Level set segmentation using image second order statistics

Transcription

1 Level set segmentaton usng mage secon orer statstcs Bo Ma, Yuwe Wu, Pe L Bejng Laboratory of Intellgent Informaton Technology, School of omputer Scence, Bejng Insttute of Technology (BIT), Bejng, P.R. hna, ABSTRAT Ths paper proposes a novel level set base mage segmentaton metho by use of mage secon statstcs an logarthmc Euclean metrc. Dfferent from prevous tensor base mage segmentaton approaches, the propose metho aopts covarance feature as regon-level escrptor rather than pxel-level one. On the bass of feature mage, we utlze secon orer statstcs of mage feature,.e., covarance matrx, to moel mage regon representaton, whch s of low menson, nvarant to unform llumnaton change, nsenstve to nose, an more mportantly prove a natural mechansm of ncorporatng fferent types of mage features by moelng ther correlatons. We moel mage segmentaton problem as one fnng the optmal segmentaton that maxmzes the covarance stance between foregroun regon an backgroun regon. Typcally, covarance matrces o not le on Euclean space. Our soluton to ths s to explot logarthmc Euclean stance as a metrc to compute the smlarty between two matrces. The expermental results show that covarance matrx as regon escrptor o form an effectve representaton for mage segmentaton problems, an the propose mage energy can be use to segment mages an extract object bounares relably an accurately. Keywors: mage segmentaton, covarance matrx, level set, actve contour 1. INTRODUTION Image segmentaton s one of funamental tasks n computer vson. It ams at separatng mage nto several perceptually homogeneous regons, whch facltates subsequent vsual analyss tasks, for example, veo cong, vsual trackng, an object recognton, etc.. For resent ecaes, a varety of algorthms have been propose by researchers n the fel. The level set metho ntrouce by Osher an Sethan [2] has been ncreasngly apple to mage segmentaton for the past ecae [4]-[9]. It s a means to mplctly propagate hyper-surfaces by evolvng an approprate hgher mensonal embeng functon, usually calle a level set functon. In the level set metho, contours are represente by the zero level set of a hgher mensonal functon. Ths metho owns many avantages over other methos, whch nclue numercal stablty, topology aaptablty an accurate contour extracton by combnng mage forces wth shape pror constrants. Due to these avantages, a large amount of actve contour moels have been evelope uner the level set framework for the mage segmentaton purpose. There are two major classes n these moels: regon-base moels [6],[10],[11] an ege-base moels [5],[12]. Among these moels, han-vese moel [6] an geoesc actve contour moel [5] are two typcal moels for two classes respectvely. Regon-base moels am to entfy the regon of nterest by a certan regon escrptor whch rves the evoluton of the actve contours. In some lteratures, the regon s often escrbe by known strbutons, or ntensty hstograms [13]. However, these features are not robust n the presence of nose whch may lea to erroneous segmentatons. Beses, algorthms usng these features are not effcent enough ue to ther hgh menson. Recently, covarance matrx as a regon escrptor has been evelope an utlze n vsual trackng an face recognton [1]. In [14] the Gabor-base regon covarance matrces escrptor s ntrouce. Both pxel locatons an Gabor coeffcents are employe to form the covarance matrxes. In ths paper, we propose a novel mage segmentaton algorthm uner level set framework that uses covarance matrx as regon level feature escrptor. Logarthmc Euclean stance [3] s aopte to measure the smlarty between fferent covarance snce covarance typcally oesn t rese n the Euclean space. Base on ths metrc, we propose an mage energy tem for mage segmentaton that maxmzes the ssmlarty between the covarance of foregroun area an that of backgroun. Dfferent from prevous tensor base segmentaton algorthm, there s no nee to extract tensor feature for each pxel, an covarance s regare as regon level feature rather than pxel level one. ompare

