Graph Cut-based Automatic Segmentation of Lung Nodules using Shape, Intensity, and Spatial Features

Transcription

1 Second Intenatonal Wokshop on Gaph Cut-based Automatc Segmentaton of Lung Nodules ung Shape, Intenty, and Spatal eatues Xujong Ye, Gaeth Beddoe and Geg Slabaugh Medcght PLC, London, UK Abstact. Ths pape pesents a new, fully automatc method of accuately extactng leons fom CT data. It fst detemnes, at each voxel, a fvedmenonal featue vecto that contans ntenty, shape ndex, and 3D spatal locaton. Then, non-paametc mean shft clusteng s appled to poduce ntenty and shape mode maps. nally, a gaph cut algothm segments the mage ung a novel enegy fomulaton that ncopoates shape, ntenty, and spatal featues. A key dffeence fom the usual gaph constucton s that we connect modes (small egons, o supe-pxels esultng fom mean shft clusteng) nstead of pxels. The ntal foegound and backgound can be automatcally obtaned by calculatng hghly sphecal egons based on the shape ndex map. The poposed method has been evaluated on a clncal dataset of thoacc CT scans that contans 100 nodules. A volume ovelap ato between each segmented nodule and the gound tuth annotaton s calculated. Ung the poposed method, the mean ovelap ato ove all the nodules s On vsual nspecton as well as ung a quanttatve evaluaton, the expemental esults demonstate the potental of the poposed method. The ch nfomaton povded by the jont spatal ntenty-shape featues povdes a poweful cue fo successful segmentaton of nodules adjacent to stuctues of mla ntenty but dffeent shape. 1 Intoducton Accuate and automatc segmentaton of medcal mages s an essental component of a compute-aded dagnos (CADx) system. Howeve, medcal mage segmentaton s typcally a dffcult task due to nose esultng fom the mage acquton pocess, as well as the chaactestcs of the object tself and ts neghbohood. Leons may be embedded n aeas of complcated anatomy; and may have vey mla ntentes to the adjacent tssues (e.g. juxta-vascula nodules). In such cases, tadtonal ntenty-based o model-based methods may fal to popely segment the object [1-3]. o example, a contast-based egon gowng appoach was ntoduced n [1]. Ths method assumes that the egon of nteest appeas as a bght o dak object elatve to the suoundng tssue. Howeve, nce adjacent blood vessels have a mla ntenty to the nodule, the segmentaton tends to nclude a pat of a blood vessel along wth the nodule. A mophologcal appoach was pesented n []. One ssue wth ths method s ts sentvty to the mophology template ze, whch makes t dffcult to choose a sutable template fo all dffeent knds of nodules. As ndcated by the authos, ths algothm s tageted fo small and hgh contast nodules; thus fo a lage nodules, especally when a blood vessel s attached to a nodule, the algothm mght fal to popely delneate the nodule.

