An Image Segmentation Method Based on Partial Differential Equation Models

Size: px

Start display at page:

Download "An Image Segmentation Method Based on Partial Differential Equation Models"

Theresa Hood
5 years ago
Views:

1 An Image Segmentaton Method Based on Partal Dfferental Equaton Models Jang We, Lu Chan* College of Informaton Engneerng, Tarm Unversty, Alar, Chna *Correspondng Author Emal: Abstract In ths paper, we propose an mage segmentaton method based on a partal dfferental equaton model. Partal Dfferental Equatons and Fuzzy Image Segmentaton algorthms have become mportant felds n mage processng research. In an mage processng system, the segmentaton process s one of the most mportant steps. More precsely, mage segmentaton s defned as the process of assgnng a label to every pxel n an mage such that pxels wth the same label share certan vsual characterstcs. Several general-purpose algorthms and technques have been developed for mage segmentaton. In order to effectvely solve an mage segmentaton problem for a specfc problem doman, these technques often have to be combned wth knowledge domans, as there s no general soluton to mage segmentaton problems. Our results show that the performance of mage segmentaton can be mproved by usng partal dfferental equaton models. Keywords - Image segmentaton method; Partal dfferental equaton; fuzzy mage segmentaton. I. INTRODUCTION Image segmentaton s one of fundamental and mportant tasks n mage analyss and computer vson. Gven an mage, the segmentaton goal s to separate the mage doman nto dssmlar regons, each of whch has a consstent trat (ntensty, color or texture, etc.) throughout that s dfferent from other regons n the mage. Once a decson s made on the desred trat, varous methods are avalable to reach the segmentaton goal. Ths paper wll focus on varaton level set methods and partal dfferental equaton (PDE) methods for mage segmentaton. The basc dea behnd these methods s explaned as follows. The segmentaton problem s formulated n terms of mnmzng (or at least fndng crtcal ponts of) an energy functonal that takes a level set functon. The level set functons evolve accordng to an evoluton partal dfferental equaton (PDE), whch s derved from the mnmzaton of the energy functonal by calculatng the L~(ordnary) gradent of energy functonal and usng contnuous gradent descent method. A sgned level set method to solve the Mumford-Shah model. The Mumford-Shah model for mage segmentaton s a powerful and robust regon-based technque; however, the numercal method for solvng the Mumford-Shah model s dffcult to mplement. Peng [1] solved a partcular case of the Mumford-Shah model usng the curve evoluton and level set method for mage segmentaton, where the bnary case of two regons was consdered. As a result, a number of generalzatons have been developed to mprove both ts applcablty and effcency. However, the Zhang s [] method based on the tradtonal level set method has slghtly some ntrnsc lmtatons. The Drac functon has to be nvolved n the assocated gradent descent equaton when mnmzng wth respect to level set functon. The rentalzng procedure s qute complcated and expensve, and s fraught wth ts own problems, such as when and how to rentalze. The numercal approxmaton of the evoluton equaton has to utlze a complex sem-mplct scheme. In L s [3] paper, he presents a sgned level set method to solve the two-phase pecewse constant case of the Mumford-Shah model for mage segmentaton, pursung the mechansm of the tradtonal level set method. The proposed method avods some ntrnsc lmtatons of solvng methods n the tradtonal level set framework, and allows for more robustness to the locatons and szes of ntal contour and more computatonal effcency. Numercal results demonstrated that the proposed method s fast enough for near real-tme bmodal segmentaton applcatons whle stll retanng enough accuracy. Varatonal level set methods for mage segmentaton based on both L~andSobolev gradents Varatonal level set methods for mage segmentaton nvolve mnmzng energy functonal over a space of level set functons usng contnuous gradent descent method n J s paper [4]. The functonal ncludes the nternal energy (curve length, usually) for regularzaton and the external energy that algns the curves wth object boundares. Current practce s n general to mnmze the energy functonal by calculatng the L~gradent of the total energy. However, the gradent s partcularly effectve for mnmzng the curve length functonal by gradent descent method n that t produces the soluton n a sngle teraton. In ths paper, we thus propose to use the gradent for the nternal energy, whle stll usng L~gradent for the external energy. The test results show that the L~plus gradent scheme has much more computatonal effcency than the methods only based on L~gradent. Implct actve contour model wth local and global ntensty fttng energy ntensty n homogenetes often occur n real-world mages and may cause consderable dffcultes n mage segmentaton n Xu s paper [5]. To handle ntensty nhomogenety effcently, some localzed regon-based models have been proposed DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

