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2 6 Pxon-Based Image Segmentaton Hamd Hassanpour, Had Yousefan and Amn Zehtaban 3 Shahrood Unversty of Technology Iran Unversty of Scence & Technology (IUST) 3 Noshrvan Unversty of Technology (NIT) Iran. Introducton The pxon concept was ntroduced by Pna and Puetter n 993. The pxon they ntroduced was a set of dsjont regons wth constant shapes and varable szes. Ther pxon defnton scheme was a local convoluton between a kernel functon and a pseudo mage. The drawback of ths scheme was that after selectng the kernel functon, the shape of the pxons could not vary. Yang and Jang presented a new pxon defnton scheme, whose shape and sze can vary smultaneously. They also used the ansotropc dffuson equaton to form the pxons and fnally they have combned the pxon concept and MRF for segmentaton of the mages. Recently, another well-behaved pxon-based mage representaton s proposed [Le Ln et al., 008]. In ther presented scheme the pxons combned wth ther attrbutes and adjacences construct a graph, whch represents the observed mage. They used a Fast QuadTree Combnaton (FQTC) algorthm to extract the good pxon-representaton. These technques ntegrated nto MRF model. The man dsadvantage of MRF-based methods s that n these algorthms the mnmzaton problem of objectve functon s very tme consumng. The most novel method whch uses pxon concept to segment the mages s ntroduced by Hassanpour et al. In ths method, frst a pre-processng step s performed whch apples the wavelet thresholdng technque. Ths step s sutable for mage smoothng due to the nose reducton property of wavelet thresholdng. To avod over-smoothed problem, the value of the threshold must be assgned properly. Then, the pxon-based algorthm s used to form and extract the pxons. Fnally, the Fuzzy C-Means (FCM) algorthm s appled to segment the mage. The advantage of usng pxons s that after formng the pxons the decson level changes from pxels to pxons and ths decreases the computatonal tme, because of the fewer number of pxons compared to number of pxels. Ths s the key aspect of pxon-based algorthms n mage segmentaton.. Pxon-based methods. Tradtonal Pxon-Based method (TPB) The TPB method s known as one of the smplest pxon-based approaches appled for mage segmentaton. The method s manly composed of two followng steps: () form the pxons, and () segment the mage.

3 496 Image Segmentaton.. Descrpton of pxon model In any pxon defnton scheme, the ablty to control the number of degrees of freedom used to model the mage s the key aspect. In other word, the pxon defnton scheme should yeld an optmum scale descrpton of the observed mage. The pxon defnton scheme whch s used n ths method can be descrbed as follows: m IP = P () j j= where IP s the pxon-based mage model; m s the number of pxons; P j s a gven pxon, whch s made up of a set of connected pxels, a sngle pxel or even a sub-pxel. The mean value of the connected pxels makng up of the pxon s defned as the pxon ntensty. Both the shape and sze of each pxon vary accordng to the observed mage. After the pxonbased mage model s defned, the mage segmentaton problem s transformed nto a problem of labelng pxons. The procedure to determne the set of pxons,. e. ther shape and sze, can be dvded nto three steps: ) obtan a pseudo mage, whch has at least the same resoluton as the observed mage; ) use an ansotropc dffuson flter to form the pxons; and 3) use a smple herarchcal clusterng algorthm to extract the pxons. Obtanng the Pseudo Image; The pseudo mage s a basc mage to form the pxons and to obtan a segmented mage, whch s derved from the observed mage. Suppose the dmenson of the observed mage s D M D N, then the dmenson of the pseudo mage can be ld M ld N, where l = n. When n = 0, the pseudo mage s the observed mage tself. When n, the pseudo mage can be obtaned by the followng teratve process n j I n,, j s even j j+ I n,,, + I n s even j s odd (, j) = () j + j I n, I n, sodd, jseven + j j j j, + I n, + I n, + I n,, j are odd 4 I n M Where = 0,,, D M and j = 0,,, D N. In the above teratve process, I corresponds to the observed mage. The essence of the process s ncreasng the resoluton through nterpolaton to descrbe the mage parts, whch have a lot of detals. Parameter n s of great mportance. If n, then the resoluton of the pseudo mage s larger than the orgnal mage and the fnally pxons formed are probable to be a sub-pxel. So, t determnes the smallest sze of the pxons. In the mage parts, where the ntenstes of nearng pxels are smlar, whch means havng lttle nformaton, the ntensty of newly nserted pxels wll be smlar wth the ntenstes of the pxels n the observed mage, from whch the new pxels are obtaned through nterpolaton. So there s a lttle dfference whether the pxons are derved from the orgnal observed mage or nterpolated pseudo mage. However, n the mage parts, where have a lot of N