2 -104- Second Intenatonal Wokshop on Image segmentaton methods based on enegy mnmzaton have been ntenvely eseached [4-10]. In patcula, gaph cut based methods, whch can acheve a global mnmum of enegy functons used n mage segmentaton, have been shown much pomse n medcal mage computng. Howeve, n most gaph-based methods, the gaph vetces ae constucted at mage pxels, and the segmentaton enegy s composed of ntenty tems. o example, Zheng et al. [5] poposed a famewok to multaneously segment and egste the lungs and nodules n CT data. o segmentaton, a D pxel based gaph cut algothm was appled on the 3D lung and nodule datasets. It s noted that, by epesentng a gaph vetex ung an mage pxel, the numbe of nodes n the gaph nceases polynomcally wth the mage ze (N D, whee N s the numbe pxels n one of the D dmenons); ths damatcally nceases the computaton tme. To mpove effcency, L et al. [6] ntoduced a gaph bult on a pe-segmentated mage ung a wateshed algothm. Howeve, the gaph cut fomulaton s solely based on the mage ntentes. It s known that pxel ntenty can be locally eoneous due to nose and othe mage acquton ssues (such as Patal Volume Effect (PVE) n CT). Thus, n these cases, nose can advesely affect the pefomance of a gaph-based segmentaton. Slabaugh et al. [7] ncopoated an ellptcal shape po nto the gaph-cut segmentaton famewok. Xu et al. [8] pesented a gaph-cuts based actve contous appoach to object segmentaton method. Zheng et al. [9] constucted a gaph Laplacan matx fo the estmaton of Gound-Glass Opacty (GGO) nodule n CT. Recently, Lu et al. [10] appled odeng constants nto an enegy smoothness tem based on an ntal labelng. A mple geometc shape po was also ncopoated n a gaph cut segmentaton. In ths pape, we popose an automatc mode-based gaph cut method fo lung nodule segmentaton. An ovevew of the appoach appeas n g. 1. At each voxel n the mage, the volumetc shape ndex (SI) s computed. The shape ndex, along wth the mage ntenty and spatal poton (x, y, z) ae concatenated nto a fve dmenonal featue vecto at each voxel. In ths fve dmenonal jont spatalntenty-shape (JSIS) featue space, mean shft clusteng s appled, poducng ntenty and shape ndex mode maps (supe-pxels). Then, a gaph s constucted ung the supe-pxels as vetces. Weghts n the gaph ae computed ung a novel enegy fomulaton that condes not only mage ntenty but also the shape featue. The use of mean-shft geneated supe-pxels poduces bette esults and mpoves speed gnfcantly compaed to a dense gaph wth vetces at evey voxel. The expemental esults on CT lung nodules demonstate the hgh pefomance of the poposed method. Input Image I (x,y,z) Shape ndex calculaton ( SI ) ve-dmenonal mean shft clusteng Intenty mode Gaph cut on mean shft supe-pxels Segmented nodule Shape mode g. 1. low dagam of the poposed gaph cut based method

3 Second Intenatonal Wokshop on Mean Shft Clusteng of JSIS featues Ou appoach fst computes the JSIS featues, whch ae then clusteed n a fvedmenonal space ung mean shft. In ths secton, we evew the shape ndex featue and ou mean shft appoach..1 Volumetc shape ndex: a 3D geometc featue The volumetc shape ndex (SI) at voxel p( x, y, z) can be defned as [11][1]: 1 1 k ( ) ( p) k ( p) SI p 1 + (1) = actan π k1( p) k ( p) whee k 1 ( p) and k ( p) ae the pncpal cuvatues at voxel p, whch ae defned as: k1 ( p) = H( p) + H ( p) K( p), k ( p) = H( p) H ( p) K( p) whee K ( p) and H ( p) ae the Gausan and mean cuvatues, espectvely. The calculaton of the Gausan and mean cuvatues ae based on the fst and second fundamental foms of dffeental geomety. A pactcal appoach fo computng these foms s to use the smoothed fst and second patal devatves the mage wth espect to x, y, z as suggested n [13]. In ths pape, po to shape ndex calculaton, a ngle-scale Gausan smoothng s employed to obtan the smoothed mage wth standad devaton of 1.5. Shape ndex epesents the local shape featue at each voxel. Evey dstnct shape, except fo the plane, coesponds to a unque shape ndex. o example, the shape ndex value of 1.00 ndcates a sphee-lke shape (e.g. nodule), and 0.75 ndcates a cylnde-lke shape (e.g. vessel). Based on the defnton, volumetc shape ndex dectly chaactezes the topologcal shape of an so-suface n the vcnty of each voxel wthout explctly calculatng the so-suface. Ths featue povdes ch nfomaton fo automated segmentaton of anatomcal stuctues o leons n medcal mages, whee the egon of nteest s wthn an aea of complcated anatomy and mage ntentes of dffeent shapes ae mla to each othe (such as an adjonng lung nodule).. Combnaton of shape ndex featue nto mean shft famewok o each voxel, 3D spatal locaton, ntenty and volumetc shape ndex featues ae concatenated n the jont spatal-ntenty-shape (JSIS) doman of dmenon d=5. Gven n data ponts p, =1,,n n a 5-dmenonal space, the multvaate kenel s defned as the poduct of thee adally symmetc kenels: s ( ) f f f () K h s h h f = ck,5k k k h s h h

