Geodesic Active Regions for Supervised Texture Segmentation

Transcription

1 Geodesc Actve egons for Supervsed Texture Segmentaton Nkos Paragos achd Derche INIA BP 9, 00, oute des Lucoles 0690 Sopha Antpols Cedex, France e-mal: Abstract Ths paper presents a novel varatonal method for supervsed texture segmentaton The textured feature space s generated by flterng the gven textured mages usng sotropc and ansotropc flters, and analyzng ther responses as mult-component condtonal probablty densty functons The texture segmentaton s obtaned by unfyng regon and boundary-based nformaton as an mproved Geodesc Actve Contour Model The defned obectve functon s mnmzed usng a gradent-descent method where a level set approach s used to mplement the obtaned PDE Accordng to ths PDE, the curve propagaton towards the fnal soluton s guded by boundary and regon-based segmentaton forces, and s constraned by a regularty force The level set mplementaton s performed usng a fast front propagaton algorthm where topologcal changes are naturally handled The performance of our method s demonstrated on a varety of synthetc and real textured frames 1 Introducton Texture segmentaton, the problem consdered n ths paper, s one of the most mportant technques for mage analyss, understandng and nterpretaton The task of texture segmentaton s to partton the mage nto a number of regons such that each regon has the same textural propertes [7] Alternatvely, ths task can be vewed as the problem of accurately extractng the borders between dfferent texture regons n an mage [1] If a prory knowledge regardng the textural propertes n a gven mage s avalable, the problem s called supervsed texture segmentaton; otherwse t s called un-supervsed Supervsed texture segmentaton requres texture analyss and modelng whch s usually performed usng two well-known technques; statstcal modelng [5, 0] and flterng theory [, 11] Addtonally, feature-based mage segmentaton s performed usng ether boundary-based methods [1, 1] or regon-based methods [1, 1] Ths work was funded n part under the VIGO research network (EC Contract No EBFMX-CT96-009) of the TM Program Durng the last years, there s a sgnfcant effort to ntegrate boundary wth regon-based segmentaton approaches [6,, ] The dffculty les on the fact that even though the two modules yeld complementary nformaton, they nvolve conflctng and ncommensurate obectves The regon-based methods attempt to captalze on homogenety propertes, whereas boundary-based ones use the nonhomogenety of the same data as a gude The most closely related work wth ths paper can be found n [], where a two-step varatonal approach s proposed that combnes the geometrcal features of a snake/balloon model and the statstcal technques of regon growng The present work has two man obectves: the frst s to propose a complete framework for texture analyss and modelng that combnes the flterng theory wth the statstcal modelng The second obectve s to combne the boundary and the regon-based texture segmentaton framework nto a coupled model, that s derved from the geodesc actve contour model The observaton set of ths framework s composed of 1 A gven set of texture pattern mages, A gven nput frame composed from these patterns Followng our prevous work [18] for supervsed texture segmentaton usng geodesc actve contours, we propose a consderable extenson that ncorporates regon-based nformaton to the exstng boundary-based nformaton under a coupled framework that can deal wth the followng problems: 1 The segmentaton of the nput frame, gven the background pattern [fg (51,5)], The extracton of regons of nterest from the nput frame, gven the correspondng patterns [fg (5,5)] The proposed algorthm s depcted n [fg 1] Intally, an off-lne step s performed that creates mult-component probablstc texture descrptors for the gven set of texture patterns, where the mult-modal data s derved usng a set of flter operators [fg 1: Learnng Phase] Then, gven the nput frame, we apply the same operators and derve an observaton set that s coherent wth the texture descrptors Then, for each pxel we estmate the probablty of beng on

