A Herarchcal Deformable Model Usng Statstcal and Geometrc Informaton Dnggang Shen 3 and Chrstos Davatzkos Department of adology Department of Computer Scence 3 Center for Computer-Integrated Surgcal Systems and Technology Johns Hopkns Unversty Emals: dgshen@cbmv.jhu.edu hrstos@parthenon.rad.jhu.edu Abstract A new deformable has been proposed by employng a herarchy of affne transformatons and an adaptve-focus statstcal. An attrbute vector s used to characterze the geometrc structure n the vcnty of each pont of the ; the deformable then deforms n a way that seeks regons wth the smlar attrbute vectors. Ths s n contrast to most actve contour s whch deform to nearby edges wthout consderng the geometrc structure of the boundary around an edge pont. Furthermore a deformaton mechansm that s robust to local mnma s proposed whch s based on evaluatng the snake energy functon on segments of the snake at a tme nstead of ndvdual ponts. arous epermental results show the effectveness of the proposed methodology.. Introducton Deformable s have been used etensvely n mage analyss especally n medcal or bologcal magng applcatons [] where the objects (or structures) to be analyzed undergo deformatons. For applcatons n whch a tranng set s avalable statstcal s ncorporatng statstcal knowledge about the object and ts varablty have been proposed [3]. On the other hand fleble Fourer contour and surface s [4] performng statstcs on Fourer parameters [5] were appled nto the segmentaton of D and 3D objects from MI volume. Herarchcal approaches usually ncrease the lkelhood of fndng the globally optmal match where the calculatons of fner detals of the deformaton are performed after matchng on a more global scale. Multscale technques have been successful n mprovng the performance of the snake [6]. In ths paper we nvestgate a herarchcal method usng a seres of global and local affne transformatons for appromatng the actual deformaton estng between the ntally provded and the desred object. The man characterstcs of our are the followng: (a) A contour s frst constructed from a tranng set. Attached to each pont on the are two knds of nformaton. Frst an attrbute vector whch reflects shape characterstcs of the 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 statstcal nformaton about the epected shape varaton whch s determned va a tranng set. Importantly the statstcal nformaton s epressed n a way that allows the actve contour to emphasze partcular aspects of the shape to be reconstructed. Ths formulaton overcomes the lmtaton of prevous statstcal shape s [3] 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. (b) The degree of smlarty between a partcular snake confguraton and the s not merely an Eucldean dstance n the D or 3D space but rather the dstance between ther correspondng attrbute vectors. (c) The snake deformaton s acheved by employng a herarchy of transformatons of a varyng degree of smoothness n order to adapt the ntal to the underlyng object n the mage. A key component of ths deformaton mechansm s that t doesn t deform snake ponts ndvdually but t deforms segments of the snake at a tme helpng the snake avod local mnma.. Geometrc descrpton of a shape In ths secton we descrbe the affne-nvarant attrbute vectors [78] as a means to characterze shape propertes around each pont of an actve contour whch wll be later used to desgn an energy term of the snake. The attrbute vector holds parameters that descrbe the shape characterstcs of the contour around each pont. In ths paper the components of the attrbute vector are the areas of trangles shown n fgure 0-7695-0737-9/00 $0.00 (c) 000 IEEE
