Active Contour Models

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1 Actve Contour Models By Taen Lee A PROJECT submtted to Oregon State Unversty n partal fulfllment of The requrements for the Degree of Master of Scence n Computer Scence Presented September Commencement June 006

2 Abstract Actve contour models have been wdely appled to mage segmentaton and analyss. It has been successfully used n contour detecton for object recognton computer vson computer graphcs and bomedcal mage processng such as X-ray MRI and Ultrasound mages. The energy-mnmzng actve contour models or snakes were developed by Kass Wtkn and Terzopoulos n 987. Snakes are curves defned n the mage doman that can move under the nfluence of nternal forces wthn the curve tself and external forces derved from the mage data. Snakes perform well on certan types of mages (such as well-defned convex shapes. There have been several mprovements proposed to the orgnal snake or actve contour model. These mprovements nclude balloon snakes adaptve snakes and GVF snakes. In ths project I revewed and mplemented ther algorthms as well as the orgnal snake model. GCBAC (Graph Cut Based Actve Contour s one of alternatve solutons to the object extracton problem. Although the GCBAC belongs to famly of actve contour models t dffers fundamentally from orgnal actve contours. In ths project I also revew and mplement the GCBAC algorthm as well.

3 Table of Contents. Introducton Orgnal Snake Actve Contour Model Internal Energy External Energy Algorthms of Snakes Lmtatons of the orgnal snake Adaptve Snakes Dscusson of Adaptve Snakes Pressure or Balloon Snakes Dscusson of Balloon snakes GVF (Gradent Vector Flow Snakes Gradent Vector Feld Generalzed GVF Snakes Dscusson of GVF Snakes GCBAC (Graph Cut Based Actve Contour Algorthm of GCBAC Dscusson of GCBAC Extractng multple objects n an mage Crtcal Ponts and Splttng and connectng operatons Removal of nvald Contours Dscusson of extractng multple objects Implementaton of Applcaton Lbrares Implementaton of the orgnal snake Internal Energy terms External Energy term Intal contour at the ntalzaton process Greedy Algorthm n the deformaton process Dynamc Programmng n the deformaton process Termnaton of the program Results of the orgnal snake Implementaton of Adaptve snakes Adaptve force Inserton and deleton of snaxels... 37

4 0.3 Results of Adaptve snakes Implementaton of GVF snakes Gradent Vector Feld Results of GVF Snakes Implementaton of GCBAC Dlaton Cost of edges s-t mnmum-cut problem Termnaton Results of GCBAC Implementaton of extractng multple objects Algorthm of extractng multple objects Removal of nvald contours Results of extractng multple objects Summary Reference Acknowledgements User manual

5 Lst of Fgures Fg. - Parametrc curve... 8 Fg..- Internal energy... 9 Fg..- Image forces wth non-flter and Gaussan flter... Fg..4- The concavty problem of the snake on the U-shape mage... 4 Fg. 3- Adaptve force... 5 Fg. 3- Step szes of the snake and Adaptve snake... 6 Fg. 3.- Detect boundary at concave regon (k = Fg. 3.- Problem of the larger step sze of the adaptve snake (k = Fg. 4- Inflatng balloon for edge detecton... 9 Fg. 5.- External forces of the orgnal snake and GVF snake... Fg. 5.- External forces at concavty regons of U-Shape object... Fg. 6.- CN (Contour Neghbor of an actve contour (sze = Fg. 6.- Connectvty of an adjacency graph... 5 Fg. 6.- Self-crossng actve contour... 6 Fg. 7.- Crtcal Ponts... 7 Fg. 7.- Creaton of an nvald contour... 8 Fg. 7.- Behavor of an nvald contour... 8 Fg Result of the orgnal snake on multple objects... 9 Fg Intal contours and ther ntervals... 3 Fg Termnatng crtera on Snake Control Panel Fg. 0- Snake Control Panel for Adaptve Snake Fg. 0.- Deleton and nserton of snaxels Fg. - Snake Control Panel for GVF Snake Fg..- Dlaton wth 5x5 and 7x7 matrx... 4 Fg..- Sze of CN on sample mage... 4 Fg. 3- Data structure of the snake Fg. 6. Screenshot of the man wndow the applcaton... 5 Fg. 6.- Screenshot of General Flter... 5 Fg. 6.- Screenshot of Gaussan Flter Fg Screenshot of Medan Flter Fg. 6.3-a Screenshot of Hstogram Equalzaton Fg. 6.3-b Screenshot of Hstogram Equalzaton after applyng Fg Screenshot of operatons Fg Screenshot of the nformaton of an mage

6 Fg. 6.4-a Screenshot of Gradent of an mage Fg. 6.4-b Screenshot of GVF feld of an mage Fg Screenshot of Snake Control Panel... 6 Fg Screenshot of GCBAC

7 . Introducton Image segmentaton s a process n whch regons or features sharng smlar characterstcs are dentfed and grouped together. Segmentaton may use statstcal classfcaton threshold edge detecton regon detecton or any combnaton of these technques to solve segmentaton problems. The technques ncludes regon-based threshold-based edge-based or connectvty-based [0]. Regon-based technques rely on common patterns n ntensty values wthn a cluster of neghborng pxels. The cluster s referred to as the regon and the goal of the segmentaton algorthm s to group regons accordng to ther anatomcal or functonal roles. The threshold-based technques rely on local pxel nformaton. It s effectve only f the ntensty levels of the objects fall squarely outsde the range of levels n the background. Snce the spatal nformaton of an mage s gnored t became problemc at blurred regon boundares. Edge-based technques rely on dscontnutes n mage values between dstnct regons and the goal of the segmentaton algorthm s to accurately detect the boundary separatng these regons. Connectvty-based technques rely on a curve known as actve contour formed by several control ponts on the mage. The actve contour then deform tself by the energes derved from an mage and locatons of control ponts. The goal of the segmentaton algorthm s to navgate an actve contour toward the boundary of a target object n an mage. In ths paper I set focus on actve contour models derved from Connectvty-based technques. I revew some of methodologes and ther approaches to the mage segmentaton problems. Actve contour models have been wdely appled to mage segmentaton and analyss. It has been successfully used n contour detecton for object recogntons computer vson computer graphcs and bomedcal mages processng such as X-ray MRI and Ultrasound mages. The prmary purpose of the actve contour models s to locate boundary of an object on an mage by usng an actve contour. The actve contour s navgated by the defned energy functons. Several energy functons have been proposed and expermented on actve contour models to acheve the objectve. One of the actve contour models known as snake was orgnally developed by Kass Wtkn and Terzopoulos []. The snake s descrbed as an energy mnmzng splne guded by external constrant forces and nfluenced by mage forces that pull t toward features such as lnes and edges. Although the orgnal snake has several lmtatons on ts performance the concept of the snake has been used to create bare-bone of the successors of the snake. Several new deas have been proposed and added to the concept of the orgnal snake. In ths project I revew some of ther methodologes as well as the orgnal snake. 6

8 Adaptve snakes dscussed n the secton 3 were proposed by Cho Lam and Su [5]. A new energy term called Adaptve forces s added nto the energy functon of the orgnal snake. The adaptve forces are used to overcome the problem of detectng concavtes n the object. Pressure snakes or Balloon snakes dscussed n the secton 4 were proposed by Cohen [7]. A new energy term known as Balloon force s added to the energy functon. The energy functons of the tradtonal snake s desgned an actve contour to converge toward the boundary of the object. Balloon force adds capabltes of nflatng/deflatng to an actve contour by assgnng postve or negatve sgns to the control ponts of the actve contour. GVF snakes dscussed n the secton 5 were proposed by Xu and Prnce [9]. A new external force known as Gradent Vector Feld s ntroduced. Smlar to the external force of the tradtonal snake the gradent vector feld s derved from ntenstes of an mage. However t fundamentally dffers from the tradtonal external force. It adds some sort of energes n homogeneous regon on an mage where there s no energy n general. In the secton 6 I revew one of alternatve solutons to the object extracton problem. It s called GCBAC (Graph Cut Based Actve Contour proposed by Xu Bansal and Ahuja [3]. The GCBAC belongs to famly of actve contour models; however t dffers fundamentally from the orgnal snake and ts successors. Unlke the metrologes of snake famly t does not use a non-parametrc curve as an actve contour. Frst the GCBAC translates an mage to an adjacency graph. An actve contour s formed by cuttng theses edges n the graph. Lastly I revew the extracton of multple objects n an mage. In general the snake s desgned for detectng a contour of a sngle object n an mage. I dscussed the one of algorthms proposed by Cho Lam and Su [5] to detect contours of multple objects n an mage n Secton 7.. Orgnal Snake Actve Contour Model The orgnal snake was developed by Kass Wtkn and Terzopoulos n 987 []. The name snake was named after ts behavor on an mage. Whle mnmzng ther energy t slthers on the mage. A snake s expressed as a planar parametrc curve n Eq. (-. The parameter s s snake control ponts known as snaxels. These snaxels are lnked together to form an actve contour (Fg.-. The snake s not a method to automatcally detect the boundary of the desred object n an mage. It requres approprate parameters settng and ntal 7