2 wth strbuton matchng metho, covarance usually has much smaller feature menson. Startng from varatonal approach, we erve the corresponng graent flow equaton for the propose segmentaton moel. The rest of ths paper s organze as follows. Secton 2 presents the new metho that uses covarance matrx regon escrptor to moel the energy functon for level set segmentaton an erve the graent flow equaton. Expermental results on synthetc an real mages are emonstrate n Secton 3, n whch the propose methoology prove to be avantageous over some exstng technques. In Secton 4 we en ths paper wth a bref concluson. 2. PROPOSED SEGMENTATION MODEL In ths paper, we use the mage secon orer statstcs an logarthmc Euclean metrc to segment mage. The propose metho aopts covarance feature as regon-level escrptor rather than pxel-level one, an then seeks optmal mage segmentaton that s acheve by maxmzng the covarance stance between the mage foregroun regon an the backgroun regon. 2.1 Segmentaton moel base on covarance matrx 2 Let be a boune open subset of R, an efne the evolvng curve n, as the bounary of the open subset of (.e., an ). In the level set metho [2], s represente by the zero level set of a level set functon : R, such that where 0(, xy) ( xy, ), (1) 0(, xy). The level set functon efnes the evolvng curve, an ( x, y) ( x, y) 0. For more etals, we refer the reaers to [2]. Let H be the Heavse functon efne n the stanar way as 1 f 0 H ( ). (2) 0 else Defne f as the W H mensonal feature mage extracte from an mage I of sze W H, where s a functon that extract mensonal mage features of I. f ( xy, ) ( Ixy,, ), (3) onsequently, the nternal regon an the external regon separate by the curve can be represente by covarance matrx of the feature ponts respectvely [1], whch are efne as an ( ) ( ) T H( )( f( x, y) ( ))( f( x, y) ( )) xy H( ) xy, (4) T (1 H( ))( f( x, y) ( ))( f( x, y) ( )) xy, (5) (1 H( )) xy

3 where ( ) H ( ) f ( x, y) xy H( ) xy an ( ) the canate object regon an the backgroun regon. (1 H ( )) f ( x, y) xy (1 H( )) xy In our metho for mage segmentaton, the functon (, I xy, ) s efne by represent the mean feature of. (6) 2 2 (, I xy, ) f(, xy) [, xyrgbi,,,, x, Iy, Ixx, Iyy, Ixy, Ix Iy ] If the mage s a gray level mage, (6) becomes. (7) 2 2 (, I xy, ) f(, xy) [, xyixy,(, ), Ix, Iy, Ixx, Iyy, Ixy, Ix Iy ] ovarance matrx, whch s a symmetrc postve efnte matrx, as a regon escrptor has several avantages. The covarance matrx proves a natural way of fusng multple features whch mght be correlate. The nose corruptng nvual samples are largely fltere out wth an average flter urng covarance computaton. In aton, compare wth hstogram escrptor, covarance usually has smaller menson. Our goal s to fn the regon such that ts covarance matrx has the maxmum stance wth the backgroun regon covarance matrx. However, the covarance matrces o not le on Euclean space. Therefore the Euclean stance s not approprate for measurng the smlarty or ssmlarty of the two gven covarance matrces. Arsgny et al. [3] propose the use of Log-Euclean stance to map the manfol of covarance matrces nto the Euclean space. In ths paper, the Log-Euclean metrc has been aopte to measure the stance between the two covarance matrces, an the mage energy moel can be efne as where s a Euclean norm on symmetrc matrces. ( ) st(, ) log ( ) log ( ), (8) In our actve contour moel apart from the above term we wll a an area-mnmzng term. The area-mnmzng term s wely use n level set segmentaton by researchers n practce, whch, as s well known, makes a curve evolve n a smooth manner, proucng the so calle balloon force. An the area term can be efne as Then, the energy functon becomes a ( ) H( ) xy. (9) ( ) ( ) ( ), (10) where 0 an 0 are fxe parameters, an controls the mage ata rven force, an controls the smoothness of the zero level set. 2.2 The ervaton of graent flow In orer to maxmze the energy functon (10), we use the stanar graent ascent metho by solvng the graent flow equaton as follows ( ) ( ) a( ) t a. (11) We frstly let log ( ) log ( ) ( ej ) 1,2, ; j1,2,, then the mage energy graent flow s