4 Pxon-Based Image Segmentaton 497 detals, t wll be better to derve the pxons from the nterpolated pseudo mage than from the orgnal observed mage. So, t s probable that a pxon s a sub-pxel to fully model the correspondng mage parts. Therefore, f the mage has many detals, t should be large, otherwse t should be small. In current mplementaton, we let n = 0. Formulaton of Pxon; To form the pxons based on the pseudo mage, let us consder the followng ansotropc dffuson equaton [Perona & Malk, 990]: Ixyt (,, ) Ixyt (,, ) Ixyt (,, ) Ixyt (,, ) Cxyt (,, ) = Cxyt (,, )( + ) + + t x y x x Ixyt (,, ) Cxyt (,, ) y y (3) where C(x,y,t)s the dffuson coeffcent, whch controls the dffuson strength. The partal dfferental equaton s used to model the heat dffuson process. In regons wth a large dffuson coeffcent, the temperature tends to be unform. Whle temperature dfferences wll be retaned n regons wth small dffuson coeffcents. We can vew the pseudo mage ntensty as the temperature of the temperature feld and the transformaton of the gradent as the dffuson coeffcent. The transformaton functon s (,, ) = c x y t I ( + ) k where K s a constant. To be convenent, the soluton of the dffuson equaton s called soluton mage. In the soluton mage, the ntensty of the regons havng less nformaton (havng fewer edges) wll tend to be unform and vce versa. So, the regons havng smlar ntensty n the soluton mage can be regarded as the pxons n our mage model. The dffuson equaton can be approxmately solved by the followng dscrete formulaton: where ( ) ( ) ( I x, y, t+δ t = I x, y, t +Δ t( d c + d c + d c + d c ) (5) n n s s e e w w (4) dn = I( x, y, t) I( x, y, t) cn = dn + ( ) k ds = Ixy (, +, t) Ixyt (,, ) cs = ds + ( ) k de = I( x+, y, t) I( x, y, t) ce = de + ( ) k dw = I( x, y, t) I( x, y, t) cw = dw + ( ) k (5)

5 498 Image Segmentaton To ensure the convergence of the above teraton process, the parameter Δ t should not be too large (here, Δ t = 0.5 ). Larger values of K ncrease the pxon sze. To descrbe the detals of the mage, K could not be too large (here K = 5 ) Extracton of the Pxons; After formng the pxons accordng to the pseudo mage, a segmentaton method s appled based on herarchcal clusterng to extract them. For ths purpose, ntally each pxel represents a cluster. Then the clusters are merged accordng to ther ntenstes and made greater pxons. The mean value of the connected pxels makng up of the pxon s defned as the pxon ntensty. Both the shape and sze of each pxon can vary accordng to the observed mage. To stop the algorthm, a threshold value, T, s assgned and the mergence process terates untl the dfference between ntenstes of two adjacent pxons would be smaller than the threshold value (here, T = 0). The pxon-based mage model s represented by a graph structure G = (Q,E), where Q s the fnte set of vertces of the graph and E s the set of edges of the graph (Fgure ). P P P 3 P P 4 P 3 P 4 P P 5 P 6 P 7 P 5 P 7 P 6 (a) Fg.. (a) Pxon model of mage, and (b) the correspondng graph structure After the pxon-based mage model s defned, the mage segmentaton problem s transformed nto a problem of labelng pxons. Whle the pxons are extracted, the mage s dvded nto a set of dsjont regons. The extracton of pxons can be consdered as a prmary segmentaton. In TPB method, to obtan the fnal segmented mage, the combnaton of pxons s contnued untl the end condton of process occurs. Ths condton s the number of segments n fnal segmentaton purpose.. MPB method Pxon-based mage segmentaton usng Markov Random Feld (MRF) model s presented by (Le Ln, et al 008). In ths method, frst an mage s expressed as a pxon-based model. As we sad before, pxons are a set of dsjont regons wth varable shape and sze. These pxons are combned wth ther attrbutes and adjacences construct a graph whch represents the observed mages. Then usng ths pxon-representaton, a Markov Random Feld (MRF) model s presented to segment the mages. In current procedure, a set of sgnfcant attrbutes of pxons and edges are ntroduced nto the pxon-representaton. These attrbutes are ntegrated nto the MRF model and the (b)

6 Pxon-Based Image Segmentaton 499 Bayesan framework to obtan a weghted pxon-based algorthm. Also, a Fast Quad Tree Combnaton (FQTC) algorthm s used to extract the good pxon representaton... Defnton of pxon representaton M Defnton. Let X = { X } be the set of all the mage pxels. A subset of X s a pxon f = n and only f all the pxels n t are connected. A pxon s then denoted by P = { X j}. j = An attrbute vector of the pxon s extracted from the observed mage = n b max mn μ σ P (,,,,, ) (6) where n s the number of pxels n P, b s the permeter of P, namely the length of the boundary between P and the other part of the observed mage, max, mn, and σ are the maxmum, mnmum, mean and varance of the observed mage ntenstes n P, respectvely. Let I(x j) denotes the mage ntensty on the pxel x j. The attrbutes of the pxon ntensty can be obtaned by j j= ( ε ) ( ε ) max = max I( X ) X P j j mn = mn I( X ) X P μ = Defnton. A set of pxons, = { } σ j j n IX ( )/ n n = (( IXj)) / n μ j= N P = P N =, s a pxon-representaton f and only f P P =, f j P j = X The above defnton shows that the pxon-representaton segments the mage nto a set of dsjont regons. A set of edges, E, can be acqured from these regons, { j, j, j } E= E P P PandP P areadjacent (9) (7) (8) where P and P j are adjacent f X k P and X j Pj, whch are neghborng pxels to each other n the mage. The strength of an edge can be defned as the length of the boundary between the two adjacent pxons, whch s denoted by b j, so b= b. An attrbute vector, j e j, s used to denote all the attrbutes of an edge. j