. For eample the area of the trangle + appromates the curvature around. Areas of larger trangles represent more global propertes of the contour. It s not hard to see that the attrbute vector correspondng to s dfferent from attrbute vectors of other ponts along the contour. We use ths fact n the deformaton process to preserve the confguraton of the snake. Let s defne a snake C snake by an ordered set of ponts commonly called snaels { = ( y ) =... }. For the th snael ( y ) ts correspondng attrbute vector can be defned as the areas of trangles: F [ ] T = f f... f [ ] [ + ] where f = y[ ] y y[ + ] s the area of a trangle formed by ( [ ] y[ ] ) ( y ) and ( [ + ] y[ + ] ) (see fgure ). otce that [ + ] = [ + + ]%. The sze of effectvely determnes the samplng resoluton of the curve for our representaton. Each attrbute vector F can be made eactly affne-nvarant by the followng normalzaton: F Fˆ = f = = where F ˆ [ ] T = fˆ fˆ... fˆ. Thus the shape nformaton of the curve C snake can be descrbed by a set F ˆ =.... of affne-nvarant attrbute vectors { } the degree of smlarty between a partcular snake confguraton and the as reflected by the Eucldean dstance between ther correspondng attrbute vectors. Ths mechansm s descrbed n Secton 3.. In Secton 3. 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 energy defnton of our snake s gven. Then the energy mnmzaton approaches are dscussed. As customary the total energy of the snake s defned as the weghted summaton of several energy terms: E = ω E = ω E + E () snake = = ( ) where ω s a weghtng parameter for the -th snael. E s composed of two terms: snake E and E whch are defned n detal n Secton 3.. The term E defnes the degree of smlarty between the snake and the. The term E defnes the eternal energy amng at deformng the snake towards a boundary n the mage. In the lterature many dfferent approaches for mnmzng the snake energy have been reported. In ths paper we wll use the greedy algorthm [9] as an optmzaton technque whch has been found to be one of the most relable fast and fleble local strateges. ( t ) + 3 + + + + +3 Fg. Schematc representaton of the concept of the attrbute vector on the -th snael. The area of a trangle formed by and + s used as the -th element of the -th attrbute vector. Here. + 3. Herarchcal snake deformaton In ths secton a herarchcal snake deformaton mechansm s proposed. In ths mechansm the segments of the seek the boundares wth the smlar shape structure n an mage edge map and are not smply nfluenced by any nearby edge. Ths s acheved by an energy term n the snake formulaton whch measures (a) (b) (c) (d) Fg. The defnton of the th snake segment and ts affne transformaton. The black arrows n (b-d) ndcate the affne transformatons of the current snake segments whle the grey arrows represent the affne transformatons already completed. 3. Snake energy defnton Accordng to the nternal energy term of the standard snake the moton of every snael ( ) s constraned only by ts two nearest neghborhoods and [ at each ] [ +] teraton. Ths can lead to two unwanted stuatons. Frst t can cause unrealstc deformatons of the snake as ndvdual ponts are pulled towards nosy or fragmented 0-7695-0737-9/00 $0.00 (c) 000 IEEE
edges. Second t does not preserve the geometrc structure of the contour.e. t can dramatcally change ts attrbute vectors. Our doesn t deform snake ponts ndvdually but t deforms segments of the snake at a tme helpng the snake avod local mnma. It also restrcts the deformaton process to be relatvely smooth. In order to acheve ths we requre that durng ts deformaton the snake should be able to preserve ts geometrc structure as reflected by the attrbute vectors. Accordngly we defne the frst term E n equaton () as the dfference between the snake s attrbute vector and the s attrbute vector whch s mathematcally gven n equaton (a). Here our s an average sample. E = = f Snk Mdl ( fˆ fˆ ) δ (a) Snk where f ˆ and ˆ Mdl are the normalzed attrbute elements (areas of trangles) respectvely for the snake and the. The parameter δ s the mportance Snk degree of the -th attrbute element f ˆ n the segment under consderaton. The th snake segment s defned as the set of + ponts { [ + ] } around the th snael. s the number of geometrc attrbutes and also used for determnng the length of snake segment. See fgure (a) for eample. The energy term s usually desgned to move the snake towards a boundary of nterest n the mage. Accordngly for every snael we can requre that n the poston of the magntude of mage gradent should be hgh and the drecton of mage gradent should be smlar to the normal