4 -106- Second Intenatonal Wokshop on whee ck, 5 s a nomalzaton constant whch assues K(f) ntegates to 1. f s s the spatal locaton, f s the ntenty and f s the shape ndex featue; h s, h and ae the kenel wndow ze fo spatal, ntenty and shape ndex kenel functon. h d / The nomal kenel s used n ths pape, whee k( f ) = ( ) exp( 1 f ) π. By ung the mean shft famewok [14], the shape ndex featue can be combned wth the ntenty fo clusteng. The mean shft vecto wth thee kenel wndows (spatal, ntenty and shape ndex) can then be calculated as: n s f f f (3) x g g g h s h h = 1 mh s h h ( p ) = p n s f f f g g g h s h h = 1 whee g ( f ) = k' ( f ). The mean shft vecto always ponts towad the decton of the maxmum ncease n the denty functon. It s noted that the mean shft algothm estmates the modes (the densest egons) of the multvaate dstbuton undelyng the featue space. The set of ponts that convege to the same mode s defned as the attacton ban. Mean shft maps all the data samples to the local maxma of the coespondng attacton ban based on fve-dmenonal featues. Supe-pxels (modes) ae fomed fo the set of pxels n each attacton ban. Each supe-pxel has a constant shape ndex and ntenty. Ths poduces mode maps, namely an ntenty mode map ( M ), and a shape ndex mode I map ( M ), and a spatal mode map ( SI M ). The spatal mode map s not used dectly S n ou enegy functon; howeve, spatal nfomaton s utlzed when defnng neghbos n ou gaph. g. shows a nodule attached to a vessel and ts coespondng ntenty and shape ndex mode maps. It can be noted that the mode maps ((c) and (d)) fom JSIS mean shft clusteng can be seen as flteed mages and ae less contamnated by outles. In the followng secton, gaph cut based segmentaton s appled on these supe-pxels. (a) (b) (c) (d) g.. One attached nodule wth ts ntenty and shape mode maps (a) Ognal CT sub-mage; (b) Shape ndex map based on Eq. (1); (c) Intenty mode and (d) shape ndex mode maps. 3 Automatc Gaph Cut based Segmentaton on Mean Shft Mode Map wth Shape eatue In ths secton, we conde the mode map as a gaph G. As mentoned above, the gaph G=(V, E) s defned wth vetces v V epesentng supe-pxels detemned

5 Second Intenatonal Wokshop on fom fve dmenonal mean-shft clusteng, and edges ε E connectng adjacent supe-pxels. A key dffeence fom the usual gaph constucton s that we connect supe-pxels nstead of the ognal pxels. As a esult, the numbe of vetces n G s geatly educed compaed to the ognal numbe of pxels n the mage. Two key ssues ae addessed n the followng two sub-sectons: ntalzaton and enegy functon defnton. 3.1 Intalzaton based on hgh sphecal concentaton In ou pevous wok, we have developed an automatc lung nodule detecton algothm [15], whch poduced a small numbe of potental nodule egons. The am of ths pape s to accuately delneate each nodule bounday. Gven that a nodule s geneally ethe sphecal o has local sphecal elements, whle a blood vessel s usually oblong, fo each potental nodule egon (o egon of nteest), a sphecal concentaton s calculated fo the automatc estmaton of ntal nodule (foegound) egon. A cluste of hgh shape ndex voxels s automatcally detemned ung hystees thesholdng [13] appled to the shape ndex map. Ths algothm fnds a sphecal egon R fomed of 3D connected voxels that all have shape ndex geate than o s equal to a elaxed theshold and contan voxels wth a shape ndex geate than o equal to a hgh theshold. The voxels n ths sphecal egon defne the foegound seeds fo the gaph cut segmentaton. To poduce the backgound seeds, the foegound egon s enlaged based on the dstance tansfom [16]. The ntal backgound egon can be obtaned by nvetng the enlaged foegound egon. Hee, the foegound egon s enlaged to ensue that the backgound seeds do not cove the nodule to be segmented. g.3 shows an example of the segmented vascula nodule ung the above automatc calculaton of foegound and backgound seeds, whee, the hgh theshold and elaxed theshold fo hystees thesholdng wee chosen to be 0.9 and 0.8 espectvely; and the ntal foegound (g.4 (a and b)) was enlaged 10 layes based on the dstance tansfom to obtan the backgound (g.4(a3 and b3)).