7 500 Image Segmentaton The pxons and edges, combned wth ther attrbute vectors, construct a graph, G { P, E} whch represents the observed mage, as shown n Fg.... Shortest pxon-representaton wth respect to a dscrmnant There are two trval pxon-representatons, P = { X} and P {{ x } x X} 0 =, =. The former takes all the mage pxels as one pxon; the latter takes each pxel as a pxon, whch s a lossless representaton. In order to represent the mage usng as few pxons as possble whle lmtng the representaton error, the shortest pxon-representaton wth respect to a dscrmnant s defned. (a) Fg.. An example of Pxon-Representaton. (a) The Pxon map, n whch the boundares between adjacent Pxons are shown; and (b) The correspondng graph, whch combnes the attrbute vectors of Pxons and edges to represent the observed mage. Defnton 3. A functon f( p) 0 of pxons s a pxon error functon f and only f ( ) = = { } f p 0, f P x, ( ) ( ) f P f P, f P P j j Defnton 4. For a gven pxon error functon, f (.), and a non-negatve constant, T, the nequalty, f. () T, defnes a pxon dscrmnant. Defnton 5. A pxon-representaton s called the shortest pxon representaton wth respect f. T, f ts number of pxons s least among all the pxon- to a gven dscrmnant, ( ) representaton satsfyng ( ) P P, f P T. In general, usng the pxon attrbute vector to descrbe the regon of the observed mage wll loss some nformaton, so a pxon error functon s used to denote the error between the pxon and the regon of the observed mage. In ths method error functon s defned as f P max mn f. T the shortest pxon-representaton ( ) =. Wth a gven dscrmnant ( ) use the least number of pxons to represent the mage, so we consder t the best pxonrepresentaton whose pxons errors do not exceed the threshold, T. (b) (0)

8 Pxon-Based Image Segmentaton Extracton of pxon-representaton The shortest pxon-representaton wth respect to a dscrmnant s not unque, as shown n Fg. 3. And t s hard to extract the shortest one from a large and complex mage. In ths secton, an approach to extract a GOOD pxon-representaton s presented, whch combnes the adjacent pxons of the lossless pxon-representaton, P = {{ x } x X}, teratvely, untl no pxons can be combned consderng the gven dscrmnant. The obtaned good pxonrepresentaton s dependent on the order of combnaton besdes the dscrmnant. (a) (b) (c) (d) Fg. 3. The non-unqueness of the Shortest Pxon-Representaton. (a) Observed mage, whose pxel ntenstes are among 00, 50, and 00; (b), (c) and (d) are three of ts shortest Pxonf P = max mn 50 s gven as a dscrmnant. The black lnes Representatons when ( ) overlappng on the mage are the boundares of Pxons...3. Combnaton of adjacent pxons The adjacent pxons n a pxon-representaton, G { P, E} =, can be combned to form a new pxon, denoted by P new = P P j, whose attrbute vector, P new, can be obtaned from P and P, j n = n + n new j b = b + b b new j j max = max( max, max ) new j mn = max( mn, mn ) new j μ = ( n μ + n μ )/ n new j j new σ σ μ σ μ μ new = n( + ) + nj( j + j )/ nnew new () where b j s the edge strength,.e. the length of the boundary between P and P. j It can be proved that P { P, Pj} + { P new} s stll a pxon-representaton. And the edge set of the new pxon-representaton can be obtaned from E by combnng the edges connectng the same two pxons after the pxon combnaton...3. Combnaton-based extracton of pxon-representaton Gven a dscrmnant, f. ( ) T Snce P {{ x } x X}, the edge error functon s defned as ( ) f E = f(p P ). E j j = satsfes all the dscrmnants, the shortest pxon-representaton

9 50 Image Segmentaton wth respect to f. ( ) T can be extracted by combnng the pxons of P, the lossless representaton, untl all error functon values of the edges are larger than T. In fact, the pxon-representaton obtaned by combnaton scheme may not always be the shortest, whch s dependent on the order of combnatons. However, t s a substtute to the shortest, for the number of pxons has been sharply cut down Fast Quad Tree Combnaton algorthm A fast Quad Tree combnaton algorthm s used to extract the shortest pxon-representaton here. Frstly, a QuadTree-based mult-resoluton pxon-representaton s constructed, as shown n Fg. 4. (a) (b) (c) (d) Fg. 4. The QuadTree-based mult-resoluton Pxon-Representaton. (a) Coarsest scale Pxon- Representaton whch uses the whole mage as one Pxon; (b), (c), (d) s the followed scale pxon-representaton, whch are obtaned by subdvdng each square of the coarser scale nto four equal squares. The square n the fnest scale only ncludes one pxel. Then a ntal pxon-representaton wth respect to f. ( ) T qt, T qt [0,T], s extracted by coarse-to-fne selectng a set of dsjont squares from the mult-resoluton pxonrepresentaton, whch satsfy f. ( ) Tqt. Fnally, the pxons connected by the edge wth the mnmal edge error are combned teratvely, untl the mnmal edge error s larger than T. If the mage regon s not a square whose edge length s the power of, the mult-resoluton pxon-representaton can be constructed as follows. Frstly, the mage s put nto a large enough square lke (a) n Fg. 4. For each scale, the pxon s then defned as the set of pxels fallng nto a square of ths scale; the squares ncludng no pxel are gnored. An example usng the fast Quad Tree combnaton algorthm s gven n Fg. 5, where the error functon s defned as ( ) f P = max mn ()..4 Image segmentaton based on pxon-representaton In ths method, a Markov random feld model-based mage segmentaton approach under Bayesan framework s used based on pxon-representaton. The nose model of the Bayesan framework n ths approach s based on the pxel ntensty...4. Bayesan framework Let I be the observed mage and S be the segmented mage. In the Bayesan segmentaton framework, the segmented mage s obtaned by maxmzng the posteror probablty,