2 recently. For example, De [6] et al. recently proposed a regon-scalable fttng (RSF) actve contour model. Very recently, Wang et al. [7] proposed a novel actve contour model drven by local mage fttng energy, whch also can handle ntensty nhomogenety effcently. However, these models easly get stuck n local mnmums for most of contour ntalzatons. Ths makes t need user nterventon to defne the ntal contours professonally. In ths study, Peng [8] propose a new actve contour model, whch ntegrates a local ntensty fttng (LIF) energy wth an auxlary global ntensty fttng (GIF) energy. II. METHODOLOGY AND FRAMEWORK OF IMAGE SEGMENTATION METHOD The LIF energy s responsble for attractng the contour toward object boundares and s domnant near object boundares, whle the GIF energy ncorporates global mage nformaton to mprove the robustness to ntalzaton of the contours. The proposed model can effcently handle ntensty nhomogenety, whle allowng for more flexble ntalzaton and mantanng the sub-pxel accuracy. The mplct actve contours have proved to be an effcent framework for mage segmentaton. Ths mplct model s derved from moton by mean curvature and uses the mage gradent to stop the evoluton process. a new formulaton of mplct actve contours based on mean curvature moton s useful for solvng ths problem. The basc process for mage segmentaton s shown n the followng fgure 1. Fgure 1. The basc process for mage segmentaton Image segmentaton, extractng the objects of nterest from mages, s a most fundamental and mportant problem n mage processng. It has always been a hot but a formdable task n an mage project. Recently, segmentaton methods based on partal dfferental equaton (PDE) have been wdely pad attenton by many researchers, due to ther varable form, flexble structure and excellent performance. The basc dea s to deform a curve, surface or mage accordng to a PDE wth ntal and boundary condtons, and obtan the desred segmentaton results as the soluton of the equaton. The evoluton PDE can be desgned drectly or ndrectly accordng to mage characterstc and user demand. However, the RSF model easly gets stuck n local a mnmum whch makes t senstve to the contours ntalzaton. Besdes, the RSF model s also senstve to hgh nose. To these ssues, we proposed an mprovement on the RSF model. Frst, the Gaussan kernel for the RSF energy s replaced wth a mollfyng kernel wth compact support. Second, the RSF energy s redefned as a weghted energy ntegral, where the weght s local entropy dervng from a grey level dstrbuton of mage. The total energy functonal s then ncorporated nto a varaton level set formulaton wth two extra nternal energy terms. The new RSF model not only handles better ntensty nhomogenety, but also allows for more flexble ntalzaton and more robustness to nose compared to the orgnal RSF model. Studyng on the ntalzaton problem of level set functon, we proposed an adaptve level set evoluton equaton startng wth a constant functon. For the segmentaton technque based on partal dfferental equatons, segmentaton can be regarded as a process of seekng the numercal soluton of a partal dfferental equaton wth ntal condton. Because the segmentaton results typcally depend on the selecton of ntal contours, most of the exstng methods need user nterventon to defne the ntal contours professonally. Ths means that they may be fraught wth the problems of how and where to defne the ntal contours. Up to now, t s stll a great challenge to fnd an effcent way to tackle the contour ntalzaton problem. Combnng the TV (Total Varaton) regularzaton, we proposed an adaptve level set evoluton equaton startng wth a constant functon. The formulaton s composed by an adaptve drvng force and a TV-based regularzng force. The adaptve drvng force makes the level set functon to have the opposte sgn along the edges at convergence and the regularzng force s used to smooth the level set functon. Due to the adaptve drvng force, the level set functon can be ntalzed to a constant functon, whch completely elmnates the need of ntal contours. Ths mples that the new formulaton s robust to ntalzaton or even free of manual ntalzaton. In addton, the evoluton PDE can be solved numercally va a smple explct fnte dfference scheme wth a sgnfcantly larger tme step. The proposed model s fast enough for near real-tme segmentaton applcatons whle stll retanng enough accuracy; n general, only a few teratons are needed to obtan segmentaton results accurately. III. THE BASIC ALGORITHM If the blurry edge, the strong nose and the ntensty nhomogenety appear n an mage, a tradtonal actve contour model fals to segment contours, especally, for a magnetc resonance mage and an ultrasound mage n medcne. Because of these reasons, we propose an actve contour based on local lnear fttng energes for a dfference mage. The actve contour model s solved by mnmzng ts energy functonal. The optmum local lnear fttng parameters n the model are obtaned. Contours for some mages wth ntensty nhomogenety are successfully extracted. Expermental results show that the method has the capacty of extractng weak edge and objects wth ntensty nhomogenety. Because only average ntensty nformaton s consdered n the local bnary fttng model, the model can successfully segment some magnetc resonance mages n medcne. However, the local bnary fttng model fals to segment an ultrasound mage wth a lot of nose that affects the DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

3 dstrbuton of ntensty. For extendng the applcaton feld of the local bnary fttng model, we propose an actve contour based on local ntenstes and local gradent fttng energy. By utlzng the level set method to solve, we successfully segment the weak edge n magnetc resonance mages and contours wth nose n ultrasound mages. Expermental results show that the proposed method has the capacty of ant-nose. The segmentaton accuracy s hgher than that of the local bnary fttng, local and global ntensty fttng models. An ultrasound mage has serous nose, and ts target edge s very weak. For solvng these problems, we propose an actve contour based on local ntensty and local Bhattacharyya dstance energy for mage segmentaton. Through usng the level set method, the weak edge successfully extracted. The proposed model weakens the nfluence of nose. Forward equatons were generated as the equaton (1): N q arg mn r ( q ) r, meas (1) q 1 A full body marker set consstng of N = 47 markers was defned to provde redundancy and robustness aganst occasonal marker dropout whch s nevtable n real-tme mage capture. After solvng (1), the estmated body pose s processed that outputs the smoothed pose q as well as the generalzed q and generalzed acceleratons q. In the nverse dynamcs processng step, a vectorc can be expressed as: c=m q q n q, q B q c ext () Where M s a square matrx, and n are terms related to Corols and centrfugal effects then we have: Nterms NDOF Ej lq n q (3) 1 j1 The k s computed analytcally by partal dfferentaton: l q Nterms NDOF Ej kj k k dqk 1 j1 d ne q So we can get the equaton (5): v q q T dl l dq k dt k qk dt (4) d q (5) The fnal processng step performed statc optmzaton s formulated as a quadratc programmng problem: N m F F= arg mn V, F 1 F max, DqF n subject to F AUC (6) n f 1 (7) 1 f The formula generates labels for each fle block. for( j ; j n 1; j ); { Wj r*( j1); T (8) c [ hw ( j)* mj] mod N}; So we can get the followng equaton (8): ( ) 1 b aib f() t f()( t dt) (1 ) a (9) jn1 1 lm f( tj)( tj) (1 ) t j Equaton (8) can be converted nto the followng form: f( y, ) f ( y, ) S( y y, ) s 1 ( y y, ) L F(y, )d y+ 1 g (1) s Jf (y, )dy Image segmentaton s a vtal mage processng technque n computer vson and mage analyss. So far, there are many ways for mage segmentaton. Among them the actve contour model based on varaton method and level set method s one of mportant mage segmentaton methods. It has emboded the superorty of partal dfferental equaton n mage segmentaton. It utlzes the dea of dynamc evoluton. Researchng actve contour model s mportant for mage segmentaton. Image segmentaton technology research based on partal dfferental equatons can promote multdscplnary cross fuson. Moreover, the flexble numercal computatonal method has better stablty durng the dscretzaton of evaluatve partal dfferental equaton, and t can meet the demand n hgh qualty mage restoraton and accurate mage segmentaton, and so on. In whch, S s cylnder cross secton, y ( x1, x), and 1 g( y y, ) kdk ( ) k ( k 1, k) (11) g( k, )exp( k (y y )) d Suppose k3, g ( k, ) can be obtaned from Equaton (8). Owng to ntroducng a shrnkage velocty term n the geodesc actve contour model, the constant velocty s set n advance. If t s chosen too large, the oversegmented result may be obtaned. If t s very small, the model fals to segment correctly deep concave boundares. In addton, whle there are multple object boundares, the method stll fals to extract all boundares. Because of these problems, a geodesc actve contour model ncludng gradent error control s used n ths model. An error functon term about gradent norm s ntroduced nto the geodesc actve contour model. For such knd of materal, general form of equaton (1) s expressed as followng equaton (1-14): DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

1 G ( k, ) [ k k k 1 1 kk k mm ] k k k k 1 1 1 e 15 k ( k, ) 11 k 11 k 1 e 15 ( k, ) m 11 k (1) g (13) In whch,, C 11 C 66 ( e15) C44 C44 11 C 44, (14) (15) A shrnkage velocty term n the geodesc

Addtonally, the geodesc actve contour model fals to segment contour whose edge s weak and blurry.

4 1 G ( k, ) [ k k k 1 1 kk k mm ] k k k k e 15 k ( k, ) 11 k 11 k 1 e 15 ( k, ) m 11 k (1) g (13) In whch,, C 11 C 66 ( e15) C44 C44 11 C 44, (14) (15) A shrnkage velocty term n the geodesc actve contour model s ntroduced. The model can segment deep concave contours. But shrnkage velocty s specfed before segmentaton. If t s set dfferently, the segmented results are dfferent. Addtonally, the geodesc actve contour model fals to segment contour whose edge s weak and blurry. Because of these problems, we propose a local adaptve parameter settng method for parameters automatze settng. It ntegrates local spatal ponts dstance and local ntensty nformaton nto the geodesc actve contour model, the orgnal geodesc actve contour model s mproved. Automatc setup parameters n the method can be acheved. It shows that the model enhances the segmentaton accuracy, and realzes the segmentaton for blurry boundares. n PC ( X) Px ( ) k 1 k C (16) PC ( X) PX ( C) PC ( ) k k k n PC ( ) PX ( C) k k 1 (17) IV. RESULTS AND DISCUSSION In the framework of level set method, we developed a nonlnear dffuson equaton drectly for mage segmentaton nonlnear dffuson equatons have receved a lot of attenton n the area of mage analyss and computer vson. However, segmentaton s ntegrated nto smoothng process whch generates a pecewse constant approxmaton to the geometrcal descrpton of mage. So, segmentaton results rely closely on the performance of smoothng. Besdes, a smoothng algorthm wth good performance n preservng mage features (edges) usually needs to desgn ntrcate dffuson term, whch may ntroduce complex computaton that may lead to the neffcency of the whole process. The pre-dsposal process for an mage s shown n the fgure. Fgure. The pre-dsposal process Based on the mechansm of nonlnear dffuson, we develop a nonlnear dffuson equaton (wth the ntal and boundary condtons drectly for mage segmentaton. The zero contour lne of level set functon startng wth a zero functon can be smoothly generated, and quckly come to a steady state whch separates object from ts background. Ths work consttutes a framework for further nvestgatons on nonlnear dffuson equatons drectly for segmentaton. The segmentaton results on the synthetc mage whch s corrupted by Gaussan nose are shown n Fg. 3. Fgure 4 shows a case study result of the whole process. Fgure 3. Segmentaton of the synthetc mage corrupted by Gaussan nose (,.1) (a) nosy mage; (b) FCM_S1; (c) FCM_S; (d) GIFP_FCM; (e) RFCM_SSI. DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

Fgure 4. A case study result of the whole process The dffuson term n our equaton s response for the smoothness of the level set functon durng the evoluton.

Ths mples that our model s robust to ntalzaton or even free of manual ntalzaton. The proposed model has four man advantages: Frst, t doesn t use mage gradent to stop the evoluton process.

Thrd, the zero-level lne of level set functon startng wth such bnary functon fnally comes to a unque steady state, thus t allows settng a termnaton crteron on the algorthm by determnng the bnary

5 Fgure 4. A case study result of the whole process The dffuson term n our equaton s response for the smoothness of the level set functon durng the evoluton. The source term s used for dentfyng object and ts background wth source and snk. The level set functon can be ntalzed to any bounded functon, e.g., a zero functon, whch completely elmnates the need of ntal contours. Ths mples that our model s robust to ntalzaton or even free of manual ntalzaton. The proposed model has four man advantages: Frst, t doesn t use mage gradent to stop the evoluton process. Second, t allows robustness to ntalzaton or even s free of manual ntalzaton snce the level set functon can be ntalzed to a bnary functon that contans both postve and negatve values. Thrd, the zero-level lne of level set functon startng wth such bnary functon fnally comes to a unque steady state, thus t allows settng a termnaton crteron on the algorthm by determnng the bnary length of zero-level lne at each of teratons. Fourth, the evoluton PDE s easly resolved numercally by the use of the sem-mplct addtve operator splttng (AOS) scheme ntroduced to nonlnear dffuson flterng, whch remans numercally stable for a large tme step and so less teraton numbers are needed to converge to the steady state soluton. The proposed algorthm has been successfully appled to both synthetc and real mages wth homogeneous ntensty regons. V. CONCLUSIONS In ths paper, the author studes on the mage segmentaton method based on partal dfferental equaton. Currently, Partal Dfferental Equaton and Fuzzy Image Segmentaton algorthms have become mportant felds n mage processng research. Gven an mage, the segmentaton goal s to separate the mage doman nto dssmlar regons, each of whch has a consstent trat (ntensty, color or texture, etc.) throughout that s dfferent from other regons n the mage. Expermental results show that the proposed method has the capacty of ant-nose. The segmentaton accuracy s hgher than that of the local bnary fttng, local and global ntensty fttng models. An ultrasound mage has serous nose, and ts target edge s very weak. For solvng these problems, we propose an actve contour based on local ntensty and local Bhattacharyya dstance energy for mage segmentaton. Once a decson s made on the desred trat, varous methods are avalable to reach the segmentaton goal. In mage processng system, segmentaton process s one of the most mportant steps. More precsely, mage segmentaton s defned as the process of assgnng a label to every pxel n an mage such that pxels wth the same label share certan vsual characterstcs. Several general-purpose algorthms and technques have been developed for mage segmentaton. In order to effectvely solve mage segmentaton problem for a specfc problem doman, these technques often have to be combned wth knowledge domans, as there s no general soluton to mage segmentaton problems. The result shows that the performance of the mage segmentaton can be mproved by usng partal dfferental equaton method. DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

6 REFERENCES [1] Bo Peng, Le Zhang, Davd Zhang. A survey of graph theoretcal approaches to mage segmentaton. Pattern Recognton, 13, pp [] Tong Zhang, Yong Xa, Davd Dagan Feng. Hdden Markov random feld model based bran MR mage segmentaton usng clonal selecton algorthm and Markov chan Monte Carlo method. Bomedcal Sgnal Processng and Control, 13, pp [3] Yangyang L, Shxa Feng, Xangrong Zhang, Lcheng Jao. SAR mage segmentaton based on quantum-nspred mult objectve evolutonary clusterng algorthm. Informaton Processng Letters, 14, pp [4] J Zexuan, Jnyao Lu, Guo Cao, Quansen Sun, Qang Chen. Robust Spatally Constraned Fuzzy C-Means Algorthm for Bran MR Image Segmentaton. Pattern Recognton, 14, pp [5] Yan Xu, Jun-Yan Zhu, Erc I-Chao Chang, Maode La, Zhuowen Tu. Weakly supervsed hstopathology cancer mage segmentaton and classfcaton. Medcal Image Analyss, 14, pp [6] Alng De, Chengan Guo. An adaptve vector quantzaton approach for mage segmentaton based on SOM network. Neurocomputng, 14, pp [7] Hongjun Wang, Ynghu Zhang, Ruhua Ne, Yan Yang, Bo Peng, Tanru L. Bayesan mage segmentaton fuson. Knowledge-Based Systems, 14, pp [8] Renbn Peng, Pramod K. Varshney. On performance lmts of mage segmentaton. Computer Vson and Image Understandng, 14, pp DOI 1.513/ IJSSST.a ISSN: x onlne, prnt

Active Contours/Snakes

Active Contours/Snakes Actve Contours/Snakes Erkut Erdem Acknowledgement: The sldes are adapted from the sldes prepared by K. Grauman of Unversty of Texas at Austn Fttng: Edges vs. boundares Edges useful sgnal to ndcate occludng