partcular, smaller and mportant features can be weghted relatvely more. The rght sde of Fg. 5 shows the effects of assgnng large weghts (fve tmes larger than others) to the landmarks of the eyes. In the frst row of the rght sde of Fg. 5, the varaton of the eyes s exactly represented by the top egenvector that corresponds to the largest egenvalue. By varyng ths face shape parameter between 3 s.d., we can capture the openng and closng of the eyes. The face database used n ths example s from the Bern face database at ftp://amftp.unbe.ch/pub/images/ FaceImages/. Besdes accountng for dfferences n the sze of a component, dfferental weghtng also provdes a means for focusng attenton to ndvdual components. For example, n segmentng basal gangla and ventrcular boundares from bran mages (Fg. 0), a pror knowledge s avalable that reflects our level of confdence n certan parts of the shape under consderaton. For example, the ventrcle usually has a stronger boundary n the bran mage and, thus, s easer to fnd. Therefore, we can ntally focus on the ventrcle by assgnng a hgher weght to t. As the ventrcular boundares are detected, our confdence n the neghborng basal gangla becomes much hgher because of ther relatvely fxed locaton relatve to the ventrcles. It should be ndcated here that a dfferent weghtng matrx employed usually leads to a dfferent subspace of tranng samplng. The reason s that a dfferent weghtng matrx makes a dfferent set of complete egenvectors and, thus, a dfferent set of truncated egenvectors used to represent the subspace of tranng samples.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 9 After some algebrac manpulatons, we arrve at the followng formula for fttng the model nstance P 0 to the pont vector P of the algned snake shape: P 0 ˆ T T 2 P T 3 ; where T ˆ W H, T 2 ˆ H T W, and 4 T 3 ˆ W W H H T S mean : Fg. 9. Examples of the poor performance of our algorthm, due to local mnma. 4. Shape Algnment and Weghtng Strategy We need to algn samples nto a common coordnate frame before performng the weghtng strategy and the statstcs on them. One tranng sample wll be selected as a standard shape and others are transformed to best ft ths standard shape. The algnment technque s obtaned from paper []. Let's defne n o x algn ;y algn j ˆ ; 2;...;N as the algned shape of the tranng shape fv j ˆ ; 2;...;Ng. Fg. 6 shows some examples of shape algnment. For smplcty, n the notaton, the algned shape s stacked and represented by a 2N-element column vector h T P ˆ x algn ;y algn ;...;x algn : N ;yalgn N Then, we can weght dfferent ponts of the snake dfferently and obtan a weghted vector S ˆ W P, by producng a dagonal weghtng matrx W before the column vector P. Here, W s a 2N 2N matrx, and S s a 2N-column vector. 4.2 Mappng the Snake to the Space Derved from the Tranng Set Now, we descrbe how we use the statstcal shape nformaton to constran the deformable model n the space of allowable (or lkely) confguraton. We frst calculate both the average vector S mean and the covarance matrx (2N 2N) from a set of the tranng vectors, fsg. 2N egenvectors of the covarance matrx can be calculated and ranked by the sze of ther correspondng egenvalues. From the statstcal theory, M egenvectors correspondng to M hghest egenvalues can be selected as the bass of the shape subspace of the tranng samples. For smplcty, n the notaton, we stack these M egenvectors as a 2N M-szed matrx H, where each column s one of egenvectors. The szes of the matrces T, T 2, and T 3 are, respectvely, 2N M, M 2N, and 2N. Once obtanng the best model nstance P 0, we can transform P 0 back to update the postons of snaxels n the current snake. 4.3 An Algorthm of Statstcal Snake Refnement The algorthm for refnng snake by the adaptve-focus statstcal model s summarzed as follows:. Algn the snake contour wth the selected standard shape by usng a algnment matrx A algn, whch s calculated by the algnment method. Afterward, stack the algned snake as a column pont vector h T ;...;x algn : P ˆ x algn ;y algn N ;yalgn N 2. Use (4) to correct the pont vector P nto a new vector P 0. 3. Update snake contour by transformng P 0 back to the orgnal coordnate space of the snake contour va the nverse matrx of A algn. 5 EXPERIMENTS The complete algorthm for our method s summarzed n Secton 5.. To evaluate our algorthm, two sets of experments are presented. The frst set of experments (Secton 5.2) demonstrates the performance of our whole model n the case of the dentcal weghts n the matrx W; ths corresponds to the nonadaptve focus model. The qualtatve and quanttatve evaluatons of our algorthm, compared to the standard snake [5] and ASM, are presented. The second set of experments (Secton 5.3) shows the performances of our adaptve-focus model. In all experments, the ntalzaton of the snake s provded by the user and the set of tranng samples s the same. There are a total of 8 samples n the tranng set of ventrcles. See Fg. 6 for some algned samples. Fg. 0. Example usng spatally varable weghtng. Larger weghts are assgned to ventrcles and smaller weghts to basal gangla. (a) Tranng samples, (b) a 00-pont model and a testng mage, and (c) fnal result.
