image segmentation based on the poincaré map method.bak

946 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 3, MARCH 2012

Image Segmentation Based on thePoincaré Map Method

Delu Zeng, Zhiheng Zhou, and Shengli Xie, Senior Member, IEEE

Abstract—Active contour models (ACMs) integrated with var-ious kinds of external force fields to pull the contours to the exactboundaries have shown their powerful abilities in object segmen-tation. However, local minimum problems still exist within thesemodels, particularly the vector field’s “equilibrium issues.” Dif-ferent from traditional ACMs, within this paper, the task of ob-ject segmentation is achieved in a novel manner by the Poincarémap method in a defined vector field in view of dynamical sys-tems. An interpolated swirling and attracting flow (ISAF) vectorfield is first generated for the observed image. Then, the stateson the limit cycles of the ISAF are located by the convergence ofNewton–Raphson sequences on the given Poincaré sections. Mean-while, the periods of limit cycles are determined. Consequently,the objects’ boundaries are represented by integral equations withthe corresponding converged states and periods. Experiments andcomparisons with some traditional external force field methods aredone to exhibit the superiority of the proposed method in cases ofcomplex concave boundary segmentation, multiple-object segmen-tation, and initialization flexibility. In addition, it is more computa-tionally efficient than traditional ACMs by solving the problem insome lower dimensional subspace without using level-set methods.

Index Terms—Active contour, dynamical system, external forcefield, limit cycle, Newton–Raphson algorithm, Poincaré mapmethod, segmentation.

I. INTRODUCTION

S egmentation for objects of interest from the image data isan important task in computer vision. Within this area, ac-

tive contour models (ACMs) or snakes [1], [2] have becomepopular by exhibiting their powerful abilities in object detec-tion and shape representation within the past two decades.

Active contours are usually defined by the curve flows mainlyderived from minimizing some variational problems or energyfunctionals [1], [3], [5]–[10], [21], [27]. Generally, there aretwo types of ACMs, i.e., parametric active contours [1], [2]

Manuscript received October 26, 2009; revised August 03, 2011; acceptedSeptember 05, 2011. Date of publication September 15, 2011; date of currentversion February 17, 2012. This work was supported in part by the Basic Re-search Program of China (973 Program) under Grant 2010CB731800, in partby the National Natural Science Foundation of China under Grant 60804051,Grant 60974072, Grant 61003170, and Grant 61103121, in part by the Re-search Fund for the Doctoral Program of Higher Education of China underGrant 20090172120011, in part by the China Postdoctoral Science Foundationunder Grant 20100740923, in part by the Guangdong Natural Science Foun-dation under Grant s2011010001936, and in part by the Fundamental ResearchFunds for the Central Universities, South China University of Technology, underGrant 2009ZM0295. The Associate Editor coordinating the review of this man-uscript and approving it for publication was Prof. Thrasyvoulos N. Pappas.

D. Zeng and Z. Zhou are with South China University of Tech-nology, Guangzhou 510641, China (e-mail: [email protected];[email protected]).

S. Xie is with Guangdong University of Technology, Guangzhou 510006,China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TIP.2011.2168408

and geometric active contours (GACs) [6], [8], [9]. The formertype usually establishes an energy functional composed of in-ternal and external energy terms. It is not intrinsic and has dif-ficulty in determining and handling the topological changes ofevolving contours. The latter one constructs an intrinsic Rie-mannian metric-based functional. Then, the active contours aretreated as the zero crossings of level-set functions [4], [5]. It cantackle topological changes elegantly at the cost of more compu-tational costs.

To our understanding and also in light of some papers in thisarea [10]–[24], there are mainly three difficulties with the aboveactive contours. First, the evolution direction of the active con-tour should be predefined. Second, they are usually sensitive toinitialization, that is, they will slowly evolve and even wronglyconverge as the initial contour is far away from or cross the de-sired boundaries. Third, the active contour usually sticks intosome local minima of the energy functional and, thus, cannotprogress further into the deep concavities of the boundaries.

Within the many efforts [11]–[14], [19]–[22], [27] workingon the above difficulties, two types of methods are discussedhere. One type is region-based methods [25]–[27], where en-ergy optimization is not driven by boundary information but byregion information of the image. In particular, for the modelpresented in [27], it is inspired (or simplified) from the Mum-ford–Shah functional [25], and optimization is done using thestatistical information both inside and outside the active con-tour. Another type of method aims to pull the active contour tothe desired boundaries by some new external force fields. More-over, the tool of the vector field was adopted herein and provento be effective, particularly by gradient vector flow (GVF) [11],[28], generalized GVF (GGVF) [12], and vector field convo-lution (VFC) [13]. These vector fields are to take the role ofthe external force fields for the evolving active contour. ForGVF/GGVF methods, a smoothed vector field is generated bydiffusing the gradient of the edge map for an observed imageto the whole image domain to enlarge the capture range of thecontour. This way, the active contour will move across the localminima, where it is usually stuck. The intrinsic model proposedin [14] as an integration of GGVF and GAC has more freedom ininitialization and can deal with topological changes. Recently, aVFC method that has been proposed in [13] seems much moreremarkable, where the authors proposed an external force fieldby convolving the edge map with a user-defined vector fieldkernel. Compared with the classic GVF methods, it is more ro-bust to noise without boundary distortion by a Gaussian filter,and it has less computational costs.