9 locatons of the snaxels accordng to the subjectve boundary. Therefore some pror knowledge about the mage s requred from hgher level system. [ x( s y( s ] [ 0] v ( s = s (- Fg. - Parametrc curve Equaton (- represents the energy functon of the snake. As the snake s defned as an energy mnmzaton splne [] t deforms tself to mnmze the energy. The energy functon s desgned for the snake to converge toward the boundary of the target. It behaves smlar to a rubber band placed outsde of the object and shrks to reach the boundary of the target. The frst two terms n the energy functon of the snake n Eq. (- are called nternal energy. The tenson and rgdty of the snake are controlled by these forces. The thrd term s called external energy. It s derved by the mage and t attracts the snake to the target contour. E snake = ( α E elastc(v(s + β E bendng(v(s + γ E mage(v(sds (- 0 - Internal Energy The nternal energy s further dvded nto two energy components: elastcty and bendng forces. The nternal energy s the sum of these forces and the energy functon s expressed as: 8

10 E dv( s dv( s nteral = α ( s + β ( s / (.- ds ds The frst and second dervatves of the contour represent these energy terms and called Elastc forces and Bendng forces respectvely. The elastc force controls the tenson of the snake. It dscourages stretch of the actve contour and t s responsble for shrnkng the contour (red arrow n Fg.-. The bendng force s defned as The bendng energy makes snake acts lke a thn plate []. It controls the rgdty of the snake. It controls only the curvature not the length of the contour. Durng the deformaton process t tres to be a smooth curve or straght lne (blue-dot arrow n Fg..-. Fg..- Internal energy The parameters α(s and β(s n front of each term represents weghtng functons. In general values of these weghtng functons are constants for all snaxels. Selectng an approprate set of these constants creates one of dffcultes of the snake. They have large mpact n snake s behavors and totally control the performance of deformaton process. Each object n an mage requres dfferent set of constants value for snake to perform well. The one way to solve ths problem s to make the snake dynamcally change these values to sutable values durng deformaton process. However t requres a computer to recognze shapes or topologes of an object n an mage automatcally. Therefore the soluton s left for further mprovement of the snake. Currently these 9

11 parameters are up for a user to select at the ntalzaton process. - External Energy The external energy s derved from the mage data. Ths mage-drven force attracts the snake to move toward the target contour. The energy term s defned n the followng equatons []. E E external external ( s = γ ( s ( I( s (.- ( s = γ ( s ( G ( s* I( s (.- ó represents gradent operator and I(s s the ntensty of the mage at s. G σ (s s two-dmensonal Gaussan functon wth standard devaton σ. The weghtng functon γ(s s used to control the mage force. Gaussan flter s appled on the orgnal mage to ncrease the capture range of the snake n Eq. (.-. Gaussan flter makes an mage blurry. Fgure.- llustrates an example of blurrng a part of an edge. The mage force dffuses from the edge (blue-dot lne to the edge (red lnes. As a result t wdens the capture range of the snake. Two snapshots of the applcaton n Fg..- show the comparson of mage forces generated wth non-flter and Gaussan flter. The red arrows n the pctures pont to the drecton of mage forces. The red dots represent the non mage force. Therefore they are seen on homogeneous area where no Fg..- Image Forces mage force exsts 0

12 Fg..- Image forces wth non-flter and Gaussan flter In general Eq. (.- s used to compute external force snce square of gradent magntude of ntensty tself has smaller capture range. The standard devaton σ plays a key role on capture range of the snake. The larger σ causes the boundares of objects become blurry. Ths s sometmes necessary to help an actve contour to move toward to the desred boundary. -3 Algorthms of Snakes The most of algorthms of the actve contour models are conssts of the followng three phases. The snakes are not an excepton. Algorthms of the actve contour models n hgher-level of vew Intalzaton process Deformaton process Termnaton of snake Intalzaton process: In the ntalzng process a user sets the ntal locatons of the snaxels around the boundary of the target object. At the same tme the set of weghtng parameters α β and γ are chosen. Afterward t starts to deform tself toward the true boundares of the object. The ntal contour must be close to the subject boundary snce the snake could move toward noses or other undesred edges or lnes on an mage f t s placed far from the true boundary.

13 Deformaton process: Durng deformaton process t deforms tself to mnmze the sum of the energy terms defned n Eq. (-. For each teraton n ths process a new locaton s searched among neghborng pxels for each snaxel. A snaxel moves to a pxel that has lower energy or t stays n the same locaton f there s none. There are two approaches to compute new locatons for snaxels. The greedy algorthm s one of the methods to fnd a new locaton for each snaxel. The other way s to use the technque of the dynamc programmng. The greedy algorthm fnds a locally optmal soluton meanwhle the dynamc programmng fnds a globally optmal soluton. The mplementaton detals of these methods are dscussed n Secton 9.4 and 9.5. The deformaton process contnues untl the snake s caught n one of the termnatng crtera. The termnatng crtera are dscussed n the followng secton. Termnaton of the snake: At the some pont of the tme we need to stop the deformaton of the snake. Naturally the snake ceases ts deformaton when all the snaxels can not fnd new locatons n the neghborng pxels or smply t converges to zero and dsappears from the sght. However t could go nto an nfnty loop when the snaxels shft along the boundary or some of snaxels oscllate. Therefore we need set a certan crteron for the termnaton of the snake. The smplest way s to set a threshold on the maxmum number of teratons executed n the deformaton process. Ths guarantees that the snake s termnated and t never goes nto an nfnty loop. It however requres a user to set the approprate the number of teratons pror to the deformaton process. Snce the number of teratons vares depends on the shape and the sze of the target object t s dffcult to estmate how many tme of teratons are needed to detect the subject contour. As a result the snake would be termnated before t reaches or close to the true boundary when the small number of the maxmum teratons was set. Another way s to set threshold on the number of snaxels moved n between teratons. For nstance we could set the threshold as 90% to the number of snaxels. Then the snake termnates ts deformaton when 90% of snaxels are not moved. In ths way the threshold does not depend on the shape or the sze of the target object. Consequently t does not need to requre a user to changng the number of teratons on each target object. However nether termnatng crtera would be useful because t s sometmes observed the snaxels shft along the boundary and the contour only moves very slghtly [5]. Wong Yuen and Tong [6] proposed other soluton for the termnaton of the snake. It s called contour length crteron CL-crteron. The CL-crteron s based on the length of the actve contour. It keeps track of the last 0 teratons and t

14 observes the changng rato of the length of the actve contour. 0 teratons are needed. Ths s due to the sudden ncrease and decrease of the rate of change of normalzed total length may cause the spkes and valley durng the teratons [5]. Cho Lam and Su [5] proposed smlar termnatng crteron called contour area crteron CA-crteron whch makes use of the normalzed total are to determne the convergence of the process. They clam that the CA-crteron has better stablty n convergence as ts fluctuaton between successve teratons s much smaller than the CL-crteron. Equaton (.3- below s used to compute the rato of CA-crteron at the k th teraton. rato A A A k k k η = (.3- k Equaton (.3- s used to calculate an area of the actve contour. It s base on the Green's Theorem n a plane. It makes CA-crteron works faster snce t does not requre square roots to obtan the lengths of the segments between all the snaxels. Area of polygon = = = n n = x x ( x k k + k y y y k + k k + x k + y k (.3- where n s the total number of the snaxels and k s at the k th teraton. Table.3- shows the result of the performance of the snake termnated by the CLand CA-crteron on the three sample mages. Table.3- Comparson wth CL- and CA-crteron on three mages (The experence s conducted on a Pentum-II 400 MHz PC [5] 3