92 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 22, NO. 8, AUGUST 2000 5. Complete Algorthm The multresoluton mplementaton of our algorthm s reached by subsamplng a set of snaxels along the snake contour and deformng ther correspondng snake segments. The complete algorthm usng both herarchcal affne transforms and adaptvefocus statstcs s as follows:. Select a small number of snake segments. The length of each snake segment s 2R snaxels, ntally, and therefore the correspondng search area s relatvely larger. 2. For every selected snake segment, determne ts best affnetransformed confguraton by mnmzng ts external energy term E Data. Here, the local affne-transformaton technque s employed to locally deform the snake segment that s consdered (detals n Secton 3.2). 3. Algn the current snake confguraton wth the standard model contour by usng the affne-transformaton matrx A algn calculated from the snake to the model. Then, stack the algned snake as a pont vector h P ˆ x algn ;y algn ;...;x algn N ;yalgn N (Secton 4.). 4. Map the pont vector P nto the new vector P 0 usng the adaptve-focus statstcal model, P 0 ˆ T T 2 P T 3 whch s descrbed n Secton 4.2. Then, transform P 0 back nto the orgnal coordnate space of the snake va the nverse matrx of A algn and update the snake. 5. Increase the number of snake segments to be consdered and decrease both the length of snake segment 2R and the sze of search area of every selected snake segment. Notce that decreasng the length of the snake segment s equvalent to decreasng R. If the length of the snake segment s 3,.e., R ˆ, then go to Step 6; otherwse, repeat Steps (2-5). 6. Use the local-curve fttng to refne the confguraton of the snake (Secton 3.3). 5.2 Expermental Results Usng Our Whole Model: The Case of Identcal Weghts A key property of the proposed method s ts tendency to mantan the geometrc shape of the snake model durng the shape deformaton procedure. Fg. 7 qualtatvely compares the performances of our method, the standard snake, and ASM n detectng ventrcles from the MRI bran mages (256 256). Intalzatons for three dfferent MRI mages are provded n Fgs. 7a, 7a2, and 7a3, where crosses ª+º are used as labels n order to track the dsplacements of ndvdual ponts. Usng the standard snake, the fnal results are shown n Fgs. 7b, 7b2, and 7b3. In addton to beng trapped by erroneous edges, the standard snake dd not preserve any anatomcal homology durng the deformaton, as reflected by the postons of crosses after the snake deformaton. ASM s able to preserve the model shape n the deformaton procedure, however, sometmes owng to local mnmum problem ts fnal results Fgs. 7c, 7c2, and 7c3 are stll unsatsfactory. On the contrary, our method gves good results, as shown n Fgs. 7d, 7d2, and 7d3. Table gves the quanttatve comparsons of our algorthm wth the standard snake and ASM, based on the results n Fg. 7. The average dstance of the fnal contour from the contour marked by experts s gven, at unts of pxel. Our algorthm has the least average dstances (:2 :7 pxels) for all three examples. Also, the maxmal dstance was calculated. Our algorthm stll has the least maxmal dstances (4:0 4:6 pxels). Obvously, n realty the ntalzaton wll not be as bad as the one used n these experments. We used ths ntalzaton to demonstrate the robustness of our algorthm to local mnma and therefore to the snake's ntalzaton, whch s known to be very mportant n most deformable models. T Fg. 8 shows more results of segmentng ventrcles from the MRI bran mages by usng our method. Gray contours denote the ntalzatons and whte contours denote the fnal results. Despte the challengng ntalzatons, the results were very good. Snce we are usng a greedy algorthm, we can't guarantee that all global mnma are found. Fg. 9 gves two examples of falure of our algorthm after convergence to local mnma. We attrbute the falure to the fact that the model and the objects are very dssmlar. In real medcal magng applcatons, one doesn't encounter cases lke the one n the left of Fg. 9, snce the model typcally s much more smlar to the shape of nterest. 5.3 Expermental Results Usng the Adaptve-Focus Deformable Model The goal n ths experment was to segment the basal gangla and ventrcular boundares from MR bran mages. In ths applcaton, the weghts of the adaptve-focus statstcal model are determned by the degrees of our confdences on the components of the shape under consderaton. Larger weghts are assgned to the landmarks of the ventrcles snce the ventrcles usually have a stronger boundary that s easer to locate. Smaller weghts are assgned to the landmarks of the basal gangla, whch have relatvely unrelable and fuzzy boundares often confounded by adjacent cortcal edge. Fg. 0b shows a 00-pont model, derved from a set of 0 samples (Fg. 0a), on the orgnal testng mage (256 256). The deformaton s drven prmarly by the ventrcular boundares of the model. As those get close to ther fnal postons, the adjacent basal gangla boundares also get close to ther fnal postons and therefore become more relable features to drve the deformaton. By usng ths strategy, a satsfactory result s obtaned and shown n Fg. 0c. Notce that the deformaton from the ntalzaton to the fnal result s large (partcularly for the lower part of ths object). 6 CONCLUSION AND FUTURE WORK In ths paper, we have proposed an adaptve-focus deformable model for segmentng 2D deformable objects from mages. The source codes for our adaptve-focus deformable model are freely avalable from the ste, http://pandora.cbmv.jhu.edu/~dgshen/ SnakeCode.htm. Several extensons of our methodology are possble. Frst, the boundary ponts are currently used to represent the object shape and also the snake segment. For speedng up our technque, the snake segment can be expressed as a B-splne. 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