10 Pxon-Based Image Segmentaton 503 (a) (b) (c) (d) Fg. 5. The fast QuadTree Combnaton Algorthm. (a) Observed mage (3689 Pxels); (b) Intal Pxon-Representaton (5 Pxons). (c) Fnal Pxon-Representaton (493 Pxons) after teratve Pxon combnaton; (d) The Pxon sze map of the fnal Pxon-Representaton, where the mage ntensty denotes the local Pxon sze. The green lnes n (b) and (c) are the boundares between adjacent Pxons where S * = arg maxp(s I) (3) s P( S I) ( PI S) P( S ). (4) We assume I = S + N, where N s ndependent Gaussan whte nose. Then the condtonal probablty s ( ) u ( I x ) P( I S) = exp( ) (5) K k k= x πσk σk K where K s the number of classes, K s the set of pxels segmented nto the Kth class, and u k s the ntensty mean of pxels n K. Let G= { P, E} be a pxon-representaton of I. Snce the characterstcs of pxels n each pxon are smlar, we assume that the pxels n one pxon wll be segmented nto the same class. So usng (7) and (5), we get The computaton of ( ) pxels. ( ) K P( I S) = exp K j ( I( xj ) uk ) k= P x P πσk σk ( μ u ) K n/ n n K + σ π σk k= P σ K [ ] = ( ) exp( ) K P I S s smplfed snce the number of pxons s far less than that of P S s the pror probablty. In ths method, the MRF model based on the pxonrepresentaton s adopted to defne the pror probablty dstrbuton as follows...4. MRF model based on pxon-representaton A neghborhood system of the graph, G { P, E} =, s defned as (6)

11 504 where ( ) { ( ) } Image Segmentaton NP = NP P P (7) ( ) { j j } NP = P e E, N (8) s the neghborhood of each pxon. Let Λ = {,, K} be the set of possble labels denotng the classes n the segmented mage and L = { l,, lk} be a famly of random varables where l Λ denotes the label of th pxon and N s the number of pxons. The segmented mage S can then be descrbed by the event, L = ω, snce we assume that the pxels n one pxon wll be segmented nto the same class. Let Ω be the set of all possble confguratons, Ω = { ω = ( ω,, ω N) ω Λ }. L s a MRF wth respect to the neghborhood, N(P), f P( L = ω) > 0, ω Ω (9) ( ) ( ) P l = ω l = ω, P P = P l = ω l = ω, P N(P ), P P and ω Ω (0) j j j j j j where P(.) and P(..) are the jont and condtonal probablty densty functons, respectvely. The confguratons of MRF obey a Gbbs dstrbuton [Hammersley & Clfford, 97] P( ω) = /Z.exp( U(ω)/T) () where Z s a normalzng constant and T s a constant called temperature. U(ω) s the energy functon, whch s a sum of clque potentals V(ω) on all possble clques,.e. In ths method, the set of clques s defned as where each pxon n G { P, E} U ( ω) V ( ω) c C c c = () { { } ( ), } C = c c = P N P P P (3) = defnes one clque. And the clque potental s defned by V = w P w b, b, b w P, P η ( ω) ( ) c c e j j p j j P N( P ) j (4) where η j s a bnary varable whch has the value f P and P j have the same label and the value 0 otherwse; wc P = n we b j, b, bj = b j /b s the normalzed edge weght; and wp P,Pj = / j s the pxon dstance weght that denotes the dfference of mage characterstcs between two pxons. s the clque weght; ( )

12 Pxon-Based Image Segmentaton 505 In all, the pror probablty s defned as..4.3 Optmzaton bj ηj PS ( ) = P( ω) = exp( n ) Z T b μ μ (5) P P P N( P ) j j From (3) and (4), the optmal segmented mage can be wrtten as * ( ) ( ) S = arg mn( ln P I S ln P S ). (6) Usng (6) and (5), the objectve functon s then obtaned, s ( ) μ u + σ [ ] bj ηj F( S) = F( ω) = [ n ( + ln σ ) + αn ] b μ μ K K k k= P σ K P N( P ) (7) K j j where α = /T s a weght of MRF model, whch denotes the tradeoff between the fdelty to the observed mage and the smoothness of the segmented mage. The constant term has been removed from the objectve functon. The class number K and the weght α are gven before optmzaton. The ntal segmented mage s obtaned usng Fuzzy C-Means (FCM) clusterng, and the ntal parameters of each class are estmated from the ntal segmented mage,.e. the means u k and varances σ k. Then the threshold T s computed, the value of T should not be too large, otherwse the pxon wll contan many pxels whch actually belong to two dfferent classes. So we usng follow emprcal functon: T = mn ( u u σ σ ) (8) 0 < k, K, j j j Fnally, the segmented mage and the parameters are optmzed, smultaneously. Let F(ω, I,new) denote the objectve functon value when the th label of ω s changed nto I,new and F(ω, I,new) denote F( ω, I,new) F(ω). The optmzaton s descrbed as follows. Intalze the number of classes K ; the total number of teraton NUM ; u,, ukand σ,, σ accordng to an ntal segmentaton, whch s obtaned usng FCM method; k compute the threshold T ; and the teraton ndex j = 0 ;. Extracton of pxon-representaton, then ntalze the pxon-based mage model: assgn a label to each pxon P, whch mnmzes the expresson u k P k l 3. Fnd the best label for each pxon, l,best, N, whch mnmzes F(ω, I,new). 4. Fnd the F(ω, I mn,best ), satsfyng ( mn,best ) (,best ) 5. If F( ω, Imn,best ) < 0 and j < NUM, go to step 4, otherwse stop teraton. F ω, I F ω, I, N 6. Update the best label of each pxon and re-estmate u k, σ k usng new ω. 7. j = j +, Go to step 3.