4 For (12), the rest of work s to ervate we have an ( ) 1 ej ej ( ) 1 ( ) 1 j1 1 j1 1 ( ) ( ) e ( ) 1 j1 ( ) an (log ( ) log ( )) ej ( ) 1 1 j ( ( ) ( ) ) j j. (12) ( ). Accorng to matrx analyss an varatonal approach, ( ) ( ) 1 ( ff T H ( ) ff T xy f T f T 2 T ) A A, (13) ( ) ( ) T 1 T T T T ( ff (1 H ( )) ff xy f f 2 ) A A, (14) where A an an A are the area of the foregroun an that of the backgroun respectvely. A H( ) xy, (15) The area energy graent flow s where () s the Drac elta functon, an ( x) H( x). x A (1 H( )) xy. (16) a ( ) ( ), (17) By substtutng (12) an (17) nto (11) an combnng the corresponng terms, we obtan the graent flow equaton. In numercal mplementaton, we use a smoothe Heavse functon H that approxmates H, whch s efne by 1 2 x H ( x) 1 arctan( ) 2, (18) where 1.5 n ths paper. Accorngly, the Drac elta functon (), whch s the ervatve of the Heavse functon H, s replace by the ervatve of H ' ( x) H( x) 1. (19) 2 2 x

5 3. EXPERIMENTAL RESULTS In ths secton, we perform several experments on both synthetc an real mages n orer to llustrate the effectveness of the propose algorthm. All the results are run on a 2.99GHz, core 2 uo P. In all the experments, we set the parameter t 1, an the contours are ntale manually aroun the objects n the mages. The mage energy tem an area tem are normalze. The mage to be segmente n Fg.1 s a mxture of two texture patterns. The curve evoluton process s epcte by showng the ntal contours (n the left), ntermeate contours (n the mle), an the fnal contours (n the rght) on the mages. In ths mage the foregroun an the backgroun ntensty strbuton are nearly the same, consequently, t s mpossble to segment ths mage base on gray level strbuton. By contrast, our metho can fuse fferent mage features, an moel ther correlatons, leang to goo segmentaton as a result. In Fg.2, we show that our metho can etect the bounary of the object of nterest n the mage ue to the avantage of the covarance escrptor. In the Soccer mage n Fg.3, the regon of nterest nclues a man an a ball. The regon changes topology as the curve evolvng. Wth the propose metho, we can see that the segment result s satsfe. The Surfer mage n Fg.4 shows changes n ntenstes wthn the foregroun, such as the change between the surfer s skn an pants. Fg.5 emonstrates the effectveness of the propose metho for the mage wth a lot of nose. In Fg.1 to Fg.5, we get accurate contours of objects n fferent scenaros that llustrate the propose metho s very promsng. Fg.1. Texture mage. The left, mle, an rght show the ntal contour (ntale manually), the ntermeate contour, an the fnal contour respectvely. ( =0.4, =0.05) Fg.2. Leaf mage. The left, mle, an rght show the ntal contour (ntale manually), the ntermeate contour, an the fnal contour respectvely. ( =1, =0.04) Fg.3. Soccer mage. The left, mle, an rght show the ntal contour (ntale manually), the ntermeate contour, an the fnal contour respectvely. ( =4, =0.05) Fg.4. Surfer mage. The left, mle, an rght show the ntal contour (ntale manually), the ntermeate contour, an the fnal contour respectvely. ( =4, =0.1)

6 Fg.5. Traffc mage. The left, mle, an rght show the ntal contour (ntale manually), the ntermeate contour, an the fnal contour respectvely. ( =1, =0.05) We compare the results of our metho wth the strbuton matchng base metho [13], whch shows the strength of the propose metho. In Fg.6, we show the comparson results for Tree mage n whch the backgroun s complex. We can see our metho can acheve better segmentaton result. In Fg.7, we present the segmentaton results for Hug mage. The Hug mage shows backgroun whch s multmoal. We can see our metho s capable of stoppng on the true object bounary, whle the strbuton matchng base metho takes a part of the backgroun as the object. Fg.6. Tree mage. The left shows the ntalzaton; the mle shows the segmentaton usng the strbuton base metho; the rght shows the segmentaton usng our metho. Fg.7. Hug mage. The left shows the ntalzaton; the mle shows the segmentaton usng the strbuton base metho; the rght shows the segmentaton usng our metho. 4. ONLUSION In ths paper, secon orer statstcs an logarthmc Euclean metrc has been aopte to get robust mage segmentaton uner level set framework. ovarance, for the frst tme, to the best of our knowlege, s utlze as regon-level escrptor for mage segmentaton. Although recently tensor has been use for level set base mage segmentaton, our current work ffers greatly from these n that we regar covarance as object or regonal representaton. The resultant graent flow s totally fferent from that usng tensor as pxel representaton n whch proxmty between pars of pxels s often requre to calculate, hence makng t prohbte to extract hgh mensonal features for every pxel. Snce covarance matrces o not le on Euclean space, we aopt logarthmc Euclean stance as a metrc to measure the smlarty between two matrces. Base upon ths metrc, we propose a novel mage energy moel for level set segmentaton, an erve ts graent flow equaton accorng to varatonal approach. Beses ths graent flow, we also ncorporate area-mnmzaton flow for better segmentaton. The experments of tests on both synthesze an real mages have proven the valty of our propose methos.

7 AKNOWLEDGEMENT Ths work was supporte n part by the Natonal Hgh-Tech R&D Program of hna uner Grant No. 2009AA01Z323, an the Natural Scence Founaton of hna (NSF) uner Grant No REFERENES [1] Tuzel, O., Porkl, F. an Meer, P., Regon ovarance: A Fast Descrptor for Detecton an lassfcaton, Proc. EV 3952, (2006) [2] Osher, S. an Sethan, J. A., Fronts propagatng wth curvature epenent spee: Algorthms base on hamltonjacob formulaton, J.omput. Phys. 79(1), (1988) [3] Arsgny, V., Fllar, P., Pennec, X. an Ayache, N., Log-euclean metrcs for fast an smple calculus on ffuson tensors, Magnetc Resonance n Mecne 56(2), (2006) [4] Paragos, N. an Derche, R., Geoesc actve regons an level set methos for moton estmaton an trackng, VIU 97(3), (2005) [5] aselles, V., Kmmel, R. an Sapro, G., Geoesc Actve ontours, Int. J. omput. Vs. 22(1), (1997) [6] han, T.F. an Vese, L.A., Actve contours wthout eges, IEEE Trans. Image Process. 10(2), (2001) [7] Zhao, H.K., han, T., Merrman, B. an Osher, S., A varatonal level set approach to multphase moton, J. omput. Phys. 127, (1996) [8] aselles, V., atte, F., oll, T. an Dbos, F., A geometrc moel for actve contours n mage processng, Numer. Math. 66 (1), 1 31 (1993) [9] L,., Huang, R., Dng, Z., Gatenby,., Metaxas, D. an Gore, J., A varatonal level set approach to segmentaton an bas correcton of mecal mages wth ntensty nhomogenety, Proc. Me. Image omput. omput. Assst. Interv. 5242, (2008) [10] Tsa, A., Yezz, A. an Wllsky, A. S., urve evoluton mplementaton of the Mumfor-Shah functonal for mage segmentaton, enosng, nterpolaton, an magnfcaton, IEEE Trans. Image Process. 10 (8), (2001) [11] Vese, L.A. an han, T.F., A multphase level set framework for mage segmentaton usng the Mumfor an Shah moel, Int. J. omput. Vs. 50 (3), (2002) [12] Vaslevsky, A. an Sq, K., Flux-maxmzng geometrc flows, IEEE Trans. Pattern Anal. Mach. Intell. 24 (12), (2002) [13] Mchalovch, O., Rath, Y. an Tannenbaum, A., Image segmentaton usng actve contours rven by the bhattacharyya graent flow, IEEE Trans. Image Process. 16 (11), (2007) [14] Pang, Y., Yuan, Y. an L, X., Gabor-Base Regon ovarance Matrces for Face Recognton, IEEE Trans. rcuts Syst. Veo Technol. 18(7), (2008)