vector of the snake. Snce we suggest deformng the whole snake segment around each snael at a tme the desgn of the energy term E for the -th snake segment results by summng ndvdual terms along the -th snake segment: r r E = δ I ( ) h( ) n( ) (b) = ( [ + ] [ + ] [ + ] ) where I( ) valued between 0 and s the normalzed magntude of the gradent on the snael ; h r ( ) s the drecton of the gradent; n r ( ) s the normal vector of the snake n a gven snael drected towards the snake nteror. From equatons () (a) and (b) the whole energy functon of the snake can be obtaned as follows: Snk Mdl Esnake = ω δ ( fˆ fˆ ) = = r r + δ ( I ( + ) n( + ) h( + ) ) (3) [ ] [ ] [ ] = 3. Snake deformaton mechansm We now descrbe a greedy deformaton algorthm for the mnmzaton of the snake energy functon n equaton (3). We have defned the energy terms E and E and suggested consderng the deformaton of a snake segment as whole. In partcular n each teraton that represents a deformaton of the snake a segment of ( + ) ponts s deformed by an affne transformaton. The energy functon n equaton (3) s evaluated by ntegratng along ths snake segment and t s optmzed va a greedy search. The reason for transformng snake segments by affne transformatons s because the value of the energy term E remans unchanged under affne transformaton of the -th snake segment snce the attrbute vector s affne-nvarant. Accordngly the new confguraton of a partcular segment can be determned drectly by mnmzng an energy term E that s found by ntegratng the ndvdual energy terms along the snake segment. Ths procedure whch greatly helps the snake avod local mnma s defned net. 3.. Affne transformaton of the snake segment The affne-transformaton of a snake segment can be descrbed by the affne transformaton of the trangle defned by snaels [ + ] [ and. Let s assume that a tentatve poston of the th pont n the greedy (t) algorthm s. Then ths tentatve selecton of the th pont wll lead to the shape change of the related trangle from [ to (fgure b). The related [ + ] ( t ) [ [ + ] deformaton procedure can be vewed as an affne transformaton A where A s an affne-transformaton matr. Suppose that all the snaels on the th segment move wth the three vertces ( [ and ] [ + ) accordng to the same affne transformaton A. Then the value of the energy term E wll reman unchanged because of the affne-nvarant nature of E. In the greedy algorthm we only need to eamne the match between the currently affne-deformed snaels and the object boundares by mnmzng E. Data A mathematcal epresson of the above affnetransformaton strategy can be detaled here by determnng the affne transform A frst and then transformng the nvolved snaels by the matr A. The matr A s gven by A ( t ) [ [ + [ [ + ( t) = y [ y y[ + y[ y y[ +. 0-7695-0737-9/00 $0.00 (c) 000 IEEE
Fgure (b) shows the deformaton of the th snake segment whle the deformatons for the upper-left and the lower snake segments can be found respectvely n fgure (c) and fgure (d). 3.. Herarchcal deformaton strategy In the defnton of the total snake energy functon E the parameter determnes the length snake ( + ) of the snake segment. The value of s typcally large n the ntal teratons and gradually reduces to. The whole herarchcal deformaton strategy suggested n Secton 3. can be summarzed as follows: () Use a large value of to determne the best affnetransformed confguraton of the snake segment around. At ths pont the number of snake segments that are consdered s small and the search area s large. () educe the value of to update the affnetransformed confguraton of the snake segment around. The number of snake segments that are consdered becomes larger whle the search area becomes smaller. (3) Fnally set equal to. Update the affnetransformed snake segment around. At ths level all snaels are consdered. 