2 Flter Pattern 1 Operator 1 Descrptor Pattern Flter Statstcal Descrptor Operator Modelng Pattern N Descrptor N Flter Operator M Boundary Extracton Input Image Boundary Boundary Boundary Data 0 1 Informaton 0 1 Processng Boundary M 0 1 Geodesc egon Texture Actve egons Informaton Segmentaton Learnng Phase [Secton ] Boundary Module [Secton 1] egon Module [Secton ] Fgure 1 Geodesc Actve egons for supervsed texture segmentaton the boundares between two dfferent texture regons Snce we deal wth mult-modal data, a probablty vector s obtaned The components of ths vector (eg boundary probabltes) are combned to a sngle frame usng some relablty measurements and provde the boundary-based texture nformaton [fg 1: Boundary Module] Besdes, usng the texture descrptors and the observaton set we determne the regon-based nformaton that s derved from the most probable temporal texture assgnment [fg 1: egon Module] Then, the segmentaton problem s stated under an mproved Geodesc Actve Contour model that ams at fndng the best mnmal length geodesc curve that preserves hgh boundary probabltes, and creates regons wth maxmum a posteror segmentaton probablty wth respect to the assocated texture hypothess We call ths model Geodesc Actve egon model, snce boundary and regon nformaton are cooperatng n a coupled actve contour model The defned obectve functon s mnmzed usng a gradentdescent method where a level set approach [1] s used to mplement the obtaned PDE Fnally, the curve propagaton problem s mplemented usng a fast front propagaton scheme, the Hermes Algorthm [17] The remander of ths paper s organzed as follows Secton deals wth the texture analyss and modelng problem whle n Secton, we ntroduce the man contrbuton of ths paper, the Geodesc Actve egon Model whch s appled to the supervsed texture segmentaton problem n Secton Fnally, expermental results and dscusson appear n Secton 5 Texture Analyss and Modelng 1 Extractng Features In many dfferent applcatons the use of lnear and nonlnear flter operators has been appled for feature extracton wth qute satsfactory results Followng ths example, we adopt a general flter bank composed of: (a) (b) (c) (d) (e) Fgure Flter ; Operator ; esponses (a) g(05), (b) LoG(05), (c) A 1 6 0, (d) A 1 0, (e) A ;1 6 The Gaussan operator fg()g [fg (a)] g(x y) = p 1 e; x +y The sotropc center-surround operator (Laplacan of Gaussan) flog()g [fg (b)], f (x y) =S 1 ; x + y e ; x +y where S s a constant scale factor Besdes, the (x y) ansotropc drectonal dervatves operators are also consdered The D Gabor operators analyze the mage smultaneously n both space [], and frequency domans [ ] G(x y )= 1 x +y e; e ;(x+y) These Gabor functons can be decomposed nto two components; the real part [G (x y )] and the magnary part [G I (x y )] The texture features are captured by the spectrum analyzer fa( )g of the Gabor components, S(x y )= p (G I)(x y) +(G I I)(x y) smoothed by a Gaussan functon [fg([c,d,e])], where (G I) denotes the convoluton operaton between the mage I and the flter G Modelng Features The texture modelng phase ams at fndng an approprate model that can be expressed usng a lmted set of parameters and preserves strong dscrmnaton power The most common model related wth flterng theory s the use of hstograms, where the flter response s dscretzed usng a lmted number of values, that affects sgnfcantly the extracted model and requres a large set of parameters (hstogram cells) We confront these problems, by adoptng a statstcal framework where the dfferent flter responses (observed hstograms) are modeled usng contnuous probabltes densty functons that are Gaussan mxtures [fg 1: Statstcal Modelng] In order to facltate the notaton, let us now make some defntons: Let F = ff : [1 M]g be the set of M preselected flter operators