6 -108- Second Intenatonal Wokshop on a1 a a3 a4 b1 b b3 b4 g.3. An example of one attached sold nodule segmentaton based on the automatc calculaton of ntal foegound and backgound. (a1-b1) 3D nodule n contnuous slces n CT; (a-b) ntal foegound based on hgh sphecal concentaton; (a3-b3) ntal backgound; (a4-b4) segmentaton esults by the poposed gaph cut based method. 3. Enegy functon The leon segmentaton poblem s fomulated as a bnay labelng poblem, so the goal s to asgn a unque label l { 0,1 } to each supe-pxel (mode) (whee 0 s backgound and 1 stands fo foegound (leon)) by mnmzng a Gbbs enegy E(L) [17]: whee ( ) ( L) = E1 ( l ) + λ E( l, l j ) E (4) V (, j) ε E 1 l s the data enegy, detemnng the enegy to asgn the label mode, and ( ) l to the E l, l j s the smoothng enegy, denotng the cost of asgnng the labels l and l j to adjacent supe-pxels and j. λ s a weghtng facto. The detals of enegy mnmzaton va the gaph cut algothm fo bnay labelng can be found n [1]. Below we focus on how to defne the two enegy tems. Data enegy (t-lnk enegy): Gven the ntal foegound { } M and m B backgound egons{ M n } whch ae automatcally calculated based on the hystees thesholdng of shape ndex map dscussed n the secton 3.1. Hee, m and n ae the supe-pxel ndces fo ntal foegound and backgound, espectvely. o each supe-pxel, the ntenty dstance of the supe-pxel to the foegound supe-pxels { M }, weghted by the shape featue, s calculated as: m () () ( SI 1.0), d = mn I M m 1 exp m σ a wheeσ a =0. 3, I() and SI() ae the th supe-pxel s ntenty and shape ndex esultng fom mean shft clusteng. Altenatvely, an exponental functon can

7 Second Intenatonal Wokshop on also be used fo the calculaton of the ntenty mlaty. The ntenty dstance to the B B B M s as: d I () M s backgound supe-pxels { } defned as: E n d = mode n B 1( l = 1) = E1( l = 0) = B B d + d d + d mn. Theefoe, ( ) d n E 1 l We note that a nodule s geneally ethe sphecal o has local sphecal elements (we defne sphecal elements as a local goupng of voxels ecognzed by hgh volumetc shape ndex values). It can be seen that d n Equaton (5) encouages a supe-pxel to have the same label as the ntal foegound supe-pxels f t has a mla ntenty to the foegound supe-pxels and also a shape featue close to one. Smooth enegy wth shape featue (l-lnk): The second tem E (, ) l l j n Equaton (4) s defned as: E ( l, l j ) = w ( E I + E SI ) + ( 1 w ) E I E (6) SI whee E s the ntenty enegy tem, denotng the ntenty dffeence between two I adjacent supe-pxels and j, whch s defned as: EI ( l, l j ) = 1 ( I( ) Imod e( j) + 1). Ths means supe-pxels wth mla ntentes have a lage E I, whch leads to asgnng the same labels to the two supe-pxels. E s the shape enegy tem, denotng the shape dffeence between two SI adjacent supe-pxels, whch s defned as ESI ( l, l j ) = 1 ( SI () SImod e ( j) + 1). Smla to the ntenty tem, the shape enegy tem captues the shape featues fo the two adjacent supe-pxels, f both the shape ndex values ae mla, E s lage, whch means wth hgh pobablty, both supe-pxels have the same label. It s noted that, both the ntenty mode value and shape mode value have been nomalzed to the same scale fo the calculaton of ntenty enegy tem and shape enegy tem. It can be seen fom Equaton (6), the ntenty tem and shape tem ae combned though a weghtng functon w whch s defned as: ( () ( )) () ( ) SI SI j, hee σ w = SI, SI j exp =0. 