3.3 Fne deformaton by local-curve fttng The deformaton mechansm we descrbed n Secton 3. s very robust but often at the epense of smoothng out the very fne detals of the boundary. In order to acheve better conformty to the shape of an object we employ a curve fttng procedure that s descrbed net. Ths procedure consttutes the fnal fne-tunng step of our algorthm. In partcular we select the poston fne of the snael from the non-zero Canny edge ponts n the fne neghborhood of the snael. Ths poston s the fne one for whch the snake segment [ ] and the [ + ] locally-connected mage edge segment are n best agreement. The mamum-compresson process employed by the Canny edge detector results n edge maps wth relatvely few non-zero ponts. Therefore the search for edges n the neghborhood of a snael can be performed fast. In fgure 3 the neghborhood of the snael s drawn as a grey dotted crcle. The snake segment [ ] s shown as a thck grey curve [ + ] connectng three consecutve snaels and [ ]. [ +] fne The snake segment [ ] resultng from a [ + ] fne tentatve placement of the -th snael at s shown as a thn black curve. The locally-connected mage edge fne segment wth the fne pont on t s shown as a thck black curve. otce here for dfferent selected poston of fne the locally-connected mage edge segments mght be dfferent. In fgure 3 the value D s used to represent the sze of the neghborhood of the fne local snake curve [ ] whch s enclosed by the [ + ] dotted lnes. The degree of smlarty between the fne etracted edge segment Seg ( ) and the local snake fne [ ] s defned as the total length of the [ + ] etracted edge segment contaned n the neghborhood of fne fne fne : Length Seg( ) ). [ ] [ + ] D ( [ ] [ + ] fne [ +] [ ] Seg( fne Fg. 3 Local-curve fttng. The fttng degree between the fne etracted edge segment Seg ( ) and the local snake curve fne [ ] s defned as the total length of the etracted [ + ] edge segment contaned n the neghborhood of the local curve fne. See the tet for detals. [ ] [ + ] The procedure of curve fttng whch determnes the fnal poston of the snael s summarzed net: fne (a) Suppose s the poston of a selected non-zero Canny edge pont n the neghborhood of the snael. (b) egard ths Canny edge pont as the seed pont and fne then track the connected edge segment Seg ( ) of ths seed pont from Canny edge map. Dfferent fne probably etracts dfferent mage edge segment. (c) Calculate the degree of smlarty fne fne Length ( Seg( ) [ ] [ + ] ) between the fne etracted edge segment Seg ( ) and the snake fne curve [ ]. [ + ] (d) The fnal fne poston of s determned by the fne best fne poston whch mamze the fttng fne fne degree Length ( Seg( ) [ ] [ + ] ) n the whole neghborhood of the snael. ) 0-7695-0737-9/00 $0.00 (c) 000 IEEE
4. Adaptve-focus statstcal In ths secton an adaptve-focus statstcal s suggested whch allows the deformable to emphasze partcular aspects of the shape to be reconstructed. Ths formulaton overcomes the lmtaton of other statstcal shape s [3] 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. Moreover our confdence n certan features or parts of features mght be relatvely hgher compared to 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. For eample n segmentng basal gangla and ventrcular boundares from bran mages (fgure 6) a pror knowledge s avalable that reflects our level of confdence n certan parts of the shape under consderaton. 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 for the neghborng basal gangla becomes much hgher because of ther relatvely fed locaton relatve to the ventrcles. It should be ndcated here that a dfferent weghtng matr employed leads to a dfferent subspace spanned by a selected set of egenvectors derved from the tranng set. The detals of our formulaton are descrbed n Sectons 4. and 4.. 4. Shape algnment and weghtng strategy Gven a set of tranng samples we frst need to algn them nto a common coordnate frame before performng the weghtng strategy and the statstcs on them. The algnment technque used n ths paper s the affnenvarant algnment method [7]. In the algnment procedure one tranng sample s selected as a standard shape and others are transformed to best ft t. Suppose that the affne transform matr relatng one tranng shape { =... } and the standard shape s A algn. Then the coordnates of the pont = y ) after shape algnment become ( algn algn algn =. y A y For smplcty n the notaton the algned shape