These vector field methods, e.g., GVF/GGVF and VFC, gen-erally perform elegantly in pulling or pushing the active con-tours to the desired boundaries. However, it has been found out

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ZENG et al.: IMAGE SEGMENTATION BASED ON THE POINCARÉ MAP METHOD 947

recently [17], [19], [23] that there still exist some difficulties fortraditional vector field methods to apply to boundary extractionwith multiple objects and complex concavities: there usuallyexist some unpreventable conflict components in these vectorfields, which will stop the active contour from progressing fur-ther to locate the exact boundaries; and the initialization flexi-bility is still restricted to a certain degree, as it is noted that theinitialization stage was declared to be also significant by somepapers [17], [18], [33].

In this paper, an interpolated swirling and attracting flow(ISAF) field is generated by extending a so-called edge tangentflow (ETF) only with a nonzero value at the boundaries to thewhole image domain. It is a static vector field. Different fromtraditional vector fields, the components in this vector field nearthe boundary are not perpendicular but tangent to the boundary.Thus, in the proposed vector field, it is possible for evolutionto be carried out along the boundaries. Then, the proposedtime-invariant vector field is considered as the right-hand-sidevector-valued function of an autonomous dynamical system. Asa result, the segmentation problem is translated to the problemof the limit cycle location by applying the related theory indynamical systems.

Section II briefly reviews several ACMs and their integrationswith the classic external force fields and then discusses some is-sues of the vector field ACMs. Section III describes in detail theproposed method, including the generation of the ISAF field andhow to handle object segmentation in the framework of dynam-ical systems. Then, experimental analysis is given in Section IV,followed by Section V to conclude this paper.

II. ON CURVE EVOLUTION

Here, we briefly summarize several kinds of ACMs accom-panied with their integrations with external force fields in curveevolution and then point out some of their shortcomings. In whatfollows, the observed image is denoted by , whereis the pixel position, and the gradient magnitude of the image(edge map) is denoted by .

A. GVF Snakes

The classic parametric active contour or snake model devel-oped by Kass et al. [1] was obtained by minimizing the fol-lowing functional:

(1)where and are positive weighting parameters to control ten-sion and rigidity of planar curve , respec-tively, and is some external constraints imposed on thecurve.

By calculus of variations and gradient descent, this minimiza-tion was achieved by calculating the steady-state solution of thefollowing curve evolution flow:

(2)

The integrations of classic snakes and GVF or generalizedGVF (GGVF) were obtained by Xu and Prince [11], [12]. Theyproposed replacing in (2) with vector field

obtained by minimizing anotherfunctional, i.e.,

(3)or

(4)

where, within each functional, the first and second terms are forregularization and conformity, respectively; , , andare weights to govern the tradeoff between regularization andconformity. A minimum of this functional will bear the propertythat the value of the vector field approximates to whenis large and slowly varies when is small.

Consequently, with the minimum of (3) or of (4),the following GVF or GGVF curve flow [11], [12] was proposedfor boundary extraction:

(5)

B. VFC Snakes

The authors in [13] proposed another external force field, i.e.,VFC, by convolving the edge map with a vector field kernel.VFC snakes have large capture ranges and are more robust tonoise.

The VFC field was given by

(6)

where is the gradient magnitude of the image, and isthe vector field kernel alternatively defined by

, whereis a small positive number.

VFC snakes is similar to GVF/GGVF snakes just by replacingin (5) with .

C. GAC and GVF

The geodesic active contour (GAC) introduced in [6]–[8] isobtained by minimizing an intrinsic functional defined by theweighted length of the curve.

The corresponding objective functional is

(7)

where is a decreasing boundary stopping function that is pos-itive such that and as .

Due to the development of level-set methods [4], where theactive contours are implicitly represented by the zero-crossings



Fig. 1. Two traditional vector field methods fail to segment the object with a cross-shaped concave boundary within two different initializations (the first andsecond rows). From left to right of each initialization (row): Initial contour, result with the GVF snake, in the vector field view of GVF, result with the VFC snake,and in the vector field view of VFC.

of a higher dimensional function, topological changes or mul-tiple boundary extraction is elegantly handled. The level-set for-mulation [7] of the GAC flow to minimize (7) is

(8)

In [14], the integration of GVF with GAC in level-set formula-tion was proposed by

(9)

where is a GVF [11] or GGVF [12] field; is thelevel-set function; is a parameter to determine the con-tribution of the curve’s smoothness; and weighting function

is defined by sign, with as a scale factor.

D. Equilibrium Issues Within the Vector Field Active Contour

In this part, some shortcomings on the above active contourintegrated with vector fields are revealed.

