neighborhood aided implicit active contours

Neighborhood Aided Implicit Active Contours

Huafeng Liu1,2, Yunmei Chen3, Wufan Chen4, and Pengcheng Shi2

1State Key Laboratory of Modern Optical InstrumentationZhejiang University, Hangzhou, China

2Department of Electrical and Electronic EngineeringHong Kong University of Science and Technology, Clear Water Bay, Hong Kong3Department of Mathematics, University of Florida, Gainesville, Florida, USA

4Key Laboratory of Medical Image Processing, Southern Medical University, Guangzhou, China

Abstract

We have developed a geometric deformable model thatemploys neighborhood influence to achieve robust segmen-tation for noisy and broken edges. The fundamental powerof this strategy rests with the explicitly combination of re-gional inter-point constraints, image forces, and a prioriboundary information for each geometric contour pointwithin its adaptively determined local influence domain.This formulation thus naturally unifies the essences of thegeometric and parametric snakes through automatic localscale selection, and exhibits their respective fundamentalstrengths of allowing stable boundary detection when theedge information is weak and possibly discontinuous, whilemaintaining the abilities to handle topological changes dur-ing front evolution. In particular, this paper presents animplementation of the method through local integration ofthe level set function and the image/prior-driven evolutionforces, where the resulting partial differential equation issolved numerically using standard finite difference method.Experimental results on synthetic and real images demon-strate its superior performance.

1. IntroductionOne of the primary goals in computer vision has been

to robustly recover the shape of objects from images ofvarious qualities. Although there exist a large amount ofdedicated methods more or less suited for particular appli-cations, a strong and persistent interest has been the gen-eral paradigm based on the parametric and geometric de-formable models [1, 5, 6, 8]. The mathematical foundationsof these deformable models consist of two integral compo-nents, the geometry for shape representation and the math-ematics/physics for constructing energy functional.

Popular explicit shape representations include paramet-ric and meshed curves/surfaces/volumes. For example, theclassical snakes are explicitly represented as parameterizedcurves in Lagrangian formulations [5]. The evolution isthen obtained by minimizing an energy measure on the en-tire contour, including both internal and external energies.The principal disadvantage of these parametric active con-tours (PACs) is that they cannot easily deal with topologi-cal changes. That is, unless specifically designed [10], thetopology of the final object has to be the same as the ini-tial one, even if it is in conflict with the image information.A different class of approaches, where active contours areimplicitly represented as a level set of a higher-dimensionalscalar function [11], has been introduced to overcome thisproblem. These geometric active contours (GACs) have re-ceived a large amount of attention in recent years becausetheir ability to naturally handle topological complexity andvariability without any additional machinery [1, 8].

Most explicit representations use basis functions with fi-nite support, and thus, give the segmentation algorithms aneasy way to perform local control of the shape. In contrast,implicit representations often do not have obvious influenceon the shape position and thus the noise in the shape rep-resentations is difficult to deal with. A desired procedureshould be capable of handling complicated topology and ge-ometry as well as robust to noise. Recently, a hybrid shapemodeling scheme based on the notion of a pedal curve hasbeen proposed. In this work, a global prior is introducedusing a parameterized model, and the local properties arefine-tuned using a geometric flow [17].

In addition, in order to achieve robust segmentationagainst varying imaging conditions, an important issuewhen using active contour algorithms is the appropriate,and often task-specific, choice and design of an image/priordriven external energy functional. Otherwise, they will face

1

Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06) 0-7695-2597-0/06 $20.00 © 2006 IEEE

difficulties in handling weak edges/gap problems [1, 15]and be very sensitive to local minima in the noisy images[18] if improper external forces are used. Considerableprogress has already been made in the designing of newexternal force (or energy) terms for geometric active con-tours. An extra stopping term for pulling back the con-tour if it passes the boundary has been investigated in [1].A weighted area functional force has been introduced tohelp the snakes be more robust with respect to small gapproblem [15]. Diffused external data forces, such as thegradient vector flow (GVF), have also been adopted [19].And boundary and region information has been integratedunder a curve-based minimization framework [12]. Theselater approaches own the benefits provided by external forcepotential field in achieving a larger capture range and ro-bustness against the boundary leakage problem [12, 19].More recently, region-aided geometric snake, which inte-grates gradient flow forces with region vector flow forcesobtained through the diffusion of the region segmentationmap, has been developed and implemented within level setplatform [18]. These integrated forces give another way tobe more robust toward weak edges. Similar region-basedstrategies have been explored by other works as well [2, 20].However, real-world images are so different from each otherthat it is difficult to expect one of the aforementioned exter-nal energies to be able to segment efficiently all of them.

We realize that, for geometric active contours, more ro-bust results can be achieved if the behavior of any individualcontour point is constrained by both local edge informationof itself and that of its neighboring points. These inter-pointrelationships provide an expanded local view, and the globalview in the extreme case when all the contour points areconsidered, of the object boundaries and thus aid in the de-lineation of noisy, diffused, and broken edges.

In this paper, we present a geometric deformable modelwith external energy that makes uses of the image and priorinformation at the adaptively determined support domainaround each point of interest, thus effectively enlarges thecapture range of each point to have a better regional under-standing of the edge information within its local neighbor-hood. It provides adaptive smoothing/averaging regularitythrough local integration of the level set function, as well asincorporates inter-point relationships through local integra-tion of the external image/prior forces. As a result, we ef-fectively obtain a hybrid active contour which combines thegeometric deformable models with the internal energy termusually appeared in the parametric snakes. Experimentalresults on synthetic and real images 1.

1We appreciate the anonymous reviewer for pointing out that, follow-ing the similar spirit, the Sobolev active contour (SAC) has attempted tominimize the energies from image and the cost of perturbing the curve[16]. Our work differs in the sense that it does local weak form averagingon both the level set function and the flow, while the SAC does so only forthe flow itself

2. Neighborhood Aided Implicit Active Con-tours

2.1. Locally Regularized Implicit Shape Represen-tation

In practice, the geometry and topology of the objects tobe segmented can be very complicated. Furthermore, due tothe imperfections during the contour evolution process, theshape data are often corrupted with noise. A good activecontour algorithm should be robust to noise, while owningthe capability of dealing with complicated geometry andtopology. There has been very active research aimed atderiving proper and efficient object shape representations,and implicit representations have attracted a lot of attentionsince they allows topological flexibility in segmentation al-gorithms.

Let contour C be implicitly represented as the zero levelset of a higher dimensional hypersurface φ (the level setfunction) [14]: C = {x|φ(x, t) = 0}, where x is a pointin space and t is the time. The higher dimensionality of therepresentation provides one of major advantages: the flexi-ble handling of changes in the topological genus. This im-plies that it can easily represent complicated shapes, splitto form multiple objects, or merge with other objects toform a single structure. This is an important property whensegmenting complex models with an unknown topologicalgenus.

Because the parameter domain of the implicit functioncould be the whole space, it usually needs to be restrictedto some bounding box, and the most basic representationis a uniform scalar grid of sampled values of φ. Never-theless, the use of a purely implicit representation is ofteninsufficient, where one of the principal challenges is how todeal with the inherent noise in the data once the level setfront has evolved. It means that the contour representationsshould be robust to noise and guarantee global consistencyof the final segmentation results.

Towards this direction, it is natural to smooth or regular-ize the level set values within a local neighborhood of anyzero set point, i.e. instead of using the level set value φ of apoint itself directly, we replace it with

φ(x, t) =∑

I

NI(x)φI = NΦ (1)

with NI the interpolating shape function and φI the level setvalue for sample point I in the neighborhood influence do-main 2. This will guarantee to yield a smooth approximationφ for any front point, on the basis of available neighboringlevel set values within its influence domain.

2Centered at each point, there is a adaptively determined surroundingregion which is called the influence domain. See more discussion later.


2.2. Neighbor-Influenced External Energy

Based on the implicit shape representation, we can arriveat the following minimization problem [1, 6]:

minφ

∫Ω

δε(φ)g(| � I(x)|)| � φ(x)|dx (2)

where δε(φ) is the Dirac measure, � denotes the gradi-ent operator, and g(.) is a monotonically decreasing func-tion for a given image I such that g(0) = 1 and limx−>∞g(x) = 0. In other words, we try to find the minimal lengthgeodesic curve that best takes into account the desired im-age characteristics.

The minimization of the objective function is done usinga gradient descent method:

∂φ

∂t=

(gκ + (�g.

�φ

| � φ| ))| � φ| (3)

where κ = div(

�φ|�φ|

)is the curvature, and | � φ| repre-

sents the norm of the gradient φ. Writing it in a brief form,we have

∂φ

∂t+ F | � φ| = 0 (4)

where F = −(gκ + (�g. �φ

|�φ| ))

.The main drawback of this segmentation model is the

potential existence of local minima in the energy functional.For example, proper segmentation often fails when the edgehas gaps or when regions are homogeneous but very noisy.The traditional way to solve this problem is to develop morecomplicated, and most likely task-specific, external energyterms. However, we argue from prior experiences with para-metric active contour [5] that, more reliable and robust re-sults can also be achieved if image/prior information fromboth the contour point itself and its neighboring points areproperly included. It is clear that neighboring image forceinformation provides useful constraints and gives better re-gional understanding of the local boundaries.

Hence, in our effort, each front point moves under the in-fluence of two forces: the data force provided by image in-formation (and prior information if available) such as GVF,and the neighborhood force due to the interaction of thepoint with other points in the influence domain. With properformulation of the neighboring interactions, front points atthe weak edges or gaps will be dominated by the neighbor-hood force such that the front would be discouraged fromleaking through the boundary. Because the neighborhoodsize, i.e. the influence domain size, is adaptively deter-mined, for front points with good data force, their move-ment is still mostly controlled by image/prior information(with very small influence domain) and thus would stick tothe object boundary exhibited in the image.

Figure 1. Generation of the influence domain for the red point.(See text for detailed procedures.)

There could be many ways to incorporate the inter-pointrelationship. In the same fashion as we compute the regu-larized level set values, one simple way of enforcing neigh-borhood influence is to replace the edge evolution functionF at a given front point by

F =∑

NIFI (5)

where FI is the edge evolution function value for samplepoint I in the neighborhood influence domain. This F func-tion can now be regarded as a smoothed version of the orig-inal edge evolution function, influenced by its neighboringpoints through the shape functions NI by assigning properweights to each point within the influence domain.

2.3. Neighborhood Aided Implicit Active Contours

Since minimizing Equation (2) is equivalent to solve thePDE Equation (4), let us modify this original formulationto incorporate the neighborhood influence we have just dis-cussed. Referring to Equation (4) as the strong form levelset equation, we wish to convert this strong form formula-tion to a corresponding local weak form formulation. Thebasic premise is that by doing so, the original PDE is nolonger exactly existed for every point of the problem do-main, but rather in a local average sense within its influencedomain for each point.

Multiplying Equation (4) by the weighting shape func-tion N and then integrate over the influence domain Ω foreach data point:

∫Ω

NT ∂φ

∂tdΩ +

∫Ω

NT F | � φ|dΩ = 0 (6)

The first term, the local integration of level set function (i.e.locally regularized implicit shape representation), focuseson holding the most relevant geometric information of theevaluation contours under the premise that the smoothnessof the final curve can be well captured. The second term,the local integration of front evolution forces (i.e. neighbor-influenced external energy), aims to provide a joint poten-tial force from the point data/prior force and its interactions


Figure 2. Segmentation results comparison between traditionallevel set (top) and neighborhood-influenced GAC (bottom) onweak edges.

with its neighboring points. Due to the combined effect ofthese two local integrations on level set values and evolu-tion forces, as well as the adaptive selection of the localinfluence domain, the local weak form level set equation(Equation (6)) posses more robust ability in segmenting ob-ject boundaries with poor contrast, high noise, and discon-tinuous edges.

A major challenge in using numerical methods for thesolutions of Equation (6) has been how to eliminate or re-duce potential front oscillations. One way is to add least-squares forms of the residuals (or the so called Galerkinleast square approximation (GLS)) to the left side of theweak form statement [4]. It plays the same role as the es-sentially non-oscillatory (ENO) schemes. Hence, the fol-lowing overall local weak form (variational) formulation isused to update the level set field:∫

Ω

NT ∂φ

∂tdΩ +

∫Ω

NT F | � φ|dΩ +∫

Ω

(F

| � φ| � NT � φ)τ(∂φ

∂t+ F | � φ|)dΩ = 0 (7)

where τ is a positive stabilization parameter. As the solutiontends to be exact, the residuals go to zero.

If we substitute Equation (1) into Equation (6), we have

Mφφ̇ + fφ = 0 (8)

with φ̇ = ∂φ∂t . More generally, considering the GLS stabi-

lizing terms at in Equation (7), we obtain:

(Mφ + MGLS)φ̇ + fφ + fGLS = 0 (9)

with

Mφ =∫

Ω

NT NdΩ (10)

MGLS =∫

Ω

(�NT · �φ

| � φ|F )τNdΩ (11)

fφ =∫

Ω

NT F | � φ|dΩ (12)

fGLS =∫

Ω

(�NT · �φ

| � φ|F )τF | � φ|dΩ (13)

Ground TruthImage

Corruptedwith Gaussian

noise(SNR:30dB)


noise(SNR:10dB)


noise(SNR:1dB)

Error:1.090714

Error:1.215299

Error:1.396611

Figure 3. Segmentation results under different noise levels. Toprow: original images. Bottom row: neighborhood-influencedGAC segmentation results on the three noisy images.

MGLS and fGLS terms are used to enhance the stabilityof Equation (6) without degrading accuracy. Mφ, acting asdiffused local internal energy, is responsible for the smooth-ness of the contour. If the influence domain reduced to aline segment, Mφ becomes the length of the contour seg-ment. This can be considered equivalent as a particularclass of internal energy in the parametric models where thecoefficient β is set to be zero. The fφ term suggests thateach front point move under the influence of two forces:the typical data force provided by image/prior informationsuch as gradient vector flow (GVF), and the neighborhoodforce due to the interaction of the point with other points inthe influence domain. Due to the combined effects of in-ternal energy and data force interactions with the neighbor-ing points, our method exhibits robustness against boundaryleakage while maintains the desired geometrical character-istics of traditional geometrical active contours.

By taking finite differences in time domain to solveEquation (9), with time step Δt, we integrate through timeusing an explicit forward-Euler updating procedure:

φn+1 = φn − Δt(Mφ + MGLS)−1(fnφ + fn

GLS) (14)

2.4. Discussions

An attractive property of the neighborhood aided implicitactive contour is that the local influence domain Ω controlsthe behavior trade-off of a front point between a PAC orGAC point. Consider the case where the size of the localinfluence domain is approaching zero, that is, the level setvalue of the point is mainly determined by itself. With Ω be-ing very small, Mφ will approach to be a unit matrix, whilefφ will become F | � φ|, and Equation (8) goes back to thestrong form φ̇ + F | � φ| = 0. On the other hand, if we


Figure 4. The ability of the neighborhood-influenced GAC to han-dle topological changes.

enlarge the influence domain to cover the entire curve, theintegration is taken over the whole contour. Perceptually,the behavior of all the points are now inter-related, and weare effectively having a parametric deformable model in-stead. Our framework thus depends crucially upon whetherthe size of influence domain has been properly determined.Influence domains of different sizes generate different Mφ

and fφ matrices, which in turn are suitable for different sit-uations. For example, large influence domains are effectivein robust segmentation of noisy images or object with bro-ken edges. On the other hand, small influence domains areneeded for object boundaries with many fine details. In thispaper, the local neighborhood is adaptively selected basedon image gradient estimation and front geometry, and willbe discussed in the following section.

The shape functions N also play an important role inour algorithm. There are several type methods that canbe used to serve as interpolation functions. The polyno-mial shape function expressed as a linear combination ofbasis functions, which is commonly used in FEM methods,can be accommodated. Moving least squares, radial basisfunctions and partition of unity, which all use the paradigmof local approximations, provide a wide range of functionchoices. However, the different choices for the shape func-tions do not change the fundamental underlying conceptsof our framework, although we realize that more sophisti-cated shape functions probably will produce more accurateresults.

The neighborhood force concept can be assigned withphysics meanings. For example, each point can be viewedas a charged particle with some electric charges computedfrom gradient-magnitude image. Thus these charges willmove under the influence of two forces: Lorenz force, re-lated to the electric field generated by its fixed charges,and Coulomb force, due to the interaction of the particlewith other particles in its influence domain. The neighbor-hood force in our framework can be viewed as the Coulombforce from this physics viewpoint. We should further pointout that the adequate form for the evolution force/energyneed to be researched further, including possible probabilis-tic formulation such that the internal energy term can be fitto the data by finding the term parameters that maximize theposterior probability.

Figure 5. Segmentation of illusory contours.

3. Numerical ImplementationsThe front evolution Equations (8) or (9) can be solved by

a variety of numerical schemes, including the classical finitedifference methods, the moving grids method, the finite el-ement method, and the meshfree point cloud method [3].Here, we present the procedures to implement the neigh-borhood influenced geometric active contours with standardfinite difference method on regular grids.

3.1. Level Set Updating Procedures

Let φ(x, t) be defined by a distance function

φ(x, t = 0) = ±d (15)

where ±d is the signed distance to the interface from thepoint x, take positive sign if x is outside and negative if x isinside. Regardless of the specific numerical schemes used,the general level set updating procedures are:

1. Initialization: Initialize φ(·, 0) to be the signed dis-tance function of the initial contour.

2. Domain Representation: The domain is represented byuniform grids. Then, find all the points in the user-specified narrow band of the current zero level set.

3. Influence Domain Generation: Generate a proper in-fluence domain for each node within the narrow band.(more details later)

4. Shape Function Construction: Choose the shape func-tion to interpolate the level set function over influencedomain Ω. (more details later)

5. Evaluation of Integrals: Calculate Mφ and MGLS , andconstruct fφ and fGLS based on image data and/orprior model constraints.

6. Updating Procedure: Update level set function φ usingEquation (14).

7. Reinitialization: Re-initialize φ(·, t + 1) to be thesigned distance function of its zero level set. 3

3To maintain numerical accuracy, reinitialization process may be per-formed every two or three time steps.


Figure 6. Segmentation of brain tumor from ultrasound image.Top row: traditional level set results. Bottom row: neighborhood-influenced GAC results.

8. Convergence Test: Set proper convergence criterion totest whether the zero level set reaches object boundary.If no, go back to step 2.

3.2. Influence Domain Determination

In the level set formulation based on distance measure,there is an entire family of isocontours of different level setvalues (although only one of which is the zero level set).For each data point in the narrow band, or a node, its all im-portant influence domain Ω is determined by a data-drivenlocal operation. And the geometry of the resulting influ-ence domain adapts to the isocontour segment to which itbelongs (see Fig. 1 for an illustration).

The level set isocontour which passed through xa, i.e.φ(x) = φ(xa), can be determined from the level set val-ues (the dotted line in Fig. 1). From xa and alongthis isocontour, one travels geodesic distance Tspan =

Tscale

2∗exp(|�I(xa)|∗|κ(xa)|) in both directions, where �I(xa)and κ(xa) is the the image gradient and curvature of theactive node, and Tscale is a scaling factor. The resultingisocontour segment defines the tangential span of the influ-ence subdomain, and the two span-defining lines are con-structed at the two end-points of the isocontour segment,perpendicular to the isocontour (the two yellow lines in Fig.1). Intuitively, the span is small when the active node is inhigh gradient location and/or it is a geometrically signifi-cant landmark such as a corner. On the other hand, the spanis large when the active node is far away from edges or isin edge gaps (if the current front is near object boundary),and/or it is part of a flat front segment.

The two green curves, which bound influence domain inthe normal directions, are each Nspan distance in level setspace from φ(xa), i.e. the two green curves are isocontoursegments for φ(x) = φ(xa)±Nspan. As long as Ω containsenough nodes in addition to the active node, the value ofNspan should be as small as possible. In our experiments,we typically set Nspan = 1.

Through the above procedures of defining the tangen-tial and normal spans for the influence domain, all the gridpoints falling within Ω are selected as nodes to construct thelocal shape function.

Figure 7. Segmentation of bone structures from CT image (earlierframes are omitted). Top row: traditional level set results. Bottomrow: neighborhood-influenced GAC results.

3.3. Shape Interpolation Functions

Theoretically, the shape functions N can be any func-tions as long they satisfy the positive and unity condi-tions, and decrease in magnitude as the distance d(xI) =|xI − xa| increases, to enforce proper local neighbor in-fluence. In our work, we construct shape function frompolynomial basis functions of order m with non-constantcoefficients.

First, we give the definition of polynomial basis function.The basis functions p(x) = {p(j)}m

j=1 have the properties:

1. p(1) ≡ 1;

2. p(j) ∈ C(s)(Ω) where C(s)(Ω) is a set of functionsthat have continuous derivatives up to order s on Ω;

3. {p(j)}mj=1 is a set of linearly independent functions

over a subset of m of the given n points in Ω.