algn algn {( y ) =... } can be stacked and represented by a -element column vector P : algn algn algn algn [ y... y ] T P =. Once all the tranng samples have been algned the statstcal can be obtaned from the set of the algned shapes {P}. Before ntroducng our adaptve focus we note that actve shape s are typcally based on statstcs on the coordnate vector P. However a much broader class of actve shape s can be obtaned by applyng a lnear transformaton to the vector P by multplyng t Fea Fea by a matr W. The purpose of W s to etract certan features from the vector P. By specally Fea desgnng W we can obtan Fourer coeffcents Wavelet coeffcents or Gabor coeffcents. Moreover a Weg weghtng matr W can be appled enhancng certan aspects of the shape. If W Fea Weg = I and W s a dagonal matr we obtan a that weghts dfferent ponts of the snake dfferently. By adaptng W Weg durng the snake s deformaton we develop an adaptve-focus whch s able to zoom n and out of ndvdual parts of an object. We contnue our development by ntroducng the feature vector S = W P Weg Fea where W = W W. Assume that M s the number of the features obtaned by the feature etracton matr Fea W. Then the sze of the matr W Fea should be Weg M the sze of W M M and the sze of matr W M. The column vector S has M elements. 4. efnement of the snake contour In ths secton we descrbe how we utlze the stattcal shape nformaton derved from the tranng set n order to constran the deformable to be n the space of allowable (or lkely) confguratons. After algnng all the tranng samples nto a common coordnate frame we can calculate both the average feature vector S and the covarance matr ( M M ). mean M egenvectors of the covarance matr can be calculated and ranked by the szes of ther correspondng egenvalues. From the statstcal theory the M egenvectors correspondng to the M hghest egenvalues can be selected as the bass of the shape subspace of the tranng samples. ote that for a dfferent transformaton matr W there wll be dfferent covarance matr and also a dfferent shape subspace spanned by the M selected egenvectors. For smplcty n the notaton we can stack these M egenvectors as a M M -szed matr H where each column s one of egenvectors. After some algebrac manpulatons we arrve at the followng formula for fttng the nstance P to the pont vector P of the algned snake shape: 0-7695-0737-9/00 $0.00 (c) 000 IEEE
P = T T P + T (4) where T = W H T H T = W and T ( W W H H T 3 = ) S. The szes of the matrces mean T T and T 3 are respectvely M M and. Once obtanng the best nstance P we can transform P back to update the postons of snaels n the current snake just by usng the nverse algn matr of A. 4.3 An algorthm of statstcal snake refnement The algorthm for refnng the set of snake snaels by the statstcal s summarzed as: () Algn the snake contour { =... } wth the selected standard shape by usng the algnment algn matr A whch s calculated by the algnment method. Afterwards stack the algned snake as a column pont vector P algn algn algn algn P = [ y ] T... y. () Use equaton (4) to correct the pont vector P nto a new vector P. (3) Update snake contour by transformng P back to the orgnal coordnate space of the snake contour algn va the nverse matr of A. 5. Eperments The complete algorthm for our method s summarzed n Secton 5.. To evaluate our algorthm two sets of eperments are presented here. The frst set of eperments (Secton 5.) demonstrates the performance of our whole n the case of the Weg dentcal weghts n the matr W ; ths corresponds to the non-adaptve focus. The second set of eperments (Secton 5.3) shows the performances of our adaptve-focus. In where we focus on demonstratng the effect of our weghtng strategy. In all eperments the ntalzaton of the snake s provded by the user. The search tme of our algorthm depends on the ntalzaton of snake and the complety of the studed mage. It ranges from second to several seconds n an SGI OCTAE workstaton. 5. Complete algorthm The complete algorthm s as follows: () a small number of snake segments (or snaels). For each snake segment ts length and the correspondng search area are both larger ntally. The length of each snake segment s ( + ) where s assgned a larger nteger value ntally. 