13 506 Image Segmentaton In fact, F( ω, I,best) can be calculated usng the correlatve terms wth the th label n F(x),.e. ( new, ) μ ul + σ nb j nb j j ηj, new F( ω, I, new) = n + ln σ lnew, + α + (9) σ l, ( ) new Pj N P b b j μ μ j.3 WPB method Pxon-based approach usng wavelet thresholdng s a recently developed mage segmentaton method [Hassanpour et al, 009]. In ths method, a wavelet thresholdng technque s ntally appled on the mage to reduce nose and to slghtly smooth the mage. Ths technque causes an mage not to be oversegmented when the pxon-based method s used. Indeed, the wavelet thresholdng, as a pre-processng step, elmnates the unnecessary detals of the mage and results n a fewer pxon number, faster performance and more robustness aganst unwanted envronmental noses. The mage s then consdered as a pxonal model wth a new structure. The obtaned mage s segmented usng the herarchcal clusterng method (Fuzzy C-Means algorthm)..3. Pre-Processng step As mentoned above, the wavelet thresholdng technque s used as a pre-processng step n order to smooth the mage. For ths purpose, by choosng an optmal wavelet level and an approprate mother wavelet, the mage s decomposed nto dfferent channels, namely lowlow, low-hgh, hgh-low and hgh-hgh (LL, LH, HL, HH respectvely) channels and ther coeffcents are extracted n each level. The decomposton process can be recursvely appled to the low frequency channel (LL) to generate decomposton at the next level. The sutable threshold s acheved usng one of the dfferent thresholdng methods and then detals coeffcents cut wth ths threshold. Then, nverse wavelet transform s performed and smoothed mage s reconstructed..3.. Wavelet thresholdng technque Thresholdng s a smple non-lnear technque whch operates on the wavelet coeffcents. In ths technque, each coeffcent s cut by comparng to a value as the threshold. The coeffcents whch are smaller than the threshold are set to zero and the others are kept or modfed by consderng the thresholdng method. Whereas the wavelet transform s good for energy compacton, the small coeffcents are consdered as nose and large coeffcents ndcate mportant sgnal features [Gupta & kaur, 00]. Therefore, these small coeffcents can be cut wth no effect on the sgnfcant features of the mage. Let X = {X,j,, j =, M} denotes the M M matrx of the orgnal mage. The two dmensonal orthogonal Dscrete Wavelet Transform (DWT) matrx and ts nverse are mpled by W and W, respectvely. After applyng the wavelet transform to the mage matrx X, ths matrx s subdvded nto four sub-bands namely LL, HL, LH and HH [Burrus et al., 998]. Whereas the LL channel possesses the man nformaton of the mage sgnal, we apply the hard or soft thresholdng technque to the other three sub-bands whch contan the detals coeffcents. The outcome matrx whch s produced after utlzng the thresholdng level s denoted as ˆL matrx. Fnally, the smoothed mage matrx can be obtaned as follows:

14 Pxon-Based Image Segmentaton 507 The bref descrpton of the hard thresholdng s as follows: γ ( Y ) ˆ X = W Lˆ (30) Y f Y > T = 0 otherrwse where Y s an arbtrary nput matrx, γ( Y ) s the hard thresholdng functon whch s appled on Y, and T ndcates the threshold value. Usng ths functon, all coeffcents less than the threshold are replaced wth zero and other coeffcents are kept unchanged. The soft thresholdng acts smlar to the hard one, except that n ths method the values above the threshold are reduced by the amount of the threshold. The followng equaton mples the soft thresholdng functon: η ( Y ) sgn( Y )( Y T) f Y > T = 0 otherrwse where Y s the arbtrary nput matrx, η( Y ) s the soft thresholdng functon and T ndcates the threshold value. The researchs ndcates that the soft thresholdng method s more desrable n comparson wth the hard one because of ts better vsual performance. The hard thresholdng method may cause some dscontnuous ponts n the mage and ths event may be a dscouragng factor for the performance of our segmentaton. Three methods are presented to calculate the threshold value, namely Vsushrnk, Bayesshrnk and Sureshrnk. The method Vsushrnk s based on applyng the unversal threshold [Donoho & Johnstone, 994]. Ths thresholdng s gven by σ logm where σ s standard devaton of nose and M s the number of pxels n the mage. Ths threshold does not adapt well wth dscontnutes n the mage. Sureshrnk s also a practcal wavelet procedure, but t uses a local threshold estmated adaptvely for each level [Jansen, 00]. The Bayesshrnk rule uses a Bayesan mathematcal framework for mages to derve subband-dependent thresholds. These thresholds are nearly optmal for soft thresholdng, because the wavelet coeffcents n each subband of a natural mage can be summarzed adequately by a Generalzed Gaussan Dstrbuton (GGD) [Chang et al., 000]..3.. Algorthm and results Our mplementatons on several dfferent types of mages show that "Daubeches" s one of the most sutable wavelet flters for ths purpose. An mage s decomposed, n our case, up to levels usng 8-tap Daubeches wavelet flter. The amount of the threshold s assgned by the Bayesshrnk rule and ths value may be dfferent for each mage. Ths algorthm can be expressed as follows. Frst mage s decomposed nto four dfferent channels, namely LL, LH, HL and HH. Then the soft thresholdng functon s appled on these channels, except on LL. Fnally the smoothed mage s reconstructed by nverse wavelet transform. Fgure 6 shows the result of applyng wavelet thresholdng on the Baboon mage. It can be nferred from ths fgure that the resulted mage has fewer dscontnutes than the orgnal mage and ts smoothng degree ncreased and wll be resulted n a fewer number of pxons. In order to obtan a better vew about pxonal mage, we ndcate the effect of pxon formng stage on an arbtrary mage. As llustrated n Fg. 7, the boundares between the adjacent pxons are sketched so that the mage segments are more proper. (3) (3)