15 -4 Lmtatons of the orgnal snake The orgnal snake works well on certan mages. However there are several lmtatons on ths method. The one of major problems comes from ts poor capture range. Although Gaussan flter s appled on the orgnal mage the external forces derved from boundares are sometmes not strong enough or wde enough to attract the snake toward true boundary of the target object. Accordngly an ntal contour must be set close to the boundary of the desred object. If the snake was ntalzed too far away from edges of the target object t wll ether move randomly due to lack of gradent force or t wll be attracted to noses or ts other near by edges lnes or ponts. Another major problem s that the snake can not progress nto the concavtes of an object. Snce there s no gradent force n the area under whte-dot-crcle on Fg..4- the snake totally depends on an nternal energy. The nternal forces of the snaxels under the regon however try to pull and straghten the snake. Consequently the snake cannot move down to the concavty of the object. The red lne n Fg..4- shows the result of the snake performed on the U-Shape object. You can see the snaxels under the concavty at the top of U-shape object ceases ts movement. Fg..4- The concavty problem of the snake on the U-shape mage 3. Adaptve Snakes Adaptve snakes proposed by Cho Lam and Su [5]. A new energy term called Adaptve forces was ntroduced. The adaptve force s added to the energy functon of 4

16 the snake n Eq. (-. The force wll be ntroduced at a snaxel f ts surroundng mage forces are smaller than a threshold [5]. The drecton of an adaptve force s perpendcular to a lne v + and v - whch s a normal vector n n Fg. 3-. Fg. 3- Adaptve force v ' = v r + k n (3- where k s the ampltude of an adaptve force Equaton (3- s used to compute a new locaton of th snaxel. An adaptve force k n wll be add to a snaxel f mage forces of the neghborng pxels are low. In the algorthm of the tradtonal snake a snaxel moves to a neghborng locaton whch s one pxel away from the current locaton. In contract Adaptve snake steps more than one pxel when k s set to more than. Fgure 3- shows the movement of the actve contour wth the step sze of 3 and 5. Therefore t makes the snake converge faster n the regon where the mage forces are weak. Furthermore t helps the snake to step over certan noses on an mage. Ths s because noses n general are low mage forces. However the larger step sze k sometmes creates a problem of makng the dstances of two adjacent snaxels uneven. Fgure 3-3 shows two examples of creatng the problems. Consequently two operatons deletons and nsertons are necessary to keep the dstances of the adjacent snaxels more constant. A new snaxel s nserted when two adjacent snaxels are far from each other. Meanwhle the current snaxel s deleted f the dstance between the next and the prevous snaxels are small. 5

17 Fg. 3- Step szes of the snake and Adaptve snake Fg. 3-3 Creatng problems of uneven spaces 3. Dscusson of Adaptve Snakes Unlke the tradtonal snake the adaptve snake can go nto the concavtes of the object. Fgure 3.- shows partal mages of Adaptve snake detectng the boundary of the object wth a concavty. The complete gf-anmatons of Adaptve snake are shown n Secton 0.3. The adaptve snake converges faster when the larger step sze s chosen. It takes 49 and 90 teratons to detect the boundary of the U-shape object when the adaptve force k s set to be 3 and 5 respectvely. Meanwhle the larger step sze creates another problem. 6

18 As I menton n the prevous secton the adaptve forces can step over some noses or undesred edges or lnes; however t may step over the true boundary as well. Fgure 3.- shows the result of the adaptve snake wth step sze of 9. Once the actve contour get nsde of the object t s dffcult to reach back to the subjectve contour snce the energy functon of the snake s desgned for an actve contour to converge. Therefore t needs some sort of forces that nflate/dverge an actve contour. In the next secton another actve contour model that utlzes such forces s dscussed. Fg. 3.- Detect boundary at concave regon (k = 3 7

19 Fg. 3.- Problem of the larger step sze of the adaptve snake (k = 9 4. Pressure or Balloon Snakes The concept of Pressure snakes or Balloon snakes was ntroduced by D. Cohen [7]. The proposed force s replaced wth the external force of the orgnal snake. The new external force s defned as: E r f balloon ( v( s = k n ( v( s P f (4- P s a new constant for the gradent magntude and k s the ampltude of a balloon force. n r s a normal untary vector at v(s. The normal untary vector s the same normal vector as llustrated n Fg. 3-. The sgn of k controls the drecton of the balloon snake. The balloon snake nflates when the sgn of the k s postve and deflates when t s negatve. It s necessary to set P slghtly larger than k so that an edge pont can stop the nflaton forces. The nflatons of the balloon forces solve the problem of the actve contour lays nsde of the subject boundary. The blue lnes n Fg.4- are the actve contour nflatng toward the target boundary by the balloon forces. Meanwhle the red-dot lnes llustrate the actve contour deflatng toward the boundary. 8

20 Fg. 4- Inflatng/deflatng balloon forces 4. Dscusson of Balloon snakes The balloon snake s more robust to the ntal poston than the orgnal snake. The orgnal snake s desgned an actve contour to shrnk to reach the target boundary. Therefore an actve contour needs to be ntalzed the outsde of the target contour. It would not reach the target boundary f the actve contour les nsde of the target contour. In the balloon model the ntal poston can le ether nsde or outsde the target contour. Ths s because the drecton of the actve contour can be control by a user. Although the requrement of an ntal locaton of an actve contour are loosen human nterventon s needed to decde whether an nflatonary or deflatonary forces on an actve contour before the energy mnmzng process starts. Ths ndcates that t requres a user to have some pror knowledge of the shape of the target object as well as the relatve poston of the ntal locaton of the actve contour and the target boundary. The balloon snake has added the flexblty of the ntalzaton of the actve contour. However t stll has not solved the problems of poor capture range. The ntal contour must be close to the true boundary of the target. GVF snakes dscussed n the next secton can solve the problem of poor capture range of the snake models. 5. GVF (Gradent Vector Flow Snakes Several new energy terms are added to the energy functon of the tradton snake to mprove the performance of the snake models. Nevertheless the problems assocated wth detectng concavtes regon of the target and ntalzng an actve contour have not been solved yet. Ths s all causes by the weak capture range of the snake. To overcome these problems t was necessary to change the concept of the external force. In ths secton I revew the GVF snake developed by Xu and Prnce [9]. 9

21 5. Gradent Vector Feld Xu and Prnce ntroduced a new external force to solve these problems. The proposed energy force s derved from a vector felds as known as Gradent Vector Feld [9]. Instead of usng the gradent magntude of an mage as an external force t uses spatal dffuson of the gradent of an edge map of an mage [8]. The GVF snake nvolves a vector feld derved by solvng a vector dffuson equaton that dffuses the gradent magntude obtaned form an mage. They defne the gradent vector feld and the energy functon as: [ u ( x y v ( x ] V ( x y = y (5.- GVF = u + u + v + v + f V E μ ( f dxdy (5.- x y x y The parameter μ s a regularzaton parameter governng the tradeoff between the frst and second terms. The value of the μ s set accordng to the amount of nose present n the mage. It s ncreased when more noses present n the mage [9]. The second term n the ntegral n Eq. (5.- s called data term. we can see the data term domnates the energy when f s large. To mnmze the energy we can set V nearly equal to f. Meanwhle when f s small the energy s domnated by sum of the square of the partal dervatves of the vector feld. The frst term n the ntegral s call smoothng term that makes vector flow V(x y varyng smoothly. In the external force of the tradtonal snake no forces are found n the homogenous regons where the gradents are nearly zero whereas t creates some forces n these regons. Ths helps to navgate the snake to move nto the concavtes of the subjectve object. 5. Generalzed GVF Snakes Although the GVF snake performs better than the orgnal snake at boundary concavtes t has some dffcultes when the boundary concavtes are long and thn [8]. To remedy the problem Xu and Prnce [] proposed the generalzed GVF snake. The both μ and f n Eq. (5.- are replaced wth more general weghtng functons; g( f and h( f. The equaton of the proposed GVF feld s defned as: V = g( f V h( f ( V f (5.- 0

22 The weghng functons g( f and h( f appled to the smoothng and data terms respectvely. Eq. (5.- s more general form of the equaton (5.-. Suppose that the weghng functons g( f and h( f are defned as Eq. (5.-. g( f = μ h( f = f (5.- Substtutng the weghtng functons above wth Eq (5.- the equaton becomes exactly the same equaton of the orgnal GVF snake defned n Eq. (5.-. Snce g( f s constant smoothng take place everywhere. Meanwhle h( f becomes larger near strong the edge. As a result t domnates at boundares. Therefore the GVF snake should provde good edges localzaton []. However t becomes problematc when two edges are close such as when there s a long and thn ndentaton along the boundary. Ths s because the GVF snake tends to smooth between opposte edges losng the forces necessary to attract the actve contour move nto the regon. To overcome ths problem we need to select the weghtng functons such that g( f gets smaller as h( f becomes larger. There are many ways to specfy pars of the weghng functons. Xu and Prnce suggest the followng weghtng functons for the generalzed GVF snake. g( f = e ( f / K h( f = g( f (5.-3 They clam that the weghng functons above wll conform to the edge map gradent at strong edges but wll vary smoothly away from the boundares. The parameter K controls the degree of tradeoff between feld smoothness and gradent conformty. 5.3 Dscusson of GVF Snakes Fgure 5.3- shows the comparson of external forces obtaned by the orgnal snake and the GVF snake. From the fgures t s obvous that the external forces derved from the GVF snake has much larger capture range than the orgnal snake. In the homogenous regons of the U-Shape object there s no external force found n the orgnal snake. Meanwhle there exst external forces that would lead an actve contour to the boundary of the object n the GVF snake.

23 Fg External forces of the orgnal snake and GVF snake Images taken from [9] The red-dot crcle on the left n Fg. 5.- shows the external forces at the concavty regons of the U-Shape object. The two fgures form rght are the close-up mages of force vectors derved from the orgnal snake (a and the GVF snake (b. We can see n Fg.5.- (a that there s no force n the regons under the blue-dot crcle. Ths s why the orgnal snake cannot move nto the concavty regon. Meanwhle there exst the forces under the blue-dot crcle n Fg.5.- (b. In addton to that the drectons of the force vectors n the regons are pontng downward. These force vectors make the actve contour move down to the concavty regons of the object. Fg External forces at concavty regons of U-Shape object Images taken from [9]

24 6. GCBAC (Graph Cut Based Actve Contour The concept of the GCBAC (Graph Cut based Actve Contour s proposed by Xu Bansal and Ahuja [3]. It s other alternatve way to extract an object boundary on an mage wth an actve contour. Unlke the tradtonal snake or ts successors an actve contour represented n the GCBAC s a non-parametrc curve. It dffers fundamentally from the tradtonal actve contour models. The GCBAC constructs an adjacency graph from an mage. Each pxel on the mage s represented as vertex of the graph. The weghts of edges between vertces are assgned accordng to the value of the gradent of an mage. Wth the adjacency graph the GCBAC enable to extract a subjectve boundary on the mage. The mplementaton detals are dscussed n Secton. 6. Algorthm of GCBAC The basc dea of the GCBAC s to fnd global mnmum wthn CN (Contour Neghbor. CN s defned as a belt-shaped neghborhood regon around a contour [3]. The yellow belt-shape n Fg. 6.- shows CN wth the wdth of 0. The wdth of a CN can be specfed by a user. Durng the deformaton process a CN s generated from the current actve contour. Wthn the CN t creates a mult-source and mult-snk mnmum-cut problem. The vertces on the nner boundary of the CN are dentfed as mult-source and smlarly the vertces on the outer boundary are dentfed as mult-snk. Fg. 6.- CN (Contour Neghbor of an actve contour (sze = 0 3

25 The mult-source and mult-snk mnmum-cut problem then s reduced to a one s-t mnmum-cut problem n the graph theory. There are several algorthms avalable to solve the s-t mnmum-cut problem. Such algorthms are Augmentng Path method by Ford-Fulkerson [4] Push-relabeled method by Goldberg and Targen [5] and Approxmaton method by Karger and Sten [6]. The soluton of s-t mnmum-cut gves a path wthn a CN whch become a new actve contour. A new CN s generated from the new actve contour. Ths process contnues untl the actve contour reaches the true boundary of the target. The followng s the GCBAC algorthm. GCBAC algorthm n hgher-level of vew Intalzaton process Deformaton process Termnaton of the GCBAC Intalzaton process: The GCBAC ntally constructs an adjacency graph from an mage. All pxels on the mage are converted to vertces on the graph. It then obtans an ntal contour around the target object from a user. The ntal contour s set to the current actve contour. Deformaton process: Durng deformaton the GCBAC takes the followng three steps. Step. Step. Step.3 Dlate the current contour to create a CN Identfy nner outer and ntermedate vertces to create mult-source mult-snk mnmum-cut problem. Reduce the problem and compute the s-t mnmum cut to obtan a new actve contour. Termnaton of GCBAC: The termnaton crteron of the GCBAC s based on the satablty of the cost of the mnmum-cut. At the end of teratons the cost of the mnmum-cut s computed and stored. The program s termnated when the costs of the mnmum-cut of the prevous and the current actve contours have small dfferences. 6. Dscusson of GCBAC The tme complexty of the GCBAC s domnated by the tme requred to solve the 4

26 s-t mnmum-cut problem. The tme complexty of the s-t mnmum-cut vares accordng to the algorthm. The fastest current algorthm of the s-t mnmum-cut problem s developed by Karger and Sten [6]. It runs O(n log 3 n tme. The algorthm I used to mplement the GCBAC s Ford-Fulkerson method [4]. It runs O(mn C tme where n s number of snaxels and m s number of edges. C represents the capacty of flows. The algorthm s dscussed n Secton. The connectvty of graph s another factor that nfluences on the runnng tme of the s-t mnmum-cut problem and the accuracy of the boundary of the target. Fg. 6.- shows 4- and 8-connectvty of the graphs. Fg. 6.- Connectvty of an adjacency graph Suppose that the costs of edges are set to and red-dot lnes are the cuttng edges. The numbers below each 4 x 4 grd are capactes of the cut. Notce that the frst two grds from left have the same costs of 6s. Ths mples that the same capacty of the cut on 4-connectvty of a graph has possbltes of representng dfferent boundares. In contrary the grds for 8-connectvty have the dstnct values for dfferent boundares. Accordngly an actve contour on 8-connectvty adjacency graph has more accuracy of representaton. The tradeoff s that the computaton tme on s-t mnmum-cut problem. Obvously the runnng tme ncreases as the number of edges ncreases. The GCBAC has several advantages over the tradtonal snake. Frst the set of weghtng parameters on each energy terms are not requred. Although the GCBAC stll needs an ntal contour around the target object from a user; however t s not necessary to set weghtng parameters requred n the orgnal snake and ts successors. Another advantage s that the GCBAC does not create a self-crossng actve contour. A soluton of s-t mnmum-cut problem returns a path that separates CN nto two parts. A new actve contour s formed wth the outer pxels of the path a red-dot lne n Fg. 5

27 6.-. Consequently t remove the problems wth self-crossng whch sometme seen n the fnal contour of the snakes. Fg. 6.- Self-crossng actve contour Snce a soluton of s-t mnmum-cut problem s a globally optmal soluton n CN the algorthm of the GCBAC has capablty to jump over local mnma wthn CN. As a result an actve contour s not attracted to the local mnma such as noses or undesred features wthn CN. The mplementaton of the GCBAC s dscussed n Secton. 7. Extractng multple objects n an mage Generally the snake actve contour or the GCBAC s desgned for detectng a boundary of a sngle object on an mage. Cho Lam and Su [5] proposed an algorthm to detect the contours of multple objects n an mage. The proposed algorthm has a number of lmtatons. The prmary lmtaton s the relatve postons of the target objects. The algorthm works merely on multple objects that have some dstances between themselves. In other words each extractng objects must not be overlapped each other such as objects n Fg Crtcal Ponts and Splttng and connectng operatons The algorthm starts searchng for crtcal ponts on the current actve contour. The crtcal ponts are all snaxels that satsfes the followng condtons: snaxel v s a crtcal f ( E( v > T ( E( v > T where T mage mage mage pont and E( v and E( v > T < T mage mage and E( v and E( v s a threshold on mage force + + > T < T mage mage or (7.- 6