3 () For every snake segment n the set of selected snake segments determne ts best affne-transformed confguraton by mnmzng ts eternal energy term E. Here the local affne-transformaton Data technque s employed to locally deform the snake segment that s consdered. (Detals n Secton 3.) (3) Algn the current snake confguraton wth the standard contour by usng the affne- algn transformaton matr A. Then stack the algned snake as a pont vector P algn algn algn algn P = [ y ] T... y. (Secton 4.) (4) Map the pont vector P nto the new vector P usng the adaptve-focus statstcal P = T T P + T. Then transform P back nto the 3 orgnal coordnate space of the snake contour va algn the nverse matr of A and update the snake. (5) Increase the number of snake segments to be consdered and decrease both the length of snake segment and the sze of search area of every selected snake segment. If the length of the snake segment s equal to 3.e. = then go to step (6); otherwse repeat steps (-5). (6) Use the fne deformaton strategy to refne the confguraton of the snake to the object boundary. (Secton 3.3) 5. The case of equal weghts In ths set of eperments we appled our method wth the equal weghts ( W Weg = I ) not adaptve for segmentng the ventrcles from the MI bran mages. Some typcal tranng eamples selected from the set of 8 are shown n fgure 4. In fgure 4 the grey contour represents the standard shape of the ventrcle. The black contours are the ndvdual shapes algned to the standard shape. A key property of the proposed method s ts tendency to mantan the geometrc shape of the snake durng the shape deformaton procedure. Fgure 5 qualtatvely compares the performances of our method the standard snake and standard actve shape (ASM) n detectng ventrcles from the MI bran mages (5656). Intalzatons for three dfferent MI mages are provded n fgures 5(aaa3) where crosses + are used as labels n order to track the deformaton of ndvdual ponts. Usng the standard snake the fnal results were shown n fgures 5(bbb3). 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 shape of the n the procedure of deformaton however sometmes owng to local mnmum problem ts fnal results (ccc3) are stll unsatsfactory. On the contrary 0-7695-0737-9/00 $0.00 (c) 000 IEEE
our method mantans the geometrc shape durng the detecton procedure and thus gves good results as shown n fgures 5(ddd3). Table gves the quanttatve comparsons of our algorthm wth the standard snake and ASM based on the eample results gven n fgure 5. In table the average dstance of the fnal contour from the contour marked by eperts s gven at unts of pel. Our algorthm has the least average dstances (.3~.7 pels) for all three eamples compared to the standard snake and ASM. Also the mamal dstance between the fnal contour and the contour marked by eperts was calculated. Our algorthm stll has the least mamal dstances (4.~4.6 pels) compared to the other two methods. 5.3 The case of dfferent weghts We have suggested usng dfferent weghts for dfferent components n our adaptve-focus statstcal. Eperment usng ths on M mages s gven net. The goal n ths eperment was to segment the basal gangla and ventrcular boundares from M bran mages. In ths applcaton the weghts of the adaptvefocus statstcal can be determned by the degrees of our confdence n the components of the shape under consderaton. See fgure 6 for eample. We can assgn larger weghts to the landmarks of the ventrcles just because 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 edges. Fgure 6(b) shows a 00-pont derved from a set of 0 basal gangla and ventrcular boundares on the orgnal testng mage (5656). Tranng samples are shown n fgure 6(a). By usng our adaptve-focus deformable a satsfactory result s obtaned and shown n fgure 6(c). It can be observed that the deformaton from the ntalzaton to the fnal result s large (partcularly for the lower part of ths object) and our s robust aganst the attracton from the boundares of corte because of usng the suggested weghtng strategy n adaptve-focus statstcal. 