In fact, after diffusing or extending the gradient of the edgemap to the whole image domain, all sets of continuous sinks inthe resulting vector field correspond to the object’s boundariesof the observed image. Only after the vector field has been gen-erated that we shall have the knowledge of the positions of theother equilibria, e.g., sources and saddles. Moreover, it is diffi-cult for us to determine the positions of these equilibria in theresulting vector field before we generate the vector fields. Con-sequently, as pointed out in [18], [19], and [23], conflict com-ponents may usually exist in the resulting vector field to preventthe active contour from further approaching the boundaries. Theabove problem is called the “equilibrium issue” of these vectorfield methods. As illustrated in Fig. 1, although the VFC snakecan progress a bit further into the boundary, neither GVF snakes

[11] nor VFC snakes [13] succeed in completely locating the ob-ject’s boundary with cross-shaped concavities in the two giveninitializations.

III. PROPOSED METHOD

Within this section, the proposed method is stated in detail.An ISAF is first proposed. Then, with the techniques of dynam-ical systems, the limit cycles corresponding to the observed ob-jects are located by the Newton–Raphson algorithm in the de-fined Poincaré section. Since the limit cycle theory, particularlythe Poincaré map method, is used in this paper, the objects to besegmented in the observed images are required to have closedboundaries.

A. ISAF Field

Inspired by the idea in [12], an ISAF field for the observedimage is generated in this part. It is composed of two com-ponents, namely, diffused ETF (DETF; swirling component)and diffused edge perpendicular flow (DEPF; attracting compo-nent). They are directly described here by the evolution equationform for simplicity.

First, DETF is givenby the steady-state solution of the following evolution equation:

(10)

where is defined as in (7), and . , which iscalled ETF, is defined by

sign (11)

In (11), is perpendicular to , i.e., .The ETF defined by (11) is a thin flow moving around the ob-jects’ boundaries with a fixed direction (see the third subfigurein Fig. 2). It has components with a nonzero vector value nearthe boundaries and with a zero vector value elsewhere. It seems

and have similar properties like , since withinthese two fields, the layers composed of the two neighboringsides of the boundaries are nonzero. However, the two swirlingdirections for the two sides of the object’s boundaries in



Fig. 2. Four kinds of vector fields of the image with a cross-shaped concave boundary. From left to right:�� ,�� , � , and DETF fields.

are reverse to each other (see the first subfigure in Fig. 2); for, these two swirling directions for the two sides are the

same, but these nonzero layers are too thick (see the secondsubfigure in Fig. 2) to cause distortion in later segmentation.For these reasons, neither of them is appropriate to define theproposed here. When is diffused by (10), a con-tinuous DETF field will be generated in the full image domain,with swirling components (eddies) around the objects’ bound-aries and nonzero components in homogeneous (flat) regions.The above analysis is further illustrated in Fig. 2. As shown inthe last subfigure in Fig. 2, the ETF field is diffused to spread tothe whole image domain, and there exists an eddy exactly at thelocation of the object’s boundary. Consequently, any particle inthe image domain would move to the boundary along the dif-fused streamlines. Then, the particle would finally rotate alongthe object’s boundary round and round. This is a key feature dif-ferent from the traditional vector field, e.g., GVF and VFC.

However, in DETF, the degree of swirl (rotation) for thestreamlines away from the boundary is too high; thus, thejourneys are so long for the streamlines start away from theboundary to move to the boundary. In fact, we just need aflow field bearing the properties that its components awayfrom the boundary would point directly to the boundary andits components near the boundary are tangent to the boundarywith a fixed direction (either clockwise or counterclockwise).

Fortunately, it also reminds us of the properties of GGVF [12]and VFC [13]. They are called DEPF here. Alternatively, DEPF

is generated by the steady-statesolution of the following evolution equation GGVF [12]:

(12)

Finally, the proposed ISAF isgenerated by interpolating the above DETF and DEPF as fol-lows:

(13)

The ISAF field of the mentioned image with a cross-shapedconcave boundary is shown in Fig. 3. Different from the tradi-tional vector fields, i.e., GVF [11], GGVF [12], and VFC [13],in which the vector field components near the boundaries pointdirectly toward the boundaries, the proposed vector field willproduce stable periodical streamlines along the boundaries. Asa result, particles far away from the objects’ boundaries wouldbe guided directly to the boundaries and then move along theboundaries round and round. Fig. 4 shows the trajectory of a

Fig. 3. Example of the proposed ISAF field of the cross-shaped concaveboundary image.

Fig. 4. Trajectory of a particle moving in the ISAF field.

Fig. 5. Examples for active contour’s failure to extract the boundary if the ISAFfield is treated as the external force field. Left column: Two kinds of initializa-tion. Right column: Corresponding results.

particle moving in the ISAF field. The trajectory of the particlefrom the time it visits the boundary corresponds to the object’sboundary. We should note that the proposed ISAF cannot be theexternal force field for the classic ACMs as the swirling compo-nents may cause the active contour to evolve to disorders easily,e.g., in Fig. 5.



B. Within the Framework of Dynamical Systems

Within the above part, an ISAF field for the observed imageis first constructed, where its eddies correspond to the objects’boundaries. Naturally, the properties of the vector field remindus of the limit cycle theory in dynamical systems. As far aswe know, a dynamical system is usually defined by an array ofordinary differential equations, e.g., , ,where , , and is thevector field. In nonlinear dynamical systems, limit cycles areisolated closed periodical trajectories of the vector field in thestate space, where the neighboring trajectory is not closed andspiral toward or away from these cycles as time goes to infinity.Thus, they share common features with the above eddies. In fact,the above eddies can be considered as limit cycles of a vectorfield that belongs to some dynamical systems. Consequently,with the knowledge of dynamical systems [30], the limit cyclesare located, including their periods and candidate points on thelimit cycles.

In particular, ISAF corre-sponds to the following autonomous dynamical system:

(14)

For convenience, it is written into the following vectorialform:

(15)

where .When an initial state is given, then the pre-

vious system is an initial-value problem (IVP), i.e.,

(16)

The integral expression of the above IVP is

(17)

Then, is the trajectory in the state space with the initialstate at and time .

Consequently, the problem of finding the point on the limitcycle is equal to locating the zeros of

(18)

where is a candidate for the period of the limit cycle. Notethat trivial solution is meaningless and beyond ourconsideration here.

Poincaré Map: In this part, the elementary knowledge of thePoincaré map is introduced.

In the state space of the dynamical system (15), every limitcycle (if there exists) has its basin of attraction, where all thestreamlines in the region flow to the limit cycle. In addition,within the state space, its Poincaré section, which is denotedby here, can be defined as a lower dimensional hyperplanetransversal to the flow of the system. In particular, it should in-tersect with its corresponding limit cycle. The “correspondinglimit cycle” means the limit cycle, the basin of attraction for

Fig. 6. Poincaré section and positive Poincaré map.

which the Poincaré section lies in. Specifically, if illustratedin Fig. 6, the section passing through can be defined by

, where is the normal of . ThePoincaré map, which is denoted by , is a diffeomorphism de-fined by , , where is the time fortrajectory emanating from to cross again to somedirection determined by normal . According to the direction towhich crosses again, the one-sided Poincaré map canbe defined. For example and also for the purpose of the proposedalgorithm, the positive-side Poincaré map, which is named ,is defined by s.t. both and

are positive. Note that we still use to denotethe first time for to come back and cross the section tothe direction such that both and arepositive.

Then, a Poincaré section for the vector field ISAF is definedin the following context.

Equivalently, its normal needs to be defined. For an initialstate , normal is defined in such a waythat

(19)

where ; and are the corresponding complexnumbers for and when the first and second compo-nents of the vector are treated as the real and imaginary parts,respectively; and Arg , with Argas the operation for the principal value of the argument. In fact,

is a deviation angle that is nonzero to make the future iterationin (29) more stable [justified in (33) and (32)]. By elementarycomputations, the normal is equivalently written into

(20)Thus, the Poincaré section is described as

(21)

Obviously, the previously defined Poincaré section will betransversal to the flow, particularly to the limit cycles moreeasily. To make the above argument clearer, an example isgiven in Fig. 7 to illustrate the proposed Poincaré section onthe above image with a cross concave boundary. However, inFig. 7, only the ISAF component near the boundary is shown,



Fig. 7. Example of the proposed Poincaré section, where the component awayfrom the boundary is screened.

whereas the component away from the boundary is screened tozero to draw the proposed section clearer.

Consequently, we have the corresponding Poincaré map de-fined as well when its section is defined and fixed in the aboveway. Then, for an initial state , we further have the fact thatperiod is a function of , which is denoted by withoutloss of generality. Eventually, period will be implicitly com-puted, and (18) is written as

(22)

where , and .Zero Computation by the Newton–Raphson Algorithm: The

zeros of (22) are computed by the Newton–Raphson algo-rithm as in [29]. As it is pointed out in [29], if the variablesof and are independently treated in the iteration of theNewton–Raphson sequence, then a poor value of may easilycause to move far away from even if is on thelimit cycle. Thus, the iteration is usually unstable. In fact, oncethe Poincaré section is fixed, depends on [see (22)]. Then,the iteration will be done w.r.t. only and be more stable. Inparticular, this iteration is to run in the Poincaré section ,which is a lower dimensional subspace of the vector field.

Differentiating both sides of (22) w.r.t. , it yields

(23)

Via IVP (16), it further yields

(24)By Appendix A, we have

(25)

Substituting (25) into (24), it then yields

(26)Lemma: For IVP (16), if is continuously differentiable

within a connected open domain , then the solution of this IVP[in (17)], i.e., , is as well continuously differentiablew.r.t. , and

(27)

(The proof may appear in some literatures of dynamical sys-tems, e.g., [30]. However, it is restated in Appendix B for thereaders to read it conveniently.)

Investigating the generation process of the ISAF by (10), (12),and (13), we have the fact that and are obviously twicedifferentiable. Thus, the conditions of the above lemma are con-sidered to be guaranteed (approximately) for weighted resultant

.Then, with (26) and (27), it further yields

(28)where is defined by (20) and

Finally, an iterative sequence to converge to the zeros ofin (22) by the Newton–Raphson algorithm [31] is

constructed as follows:

(29)

where is implicitly computed (not by an iterativecomputation) at each iteration by directly searching when a tra-jectory starting at hits section to have , and theintegral in (28) is approximated by a summation.

C. Multiple-Object Segmentation

In this part, without using the level-set method [4], [9], theproposed method is applied to multiple-object segmentation.

Concretely, the corresponding ISAF field for an observedimage is constructed first. Then, distinct initial states, e.g.,

, are randomly placed in the image domain. Theyroughly correspond to the number of objects in the observedimage. However, usually, is chosen more than the number ofdesired objects. Then, these states will move along the flowto generate trajectories within the flow field, and finally, theywill move to the orbits along the objects’ boundaries round andround to finish the object location.



TABLE IEXPERIMENTAL DETAILS FOR THE FOUR SYNTHETIC IMAGES IN FIG. 8

For each of the initial states, the corresponding Poincarésections are defined. Then, with the Newton–Raphson sequenceconstructed in (29), we compute the fixed points of the corre-sponding proposed Poincaré maps and, at the same time, findthe corresponding periods implicitly.

In particular, with the iteration of (29), converged states onthe boundaries are denoted by s.t. as

, where . Furthermore, the correspondingperiods are denoted by , .

Consequently, the representation for the objects’ boundarieswith converged states is

(30)

where

(31)

It is possible that some converged states may be on the sameorbit to cause these orbits in the above representation to coin-cide. However, the corresponding periods are close to each otherin this case. Utilizing this fact, one may have a rough judge-ment of which orbits coincide. To be more precise, one maycompute the distance between the orbits with close periods, e.g.,

and , whereand .

IV. EXPERIMENTAL RESULTS

Here, the proposed method is applied to various observed im-ages to verify its performance. Because the digital images aredefined only on a discrete grid of integers, within these applica-tions, the iterations of states are done in continuous real spaceby linearly interpolating the obtained vector field (ISAF) to en-sure convergence, not by rounding off to integers at each itera-tion. In addition, the implementations of the proposed methodare given together with some comparisons with the traditionalmethods that were mentioned in the beginning of this paper.

A. Experiments on Synthetic Images

The proposed method is applied to four synthetic images (seeFig. 8), including the objects with a deep concave boundary,with a cross-shaped concave boundary, with four discs, and withnested boundaries. Within each experiment, the initial states(star-shaped points) are randomly placed in the image domain.

Fig. 8. Experiments with the proposed method on four synthetic images. Leftcolumn: Initial states (white star-shaped points) and converged states (whitesquare points) of Newton–Raphson sequences. Right column: Boundary rep-resentation.

Then, the iteration sequence generated by (29) are convergedto the converged states (white square points) on the limit cycles(boundaries). Meanwhile, the periods for every converged statesare found, which correspond to the boundary length of the ob-jects. In the end, with these converged states and the periods athand, the objects’ boundaries are represented one by one with(30). Furthermore, the corresponding experimental details arepresented in Table I. Naturally, it is found that the periods of theorbits that coincide are close.

In Fig. 9, another experiment is done on a synthetic fish withcomplex deep concave boundaries by drawing its ISAF fieldin detail. As shown in the subfigure of the ISAF, the swirling



Fig. 9. Experiments with the proposed method on a synthetic fish with complexconcave boundaries. From top to bottom, left to right: Observed image, ISAFfield, 12 initial states (white star-shaped points) and converged states (whitesquare points) of Newton–Raphson sequences, and boundary representation.

Fig. 10. The proposed method is applied to a four-island image for two initial-izations. Left column: Initial states (white star-shaped points) and convergedstates (white square points) of Newton–Raphson sequences. Right column:Boundary representation.

components are still immune from the deep concave boundaries,although the boundaries are more complex.

B. Experiments on Real Images

The proposed method is applied to more real images in thispart. First, it is applied to two aerial photos with two differentinitializations, e.g., four islands in Fig. 10 and a lake in Fig. 11,and later to three real medical images in Fig. 12. For the appli-cations in Fig. 10, the initial states are randomly placed in theimage domain. For the other experiments, the initial states aremanually placed in the basins of attraction of the correspondingboundaries, and then, the converged states are also located bythe iteration of (29). Consequently, the segmented boundariesare presented in Figs. 10 –12 (right columns). From these exper-iments, it is noted that the proposed method performs well to lo-cate multiple, concave, and inhomogeneous boundaries (see the

Fig. 11. The proposed method is applied to a lake image for two initializa-tions. Left column: Initial states (white star-shaped points) and converged states(white square points) of Newton–Raphson sequences. Right column: Boundaryrepresentation.

Fig. 12. The proposed method is applied to three medical images. Leftcolumn: Initial states (white star-shaped points) and converged states (whitesquare points) of Newton–Raphson sequences. Right column: Boundaryrepresentation.

third experiment in Fig. 12), where some parts of the bound-aries are weak. Moreover, the proposed method is applied tosome more real-life images in Fig. 13 to further validate itseffectiveness.

C. Initialization Sensitivity

In this part, some analysis on the sensitivity of the proposedmethod to initialization is given by the experiments in Fig. 14.There are five pieces of boundaries of the two spanners in theobserved image. The experiment is repeated by different num-bers of initial states randomly placed in the image domain. Eachinitial state belongs to only one basin of attraction, leading toone piece of boundary. In the first three applications, eight ran-domly placed initial states are given, and they all converged tothe corresponding pieces of boundaries and construct them. Toconstruct the holes inside the spanners, in the fourth application,50 initial states are randomly given. They converge and con-struct all of the five pieces of boundaries. Consequently, eachstate is able to construct the complete piece of boundary to givea meaningful result. In this sense, the proposed algorithm has



Fig. 13. The proposed method is applied to three more real images. Leftcolumn: Initial states (white star-shaped points) and converged states (whitesquare points) of Newton–Raphson sequences. Right column: Boundaryrepresentation.

good flexibility to initialization, and further justification will befound in the following part of comparisons with some traditionalmethods.

D. Comparisons to Some Traditional Vector Field Methods

To exhibit the superiority of the proposed method, somefamous traditional vector field integrated methods (some is inthe framework of level-set methods), including GGVF snakes[12], VFC snakes [13], GAC methods [8], and GGVFGACmethods [14], are applied to the two following images by twodifferent initializations for comparisons in Fig. 15. In the firstinitialization, where almost all the boundaries are surroundedby the initial curves, for the image of spanners, VFC snakesperform better than GGVF snakes; however, for the imageof a synthetic fish with very complex concave boundaries,they all poorly behave. In addition, these curves do not adapttheir topology to multiple-object segmentation. In the secondinitialization, where the initial contours cross the boundaries,both GGVF and VFC snakes poorly perform in these twoimages. For the spanner image, GAC, where the level-setmethod is utilized, shows a relatively good result in the firstinitialization but cannot locate the nested boundaries inside thespanners; it works worse in the second initialization, whereit fails to find any piece of complete boundaries. However,for the synthetic fish image, GAC still poorly performs forthe two initializations. It seems that different initializations ofthese three methods will give birth to different results, and veryoften, they poorly perform in these two images. To reduce theinitialization sensitivity, GGVFGAC, which combines GAC

with GGVF to generate a balloon force, can guide the contourto the boundaries from both sides. However, GGVFGAC stillfails to segment the boundaries in the given images sincethe application of the vector field will usually give birth tooverwhelmed power to stop the contour from splitting.

V. CONCLUSION AND DISCUSSIONS

In this paper, object segmentation is achieved in a novelmanner by the Poincaré map method in the field of dynamicalsystems. First, for an observed image, an ISAF vector field isproposed, where there exist swirling components (with fixeddirections) near the object’s boundaries. This is a key feature ofthe ISAF compared with the traditional vector field utilized inthe ACM method. These swirling components treated as limitcycles in view of dynamical systems correspond to the desiredobjects.

Then, the Poincaré section and the corresponding Poincarémap for the ISAF are defined. Consequently, given some ini-tial states in the vector field, they naturally belong to the basinsof attraction of the corresponding limit cycles. After that, theNewton–Raphson algorithm is utilized to locate the limit cyclesvia locating one point on each limit cycle. In the end, the objects’boundaries are represented by integral equations. Without usingthe time-consuming level-set methods like most of the ACMs,the proposed algorithm can achieve multiple-boundary extrac-tion by placing some initial states in the vector field. In addition,it runs more efficiently since the Newton–Raphson algorithm iscarried out in the Poincaré section—a lower dimensional sub-space of the image domain—while the traditional ACMs evolvethe contour in the whole image domain.

The proposed method elegantly performs in some localminima problems, e.g., multiple and complex concave bound-aries extraction, and it has much more flexibility in initializa-tions.

Although most traditional literatures tackle the local minimaproblem either by establishing new functionals or by evolvingcurves in new external force fields, this paper succeeds in rep-resenting the closed boundaries of the objects with the beautifultool of the Poincaré map method. However, we should admitthat the proposed method can only handle closed boundaries.For open-boundary extraction, the limit cycle theory may not beapplied. Moreover, there are still several further issues we mightcare about: 1) Different from most traditional ACMs where usu-ally the smoothness of the final curves to represent the bound-aries is required to be or even higher, the proposed methoddoes not explicitly incorporate regularization constraints intothe representation curves, which are determined by the Poincarémap method and are only . That may mostly accord withthe fact that the boundary is usually only continuous. 2) It maybe future work when the proposed method is done in a mul-tiscale framework to enhance robustness. 3) The extension ofthe proposed method to 3-D object segmentation may be pos-sible but more challenging when one needs to generate moredelicate ISAF fields in 3-D space. 4) User interaction, like mostimage-editing methods, can as well be added to refine the di-rection of the Poincaré section, in addition to placing the initialstates in the corresponding basins of attraction.



Fig. 14. The proposed method is run on the image of spanners among different random initializations. From the left column: The number of initial states in thefirst three applications is 8, whereas the number of initial states in the fourth application is 50.

Fig. 15. Comparisons with some traditional methods for two different initializations on two images of synthetic fish and real spanners. From the left column: Twoinitializations, GGVF snake result, VFC snake, GAC result, and GGVFGAC result.

APPENDIX ACOMPUTATION OF

Because from (22) we have , it equiva-lently means that . Differentiating w.r.t.

, it yields

(32)

By the definitions of (19) and (21), it holds that

(33)

Then, from (32), it yields

(34)

APPENDIX BPROOF OF THE LEMMA

First, we state a notation below. Given a vector-valuedfunction ,



, and an increase. A notation is given

by

(35)

where , andis the element of the matrix.

For the IVP in (16), given an initial state and an increase, it yields

(36)

By the notation in (35), it then yields

(37)

Since is continuously differentiable, by Taylor expansionat , one obtains

(38)

It as well implies that is Lipschitz in , and with thetheory of continuous dependence on initial conditions and pa-rameters [32], if , then

, and further,.

Let , then thefollowing integral equation is obtained:

(39)

When is treated as a parameter of the IVP in (16), thenis continuously dependent on [32]. In addition, by

investigating integral expression (17), is also continu-ously dependent on . In fact, is finite when iscontinuously differentiable, and it yields

, where is a constant. Consequently, the integrand in(39) is continuous w.r.t. and . Passing limit, it yields f

(40)

Let , then from (40), it yields

(41)

It corresponds to the following IVP:

(42)

Then, we have

(43)

ACKNOWLEDGMENT

The authors would like to thank all the anonymous reviewersfor their valuable comments.

REFERENCES

[1] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contourmodels,” Int. J. Comput. Vis., vol. 1, no. 4, pp. 321–331, 1987.

[2] D. Terzopoulos and K. Fleischer, “Deformable models,” Vis. Comput.,vol. 4, no. 6, pp. 306–331, 1988.

[3] M. Gastaud, M. Barlaud, and G. Aubert, “Combining shape prior andstatistical features for active contour segmentation,” IEEE Trans. Cir-cuits Syst. Video Technol., vol. 14, no. 5, pp. 726–734, May 2004.

[4] S. Osher and J. A. Sethian, “Fronts propagating with curvature-depen-dent speed: Algorithms based on Hamilton–Jacobi formulations,” J.Comput. Phys., vol. 79, no. 1, pp. 12–49, Nov. 1988.

[5] R. Malladi, J. A. Sethian, and B. C. Vemuri, “Shape modeling with frontpropagation: A level set approach,” IEEE Trans. Pattern Anal. Mach.Intell., vol. 17, no. 2, pp. 158–175, Feb. 1995.

[6] V. Caselles, F. Catte, T. Coll, and F. Dibos, “A geometric model foractive contours,” Numer. Math., vol. 66, no. 1, pp. 1–31, Oct. 1993.

[7] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi,“Gradient flows and geometric active contour models,” in Proc. IEEEInt. Conf. Comput. Vis., 1995, pp. 810–815.

[8] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,”Int. J. Comput. Vis., vol. 22, no. 1, pp. 61–79, Feb./Mar. 1997.

[9] N. Paragios and R. Deriche, “Geodesic active contours and level setsfor the detection and tracking of moving objects,” IEEE Trans. PatternAnal. Mach. Intell., vol. 22, no. 3, pp. 266–280, Mar. 2000.

[10] D. Adalsteinsson and J. A. Sethian, “A fast level set method for propa-gating interfaces,” J. Comput. Phys., vol. 118, no. 2, pp. 269–277, May1995.



[11] C. Xu and J. L. Prince, “Snake, shapes and gradient vector flow,” IEEETrans. Image Process., vol. 7, no. 3, pp. 359–369, Mar. 1998.

[12] C. Xu and J. L. Prince, “Generalized gradient vector flow externalforces for active contours,” Signal Process., vol. 71, no. 2, pp. 131–139,Dec. 1998.

[13] B. Li and S. T. Acton, “Active contour external force using vector fieldconvolution for image segmentation,” IEEE Trans. Image Process., vol.16, no. 8, pp. 2096–2106, Aug. 2007.

[14] N. Paragios, O. Mellina-Gottardo, and V. Ramesh, “Gradient vectorflow fast geometric active contours,” IEEE Trans. Pattern Anal. Mach.Intell., vol. 26, no. 3, pp. 402–407, Mar. 2004.