15 508 Image Segmentaton (a) (b) Fg. 6. Result of applyng wavelet thresholdng technque on Baboon mage: (a) Orgnal mage, and (b) smoothed mage (a) (b) Fg. 7. The effect of applyng the pxon formng algorthm to the baboon mage: (a) The orgnal mage, (b) the output mage wth boundares between pxons.3. Image Segmentaton usng pxon method In ths approach the wavelet thresholdng technque s used as a pre-processng step to make the mage smoothed. Ths technque s appled on the wavelet transform coeffcents of mage usng the soft thresholdng functon. The output of pre-processng step s then used n the pxon formulaton stage. In TPB algorthm, after obtanng the pseudo mage, the

16 Pxon-Based Image Segmentaton 509 ansotropc dffuson equaton was used to form the pxons. In WPB algorthm, utlzng the wavelet thresholdng method as a pre-processng stage elmnates the necessty of usng the dffuson equatons. After formng and extractng the pxons, the Fuzzy C-Means (FCM) algorthm s used to segment the mage. The FCM algorthm s an teratve procedure descrbed n the followng [Fauz & Lews, 003]. Gven M nput data {x m ;m =,...,M}, the number of clusters C ( C < M), and the fuzzy weghtng exponent w, < w<, ntalze the fuzzy membershp functons u (0) c,m wth (0) c =,...,C and m =,...,M whch are the entry of a C M matrx U. The followng procedure s performed for teraton l =,,... :. Calculate the fuzzy cluster centers. Update (l) U wth product nduced norm. (l) = l v c wth c,m w,m M M w w c = c,m m c,m m= m= v (u ) x / (u ) C d uc,m = / ( ) where (d,m) = xm v and. s any nner d (l ) U + (l+ ) (l) 3. Compare U wth n a convenent matrx norm. If U U ε stop; otherwse return to step. The value of the weghtng exponent, w determnes the fuzzness of the clusterng decson. A smaller value of w,.e. w s close to unty, wll gve the zero/one hard decson membershp functon, and a larger w corresponds to a fuzzer output. Our expermental results suggest that w = s a good choce. Fgure 8 llustrates ths method block dagram. 3. Evaluaton of the pxon-based methods In ths secton the pxon-based mage segmentaton methods are appled on several standard mages and the results of these mplementatons are extracted. For ths purpose, commonly used mages such as baboon, pepper and cortex are selected and the performance of applyng the mentoned methods on them s compared. In order to evaluate these methods numercally, several experments have been carred out on dfferent standard mages and some crtera such as number of the pxons n mage, pxon to pxel rato, normalzed varance and computatonal tme are used whch are ntroduced n followng. 3. Measurements Computatonal tme; In most applcatons, the tme whch s consumed to perform algorthms s an mportant parameter to evaluate them. So, researchers always seek to decrease the computatonal tme. Number of pxons and pxon to pxel rato; As expressed prevously, after formng the pxons, the mage segmentaton problem transformed to labelng the pxons. So, decrement n the number of pxons and related pxon to pxel rato results n a decrement n computatonal tme. Certanly t should be noted that the detals of the mage do not elmnate n ths way. Varance and Normalzed Varance; One of the most mportant parameters used to evaluate the performance of mage segmentaton methods s the varance of each segment. The smaller value of ths parameter mples the more homogenety of the regon and consequently the better segmentaton results. Assume that after the segmentaton process,

17 50 Image Segmentaton Fg. 8. The block dagram of the proposed method the mages are dvded nto K segments wth dfferent average values whch we have called these segments as Classes. In addton to the typcal varance, the normalzed varance of each mage can be calculated. If N k and V(k) denotes the number of the pxels and the varance of each class respectvely, the normalzed varance of each mage can be determned as below: where and V V V = (33) V * K k= Nk V( k) = (34) N V k= ( ) M K ( I x, y ) = (35) N In the above equatons, k denotes the number of classes, I( x,y ) s the gray level ntensty, M and N are the averaged value and the number of pxels n each mage respectvely. 3. Expermental results In ths secton, results of applyng the TPB, MPB and WPB methods on several standard mages are consdered. Fgs. 9(a), 0(a) and (a) are the Baboon, Pepper and Cortex mages used n ths experment. Fgs. 9(b), 0(b), (b) and 9(c), 0(c), (c) show the segmentaton