28 Fgure 7.- llustrates two pars of the crtcal ponts on the actve contour. Snce the dstance of the par of the crtcal ponts on the left are further than the pre-defned dstance threshold D T these crtcal ponts are gnored. Meanwhle the dstance of the par of crtcal ponts on the rght s lower than the dstance threshold. Therefore at these crtcal ponts on the rght the splttng and connectng operatons are performed to form two actve contours. The red-dot lnes n Fg. 7.- represent new connectons after the splttng and connectng operaton. The algorthm s dscussed n Secton 3. Fg. 7.- Crtcal Ponts 7. Removal of nvald Contours The operaton of splttng and connectng sometmes creates nvald contours as llustrated n Fg Therefore these nvald contours must be removed at the between teratons n the deformaton process. To dentfy an nvald contour an area of each actve contour s calculated at the end of teraton. In general an nvald contour converges to a certan pont by nternal forces of the snake due to no external force nsde of the contour. Some nvald contours stop convergng to a pont and cease ther movement (Fg. 7.-a. Other contours grow n a reverse drecton. Ths s because snaxels are too close each other some snaxels skp over other snaxels (Fg. 7.-b. The area of contours shown n Fg. 7.- becomes zero. Meanwhle the area of contours shown n Fg. 7.- becomes negatve. Snce the areas of these contours sgnfy the nvaldty of the actve contours the threshold s set on the area of contour at end of 7

29 teraton for the removal of the nvald contours. Fg. 7.- Creaton of an nvald contour Fg. 7.- Behavor of an nvald contour 7.3 Dscusson of extractng multple objects The proposed algorthm by Cho Lam and Su performs well wth snakes such as the adaptve or the GVF snake. However t won t works wth the orgnal snake. Fgure 7.3- shows the result of the orgnal snake performed on the mage wth two objects. Snce the orgnal snake dose not have enough forces to move the contour nward (blue arrows the two pars of crtcal ponts (yellow crcles won t be close enough to fall under the dstance thresholds. For ths reason the orgnal snake can not extract multple objects wth ths algorthm. 8

30 Fg Result of the orgnal snake on multple objects 8. Implementaton of Applcaton I mplemented an applcaton that helps to understand algorthms of actve contour models. Besdes these algorthms I ve mplemented some other functons that help to understand and analyze an mage. Table 7- below shows the lst of avalable functonaltes of the applcaton. The screenshots of the applcaton are provded n Secton 6. Flters Hstogram Property ACM Functons Gaussan Flter Medan Flter User-defned Flter Hstogram equalzaton Hstogram operatons Show the nformaton of an mage Show gradents of an mage Show GVF fled of an mage Orgnal snake Adaptve snake GVF snake Screenshots Fg.6.- Fg.6.- Fg.6.-3 Fg.6.3- Fg.6.3- Fg.6.4- Fg.6.4- Fg.6.4- Fg.6.5- Fg.6.5- Fg.6.5-9

31 GCBAC (Graph Cut Based Actve Contour Fg.6.5- Table 8- Lst of the avalable functons n the applcaton 8. Lbrares I used QT lbrary from Trolletch [] to buld framework and GUI components of the applcaton. QT s a complete C++ applcaton development framework that nclude a class lbrary and tools for cross-platform development and nternatonalzaton []. Qt has rch class lbrary for mage manpulaton and GUI graphcal user nterface components. Rapd Applcaton Development known as RAD s a method of quckly developng software. RAD program allow you to create GUI n mnutes. A user smply drags GUI components to proper poston onto an empty form and the RAD tool takes care of the source code for you. Qt provdes such a tool called QT Desgner whch handles such task for you. I also used OpenGL lbrary [] to vsualze prmary on showng the gradent vector and the gradent vector feld derved from mage. OpenGL provdes a cross-platform API for wrtng applcatons that produce D and 3D computer graphcs. The nterface conssts of about 50 functon calls to draw complex D or 3D scenes from smple prmtves. 9. Implementaton of the orgnal snake The mplementaton of the snake nvolves energy terms approaches of deformaton process program termnaton and obtanng an ntal contour from a user. Each of them s dscussed n the followng sectons. 30

32 Fgure 9- shows partal mage of the Snake Control Panel. The slders at the top control the weghts of nternal and external forces of the snake. The box n the mddle s set the ntervals of snake control ponts. The approaches of the snake algorthm are selected at the algorthm secton dscussed n Secton 9.4 and 9.5. The button Roll back moves the current actve contour to the prevous locatons. The Save Contour saves the current contour and the Retreve Contour retreves the saved contour. The detals of the Snake Control Panel are shown n Secton 6.5. Fg.9- Snake Control Panel 9. Internal Energy terms The nternal energy terms are defned n Eq. (.-. They are the frst and second dervatve of the parametrc curve of the snake. Dscretng each energy term of the nternal energy we have the followng formulas. E elastc ( s = α ( s v α ( s dv( s ds v (9.- E bendng ( s = β ( s v dv( s ds β ( s ( v v ( v v β ( s v + v + v = ( x y = 0 K N where N + s number of snaxels (9.- The elastc force nvolves two adjacent snaxels. The dstances of two adjacent snaxels are used as an elastc energy term. Meanwhle the bendng force nvolves three adjacent snaxels. The curvatures of these three adjacent snaxels are computed and used as a bendng energy term. 9. External Energy term 3

33 The external energy s derved from the mage data. The gradent magntude of ntenstes of an mage s used as external energy. Equaton (9.- s used to compute external energy at each snaxel [3]. In general the Gaussan flter s appled on the mage to ncrease capture range. Thus the orgnal mage s fltered out pror to the computaton of the gradent magntude. f dv ( x y mag( f dx = = dv dy = dv ( dx dv + ( dy ( Intal contour at the ntalzaton process At the ntalzaton process a user drags mouse to create an ntal contour. The ponts under the contour are stored n a lst. The snaxels are chosen from some of ponts n the lst accordng to the nterval specfed by a user. Snce snake works well when each segment of snaxels has equal length the nterval gves even spaces between snaxels. Fgure 9.3- shows the ntal contours wth ntervals of 5 and 40. Fg Intal contours and ther ntervals 9.4 Greedy Algorthm n the deformaton process 3

34 The greedy algorthm for the snake s relatvely smple and easy to be mplemented. Durng the teratons the sum of the nternal and external energes s computed at a snaxel and ts 8 neghborng pxels. A locaton that has the smallest energy s set to be a new locaton. Therefore a snaxel moves to one of the eght possble neghborng pxels the red arrows n Fg 9.4- left or stay the same locaton f t cannot fnd the smaller energy than that the current snaxel has. The faster greedy algorthm s proposed by Lam and Yan [3]. Instead searchng all neghborng locatons t searches 4 possble pxels (blue-dot arrows n Fg 9.4- rght. They clam that the neghbors of the locaton havng the smallest value of the energy functon also have the small values. Therefore the computaton requred n searchng for a new locaton can be greatly reduced by searchng the neghbors n alternate patterns n Fg By alternatng the two search patterns I and II the whole space searched by the greedy algorthm. Fg Possble movements at a snaxel durng teraton The expermental results of comparson of two algorthms n Table 9.4- are provded by Lam and Yan [3]. The experment was conducted on San Sparc workstaton. 33

35 Table 9.4- Expermental results: Number of teratons and executng tme for the greedy and the fast greedy algorthms [3] The tme complexty of both the greedy and the fast greedy algorthms s O (n t tme where n s the number of snaxels and t s the number of teratons executed n the deformaton process. Hence ths s nexpensve operaton. However no matter the greedy or the fast greedy algorthms nether algorthm would guarantee the optmal soluton. Durng teratons the snake searches locally optmal choces from neghborng pxels at each snaxel. To compute global optmal soluton all possble combnatons of snaxels must be computed. Next secton shows how to solve t effcently. 9.5 Dynamc Programmng n the deformaton process Unlke the greedy algorthm the dynamc programmng fnds a globally optmal soluton. The dea of the dynamc programmng s to break a large problem down nto ncremental steps so that at any gven stage the optmal solutons are known to sub-problems. When the technque s applcable ths condton can be extended ncrementally wthout havng to change prevously computed optmal solutons to sub-problems. It removes redundant computaton tme of the algorthm. For the energy-mnmzng algorthm t normally takes O(9 n tme to compute all combnatons of the possble moves of the snaxels where n s the number of snaxels on the actve contour. Wth the technque of the dynamc program t can be reduced the computaton tme sgnfcantly. Snce the nternal energy nvolves three consecutve snaxels the equaton of the total energy of the snake s defned as: 34

36 E snake ( v v v 3 L v n = E( v v v3 + E ( v v3 v 4 + L + E ( v v v n n n n (9.5- where E ( v v v + = Eexternal ( v E nternal ( v v v + + In order to apply the dynamc programmng to Eq. (9.5- a stage varable S s defned as follows: S v v mn S + E ( v E ( v v v (9.5- ( + = external + nternal + v For each stage a new energy and the energy n the prevous stage S - are added and stored n the current stage S. The total accumulated energes are stored at S n-. The trace back starts wth the mnmum energy found at S n-. It takes O(9 tme to update stage varable for each neghborng snaxels. Therefore the total tme needed for a snaxel become O(9 9 tme. Snce the number of snaxels s n the tme complexty of the snake algorthm wth the dynamc programmng becomes O(n 9 3 tme. Comparng to the O(9 n tme wthout the dynamc programmng t makes sgnfcant mprovement. 9.6 Termnaton of the program At some pont of the tme the program has to be termnated. Several termnatng crtera are dscussed earler. I employed the followng three termnatng crtera. Termnatng Crteron Threshold # of snaxels moved 80% 80% of snaxels stays the same locaton. CA-Crteron 0.00% Rato of change of the area n two consecutve teratons CL-Crteron 0.00% Rato of change of the length n two consecutve teratons Table 9.6- Termnatng crtera and ther thresholds 35

37 Fg Termnatng crtera on Snake Control Panel Fgure 9.6- s the snapshot of the termnaton crtera selecton on the Snake Control Panel. The snake s termnated by one of the crtera n the lst box. The threshold for Snake Moved s set to be 80%. The snake s termnated when 80% of snaxels are stable. Meanwhle the thresholds for CA- and CL-Crteron are set to 0.00%. Ths means that the snake s termnated when the rato of change n area or length of the snake s less than 0.00 n two consecutve teratons. In addton to these crtera the snake s termnated n every 50-teraton. Ths prevents the snake to go nto an nfnty loop snce the snake sometme doesn t fall nto any of these termnatng crtera. 9.7 Results of the orgnal snake The results of the orgnal snake are shown n the followng lnk. Lnk : Lnk to Image Gallery 0. Implementaton of Adaptve snakes The mplementaton of the adaptve snake nvolves computaton of an adaptve force at snaxels and nserton or deleton of snaxels durng the deformaton process. These functonaltes are ntegrated to the algorthm of the orgnal snake. The mplementaton detals are dscussed n the followng sectons. Fgure 0- shows parameter settng for Adaptve snake on the Snake Control panel. A user can set the strength of the ampltude of the adaptve force at box under Adaptve Force. The adaptve force s appled on a snaxel only f the snaxel has lower mage force than the threshold specfed n the box under Threshold. The default value for the adaptve force s set to 3 pxels and the threshold of the mage force s

38 Fg. 0- Snake Control Panel for Adaptve Snake 0. Adaptve force To compute an adaptve force on th snaxel a unt normal vector n r at v need to be calculated. Snce the normal vector v r ' at v s perpendcular to the vector (v + v - t satsfy Eq. (0.-. Then the normal vector s normalzed Eq (0.- to obtan the unt normal vector n r at v. A new locaton of a snaxel s computed n Eq. (0.-3 v r ' ( v + v = 0 (0.- r v r ' n = r (0.- v' r v '( new locaton = k n + (0.-3 v 0. Inserton and deleton of snaxels Snce the nature of the algorthm of the adaptve snake t sometmes creates uneven spaces between snaxels. Ths s not a desrable phenomenon for the snake to work well. Therefore two operatons nserton and deleton are need to be appled after a new actve contour s generated. The condtons of the operatons and ther thresholds are lsted n Table 0.- and 0.-. Fgure 0.- shows some stuatons that the operatons of nsertons and deletons are needed. Delete the snaxel v Operatons Insert a new snaxel n the mddle of v and v + Condton d < t d d < t d < t 3 d d * d s an average dstance of the snaxels Table 0.- Condtons of deletons and Insertons Thresholds Default value T 0.8 T 0.8 T 3.8 Table 0.- Dstance thresholds 37

39 Fg. 0.- Deleton and nserton of snaxels 0.3 Results of Adaptve snakes The results of Adaptve snake are shown n the followng lnk. Lnk : Lnk to Image Gallery. Implementaton of GVF snakes The mplementaton of the GVF snake s prmary focused on the creatng the GVF feld derved from the gven mage. The algorthm of the GVF snake s the same as the orgnal snake. The only dfference s that the GVF snake uses external force form the GVF feld nstead of the gradent magntude whch s used n the orgnal snake. Fgure - shows parameter settng for GVF snake on Snake Control panel. A user can set a number of teraton to compute a gradent vector feld and a value of the mu The mu s set accordng to the amount of nose present n the mage. The button Update Feld update the GVF feld only f the number of the teraton has been changed. Fg. - Snake Control Panel for GVF Snake 38

40 39. Gradent Vector Feld The gradent vector feld s derved from an mage. The gradent vector feld s computed wth the tradtonal GVF snake. The followng equatons are used to compute the gradent vector feld. Usng the calculus of varatons [4] the GVF feld can be computed by the followng Euler equatons []. 0 ( ( = + y x x f f f u μ (.-a 0 ( ( = + y x y f f f u μ (.-b where s the Laplacan operator Equaton (.- can be solved by treatng u and v as functons of the tme and solvng []. ( ( ( ( ( y x c t y x u y x b t y x u t y x u t + = μ (.-a ( ( ( ( ( y x c t y x v y x b t y x u t y x v t + = μ (.-b ( ( ( ( ( ( ( ( ( y x f y x b y x c y x f y x b y x c y x f y x f y x b where y x y x = = + = To set up the nteractve soluton let the ndces j and n correspond to x y and t respectvely and let the spacng between pxels be Δx and Δy and the tme step for each teraton be Δt. Then the requred partal dervatves can be approxmated as 4 ( 4 ( ( ( j j j j j j j j j j n j n j t n j n j t v v v v v y x u u u u u u y x u v v t v u u t u = = = = Δ Δ Δ Δ Δ Δ Substtutng these approxmaton nto Eq. (.- gves the followng the

41 nteractve soluton. u Δ + (.-3 n+ n n n n n n j = ( b j t u j + r ( u+ j + u j+ + u j + u j 4u j c j v Δ + (.-3 n+ n n n n n n j = ( b j t v j + r ( v + j + v j+ + v j + v j 4 v j c j where r = μδt ΔxΔy In my programmng code I set the spacng Δx and Δy and the tme step Δt to be. The followng codes are used nsde the teraton process. for ( k = 0; k < number of teraton; k + + { L for ( = 0; < wdth* heght; + + { L du[ ] = ( b[ ] * du[ ] + mu lapu[ ] + c[ ]; dv[ ] = ( b[ ] * dv[ ] + mu lapv[ ] + c[ ]; L } L } where dv du b c and c are double arrays Fgure.- shows the comparson of capture ranges of the orgnal snake and the GVF snake under the blue crcle on the star object. The GVF feld n Fg..-(b s computed wth 60 teratons. It s observed that the external forces generated by the orgnal snake have narrow capture range. 40

42 Fg..- Capture ranges of the orgnal snake (a and the GVF snake (b.. Results of GVF Snakes The results of the GVF snake are shown n the followng lnk. Lnk : Lnk to Image Gallery. Implementaton of GCBAC Smlar to the snake the GCBAC requres a user to draw an ntal contour around the subject object. It takes the same procedure of the snake to obtan an ntal contour. The keys components of the mplementaton of the GCBAC are dlaton cost of an edge and the s-t mnmum-cut problem. Each component s dscussed n the followng sectons.. Dlaton The dlaton process generates a CN (Contour Neghbor of the current contour. The red ponts n Fg..- are the ponts on the ntal contour. A CN s generated wth a matrx centered of these ntal ponts. The sze of a matrx s specfed by a user. Fg..- shows CN computed wth sze of 5 and 0. The regons wth yellow color are CN of the actve contour. The number of vertces and edges wthn the CN are lsted on the table on Fg..-. 4

43 Fg..- Dlaton wth 5x5 and 7x7 matrx Fg..- Sze of CN on sample mage. Cost of edges Equaton.- s the formula to compute the costs of the edges on the graph. The C ( j s the cost of the edge between vertces and j. 4

44 ( g( j + g( j C( j = grad j ( g( j = exp( max k ( grad j ( k note : grad j ( k s the gradent at drecton of j C( j == C( j 6 locaton k n the (.- Equaton (.- s desgned to assgn the hgh value to the edges on the hgh gradent. These edges on the hgh gradent are most lkely boundares of the objects or noses n the mage. Meanwhle the equaton assgns the low value to the edges on the homogenous area. Generally the background or the surfaces of the objects on the mage have no varance of gradents; therefore these areas become homogenous. Snce the algorthm of the s-t mnmum-cut problem dscussed n the next secton starts to elmnate form the edges wth low value the soluton of the mnmum-cut would be the edges wth hgh value. Ths s desrable because the boundary of the target object exst on the edges on hgh gradent. As a result the actve contour formed by these edges reaches the contour of the target object on the mage..3 s-t mnmum-cut problem As I mentoned earler there are several algorthms avalable to solve the s-t mnmum-cut problem. I used Ford-Fulkerson algorthm based on augmentng path method n my mplementaton of the GCBAC. The basc flow of the algorthm takes the followng steps. Step : Create the s-t mnmum-cut problem form the gven CN. Frst t creates a source and a snk node. Then the source node s connected to vertces on outer boundary of the CN. Smlarly the snk node s connected to the vertces on the nner boundary of the CN. The costs of the edges between these connectons are assgned the maxmum values. Ths s because we want to fnd the cuttng edges wthn the CN. Step : Fnd a path form the source to snk node. The BFS (Breath Frst Search algorthm s used to fnd a path from source to snk node. Once a path s found t determnes the mnmum capacty c on the path. 43

45 Step 3: Flow the mnmum cost to the path and update the resdual graph It updates the resdual graph by flowng c found n the step to all the edge on the path. Ths guarantees at least one edge are cut. Step 4: Repeat untl no path from the source to snk node s found. It contnues step and 3 untl there s no path from the source to the snk node. Once no path s found the program computes a new actve contour determned by the cuttng edges.4 Termnaton The cost of the mn-cut s computed each teraton n the deformaton process. The current cost s compared wth the prevous cost of the mn-cut. The program s termnated when the dfference of these costs s lower than pre-defned threshold..5 Results of GCBAC The results of the GCBAC are shown n the followng lnk. Lnk : Lnk to Image Gallery 3. Implementaton of extractng multple objects Snce extractng multple objects needs to keep track multple actve contours smultaneously the data structure must be change. Fgure 3- llustrates the data structure for extractng sngle and multple objects. Instead of usng a lst of snaxels snake (std::lst a lst of lsts of snaxels snakelst (std::lst of std::lst s used for multple objects extracton. 44

46 Fg. 3- Data structure of the snake 3. Algorthm of extractng multple objects The followng three steps are performed to extractng multple objects n an mage. These steps are taken at the end of each teraton n the deformaton process of the snake algorthm. Step: Identfy all crtcal ponts Each snaxel on the current actve contour s searched and determned f t s a crtcal ponts. Any snaxel that satsfed the condtons of a crtcal pont defned n Eq. (7.- s marked as a crtcal pont and pushed back to cplst. The cplst s defned as: cplst ( crtcal pont lst = { c ( s = 0 K n } (3.- Step: Fnd pars of the crtcal ponts In ths step pars of crtcal ponts are determned n the cplst. The table 45

47 below shows algorthm to fnd a par of crtcal ponts n the cplst. Algorthm to dentfy a par of crtcal ponts [5] Step Set = 0 as the startng crtcal pont C. Step Set j = + for the other crtcal pont C j. Step 3 Compute the dstance d j between the two crtcal ponts C and C j. Step 4 If d j s less than a threshold C and C j are marked as a par of connected ponts. Step 5 If j < n then j = j + and go to Step 3. Otherwse go to Step 6. Step 6 If < n then = + and go to Step. Otherwse go to Step 7. Step 7 END Fgure 3.- llustrates example of how the pars of crtcal ponts are searched for the frst two crtcal ponts C 0 and C n the cplst. The Black arrows from left to rght are search path for C 0 and the red-dot arrows are for C. Once a par of crtcal ponts s found t s added to pcplst. The pcplst s defned n Eq. (3.-. Fg.3.- Searchng paths for the frst two crtcal ponts C 0 and C pcplst ( pars of crtcal pont lst = { P ( v v < j < n k m} (3.- k j < Step3: Perform Splttng and Connectng operaton All the pars of crtcal ponts found at the prevous step are connected. A new contour s pushed back to the snakelst. 46

48 3. Removal of nvald contours The nvald contour are remove at the between teratons. The threshold on the area dscussed n Secton 7. s set to 0. as default value. Any contours below the threshold are removed from the snakelst. 3.3 Results of extractng multple objects The results of extractng multple of objects are shown n the followng lnk. Lnk : Lnk to Image Gallery 4. Summary In ths paper I have revewed and dscussed some of the mythologes of actve contour models; the orgnal snake and ts successor and the GCBAC. I have mplemented these metrologes to understand more about mechansm of ther algorthms. Many mprovements based on the tradtonal snake or alternatve approaches to detect boundares of object(s have been proposed and made. The performance of each algorthm s superor to others n terms of tme complexty and accuracy on certan mage. However they ether solve one or more problems but creatng new dffcultes. The prmary goal of the actve contour models s to detect the boundares of the target object(s n the mage. The one of ultmate goals of actve contour model s to acheve the goal wth less human nterventons. Snce all the approaches I have revewed requres a user to nput one or more parameters a user need to have pror understand of the mage. The wrong parameter settngs could end up wth panful results. Therefore automatcally fndng or predct the values of these parameters would be the next step. 47

49 5. Reference [] M. Kass A.Wtkn and D. Terzopoulos. "Snakes: Actve contour models" Internatonal Journal of Computer Vson (4: [] Ivns J; Porrll J: 994. "Statstcal Snakes: Actve Regon Models" Ffth Brtsh Machne Vson Conference (BMVC'94; York England: vol pp [3] K.M. Lam and H.Yan: Electroncs Letter 6th January 994 Vol.30 No. [4] C. Xu A. Yezz Jr. and J. L. Prnce "On the Relatonshp between Parametrc and Geometrc Actve Contours" n Proc. of 34th Aslomar Conference on Sgnals Systems and Computers pp October 000. [5] Wa-Pak Cho Kn-Man Lam and Wan-Ch Su "An adaptve actve contour model for hghly rregular boundares" Pattern Recognton Vol. 34 pp [6] Y.Y. Wong P.C. Yuen C.S. Tong Contour length termnatng crteron for snake model Pattern Recognton 3 ( [7] L. D. Cohen. "On actve contour models and balloons" CVGIP: Image Understandng 53(:-8 March 99. [8] C. Xu and J.L. Prnce "Gradent Vector Flow: A New External Force for Snakes" Proc. IEEE Conf. on Comp. Vs. Patt. Recog. (CVPR Los Alamtos: Comp. Soc. Press pp June 997. [9] C. Xu and J. L. Prnce "Snakes Shapes and Gradent Vector Flow" IEEE Transactons on Image Processng 7(3 pp March 998. [0] C. Xu and J. L. Prnce "Global Optmalty of Gradent Vector Flow" Proc. of 34th Annual Conference on Informaton Scences and Systems (CISS'00 Prnceton Unversty March 000. [] C. Xu and J. L. Prnce "Gradent Vector Flow Deformable Models" Handbook of Medcal Imagng edted by Isaac Bankman Academc Press September 000. [] C. Xu and J. L. Prnce "Generalzed Gradent Vector Flow External Forces for Actve Contours" Sgnal Processng --- An Internatonal Journal 7( pp December 998. [3] Nng Xu Rav Bansal and Narendra Ahuja "Object Segmentaton Usng Graph Cuts Based Actve Contour" IEEE Internatonal Conference on Computer Vson and Pattern Recognton June 003 [4] L. R. Ford Jr. and D. R. Fulkerson "Maxmal flow through a network" Canadan Journal of Mathematcs vol.8 pp [5] A. Goldberg and R. Tarjan. A new approach to the maxmum flow problem. Journal of the Assocaton for Computng Machnery 35(4: October 988. [6] D. R. Karger and C. Sten A new approach to the mnmum cut problems J. 48