3 Let T = ft : [1 N]g be the set of N texture patterns, and D = fd : [1 N]g be the assocated data set And, let D(A) =fa : [1 N] [1 M]gg be the set of flter operator responses to the nput data set, where A s the response of f to D We assume that each flter response can be modeled usng low-level statstcs, where ts observed densty functon s assumed to be condtonal probablty Let p (:) be the condtonal probablty densty of the data component A If we assume that ths probablty densty functon s homogeneous, e ndependent of the pxel locaton, then t can be decomposed nto many dfferent Gaussan components; p (x )= C X k=1 P k pk (xk k ) where C s the number of mxture components, P k be the a pror probablty of the component k, and s the vector of the unknown mxture parameters: = f(p k k k ):k [1 ::: C ]g The component number s derved automatcally form the observed data [18], whle the estmaton of the unknown parameters s done usng the Maxmum Lkelhood Prncple The output of ths operaton s a powerful probablstc texture descrpton model where each pattern s assocated wth a vector of probablty densty functons p (x) =(p 1 (x 1 ) ::: p M (x M )) that characterzes ts behavor wth respect to the dfferent flter operators [fg 1: Texture Descrptors] Geodesc Actve egons The Geodesc Actve Contour model has been ntally proposed n [, 8, 10] and successfully appled to a wde varety of computer vson applcatons These methods are based on boundary-based nformaton, and am at fndng the best mnmal-length smooth curve for a measure derved from the propertes of the mage Besdes, motvated by the work proposed n [, ], the Geodesc Actve egon model has been ntally ntroduced n [16] to deal wth the problem of supervsed texture segmentaton and has been successfully exploted n [19] to deal wth the trackng problem Ths model s a consderable extenson to the geodesc actve contour model snce t ncorporates regon-based nformaton and ams at fndng a partton where the nteror as well as the exteror regon preserves the desred mage propertes We are gong to ntroduce ths model for a smple segmentaton case wth two possble decsons In order to facltate the notaton, let us make some defntons: Let I :!be the nput frame Let P() = f A B g be a partton of the frame doman nto two non-overlappng regons f A \ B = g, where A s the regon of nterest (hypothess h A ) And, let f@ A g be the A regon boundares A A A A 01 B (a) (b) (c) (d) Fgure Geodesc Actve egon Model: (a) the nput, (b) the boundary-based nformaton,(c,d) the regon-based nformaton [proportonal to the frame ntensty] for hypothess h A [c] and for hypothess h B [d] If we assume that for the gven frame [fg (a)] some nformaton regardng the real regon boundares s avalable [fg (b)], then the extracton of the regon of nterest can be vewed as the problem of accurately extractng ts boundares Let [p(i(s)b)] be the condtonal boundary densty functon that measures the probablty of a gven pont beng at the real boundares of A Then, the regon of nterest can be obtaned usng the geodesc actve contour framework, thus mnmzng E(@ A )= Z 1 g(p(i(@ A _ A (p) 0 dp A (p) :[0 1]! s a parameterzaton of the regon boundares n a planar form, and g() s a postve monotoncally decreasng functon, such that g(0) = 1 and g(x)! 0 as x! 1 The soluton of the segmentaton problem s equvalent wth fndng the geodesc curve of mnmal length that best takes nto account the desred mage characterstcs (mportant boundary probabltes)[9] Let us now assume that an a pror knowledge about the desred ntensty propertes of the dfferent regons s avalable; the condtonal probablty densty functons [p A (I(s)) p B (I(s))] wth respect to the hypothess h A and h B [fg (c,d)] Then, the extracton of the regon of nterest s equvalent to creatng a consstent frame partton between the observed data, the assocated hypothess and ther expected propertes Ths partton can be vewed as an optmzaton problem wth respect to the a posteror segmentaton probablty, gven the observaton set Let [p(p()i)] be the a posteror segmentaton densty functon wth respect to the dfferent parttons P() gven the nput data I Ths densty functon s gven by the Bayes rule as: p(p()i) = p(ip()) p(p()) p(i) If we assume that all the parttons are a pror equally possble then we can gnore the constant terms p(i) p(p()) and we can rewrte the densty functon as: p(p()i) =p(if A Bg) = p([i A A] \ [IB B]) = p(ia A) p(ib B) Besdes, f we assume that the ponts wthn each regon are ndependent, we can replace the regon probablty wth: p(i X X )= Y s X p X (I(s))