3 σ It s noted that, when the shapes between two adjacent supe-pxels ae the same, w =1, and the enegy depends on the fst tem of Equaton (6). Howeve, when two adjacent supe-pxels have vey dffeent shapes, w s small, so the Equaton 6 depends on the second tem. g.4 shows a juxta-vascula nodule segmentaton by ung dffeent smooth enegy functons. Due to PVE n CT magng, pat of the nodule s pxels (e.g. g.4(a3)) have elatvely low ntentes, compaed to that on the othe slces. By ung the fst tem only n Equaton (6), those pxels wth low ntenty (but mla shape featue) can stll be coectly dentfed as beng pat of the nodule object as seen n g.4 (b3). Howeve, some small amounts of vessel (mla ntenty but dffeent shape featue) ae also ncluded nto the nodule object, as seen n g. 4(b1-b3). Ths s because the fst tem equally condes the mlaty fo both of the ntenty and shape featues. g.4(c1-c3) ae the esults by ung the second tem only. It can be seen that, the shape featue s only used as a weghtng to SI (5)

8 -110- Second Intenatonal Wokshop on the ntenty featue. Dffeent shapes gves lowe weghtng to ntenty tem, ths s why pat of nodule can be popely sepaately fom the adjonng vessel wth mla ntenty, as shown n g.4(c1-c). Howeve, the pxels wth lowe ntenty due to PVE ae wongly dentfed as backgound due to the dffeent ntentes, compaed to that on the othe slces, as shown n g.4(c3). g. 4(d1-d3) ae the esults by combnng both of tems as n Equaton (6), n whch the nodule bounday can be popely delneated despte the PVE and the pesence of vessels wth mla ntenty. a1 a a3 b1 b b3 c1 c c3 d1 d d3 g.4. An example of one attached sold nodule segmentaton by ung dffeent smooth enegy functon. (a1-a3) 3D nodule n 3 contnuous slces n CT; (b1-b3) nodule segmentaton by ung the fst tem only n Equaton 6; (c1-c3) esults by ung the second tem only n Equaton 6; (d1-d3) nodule segmentaton by ung the smooth enegy n Equaton 6. 4 Expemental Results and Dscuson The poposed algothm has been evaluated on CT lung data. The thee kenel wndow zes (spatal h s, ntenty h and shape ndex h ) n the fve-dmenonal mean shft clusteng wee set to be 3.0, 6.5 and 3.0, espectvely. The poposed gaph cut algothm was appled to the mean-shft supe-pxels ung Equatons 4, 5, and 6, whee the weghtng facto λ was set to be 100. g.5 shows an example of the poposed method on one Gound-Glass Opacty (GGO) nodule. It s known that GGO nodules ae usually wth fant contast, egula shape and fuzzy magns, t s challengng to popely segment the nodule bounday. The poposed method demonstates good pefomance on ths GGO nodule segmentaton. o compason, the segmentaton esults wthout the shape featue ae gven n g. 5(a3) and (b3), whee, fou-dmenonal mean shft wth spatal and ntenty featues wee used, also n the defnton of smooth enegy tem (6), only the ntenty enegy tem was condeed. It can be seen that, by condeng the shape ndex featue n both the mean shft clusteng and the defnton of enegy fomulaton fo gaph cut, the nodule bounday can be popely delneated fom the backgound despte the pesence of othe non-taget stuctues (such as vessels). The pefomance of ou mode-based (mode map) gaph cut algothm was also compaed wth that of pxel-based method, whee, n the gaph constucton, each vetex epesents one pxel. g.6 shows the compason esults on anothe GGO nodule mage, whee, vetces (pxels) was constucted n the pxel-based gaph, compaed to 946 vetces fo the supe-pxel based method. Testng was pefomed on a system wth a.39ghz CPU and GB memoy. Constucton of the

9 Second Intenatonal Wokshop on gaph and enegy mnmzaton equed 16 seconds fo the pxel-based method, and 1. seconds fo the supe-pxel based method (ncludng the mean shft). The majoty of the computaton tme s n the gaph constucton, whch ncludes the calculaton of the both enegy tems fo each vetex. On the supe-pxel based gaph, the fewe vetces esults n a much faste un-tme. Snce the ntenty mode map fom the fve-dmenonal jont spatal-ntentyshape ndex mean shft algothm expesses the local stuctue of the data, t can be seen that ou poposed supe-pxel-based method poduces bette esults and mpoves the speed gnfcantly. o a quanttatve evaluaton, the poposed method has been tested on a database of clncal chest CT scans, contanng 100 nodules (sold and mxed-sold nodules) wth a slce thckness angng fom 0.5mm to.0mm. The ze of the nodules anged between 5mm to 0mm n damete. To poduce the gound tuth, each nodule bounday was manually delneated by expeenced adologsts. An ovelap ato between the segmented nodule and the gound tuth annotaton s calculated. g. 7 shows the ovelap atos based on the poposed method wth and wthout shape ndex featue. It s noted that, wthout shape featues, the mean ovelap ato fo the whole dataset s 74% wth standad devaton (std) of Howeve, the mean ovelap ato has been nceased to 81% wth the std deceang to by ung the poposed method. Ths ndcates the segmentaton based on ou poposed method s stable and accuate fo dffeent types nodules. In ths pape, the paametes fo thee kenel wndow ze (spatal h s, ) n Equaton and the weghtng facto ( λ ) n ntenty h and shape ndex h Equaton 4 ae chosen expementally. As t s noted that, kenel wndow zes depend on the data stuctue. To mpove the pefomance, we ae cuently analyzng the sentvty of the segmentaton esults to those paametes and also a vaable wndow ze needs to be futhe nvestgated. Geneally speakng, t s challengng to segment GGO bounday due to ts egula shape and fant contast. In ths secton, g.5 and g.6 show good examples of GGO nodule segmentaton by ung the poposed method. Ths s because, fstly, the method clustes pxels unde mean shft famewok by takng nto account the jont spatal-ntenty-shape featue. The esultng mode map gnfcantly educes the vaance of both of the ntenty and shape featues. Then, the supe-pxels fom mean shft clusteng ae futhe meged by ung gaph cut algothm. Howeve, ths s a plot study fo the GGO nodule segmentaton. uthe expements (e.g. quanttatve evaluaton) ae needed befoe the method can be appled n clncal pactce.

10 -11- Second Intenatonal Wokshop on a1 a a3 a4 b1 b b3 b4 g.5. Example of one GGO nodule segmentaton. (a1-b1): 3D GGO n two contnuous slces; (a-b): shape ndex mode map fom fve-dmenonal mean shft; (a3-b3): segmentaton wthout shape featue; (a4-b4): segmentaton esults based on the poposed method. g.6. Segmentaton esult on one GGO (left) based on pxel-based gaph-cut algothm (mddle) and ou poposed mode-based gaph-cut algothm (ght). Volume ovelap ato Poposed 5d MS and GC wth shape featue featue 4d MS and GC wthout shape featue ato Nodule numbe g.7. Volume ovelap ato based on the two dffeent methods

11 Second Intenatonal Wokshop on Concluon We have pesented a new automatc method of extactng lung nodules fom CT data. A fve dmenonal JSIS mean shft clusteng s fstly used to poduce both of ntenty and shape ndex mode maps. A gaph cut algothm s then appled to the mode map ung a novel enegy fomulaton whch condes not only mage ntenty but also the shape featue. The jont JSIS featue povdes ch nfomaton fo leon segmentaton. Both by vsual nspecton on both sold nodules (such as g.3 and 4) and GGO nodules (such as g.5 and 6), as well as ung a quanttatve evaluaton on 100 nodules (sold and mxed-sold) demonstates the potental of the poposed method. The method can not only successfully segment nodules adjacent to stuctues of mla ntenty but dffeent shape, but also can coectly dentfy some pat of nodules wth dffeent ntenty (due to PVE n CT magng) but mla shape. Refeences 1. S.A.Hjjatoleslam and J.Ktte, Regon gowng: A new appoach, IEEE Tans. Image Pocess., vol.7, no.7, pp , (1998).. W.J.Kosts, A.P.Reeves, D..Yankelevtz, and C.I.Henschke, Thee dmenonal segmentaton and gowth-ate estmaton of small pulmonay nodules n helcal CT mages, IEEE Tans. Med. Imag., vol., no.10, pp , (003). 3. W.Mullally, M. Betke, J.Wang, and J.P.Ko, Segmentaton of nodules on chest computed tomogaphy fo gowth assessment, Med. Phys., vol.31, no.4, pp , (004). 4. Y.Boykov, O.Veksle, R.Zabn, ast Appoxmate Enegy Mnmzaton va Gaph Cuts, IEEE Tans. Patten Analys and Machne Intellgence, vol.3, No.11, (001). 5.Y.Zheng, K.Stene, T.Baue, J.Yu, D.Shen, Lung nodule Gowth Analys fom 3D CT Data wth a Coupled Segmentaton and Regstaton amewok, ICCV, (007). 6. Y. L, J.Sun, C.-K.Tang and H.-Y.Shum. Lazy snappng, ACM Tans. Gaphcs 3(3): , (004). 7. G.G.Slabaugh and G.B.Unal. Gaph cuts segmentaton ung an ellptcal shape po. In ICIP(), pp.1-15, (005). 8. N.Xu, N.Ahuja, R.Bansal, Object segmentaton ung gaph cuts based actve contous, Compute Von and Image Undestandng, 107, pp.10-4, (007). 9. Y.Zheng, C.Kambhamettu, T.Baue and K.Stene, Estmaton of Gound-Glass Opacty Measuement n CT Lung Images, MICCAI, pp.38-45, (008). 10. X.Lu, O.Veksle and J.Samaabandu, Gaph Cut wth Odeng Constants on Labels and ts Applcatons, CPVR, Alaska, (008). 11. O.Monga and S. Benayoun, Ung patal devatves of 3D mages to extact typcal suface featues, Compute Von and Image Undestandng, vol.61, pp , (1995). 1. O.augeas, Thee-dmenonal compute von: A geometc vew-pont, Cambdge, MA: MIT pess, (1993). 13. H.Yoshda and J. Napp, Thee-dmenonal compute-aded dagnos scheme fo detecton of colonc polyps, IEEE Tans. Medcal Imagng, 0(1), pp , (001). 14. D.Comancu and P.Mee, Mean shft: A obust appoach towad featue space analys, IEEE Tans. on Patten Analys and Machne Intellgence, vol.4, pp , (00). 15. X.Ye, X.Ln, J.Dehmeshk, G.Slabaugh, G.Beddoe, Shape-based compute-aded detecton of lung nodules n thoacc CT mages, IEEE Tans. Bo-medcal engneeng, 56(7), , (009). 16. R.C, Gonzalez and R.E. Woods, Dgtal Image Pocesng, Addson Wesley, (00). 17.S.Geman and D.Geman, Stochastc elaxaton, gbbs dstbutons, and the bayean estoaton of mages. Tans. Patten Analys and Machne Intellgence, 6: (1984).