50 ACM vol.43 no.4 (996 pp [7] J. Hao and J. B. Orln. "A Faster Algorthm for Fndng the Mnmum Cut of a Graph" In Proc. SODA pages [8] Zeyun Yu and Chandrajt L. Baj "Normalzed Gradent Vector Dffuson and Image Segmentaton" Lecture Notes In Computer Scence; Vol. 35. Proceedngs of the 7th European Conference on Computer Vson-Part III Pages: Year of Publcaton: 00. [9] C. Xu D. L. Pham and J. L. Prnce "Medcal Image Segmentaton Usng Deformable Models" SPIE Handbook on Medcal Imagng -- Volume III: Medcal Image Analyss edted by J.M. Ftzpatrck and M. Sonka May 000. [0] Raman Pchuman "Survey of Current Technques" tml Jul 997. [] Trolltech Qt lbrary [] OpenGL Archtecture Revew Board OpenGL Lbrary [3] Rafael C. Gonzalez Rchard E. Woods Dgtal Image Processng nd edton 00 [4] R. Courant and D. Hbert "Methods of Mathematcal Physcs". vol.. New York: Interscence

51 6. Acknowledgements I wsh to acknowledged and thank those people who contrbuted to ths project. I must express my deep grattude to my advsor Prof. Erc N. Mortensen. Frst of all I thank hm for beng my major professor and gvng me the drecton to my goal. Hs valuable feedback contrbuted greatly to ths project. I thank Dr. Tmothy A. Budd and Dr. Thnh Nguyen for beng my commttee members and sparng tme to read my document. Lastly I thank my parents for supportng me over years of my school tme. Wthout ther fnancal support and encouragements I would not have been acheved my goal. 50

52 7. User manual 6. General Fg.6. s the screenshot of the man wndow of the applcaton. The applcaton provdes several mage tools that apply on an mage. Although the functonaltes of the applcaton are prmary focused on the actve contour mode t provdes functons to analyss an mage as well. The followng sectons present bref explanatons these functonaltes along wth ther screenshot. Fg. 6. Screenshot of the man wndow the applcaton 6. Flters A user can apply flters on an mage. A flter s selected by clckng the tab bar at top of the flter control panel. A tab bar can swtch between the parameter wndows of General Gaussan and Medan flters. A flter s appled by clckng Apply Flter button on the Flter Control Panel. The operaton can be undone by selectng the Undo button. 5

53 . General Flter A user can apply a user-defned flter by drectly typng the coeffcents of the mask of the flter n the edt-lne box n the mddle of the Flter Control Panel. It can retreve the saved flter. Fg. 6.- Screenshot of General Flter. Gaussan Flter A user can apply a Gaussan flter on the mage. A sze of the mask of Gaussan flter s set by a user. The sze must be postve and odd nteger. The coeffcents of a Gaussan mask are vewed by clckng Vew Flter button. The popup wndow shows the matrx of the coeffcents. It can closed by double-clck anywhere on the popup wndow. The standard devaton of the Gaussan functon s fxed to.0. 5

54 Fg. 6.- Screenshot of Gaussan Flter 3. Medan Flter A user can apply a Medan flter on an mage. A sze of the mask of a Medan flter s set by a user. The sze must be postve and odd nteger. The Medan flter can apply only on a gray-scale mage. 53

55 Fg Screenshot of Medan Flter 6.3 Hstogram A user can vew the hstogram of an mage. The plot at the top of the hstogram control panel shows the frequency of the ntenstes of the mage.. Hstogram Equalzaton Fg.6.3-a and 6.3-b show before and after the hstogram equalzaton s appled on the mage. 54

56 Fg. 6.3-a Screenshot of Hstogram Equalzaton 55

57 Fg. 6.3-b Screenshot of Hstogram Equalzaton after applyng. Hstogram Operatons The two operatons related the hstogram are provded. The Shft operaton shfts the ntenstes of an mage. A user can specfy the range and the amount of the ntenstes of the mage to sht. The Cut operaton cuts the ntensty of the mage. A user can set the range of the ntenstes of the mage. 56

58 Fg Screenshot of operatons 6.4 Image Propertes. Informaton of an mage The Image Informaton wndow (Fg.6.4- shows the nformaton of an mage. It shows the nformaton such as name locaton color depth and the frequency of ntenstes of the mage. The Image Property wndow s shown by selectng the menu tem Image Property under edt on menu bar. 57

59 Fg Screenshot of the nformaton of an mage. Gradent and GVF feld of an mage Fg.6.4-a and 6.4-b show the OpenGL wndow and the OpenGL Control Panel. The lst of check boxes on the control panel controls the vewng layers on the OpenGL wndow. Each component s shown or hdden by markng or unmarkng a check box next to t. The OpenGL wndow and ts control panel for Gradent s shown by selectng the menu tem Show OpenGL dalog n the rght-clck menu on the man wndow. Meanwhle the OpenGL wndow and ts control panel for the GVF feld s shown by selectng the menu tem Show OpenGL dalog n the rght-clck menu on the actve contour models wndow. The center of the OpenGL wndow s the locaton of the mouse on the mage when the menu tem Show OpenGL dalog s clcked. 58

60 Fg. 6.4-a Screenshot of Gradent of an mage Fg. 6.4-b Screenshot of GVF feld of an mage 59

61 6.5 Actve Contour Models A user can select a method of the actve contour models. Two types of actve contour models are provded; the snake actve contour models and the GCBAC. The user can select one of them by selectng the menu tems under Actve Contour Model on the menu bar.. Snake actve contour models The control panel n Fg s used to set parameters for the orgnal snake the adaptve snake and the GVF snake. Table 6.5- lsts functonalty of each secton on the Snake Control Panel. Sectons Coeffcents Snaxels Snake Orgnal snake Adaptve snake GVF snake Termnaton Crtera Other [Rght bottom of the Functons Adjust weghts of the nternal and external energes of the snake. Set nterval of the snake control ponts on the contour. Select an algorthm of snake Rollback the prevous teraton of the snake Save the current contour Retreve the saved contour Perform the orgnal snake. Set the adaptve force n pxel. Adjust threshold on the mage force. Perform the Adaptve snake. Set the value of the mu Set the number of teraton to compute GVF feld Perform the GVF snake. Update GVF feld. It updates GVF feld only f the number of teraton has been changed. Select a termnaton crteron. Show the current number of teratons. Show the message how the algorthm of the snake has been termnated. Trace back an actve contour. Perform the deformaton process by sngle teraton. Extract multple objects 60

62 Snake Control Panel] Close the Snake Control Panel Table 6.5- Lst of functonaltes on the Snake Control Panel Fg Screenshot of Snake Control Panel. GCBAC Fg.6.5- shows the GCBAC wndow. The sze of CN (contour neghbor s specfed by a user at the bottom of the GCBAC wndow. 6

63 Fg Screenshot of GCBAC 6

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