[15] G. Papandreou and P. Maragos, “Multigrid geometric active contourmodels,” IEEE Trans. Image Process., vol. 16, no. 1, pp. 229–240,2007.

[16] M. Sakalli, K. M. Lam, and H. Yan, “A faster converging snake algo-rithm to locate object boundaries,” IEEE Trans. Image Process., vol.15, no. 5, pp. 1182–1191, May 2006.

[17] C. Li, J. Liu, and M. Fox, “Segmentation of edge preserving gradientvector flow: An approach toward automatically initializing and splittingof snakes,” in Proc. IEEE CS Int. Conf. Comput. Vis. Pattern Recog.,2005, pp. 162–167.

[18] C. Li, J. Liu, and M. D. Fox, “Segmentation of external force field forautomatic initialization and splitting of snakes,” Pattern Recognit., vol.38, no. 11, pp. 1947–1960, Nov. 2005.

[19] X. Xie and M. Mirmehdi, “MAC: Magnetostatic active contour model,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 30, no. 4, pp. 632–646,Apr. 2008.

[20] L. D. Cohen and I. Cohen, “Finite-element methods for active contourmodels and balloons for 2-D and 3-D images,” IEEE Trans. PatternAnal. Mach. Intell., vol. 15, no. 11, pp. 1131–1147, Nov. 1993.

[21] L. D. Cohen and R. Kimmel, “Global minimum for active contourmodels: A minimum path approach,” Int. J. Comput. Vision, vol. 24,no. 1, pp. 57–78, Aug. 1997.

[22] G. Sundaramoorthi, A. Yezzi, and A. Mennucci, “Sobolev active con-tours,” Int. J. Comput. Vis., vol. 73, no. 3, pp. 345–366, Jul. 2007.

[23] S. L. Xie, D. L. Zeng, Z. H. Zhou, and J. Zhang, “Arranging and inter-polating sparse unorganized feature points with geodesic circular arc,”IEEE Trans. Image Process., vol. 18, no. 3, pp. 582–595, Mar. 2009.

[24] H. Li and A. Yezzi, “Local or global minima: Flexible dual-front activecontours,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 29, no. 1, pp.1–14, Jan. 2007.

[25] D. Mumford and J. Shah, “Optimal approximation by piecewisesmooth functions and associate variational problems,” Commun. PureAppl. Math., vol. 42, no. 5, pp. 577–685, Jul. 1989.

[26] S. Zhu and A. Yuille, “Region competition: unifying snakes, regiongrowing, and Bayes/MDL for multiband image segmentation,” IEEETrans. Pattern Anal. Mach. Intell., vol. 18, no. 9, pp. 884–900, Sep.1996.

[27] T. F. Chan and L. A. Vese, “Active contours without edges,” IEEETrans. Image Process., vol. 10, no. 2, pp. 266–277, Feb. 2001.

[28] J. Tang and S. T. Acton, “Vessel boundary tracking for intravitalmicroscopy via multiscale gradient vector flow snakes,” IEEE Trans.Biomed. Eng., vol. 51, no. 2, pp. 316–324, Feb. 2004.

[29] T. S. Parker and L. O. Chua, Practical Numerical Algorithms forChaotic Systems. New York: Springer-Verlag, 1989.

[30] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systemsand Linear Algebra. New York: Academic, 1974.

[31] P. Deuflhard, Newton Methods for Nonlinear Problems. Berlin, Ger-many: Springer-Verlag, 2004.

[32] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Pren-tice-Hall, Dec. 28, 2001.

[33] B. Li and S. T. Acton, “Automatic active model initialization viaPoisson inverse gradient,” IEEE Trans. Image Process., vol. 17, no. 8,pp. 1406–1420, Aug. 2008.

Delu Zeng received the B.S. and M.S. degrees in ap-plied mathematics and the Ph.D. degree in electronicand information engineering from South China Uni-versity of Technology, Guangzhou, China, in 2003,2005, and 2009, respectively.

His research interests include partial differentialequations, machine learning, and their applications inimage/video processing, i.e., image segmentation andobject tracking.

Zhiheng Zhou received the B.S. and M.S. degreesin applied mathematics and the Ph.D. degree inelectronic and information engineering from SouthChina University of Technology, Guangzhou, China,in 2000, 2002, and 2005, respectively.

He is currently an Associate Professor with SouthChina University of Technology. His research inter-ests include image processing and image and videotransmission.

Shengli Xie (M’01–SM’02) received the M.S.degree in mathematics from Central China NormalUniversity, Wuhan, China, in 1992 and the Ph.D.degree in control theory and applications from SouthChina University of Technology, Guangzhou, China,in 1997.

He is currently a Full Professor with GuangdongUniversity of Technology, Guangzhou, China. He isthe author or coauthor of two books and more than 70scientific papers in journals and conference proceed-ings. His research interests include automatic control

and blind signal processing.


image segmentation based on the poincaré map method.bak

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