18 Pxon-Based Image Segmentaton 5 results of TPB and PMB methods, respectvely. The segmentaton results of WPB method are llustrated n Fgs. 9(d), 0(d) and (d). As shown n these fgures, the homogenety of regons and the dscontnuty between adjacent regons, whch are two man crtera n mage segmentaton, are enhanced n WPB method. (a) (b) (c) (d) Fg. 9. Segmentaton results of the Baboon mage: (a) Orgnal mage, (b) TPB's method, (c) WPB's method, and (d) WPB's method In addton, several experments have been carred out on the dfferent mages and the average results are drawn n several tables. In Table, the number of pxons and the rato of Pxon-Pxel n the three methods are shown. As can be seen from ths table we can fnd that these parameters are decreased sgnfcantly n WPB method n comparson wth two other methods whch resulted from applyng wavelet thresholdng technque before formng pxons. Table shows the computatonal tme requred of the three methods (Intel(R) Core(TM) Duo CPU.0 GHz processor, wth MATLAB 7.4). By usng pxon concept wth

19 5 Image Segmentaton wavelet thresholdng technque n the WPB method, the computatonal cost s sharply reduced. Snce the MRF technque, because of ts complcated mathematcal equatons, s a tme consumng process, the MPB method expends much tme compared to TPB method. In ths experence, after the segmentaton process, the mages are dvded nto three segments or Classes. The varance and average of each class are lsted n Tables 3-5, for mentoned mages. In most cases, the varance values of the classes of dfferent mages n WPB method are smaller n comparson wth the other methods. In order to nvestgate the performance of methods more exact, the normalzed varance of each mage after applyng the (a) (b) (c) (d) Fg. 0. Segmentaton results of the Pepper mage: (a) Orgnal mage, (b) TPB's method, (c) MPB's method, and (d) WPB's method

20 Pxon-Based Image Segmentaton 53 (a) (b) (c) Fg.. Segmentaton results of the Cortex mage: (a) Orgnal mage, (b) TPB's method, (c) MPB's method, and (d) WPB's method three methods are calculated too. The normalzed varance results llustrated n the tables demonstrate that n the pxon-based approach whch used wavelet (WPB method), the amount of pxels n each cluster s closer to each other and the areas of mages are more homogenous. Images (Sze) Baboon (56 56) Pepper (56 56) Cortex (8 8) The number of pxels TPB's method The number of pxons MPB's method WPB's method (d) The rato between the number of pxons and pxels TPB's MPB's WPB's method method method % 3.4 % 9.79 % % 9.43 % 5.04 % % 0. % 9.3 % Table. Comparson of the number of pxons and the rato between the number of pxons and pxels, among the three methods

21 54 Image Segmentaton Images TPB's method (ms) MPB's method (ms) WPB's method(ms) Baboon Pepper Cortex Table. Comparson of the computatonal tme, between the three methods Method Parameter class class class 3 average TPB's varance method MPB's method WPB method Normalzed Varance average varance Normalzed Varance average varance Normalzed Varance 0.0 Table 3. Comparson of varance values of each class, for the three algorthms (Baboon). Method Parameter class class class 3 average TPB's varance method MPB's method WPB's method Normalzed Varance average varance Normalzed Varance 0.05 average varance Normalzed Varance 0.07 Table 4. Comparson of varance values of each class, for the three algorthms (Pepper). Method Parameter class class class 3 average TPB's method varance Normalzed Varance 0.03 average varance MPB's method WPB's method Normalzed Varance 0.09 average varance Normalzed Varance 0.00 Table 5. Comparson of varance values of each class, for the three algorthms (Cortex).

22 Pxon-Based Image Segmentaton Concluson Ths chapter provded an ntroducton to the pxon-based mage segmentaton methods. The pxon s a set of dsjont regons wth varable shapes and szes. Dfferent algorthms were ntroduced to form and extract the pxons. Pxon-based methods were dvded nto three classes: TPB method, whch used from the tradtonal pxon defnton to segment the mage; MPB method, whch combned the pxon concept and MRF to obtan the segmented mage; and WPB method, whch segmented the mage by a pxon-based approach utlzng the wavelet thresholdng algorthm. The chapter was concluded wth llustraton of expermental results of applyng these methods on dfferent standard mages. 5. References Andrey P. & Tarroux, P. (998). Unsupervsed segmentaton of Markov random feld modeled textured mages usng selectonst relaxaton, IEEE Trans. Pattern Anal. Machne Intell., vol. 0, pp Bonnet, N. ; Cutrona, J. & Herbn, M. (00). A no-threshold hstogram-based mage segmentaton method, Pattern Recognton, Volume 35, Issue 0, pp Burrus, C. S. ; Gopnath, R. A. & Guo, H. (998). Introducton to Wavelets and Wavelet Transforms, Prentce Hall,New Jersey. Comancu, D. & Meer, P. (00). Mean shft: a robust approach toward feature space analyss, IEEE Trans. Pattern Anal. Mach. Intell. 4 (5), pp. 8. Donoho, D. L. &. Johnstone, I.M. (994). Ideal spatal adaptaton va wavelet shrnkage, Bometrca, Vol. 8, pp Elfadel I. M. & Pcard, R. W. (994). Gbbs random felds, cooccurrences, and texture modelng, IEEE Trans. Pattern Anal. Machne Intell., vol. 6, pp Fauz M. F. A. & Lews, P. H. (003). A Fully Unsupervsed Texture Segmentaton Algorthm, Brtsh Machne Vson Conference 003, Norwch, UK. pp.0-06 Francsco de A.T. de Carvalho, (007). Fuzzy c-means clusterng methods for symbolc nterval data, PatternRecognton Letters, Volume 8, Issue 4, pp Gonzalez, R. C. & Woods, R.E. (004). Dgtal Image Processng, Prentce Hall, Gupta,S. & kaur, L. (00). Wavelet Based Image Compresson usng Daubeches Flters, 8th Natonal conference on communcatons, I.I.T. Bombay, pp Hassanpour, H & Yousefan, H. (00). A Pxon-Based Approach for Image Segmentaton Usng Wavelet Thresholdng Method, Internatonal Journal of Engneerng(IJE), Vol. 3, pp Jansen, M. (00). Nose Reducton by Wavelet Thresholdng, Sprnger Verlag New York Inc., Pages Chang, S. G. ; Yu, B. & Vetterl, M. (000). Adaptve Wavelet Thresholdng for mage Denosng and compresson, IEEE Trans. Image Processng, Vol.9, pp Kato, Z. ; Zeruba, J. & Berthod, M. (999). Unsupervsed parallel mage classfcaton usng Markovan models, Pattern Recognt., vol. 3, pp Km, I.Y. & Yang, H.S. (996). An ntegraton scheme for mage segmentaton and labelng based on Markov random feld model, IEEE Trans. Pattern Anal. Mach. ntell. Vol.8 No.. pp