4 The maxmzaton of a posteror probablty s equvalent wth the mnmzaton of the [-log()] functon of ths probablty, E(@P()) = ;log = ; A Y Y p A(I(s)) p B(I(s) s A s B log [p A(I(x y))] dxdy ; B 5 log [p B(I(x y))] dxdy We fuse the two dfferent segmentaton models by defnng the Geodesc Actve egon obectve functon as Z 1 E(@ A)=(1; ) g(p(i(@ A(p))))@ _ A(p)dp ; 8 < : A 0 log [p A(I(x y))] dxdy + log [p B(I(x y))] dxdy The mnmzaton of ths functon s performed usng a gradent descent method If u = (x y) s a pont of the ntal curve (that can belong ether to A or to B ) and we compute the Euler-Lagrange equatons [], then we should deform the curve to ths pont usng the followng equaton: du dt = log pb(i(u)) p A(I(u)) B {z } regon term (1 ; )(g(u)k(u) ;rg(u) N(u))] {z } boundary + term N (u) where K s the Eucldean curvature A and N s the unt nward normal A The obtaned PDE moton equaton has two knd of forces actng on the curve, both n the drecton of the normal The regon force ams at shrnk or to expand the curve n the drecton that maxmzes the a posteror segmentaton probablty Thus, f u s a pont of h B [p B (I(u)) >p A (I(u))] then ths force acts to shrnk the h curve pb (I(u)) log p > 0 A(I(u)) otherwse acts to expand t Besdes, the boundary force contans two sub-terms; one that moves the curve towards the regon boundares constraned by the curvature effect and one that attracts these boundares (refnement term) Texture Segmentaton We vew the segmentaton as a frame partton problem [defned by a curve] nto non-overlappng regons that preserve homogeneous textural propertes and characterstcs Some complementary defntons are requred: Let I be the textured nput frame and let D(I) =fi : [1 M]g be the set of flter responses to ths frame Let P() =f : [0 N ]g be a partton of frame doman nto f N +1g non-overlappng regons, where 0 s the regon that corresponds to the background pattern = f@ : [1 N ]g be the regon boundares of the partton P() And, let t be the texture pattern that corresponds to the regon 9 = N (a) N L Orentaton: 0 Orentaton: π (b) π Orentaton: Orentaton: π Fgure (a) Neghborhood partton that ndcates a boundary pont, (b) Possble parttons 1 Defnng the Boundary Informaton It s well known that the extracton of boundary nformaton for textured mages s a very tougher task We propose a probablstc method to determne ths nformaton [18, ] Let N (s) and N L (s) be the regons assocated wth a neghborhood partton, and let D(N (s)) be the correspondng data Under these assumptons and usng the Bayes rule, the probablty that s les on the boundares between two regons p(bd(n (s))) s gven by: p(bd(n (s))) = p(d(n (s))b) p(d(n (s))b [ NB) p(b) where p(d(n (s))b) (resp p(d(n (s))n B)) s the condtonal boundary (resp non-boundary) probablty and p(b) s the a pror boundary probablty whch s a constant scale factor and can be gnored The condtonal boundary and (resp non-boundary) probablty can be estmated drectly from known quanttes snce f s s a boundary pont, then there s a partton [N L (s) N (s)] where the most probable texture assgnment for N L (s) s the background pattern ft 0 g and for N (s) s a dfferent pattern ft r g or the opposte Besdes, f s s not a boundary pont, then the most probable texture assgnment for N L (s) as well as for N (s) s ether ft 0 g or ft r g These probabltes are gven by [16], p(i(n (s))b) =p t0 (I(N (s)))ptr (I(N L(s))) + p tr (I(N (s)))pt0 (I(N L(s))) p(i(n (s))nb)=p t0 (I(N (s)))pt0 (I(N L(s))) + p tr (I(N (s)))ptr (I(N L(s))) Snce we deal wth mult-modal data, each data component can provde boundary-based measurements for a gven pxel s Assumng that the neghborhood partton s known, the boundary probablty p B (sfi N t r g) for s wth respect to the data component I s gven by p (I(N (s))b) p B (sfi N t r g)= p (I(N (s))b) +p (I(N (s))nb) where p (I(N (s))b) (resp p (I(N (s))nb)) s derved (1)

5 from [eq (1)] Ths probablty s defned gven a neghborhood partton as well as a texture assgnment for the second local regon, thus the next problem s to defne ths partton We consder four possble neghborhood parttons (the vertcal, the horzontal and the two dagonals) [fg (b)], obtaned by assumng four orentatons = 0:0 : 1 : : ; besdes, t has been found expermentally that the llustrated n [fg (b)] neghborhood sze gves very satsfactory results Fnally, to estmate ths probablty, we need a texture assgnment for the second neghborhood regon To overcome ths problem, we estmate the boundary probablty for all possble parttons and for all possble assgnment by generatng the matrx 6 6 P B (s) = p B (s0 t 1) p B (s1 t 1) p B (s t 1) p B (s t 1) p B (s0 t k ) p B (s1 t k ) p B (s t k ) p B (s t k ) p B (s0 t N) p B (s1 t N) p B (s t N) p B (s t N) where the lnes correspond to the possble texture assgnments ft 1 ::: t t0 ;1 t t0 +1 ::: t N g and the columns to the dfferent neghborhood parttons f0 1 g The element (m n) of ths matrx, corresponds to the boundary probablty of partton n f the second local regon s assgned to the texture hypothess t m Ths operaton provdes fmg boundary probablty frames (one for each data component fi g) that have to be combned to a sngle frame Ths can be done usng the mean value between the dfferent frames, but t s not the most proper soluton snce the qualty of the boundary maps dffers from data component to data component We ntroduce a relablty measurement for each boundary map, whch s assocated wth the dscrmnaton power of the correspondng flter wth respect to the background pattern At the end of the analyss and modelng phase, we have assocated to each pattern, a descrptor, whch gven a value and ts nature (data component) returns the probablty of beng part of ths descrptor A flter operator has strong dscrmnaton power, f all the observed values of the correspondng background data component are beng correctly classfed, (eg the probablty wth respect to the background pattern s superor to the probablty wth respect to any other pattern) The probablty of beng correct P (C) wth respect to the flter f s equvalent to an observaton x beng classfed as the background texture pattern t 0, where the true state of nature s t 0 As a consequence, for each flter operator f we have P (C) =P x D t0 \ D t0 p t0 (D t0 (x y))h hp t0 (x) p (x) :8 [0 M] D t0 (x y) dxdy where H(x ) : [0 N]! s a bnary functon gven by, H(a ) = 1 f p t0 (a) p (a) 8 [0 N] 0 otherwse We normalze the relablty measurements h [P (C)] wth respect to the dfferent flter operators w (C) P = P Mk=1 P k (C) and we use them to generate a global boundary matrx that combnes the dfferent flter responses: PB(s) = 6 P M P w p(s0 t1) P M P w p(s0 tn) M w p(s t1) M w p (s tn) The boundary probablty p B (s) for the pxel s s then provded by the hghest element of the matrx P B(s) Defnng the egon Informaton Let p(p()d()) be the a posteror segmentaton probablty wth respect to the partton P() Snce the a posteror regon probabltes p (D( )t ) are ndependent, the global a posteror segmentaton probablty s gven by, p(p()d(i)) = p ; \ N =0 [D( )t ] = Y N =0 7 5 p (D( )t ) where D( ) s the mult-modal data assocated wth the regon The use of mult-modal data drves to mult-varate condtonal probabltes If we assume ndependence between the dfferent flter responses, then the a posteror segmentaton probablty s gven by p(p()d(i)) = Y N MY =0 =1 p (I ( )t ) where p (I ( )t ) s the a posteror segmentaton probablty for the regon f g wth the respect to the data component fi g Settng the Energy Although we made the assumpton that the dfferent flter responses are ndependent, they have some uncertanty measurements, snce we use a global statstcal model to descrbe ther behavor These uncertantes are expressed from the dscrmnaton power of the correspondng flters fw g The geodesc actve regon functonal for supervsed texture segmentaton conssts of mnmzng E(@P()) = (1 ; ) ; X N Z 1 =1 X N =0 0 g(pb(@ (p)))@ṙ(p)dp MX =1 where g() s a Gaussan functon w log hp t (I (x y)) dxdy

6 Mnmzng the Energy Let u =(x y) be a pont of the ntal curve Ths pont can ether be at regon 0 or at regon k Based on ths hypothess, we compute the Euler-Lagrange equatons [, ] (Secton ), and we derve the followng moton equaton for u: " du dt = MX =1 w log! p t0 (I (u)) + p tk (I (u)) (1 ; )(g(pb(u))k(u) ;rg(pb(u)) N(u))] N (u) The nterpretaton of the above PDE s obvous Gven a ntal curve, t creates a partton of the mage [determned by a curve that attracts the regon boundares] where the exteror curve regon corresponds to the background pattern whle the nteror regons correspond to the other patterns The obtaned PDE can be mplemented usng a Lagrangan approach, that s lmted snce t cannot deal wth topologcal changes of the movng front and suffers from nstablty n the doman of numercal approxmatons Ths can be avoded by ntroducng the work of Osher and Sethan [1] n our scheme The central dea s to represent the movng as the zero-level set f =0g of a functon Ths representaton s mplct, parameter-free and ntrnsc Addtonally, t s topologyfreeit s easy to show, that f the embeddng functon deforms accordng to d (p t) =F(p) r(p t) dt then the correspondng movng front evolves accordng to: d C(p t) =F(p)N dt Thus, the mnmzaton of the proposed geodesc actve regon obectve functon s equvalent to searchng for a steady-state soluton of the followng equaton: "! d MX p t0 dt (u) = (I (u)) w log +(1; ) p tk (I (u)) =1 g(pb(u))k(u) +rg(pb(u)) # r(u) r(u) r(u) where the geometrc propertes are estmated drectly from the level set frame The Level Set Equaton s mplemented usng the Hermes algorthm [17, 15] that proposes a fast way to deform the ntal curve locally towards the mnmum of the obectve functon 5 Implementaton Issues The proposed method can be used to segment a gven texture frame, n the case where the background texture pattern s known [fg (51,5)] Ths method can be easly extend to extract some specfc regons of nterest determned by the correspondng preferable patterns [fg (5,5)] In both cases, the curve propagaton requres a texture assgnment for the gven pont that has to be compared wth the preferable assgnments Ths ssue s confronted by assumng that the temporal segmentaton map s derved by the most probable texture assgnments Thus for a gven curve pont, we assume that t s located between the preferable regon and the regon that corresponds to the most probable assgnment (whch s derved from the observed data) 5 Conclusons, esults Synthetc data [fg (51)], as well as real-word data [fg (5, 5, 5)] have been used to test and valdate the proposed approach 1 Summarzng, we have consdered a curve propagaton approach for supervsed texture segmentaton The man contrbuton of our approach s the proposton of a coupled varatonal energy framework whch ntegrates boundary-based and regon-based nformaton modules and connects the mnmzaton of the obectve functon wth the curve propagaton theory, namely the Geodesc Actve egon framework Ths framework was successfully appled to the supervsed texture segmentaton problem, where the boundary nformaton s determned usng a probablstc framework, whle the regon-based nformaton s expressed drectly va condtonal probabltes The qualty of ths nformaton s ensured by the use of powerful probablstc texture descrptors (learnng phase) that combne flterng theory and statstcal modelng The proposed model preserves robustness, and s ndependent from the ntalzaton step thanks to the level set mplementaton and to the regon-based term whch creates data-dependent postve and negatve propagaton forces The proposed model s not lmted to texture segmentaton, but t can be used to deal wth a wde varety of computer vson applcatons that can be reformulated as frame partton problems The future drecton of ths work s to valdate the proposed model to other computer vson domans An extended verson of ths paper can be found n [16] Varous expermental results (n MPEG format), ncludng the ones shown n ths artcle, can be found at: The segmentaton performance of our method s demonstrated n [fg (51, 5)] Addtonally, the performance wth respect to the extracton of regon of nterest s demonstrated n [fg (5, 5)] where the patterns of nterest are gven (zebra, chta) Each demonstraton contans nformaton about the modelng phase (patterns and flter operators) The flter operators are selected manually, and ther sze s ether 7x7, or 9x9, or 11x11 The ndependence of our method wth respect to the ntalzaton step s clearly demonstrated As t concerns the curve propagaton, n the case of the absence of curvature, t follows the normal drecton (the boundary force s not vald, and the regon force s not affected by the curvature), whle the presence of curvature ams at creatng a smooth curve propagaton sequence Fnally, the computatonal cost of our approach s related wth the ntalzaton step, and vares between and 5 seconds usng an ULTA 10, 99 MHz (the learnng phase s not ncluded)

7 eferences (1) () () () Fgure 5 (1) Patterns: 5, flters: 5 S(1), LoG(1), A[(1 ), A(1 6 0), A(1 ) () Patterns:, flters: 6 S(1), dervatves, A[(1 ), A(1 6 0), A(1 0), A(1 0 )] () Patterns:, flters: 8 S(1), GsA[(1 0), A(1 6 0), A(1 A(1 0 0), ), A(1 0 6 ), A(1 6 ), A(1 )] () Patterns:, 6 flters: 9 S(1), LoG(1), A[(1 0), A(1 6 0), A(1 A(1 0 0), ), A(1 0 6 ), A(1 ), A(1 6 6 )] [1] Adams and L Bschof Seeded egon Growng IEEE PAMI, 16:61 67, 199 [] A Bovk, M Clark, and W Gester Multchannel texture analyss usng localzed spatal flters IEEE PAMI, 1:55 7, 1990 [] V Caselles, Kmmel, and G Sapro Geodesc actve contours IJCV, :61 79, 1997 [] A Chakraborty, H Stab, and J Duncan Deformable Boundary Fndng n Medcal Images by Integratng Gradent and egon Informaton IEEE Transactons on Medcal Imagng, 15(6): , 1996 [5] P Chen and T Pavlds Segmentaton by texture usng correlaton IEEE PAMI, 5:6 69, 198 [6] J Haddon and J Boyce Inage Segmentaton by Unfyng egon and Boundary Informaton IEEE PAMI, 1:99 98, 1990 [7] A Jan and F Farrokhna Unsupervsed texture segmentaton usng Gabor flters Pattern ecognton, : , 1991 [8] S Kchenassamy, A Kumar, P Olver, A Tannenbaum, and A Yezz Gradent flows and geometrc actve contour models In IEEE ICCV, pages , Boston, USA, 1995 [9] S Kchenassamy, A Kumar, P Olver, A Tannenbaum, and A Yezz Conformal curvature flows: from phase transtons to actve vson Archve of atonal Mechancs and Analyss, 1:75, 1996 [10] Mallad, J Sethan, and B Vemur Shape modelng wth front propagaton: A level set approach IEEE PAMI, 17: , 1995 [11] S Mallat Multresoluton approxmatons and wavelet orthonormal bases of L () Trans Amer Math Soc, 15:69 87, 1989 [1] B Manunath and Chellapa A Computatonal Approach to boundary detecton In IEEE CVP, pages 58 6, 1991 [1] B Manunath and Chellapa Unsupervsed texture segmentaton usng Markov andom Feld models IEEE PAMI, 1:78 8, 1991 [1] S Osher and J Sethan Fronts propagatng wth curvaturedependent speed : algorthms based on the hamlton-acob formulaton Journal of Computatonal Physcs, 79:1 9, 1988 [15] N Paragos Geodesc Actve egons and Level Sets: Contrbutons and Applcatons n Artfcal Vson PhD thess, Unversty of Nce/Sopha Antpols, France, November [16] N Paragos and Derche Geodesc Actve egons for Texture Segmentaton esearch eport 0, INIA, France, [17] N Paragos and Derche A PDE-based Level Set approach for Detecton and Trackng of movng obects In IEEE ICCV, pages , Bombay, Inda, 1998 [18] N Paragos and Derche Geodesc Actve Contours for Supervsed Texture Segmentaton In IEEE CVP, Colorado, USA, 1999 [19] N Paragos and Derche Unfyng Boundary and egon-based Informaton for Geodesc Actve Trackng In IEEE CVP, Colorado, USA, 1999 [0] M Unser Local lnear transforms for texture measurements Sgnal Processng, 11:61 79, 1986 [1] S Yhann and T Young Boundary localzaton n texture segmentaton IEEE IP, :89 856, 1995 [] X Zeng, L Stab, Schukz, and J Duncan Volumetrc Layer Segmentaton Usng Coupled Surfaces Propagaton In IEEE CVP, pages , Santa Barbara, USA, 1998 [] S Zhu and A Yulle egon Competton: Unfyng Snakes, egon Growng, and Bayes/MDL for Multband Image Segmentaton IEEE PAMI, 18:88 900, 1996