6. Concluson and future work In ths paper we have proposed an adaptve-focus deformable for segmentng D deformable objects from mages. The source codes for our adaptve-focus deformable are freely avalable from the ste http://pandora.cbmv.jhu.edu/~dgshen/snakecode.htm. Several etensons of our methodology are possble. Currently the boundary ponts are used to represent the object shape and also the snake segment. For speedng up our technque the snake segment can be epressed as a B-splne. Then the deformaton of the snake segment can be vewed as a B-snake under affne-transformaton constrant [0]. Etenson to our technque for 3D s also possble. In ths paper we partcularly focused on desgnng the D verson of our deformable. We are now etendng our D technque to 3D by redefnng the geometrc features and the deformaton strateges. eference:. T. McInerney and D. Terzopoulos Deformable s n medcal mage analyss: a survey Medcal Image Analyss (): 9-08 996.. C. Davatzkos Spatal Transformaton and egstraton of Bran Images Usng Elastcally Deformable Models Comp. s. And Image Understandng Specal Issue on Medcal Imagng 66(): 07- May 997. 3. T.F. Cootes D. Cooper C.J. Taylor and J. Graham Actve shape s-ther tranng and applcaton Computer son and Image Understandng ol. 6 o. pp. 38-59 Jan. 995. 4. G. Szekely A. Kelemen C. Brechbuhler G. Gerg Segmentaton of -D and 3-D objects from MI volume usng constraned elastc deformatons of fleble Fourer contour and surface s Medcal Image Analyss (): 9-34 996. 5. L.H. Stab and J.S. Duncan Boundary fndng wth parametrcally deformable s IEEE Trans. on PAMI 4():06-075 99. 6. Jula A. Schnabel Smon. Arrdge Actve shape focusng Image And son Computng (7):5-6 pp.49-48 999. 7. Dnggang Shen W. H. Wong and Horace H. S. Ip Affne nvarant mage retreval by correspondence matchng of shapes Image and son Computng 7(7): 489-499 May 999 8. Horace H. S. Ip Dnggang Shen An affne-nvarant actve contour (AI-snake) for -based segmentaton Image and son computng 6(): 35-46 998. 9. D. J. Wllams and M. Shah A fast algorthm for actve contours and curvature estmaton Computer son Graphcs Image Processng 55:4-6 99. 0. Yue Wang Eam Khwang Teoh and Dnggang Shen Lane detecton usng B-snake IEEE Internatonal Conference on Informaton Intellgence and Systems Washngton DC ov. -3 999.. T.F. Cootes C.J. Taylor A. Lants Actve shape s: evaluaton of a mult-resoluton method for mprovng mage search n Proc. Brtsh Machne son Conference pp.37-336 994.. M. Chen T. Kanade D. Pomerleau J. Schneder 3-D deformable regstraton of medcal mages usng a statstcal atlas MICCAI Sept. 999. 0-7695-0737-9/00 $0.00 (c) 000 IEEE
Fg. 4 Some typcal tranng samples algned and used n our statstcal. The grey contour represents the standard shape whle the black ones denote the ndvdual shapes algned to the standard shape va affne transformatons. There are totally 8 samples n the tranng set of ventrcles. ventrcle (a) basal gangla (a) (a) (a3) ventrcle (b) (b) (b3) (b) (c) Fg. 6 Eample usng spatally varable weghtng. Here larger weghts are assgned to ventrcles because they have a stronger boundary and are easer to locate. Also smaller weghts are assgned to the landmarks of basal gangla because basal gangla have unrelable and fuzzy boundares often confused wth adjacent cortcal edges. The deformaton s drven prmarly by the ventrcular boundares of the. 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. (a) Tranng samples (5656); (b) a 00-pont derved from a set of samples s placed on the orgnal testng mage; (c) fnal result. (c) (c) (c3) (d) (d) (d3) Fg. 5 Qualtatve comparsons of our algorthm wth the standard snake and ASM. (aaa3) the manual ntalzatons (bbb3) the results of the standard snake (ccc3) the results of ASM and (ddd3) the results from our. Table Quanttatve comparsons of our algorthm wth the standard snake and ASM usng the results already gven n fgure 5. Fg.5(a) Fg.5(a) Fg.5(a3) average dst. ma dst. average dst. ma dst. average dst. Ma dst. Standard snake 6.9 8.3 6.8 9.5 3.3 8.4 ASM.5 8.0.8.7 5.9.8 Our algorthm.3 4.. 4.0.7 4.6 0-7695-0737-9/00 $0.00 (c) 000 IEEE