23 56 Image Segmentaton Lakshmanan, S. & Dern, H. (989). Smultaneous parameter estmaton and segmentaton of Gbbs random felds usng smulated annealng, IEEE Trans. Pattern Anal. Machne Intell., vol., no. 8, pp Ln, L. ; Zhu, L. & Yang, F. & Jang, T. (008). A novel pxon-representaton for mage segmentaton based on Markov random feld, Image and Vson Computng journal of ELSEVIER, Vol.6, pp Papamchal, G.P. & Papamchal, D.P. (007). The k-means range algorthm for personalzed data clusterng n e-commerce, European Journal of Operatonal Research, Volume 77, Issue 3, pp Pña, R. K. & Pueter, R. C. (993). Bayesan mage reconstructon: The pxon and optmal mage modelng, P. A. S. P., vol. 05, pp Perona P. & Malk, J. (990). Scale-space flterng and edge detecton usng ansotropc dffuson, IEEE Trans. Pattern Anal. Machne Intell., Vol., No. 7, pp Puetter, R. C. (995). Pxon-based multresoluton mage reconstructon and the quantfcaton of pcture nformaton content, Int. J. Imag. Syst. Technol., vol. 6, pp Sh, J. & Malk, J. (000). Normalzed cuts and mage segmentaton, IEEE Trans. Pattern Anal. Mach. Intell. (8) pp Yang, F. & Jang, T. (003). Pxon-based mage segmentaton wth Markov random felds, IEEE Trans. Image Process. (), pp Zhu, S.C. & Yulle, A. (996). Regon competton: unfyng snakes, regon growng, and byes/mdl for mult-band mage segmentaton, IEEE Trans. Pattern Anal. Mach. Intell. 8 (9),pp

24 Image Segmentaton Edted by Dr. Pe-Gee Ho ISBN Hard cover, 538 pages Publsher InTech Publshed onlne 9, Aprl, 0 Publshed n prnt edton Aprl, 0 It was estmated that 80% of the nformaton receved by human s vsual. Image processng s evolvng fast and contnually. Durng the past 0 years, there has been a sgnfcant research ncrease n mage segmentaton. To study a specfc object n an mage, ts boundary can be hghlghted by an mage segmentaton procedure. The objectve of the mage segmentaton s to smplfy the representaton of pctures nto meanngful nformaton by parttonng nto mage regons. Image segmentaton s a technque to locate certan objects or boundares wthn an mage. There are many algorthms and technques have been developed to solve mage segmentaton problems, the research topcs n ths book such as level set, actve contour, AR tme seres mage modelng, Support Vector Machnes, Pxon based mage segmentatons, regon smlarty metrc based technque, statstcal ANN and JSEG algorthm were wrtten n detals. Ths book brngs together many dfferent aspects of the current research on several felds assocated to dgtal mage segmentaton. Four parts allowed gatherng the 7 chapters around the followng topcs: Survey of Image Segmentaton Algorthms, Image Segmentaton methods, Image Segmentaton Applcatons and Hardware Implementaton. The readers wll fnd the contents n ths book enjoyable and get many helpful deas and overvews on ther own study. How to reference In order to correctly reference ths scholarly work, feel free to copy and paste the followng: Hamd Hassanpour, Had Yousefan and Amn Zehtaban (0). Pxon-Based Image Segmentaton, Image Segmentaton, Dr. Pe-Gee Ho (Ed.), ISBN: , InTech, Avalable from: InTech Europe Unversty Campus STeP R Slavka Krautzeka 83/A 5000 Rjeka, Croata Phone: +385 (5) Fax: +385 (5) InTech Chna Unt 405, Offce Block, Hotel Equatoral Shangha No.65, Yan An Road (West), Shangha, 00040, Chna Phone: Fax: