Snake-Based Segmentation of Teeth from Virtual Dental Casts

Transcription

1 1 Snake-Based Segmentaton of Teeth from Vrtual Dental Casts Thomas Kronfeld, Davd Brunner and Gudo Brunnett Chemntz Unversty of Technology, Germany, {tkro, brunner, ABSTRACT Durng the past years several nnovatve and technologcal developments have been made n oral surgery. Today, dgtzed dental casts are common for smulaton and plannng of orthodontc nterventons. In order to work wth these models, knowng the exact poston of the teeth s of hgh mportance. In ths paper, we present a new method for tooth segmentaton wth mnmal user nteracton. At the begnnng, an ntal estmate for the separatng curve between the teeth and the gum s computed and optmzed by use of an actve contour. The second step calculates the dental arch and the nterstces between the teeth. In order to detect each tooth surface exactly, we fnally poston a snake around the cusp of each tooth. Keywords: actve contours, snakes, tooth segmentaton, mesh segmentaton. DOI: /cadaps.2010.xxx-yyy 1 INTRODUCTION In orthodontcs surgery, dental plaster casts of the patent s denture are made to smulate and plan nterventons (nlays, brdges, malocclusons). Tradtonally, clncans analyze these plaster casts manually. In the past two decades, computerzed systems became an mportant factor n dental surgery due to the mprovement of threedmensonal scannng devces (for example [5, 6, 15]). In ths paper, three-dmensonal dental models are represented as trangular meshes. As well as n the tradtonally case, the frst step s the accurate separaton of the teeth from the dgtzed dental model. After the exact poston of each tooth s determned, one s able to measure orthodontc features and smulate orthodontc procedures such as tooth rearrangement. Tooth segmentaton, n general, s a dffcult task, snce teeth occur n dfferent shapes and ther arrangements vary substantally from one ndvdual to another. Furthermore, when dealng wth models wth severe maloccluson, nterstces may be mssng. A lot of automatc methods have already been presented (for a recent survey see [21]). None of these methods have been desgned to specfcally deal wth teeth and usng them drectly for tooth segmentaton wll produce bad results. Algorthms whch were specfcally desgned to deal wth dental models provde better results. Most of them, however, requre tme-consumng and error-prone user nteracton. The method proposed by Kondo et al. [12] s the only hghly automated tooth segmentaton method we know. At frst, the user selects four reference ponts on the surface of the mesh. After that, the algorthms computes two range mages: the plan-vew and the panoramc range mage. The plan-vew range mage s used to determne the dental arch, whch s used as a reference to calculate the panoramc range mage. Both mages are processed separately and composed nto one ndcator functon, whch s used to detect the nterstces between teeth. Unfortunately, the method produces poor results, when processng models wth severe malocclusons. The approach of L et al. [13] uses a modfed watershed algorthm for manual segmentaton. At frst, the user selects some ponts on the surface of each tooth. Startng at the selected markers, the method performs a regon growng

2 2 method untl t reaches vertces wth negatve mean curvature. As stated by them and many other researchers, the boundary between tooth and gum s determned by vertces wth negatve mean curvature values. For example, Zhao et al. [28] descrbe a method based on the calculaton of mean curvature, drectly. Intally, the user specfes a certan threshold as an upper curvature bound. Vertces wth smaller mean curvature are grouped nto connected regons. Then they apply morphologcal operatons to avod multple branchng and extract the skeleton of each regon to obtan thnner boundary lnes. In order to acheve a complete segmentaton, the user has to remove nosy lnes and to connect open boundary lnes manually. A smlar method was suggested by Tanran et al. [23]. Snthanayothn and Tharanont [22] propose two dfferent manual segmentaton methods. The frst one s based on the selecton of landmarks that are located at the transton between tooth and gum. These landmarks are then nterpolated usng a cubc splne to complete the boundary. Tooth segmentaton usng cuttng planes s the second descrbed method. Here, the user needs to poston cuttng planes between the teeth, so that each tooth s enclosed by two planes. Besde the methods mentoned above there are some commercal products [1, 3, 4] but the detals of ther algorthms have not been publshed. The workflow s as follows: the dentst has to send the dental plaster cast to the company. After a few days the dentst receves a data fle whch contans the dgtzed and segmented dental cast. Now, one can rearrange each tooth or ft n nlays, but wthout the possblty to change the segmentaton. In ths paper, we follow a completely dfferent approach. Our method for tooth segmentaton s based on actve contour models (also known as snakes). The actve contour model was orgnally developed by Kass et al. [11] to detect salent features n two-dmensonal mages. The am of ther research was to fnd connected boundares based on low-level nformaton such as the gradent magntude of an mage. They postulated that any object boundary could be descrbed mathematcally usng a parameterzed curve. The curve s ntally placed outsde the object and evolves under the nfluence of nternal und external energes untl t algns wth features of nterest wthn an mage. The nternal energy derves drectly from the curve and serves as a smoothness constrant, whle the external energy derves from the mage and forces the curve to lock on salent mage features. The overall evoluton process s obtaned by energy mnmzaton. Mlroy et al. [14] extends the orgnal snake model to three-dmensonal surface for segmentaton purposes n reverse engneerng. In ther approach, the snake evolves drectly on the surface. The user specfes a small closed contour wthn the regon to be segmented, whch then grows under the nfluence of an nternal pressure force untl t reaches vertces wth local curvature maxma. Each vertex of a snake s moved to one of two adjacent postons on the surface of the mesh, perpendcular to the snake curve. The energy model, however, requres a locally contnuous surface n the neghborhood of each vertex. For that purpose, the regon surroundng each vertex s approxmated by a second order surface. A smplfed model for trangle meshes was proposed by Jung et al. [9]. Here, snake elements are located on adjacent vertces. Durng an teraton, each vertex of the snake moves to one of ts neghborng vertces on the mesh, n order to reduce ts energy as much as possble. The evoluton s carred out accordng to the greedy prncple, whch was frst ntroduced for actve contours by Wllams and Shah [25]. Fg. 1: Snake-based segmentaton process. The rest of ths paper s organzed as follows. In secton 2 we revew the mathematcal background of the proposed method. At frst we calculate possble tooth boundares (termed: features) and mprove them by suppressng nose (secton 2.1). After that, the cuttng plane between the teeth and the gum s estmated (secton 2.2) and converted nto one or more snakes. The energy functonal as well as the evoluton equaton of these snakes are outlned n secton 2.3. Our newly defned external energy functonal s presented n secton 2.4. After the snake evoluton converged, we calculate the exact poston of nterstces between neghborng teeth by usng

3 3 the edge tangent flow (secton 2.5). As a last step, each tooth s separated ndvdually. Therefore, we ntalze a snake at the crown of a tooth, whch then grows untl t reaches the transton between tooth and gum. A complete overvew of our approach, as well as a dscusson on mplementaton detals s presented n secton 3 (see also fgure 1). Results are demonstrated n secton 4, and conclusons are drawn n secton 5. 2 MATHEMATICAL BACKGROUND The three-dmensonal dental model s represented usng a trangular mesh M= ( VT, ), whch conssts of a set of vertces V = ( v,, v ) 1 n and a set of ndexed trangles T = ( t,, t ) 1 nt where each vertex v V represents a unque poston n V 3 E and each trangle t T s defned by three vertces t= {( v, v, v) v, v, v V ; v v v}. The set of edges E s gven by E= {( v, v) v, v V ;0 j, n } j k j k j k where v v belong to the same trangle t T. At each vertex j length nv ( ) = 1. A vertex w V s called neghbor of a vertex neghbors of a gven vertex v V s defned by N ( v). 2.1 Feature regon j j v V a normal vector nv ( ) s defned wth unt v V ff the edge (, vw) E exsts. The set of In order to detect regons of nterest on the mesh by usng snakes, one has to assgn an attrbute to each vertex of the mesh. In general, surface features are defned usng curvature nformaton, snce curvature ntrnscally descrbes the local shape of a surface. Consder a contnuous, orented, and unbounded surface S and a pont v whch les on S along wth ts unt normal vector nv ( ). One can construct a plane Pv ( ) that contans v and nv ( ). The ntersecton of ths plane wth S results n a curve c on the surface. Ths curve has an assocated normal curvature κ ( v ) at the n vertex v. The curvature value does not however specfy the surface curvature of S at v, snce the constructed plane s not unque. If we rotate the plane Pv ( ) around the unt normal vector nv ( ), we receve a new contour along wth ts own curvature value. Among all these curves, there s one wth mnmal curvature κ ( v ) and one 1 wth maxmal curvature κ ( v ). These two curvatures are also known as the prncpal curvatures of S at pont v. 2 V Fg. 2: Color-coded mean curvature values for each vertex negatve curvature values are colored n blue. v V. Postve curvature values are shown n red whle In order to deal wth pecewse smooth surfaces such as trangle meshes, we have to estmate dscrete curvature values. We calculate these dscrete curvature values accordng to the method proposed by Rusnkewcz [20]. For each vertex v V the prncpal curvature values are gven by κ( v ) and κ ( v ). Based on these values, we 1 2 κ( v) + κ( v) 1 2 compute the mean curvature κ ( v ) = for each vertex v V. The values κ ( v) are then mapped to H H 2 the range[ 1,1] (see [24] for mplementaton detals). Fgure 2 shows the dstrbuton of mean curvature on the mesh.

4 4 (a) (b) (c) Fg. 3: Results for the vertex set D (blue) accordng to the threshold ε= 0.1 (a) and ε= 0.5 (b). The colorcoded vertces belongng to the vertex set P (red) wth η= 0.8 are shown n (c). State of the art segmentaton methods mostly defne part boundares accordng to the mnma rule [7]. Ths rule states that boundares between parts could be found along lnes of negatve mnma curvature. Fgure 2 shows an example of the mean curvature dstrbuton and n fgure 3(a) vertces wth κ H 0.1 are hghlghted. These examples show that vertces wth negatve curvature are located at the transton between tooth and gum, so we could apply the mnma rule as well. Unfortunately, there are many nosy regons due to the dgtzaton and the natural shape of the gum. Therefore, we have to reduce the number of consdered vertces. At frst, a user defned threshold ε s used as an upper bound. Based on ε the feature vertex set D= { v V κ ( v) ε} (2.1) s generated. Vertces belongng to D are grouped nto one or more separate regons. Two vertces same regon, f they are drectly connected by an edge 0 1 k 1 k H uw, D belong to the ( uw, ) E or f there exsts a path ( u=v, v,, v, v =w ) betweenu andw whose vertces belong to the set v D (wth 0 k ). The choce of a threshold value ε s complcated. Usng the value ε 0.5 results n too few vertces n D to obtan closed regons at the tooth boundares (fgure 3(b)). On the other hand, when usng a threshold of ε = 0.1 or above, many undesred regons reman (fgure 3(a)). Our experments showed that ε= 0.3 s a good choce. After that, most nosy regons have been fltered, but there are stll small feature regons left. Thus regons wth less than 0.01 D vertces are deleted. Fnally we close small holes between and wthn the remanng regons, by applyng morphologcal operatons [19]. In the followng sectons, we assume that D specfes the enhanced vertex set. Snce these feature regons along wth ther curvature values are part of the external snake energy, we defne a new curvature functon for each vertexv V : κ ( ) f κ( ) H v v D v = (2.2) 0 otherwse The fnal segmentaton result s obtaned by startng an actve contour at the surface of each ndvdual tooth crown. Vertces belongng to the crown are characterzed by strong postve mean curvature. So we construct a new vertex set: P= { v V : κ ( v) > η} (2.3) where η s a user defned threshold. Experments showed that a value of η= 0.8 s suffcent. An example of ths set s shown n Fgure 3(c). Unfortunately, we cannot use these vertces drectly to segment the teeth, because there are many vertces wthn P that are located at the surface of the gum and at the base of the dental model. In that way, we wll obtan a severe over segmentaton, whch cannot be reduced wthout expert knowledge. Therefore, we have to thn out the set P untl only vertces at tooth crowns reman. H

5 5 (a) (b) (c) Fg. 4: Best fttng plane resultng from the prncpal component analyss of the vertex setd. 2.2 Prncpal Component Analyss Prncpal component analyss s a common tool n statstcal analyss. The central dea s to reduce the dmensonalty of a set of samples, whle retanng as much as possble of the varaton present n the set. Ths s acheved by transformng the orgnal set to a new set of varables, whch are called components. One could sort the components n a way that the frst few of them retan most of the varaton present n all of the orgnal varables. Components that are derved n ths way are called prncpal components. The earlest descrpton of ths technque was gven by Pearson [17] and Hotellng [8]. Pearson [17] developed a method for fndng a lne or plane that best ft a set of ponts n p-dmensonal space. The term best ft, however, s mathematcally mprecse. Consder a set of ponts n three-dmensonal space and ther perpendcular dstance to a set of planes. The plane whch mnmzes the sum of dstances s called best fttng plane. Pearson [17] also showed that ths plane contans the centrod of the pont-set. In our case, we are lookng for a frst estmate for the boundary between teeth and gum. Vertces belongng to the vertex set D are located n the vcnty of the searched boundary, as shown above. Therefore we use prncpal component analyss to calculate the plane that best fts to all vertces n D. The calculaton s based on egenvalue decomposton of the covarance matrx, whch s gven by 1 C= ( v m) ( v m) T (2.4) D where v D 1 m= v D s the barycentrc coordnates of the vertex set D. The square matrx C s symmetrc and real-valued. Thus, the egenvalue decomposton, calculated by the Jacob egenvalue algorthm [18] results n three real-valued egenvalues. The egenvector correspondng to the largest (frst) egenvalue ponts n drecton of the maxmal varance n the data-set. The plane we search for s spanned by the egenvectors, correspondng to the frst and second egenvalue and contans the barycenter m (fgure 4). 2.3 Snake Model The actve contour model was frst proposed by Kass et al. [11] for representng mage contours. Ther basc snake model s an energy mnmzng curve, whch evolves under the nfluence of several forces. The contour s represented as a parametrc curve c () s = ((), xs ys ()) where s [0,1] and the energy functonal to be mnmzed s wrtten as v D 1 1 ( ) * = = + + snake snake nt mage con 0 0 (2.5) E E (())d cs s E (()) cs E (()) cs E (())d. c s s (2.6) E represents the nternal energy of the contour, nt constrant force. The nternal energy E nt E refers to the mage forces and mage 2 2 α() s c() s + β() s c () s s ss = 2 E s the external con (2.7)

6 6 conssts of the frst and second-order dervatve of the parametrc curve, where α () s and β () s are user defned weghts. The frst term wll have large values at gaps n the curve, whle the second term ncreases when the curve bends rapdly. The fnal locaton of the snake corresponds to local mnma of the energy functonal. The functonal optmzaton s solved usng calculus of varaton, whch nvolves dervng a varatonal ntegral and solvng the correspondng Euler-Lagrange partal dfferental equaton teratvely. Therefore mage forces and constrant forces need to be dfferentable to guarantee convergence. Also the computed contour coordnates are real numbers, allowng the ponts to fall between the dscrete pxel coordnates of the underlyng mage. In order to overcome these drawbacks, as well as to speed up computaton, Wllams and Shah [25] proposed an evoluton method based upon the greedy prncple. Frst of all, they use a sampled verson of the contour c= ( c,, c) where each sample c (where 0 n ) s related to a dscrete pxel coordnate. Durng an 0 n teraton, the energy functon s computed at c and each of ts eght neghbors. Therefore, the energy functonal to be mnmzed s wrtten as: n n * = = + + snake snake nt mage con = 0 = 0 ( ) ( ( ) ( ) ( )) E E c E c E c E c (2.8) and the nternal energy s approxmated usng fnte dfferences. Mlroy et al. [14] extend the snake evoluton to three-dmensonal wreframe models. Ther evoluton method s based on the greedy prncple as well. Durng each teraton, each vertex of the snake s moved to one of two nearby postons on the surface, perpendcular to the snake contour. The dstance of these ponts s specfed by the user. Therefore, the regon surroundng each vertex s approxmated by a second-order surface. Jung et al. [9] present a smpler evoluton scheme to ensure that the snake does not leave the surface. The snake elements are always located at vertces of the mesh. In each teraton, the snake elements move to one of ther neghborng vertces. (a) Fg. 5: For each snaxel s the unt normal vector and unt tangent vector are approxmated (a). At each teraton step, two adjacent vertces are selected (b). (b) The actve contour model used n ths artcle conssts of an ordered set of vertces S = ( s,, s) 1 n wth S V. Furthermore, successve snake vertces ( s, s ), = 0,, n 1 are restrcted to le on adjacent vertces + 1 s N ( s ). We wll call snake vertces snake elements or for short snaxel. In ths paper, we only consder 1 + smple and closed actve contours ( s 0 = s n whereas s s for j: j, = 0,, n 1 ). On that note all subscrpt j arthmetc s modulo n. Vertces whch are enclosed by the snake are called nteror vertces. At each snaxel s the unt tangent vector ts ( ) = s s s s s approxmated, as shown n fgure 5(a). The snake wll now be orented such that the unt snaxel normal vector bs () at each snake element s S ponts away from the nteror vertces of the contour. The unt snaxel normal s defned as the outer product between the unt tangent vector and the unt surface normal vector: (2.9)

7 7 bs () = ts () ns () s S. (2.10) In each teraton, a snaxel s S can only move to a vertex wthn ts neghborhood N ( ). We further restrct the possble canddates n the same way as proposed by Mlroy et al [14]. Each snaxel s moved to one of two adjacent vertces s, s N ( s ), perpendcular to the snake curve, or remans at ts current poston s = s. Fgure 5(b),1,2,0 shows an example for the choce of canddates. The energy of the snake element s S s defned as: where e s the nternal energy and nt es ( ) = e ( s ) + e ( s ) (2.11) nt ext e s the external energy. Note that ext s nt ext 3 e,e,e R are column vectors es ( ) = ( e( s), e( s), e( s )) T. The value of the topmost element of these vectors corresponds to the contour energy at the snake element s S. Durng an teraton, each element s moved to the vertex correspondng to the mnmal value of es ( ). The energy functonal to be mnmzed s wrtten as: E * snake n ( S ) = mn{ e( s)} (2.12) 0 j 2 = 0 and the evoluton s carred out accordng to greedy prncple [25]. The nternal energy generally conssts of the frst and second order dervatve of the curve, whch are also called tenson and bendng energy, respectvely. Here, we approxmate the dervatves by fnte dfferences. The nternal energy term for each element ent,( s ) of the vector e ( s ) s therefore defned as follows: j nt j e ( s) = αs s + βs 2s + s (2.13) nt, j j, j, 1 where α and β are user defned parameters. Large values of α shrnk the snake, whle large values of β reduce the curvature of the snake. The external energy e can take varous forms; n general, t s derved from nformaton of the surface ext curvature. At an explored feature vertex the value of the external snake energy wll be a local mnmum. Here, the vertex set D contans the features to be detected and the external energy s derved from the curvature functon κ, defned n equaton (2.2). Unfortunately, we cannot use ths curvature functon drectly, snce ther value s only nonzero at feature vertces. Ths means, that the external energy would be constant on bg parts of the mesh, and so the evoluton of the snake depends completely on the nternal energy. As one can see n fgure 3, the relatve poston between the cuttng contour obtaned by the prncple component analyss and the transton between teeth and gum, changes completely from denture to denture. In fgure 3(a) the contour les completely above the searched feature, whle n fgure 3(b) the contour les completely below. For ths purpose, the external energy e ( s) = e ( s) + e ( s ) (2.14) ext feat press s composed of two newly defned energy functonals, the feature attracton energye feat and the pressure energy e. The feature attracton energy, whch wll be explaned n secton 2.4, expands the scope of feature regons press by usng a pre-calculated vector feld, the feature attracton flow. The second energy term, the pressure energy e, s smlar to the balloon force proposed by Cohen [2]. press Hs am was to modfy the external forces n such a way as to obtan more stable results for mage snakes. In order to archve ths goal, he ntroduced an nflaton force by whch the snake behaves lke a balloon. Ths force s wrtten as E = k n () s where n () s s the unt outward normal of the curve wth arc-length s and k determnes ball the magntude of the force. Ths avods that the curve s trapped by spurous nosy feature ponts and makes the result much more nsenstve to the ntal poston. In our case, the pressure force s used as a tool to accelerate the movement of the snake and to reduce the nfluence of swrlng effects n the feature attracton flow. It also affords the opportunty to control the evoluton

8 8 drectly, for example when the snake s used as a semautomatc segmentaton tool and the ntalzaton s done by the user. The energy term for each element e, n the vector e s wrtten as the nner product: press,j press e press, j s s ( s) = δ bs ( ), s s j, j, where bs ( ) s the outward unt normal of S at the snake element s and s s ts canddate vertex. The weght j, δ determnes the strength of ths force, whereas the sgn of δ specfes the drecton n whch the force acts. Negatve values let the snake grow, whle a postve value shrnks the snake. In general, the magntude ofδ should be small, otherwse the snake may not converge. (2.15) 2.4 Feature Attracton Energy Fg. 6: Examples of the derved feature attracton flow. As mentoned above, the relatve poston between the cuttng contour and the transton between teeth and gum could not be determned n the frst place. Therefore, especally for automatc segmentaton, we have to expand the capture range of features to attract the snake from an arbtrary poston. Ths problem was frst addressed by Xu and Prnce [26, 27]. They proposed the gradent vector flow as a new external force for parametrc mage snakes. In order to compute such a feld, one has to defne a scalar feld as an overlay of the mage. After that, the gradent of ths scalar feld s calculated. The fnal vector feld s obtaned by dffuson of the gradent nformaton over the entre mage. One can nterpret ths feld as the drecton to be followed to reach the nearest object boundares. In our case, the computaton s based on the regon ncluded n the set D and the above defned curvature functon κv ( ). The gradent of a scalar feld s defned as a vector feld whose components are the partal dervatves of the underlyng scalar feld. Here we deal wth a dscrete and mostly rregular grd, so we have to approxmate the gradent vectors. Therefore, the gradent s determned usng fnte dfferences: 1 u v ( v) = ( κ( v) κ( u)). (2.16) N( v) u v u N( v) Three general propertes of the gradent vectors are essental for the followng calculatons. Frst of all, the vector v ( ) at each vertex v V ponts n the drecton of the nearest feature vertex. Second, these vectors generally have large magntudes only n the mmedate vcnty of a feature regon. Thrd, n regons wthout features v ( ) s nearly zero. We defne the feature attracton flow to be the vector feld fv, ( ) where for each v V the vector of ths feld ponts towards the nearest feature regon. The feature attracton flow s ntalzed wth the derved curvature gradent f ( v ) = ( v ). After that, an teratve dffuson process s performed, whch dstrbutes the gradent 0 nformaton over the entre mesh: 1 f( v) = (1 λ) ( ) λ ( ),. t + f u t 1 v v V ( ) N v u N( v) (2.17)

9 9 After t teratons we obtan a smooth vector feld f, whch has non-zero magntude at each vertex of the mesh. t Fgure 6 shows an example of the feature attracton flow. In partcular, at vertces v V wth large magntude v ( ) the second term of equaton (2.14) domnates the sum, whch keeps the vector f( v ) nearly equal to the ntal gradent vector. The frst term of equaton (2.14) domnates at vertces t v V where v ( ) s nearly zero, yeldng a slowly varyng feld. The parameter λ s a regularzaton parameter controllng the tradeoff between the frst term and the second term n the sum. Wth small values, one can reduce the nfluence of nose. After each teraton we perform an mmedate step where the length of each vector s clamped to the range [0,1]. Ths prevents the magntude of vectors near or wthn feature regons from rsng rapdly. The feature attracton energy s then defned by ( ) j, j, e ( ) δ fs s s s = ( ), fae, j fs j, (2.18) ( ) fs s s j, j, where fv ( ) s the feature attracton flow after t teratons. The mnuend ensures that the snake moves n drecton of ncreasng feature attracton flow vectors. Due to the mnmzaton of the subtrahend the snake s movng n drecton of the nearest feature regon and converges there. 2.5 Edge Tangent Flow In dgtzed dental casts, often nterstces between adjacent teeth vansh due to severe maloccluson or low resoluton scanners. In cases such as shown n fgure 7(a-b), automatc segmentaton may fal. Therefore, we have to trace and reconnect the nterstces, pror to the fnal segmentaton. Kang et al. [10] frst solved ths problem for two-dmensonal mages by usng edge tangents. Ther am was the development of an enhanced edge detector to produce hgh-qualty lne drawngs. In general, the task of creatng lne drawngs s dffcult, snce salent edges often appear as pecewse unconnected lnes n natural photographs. Only n an overall vew one can decde whether there s a salent edge or not. In order to maxmze the lne coherence, they construct a smooth vector feld wth whch one s able to follow the shape of lnes. For ths purpose, they defne the edge tangents, whch are perpendcular to the mage gradent. In ths paper, we extend the defnton of the edge tangent flow to three-dmensonal trangle meshes. We defne the edge tangent vector, denoted t ( v ) for each vertex v V, as the outer product between curvature ETF gradent v ( ) and surface normal nv ( ) : An example s shown n fgure 7(c). t ( v ) = ( v ) nv ( ) (2.19) ETF (a) (b) (c) Fg. 7: Unconnected feature regons due to low resoluton scanners (a) and severe maloccluson (b). An example of the edge tangent flow s shown n (c).

10 10 3 TOOTH SEGMENTATION An overvew of the proposed segmentaton algorthm s shown n fgure 1. The algorthm starts wth estmatng the mean curvature for each vertex v V. Now, the user defnes the threshold ε to obtan the vertex set D, as well as the threshold η to obtan the vertex set P, accordng to the method descrbed n secton 2.1. The vertces belongng to the set D are located at the transton between the teeth and the gum. In order to obtan an ntal estmate of ths regon, we perform the prncpal component analyss on the vertces of D. As stated n secton 2.2, the best fttng plane s defned by the vectors belongng to the two larger egenvalues. By cuttng the dental cast wth that plane, we obtan we obtan one or more closed and orented contours (fgure 3). Each of them s transformed nto a snake and evolves ndependently. After each snake element s moved, n order to mnmze the energy functonal, t may happen that adjacent snake elements are no longer located on adjacent vertces. In those cases we have to reconnect them. Durng evoluton, a snake may shrnk untl t encloses only a sngle trangle. These snakes wll be dscarded. On the other hand, two snakes can expand and collde wth each other, and then they are merged. At the end, the remanng snake encloses the teeth completely and s called ntal snake. Now, we use the edge tangent flow to detect and reconnect the nterstces. For each snake element, the neghborng vertces are examned. If the edge tangent flow vectors at these vertces are orthogonal to the tangent vector of the current snaxel, then t s most lkely that there s an nterstce. Frst, the correspondng snaxel at the other end of the nterstce s estmated. After that the shortest path n drecton parallel to the edge tangent flow vectors and wthn the nteror of the ntal snake to the correspondng snaxel at the other end of the nterstce s searched. If the feature regon along the searched path s unconnected, then these gaps wll be closed as long as they are smaller then fve percent of the Eucldean dstance between these correspondng snaxel. The remanng task s to detect the surface of each ndvdual tooth exactly. Unfortunately, we cannot ensure that all snake elements of the ntal snake are located at the transton between teeth and gum. As a consequence, some nterstces have not been detected yet. But besdes that, we have reduced the consdered surface of the mesh sgnfcantly. The remanng parts are the surface of the teeth and maybe some small parts of the gum. Therefore, we can be sure that all regons ncluded n the vertex setp and at the nteror of the ntal snake, are cusps of teeth. Each of these regons wll be used to ntalze an ndvdual snake. Under the nfluence of the feature attracton flow and the pressure force, each snake expands untl t reaches the tooth boundary. In order to avod expanson, when tooth boundares are mssng, we ntroduce the hard constrant that the snake has to stop when t reaches the ntal snake or detected nterstces. 4 EXPERIMENTAL RESULTS The proposed method was tested on ten dfferent dental casts and performs constantly well. The resultng segmentaton s shown n fgure 10 and the overall runtme for these meshes s lsted n table 1. The nternal energy was weghted by settng α= β= 0.6 throughout the tests. The expermental results showed that weghtng the feature attracton energy wth γ= 0.5 s suffcent. Snce the pressure energy accelerates the movement of the snake, we set δ = 0.2 for the ntal snake and δ= 0.2 to weght the snakes started wthn the teeth. Model (number of vertces) Model 1 (142k) Model 2 (208k) Model 3 (155k) Model 4 (209k) Curvature estmaton 1.34 s 1.9 s 1.4 s 1.85 s PCA and snake ntalzaton 1.15 s 1.7 s 1.3 s 1.7 s Feature attracton flow 3.2 s 4.86 s 3.6 s 4.76 s Contour calculaton 0.22 s 0.3 s 0.24 s 0.2 s Outer snake optmzaton 4.4 s 3.3 s 3.5 s 5.2 s Sngle tooth segmentaton 0.09 s 0.1 s 0.14 s 0.09 s Total tme 12 s 14 s 13 s 15 s Tab. 1: Runtme statstc for all models llustrated n fgure 11.

11 11 Nevertheless n dental casts, where the boundary between tooth and gum s very smooth, the segmentaton fals. The same error occurred, when neghborng teeth overlap each other due to severe maloccluson. In these cases, the user needs to segment the tooth manually. 5 CONCLUSION We have presented a hghly automated segmentaton method for separaton of teeth from trangular meshes. The algorthm s based on actve contours, wherefore t s robust even n the presence of heavy nose. Through our experments, we found that our approach s fast and requres mnmal user nteracton. Hence t enables orthodontcs to plan dental treatments much faster. The proposed snake model s not restrcted to operate on ths specal type of three-dmensonal models. Therefore, as future work, we wll nvestgate ther behavor for the segmentaton of several other models, lke dgtzed archeologcal models. (a) (b) (c) (d) (e) (f) Fg. 10: The dental model s cut by the best fttng plane (a). Vertces belongng to the cut trangles are transformed nto the ntal snake, whch s then optmzed (b). The resultng segmentaton for model 1 s shown n (c). The segmentaton results for model 2-4 are shown n (d)-(f). 6 ACKNOWLEDGEMENTS Ths work was partally supported by the Sächssche Aufbaubank and fnanced from budget of the European Unon and the Free State of Saxony. We lke to thank Image Instruments for provdng the dental models.

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