Typically, the basis functions consist of monomials of thelowest orders to ensure minimum completeness: p(x) ={xα} with α = 0, 1, 2, ...,m.

The MLS approximation for the level set value φ of eachpoint in the influence domain is defined by 4

φh(x) =m∑

j=1

pj(x)aj(x) ≡ pT (x)a(x) (16)

where p(x) is the polynomial basis functions of order , anda(x) are their coefficients. These coefficients can be ob-tained by performing a weighted least squares fit:

J =n∑I

w(x − xI)[pT (xI)a(x) − φI ]2 (17)

with w(x − xI) a weight function, and n the number ofpoints xI in the influence domain of x. The minimizationcondition requires ∂J

∂a = 0, and leads to:

a(x) = A−1(x)B(x)Φ (18)

4It is exactly the same procedure for the front evolution function F


where

A(x) =n∑I

w(x − xI)p(xI)pT (xI) (19)

B(x) = [B1,B2, ...,Bn] (20)

with BI = w(x − xI)p(xI) and ΦT = {φ1, φ2, ..., φn}.Note that we require n >> m which prevents the singu-larity of the matrix A and ensures the existence of A−1.Substituting the results into Equation (16), the dependentvariable φh can be expressed as

φh(x) =n∑I

NI(x)φI ≡ NΦ (21)

where the MLS-derived shape function NI(x) is

NI(x) =m∑j

pj(x)(A−1(x)B(x))jI = pT A−1BI (22)

and N(x) = [N1(x), N2(x), ..., Nn(x)] = pT A−1B. Fur-ther, the derivatives of the shape functions that are necessaryto compute the gradients of the approximations are:

N′(x) = (pT )′A−1B + pT ((A−1)′B + A−1B′) (23)

It can be shown that if the weight functions w(x − xI) iscontinuous up to its first k derivatives, then the shape func-tion is also continuous up to its first k derivatives [7].

The weight functions w(x − xI) play important roles inconstructing the MLS shape functions. Theoretically, theweighting functions can be any functions as long they arepositive and continuous together with their derivatives upto the desired degree, and they satisfies the positivity, com-pactness, and unity conditions. Defining dI = |x−xI|, andr = dI/dmI , where dmI is the size of influence domain ofthe Ith node, the weight function can be written as a func-tion of normalized radius r. In our implementation, we usethe cubic spline function:

w(r) =

⎧⎨⎩

23 − 4r2 + 4r3 for r ≤ 1

243 − 4r + 4r2 − 4

3r3 for 12 < r ≤ 1

0 for r > 1(24)

For two-dimensional case, tensor product concepts areemployed to the construction of the cubic spline weightingfunctions. The tensor product weight function at any pointis given by

w(x − xI) = w(rx).w(ry) = wx.wy (25)

where w(rx) or w(ry) is given by Equation (24) with r re-place by rx or ry respectively; rx and ry are given by

rx =|x − xI |dmIx

, ry =|y − yI |dmIy

(26)

Figure 8. Neighborhood-influenced GAC segmentation of naturalscene images.

where dmIx (or dmIy) is the size of influence domain in x(or y) direction.

Integrating over the influence domain can be carried outthrough numerical techniques which approximate a contin-uous integral over Ω into a discrete sum:

∫Ω

f(ξ)dΩ =nq∑l=1

NIf(ξl) (27)

where nq is number of grid points in the influence domain.

4. Experimental EvaluationsWe have conducted several experiments on synthetic and

real images with our neighborhood-aided implicit activecontour (WF-GDM), and compared the segmentation re-sults with those from the traditional geometric active con-tour (TLS-FD) with the same original force term. 5 Thesame force term is used for all experiments [13]: F =βgκ− (1−β)g(G.�φ)/|�φ|, g = 1

1+|�G∗I| , with g theimage gradient force, κ the contour curvature, β a constantweighting parameter, and G image gradient vector flow.

In Fig. 2, comparison is made between TLS-FD and WF-GDM (bottom) on dealing with weak edge leakage problem.The test object contains two blurred areas on the boundary.We add Gaussian noise with SNR=9dB to the original im-age and then perform TLS-FD and WF-GDM segmentationon the noisy one. Clearly, the traditional level set curvekeeps shrinking and leaks through the weak edges. The WF-GDM does not suffer from the leakage problem, and con-verges close to the true boundary since the neighborhoodpoints information offers useful global view on the imageboundaries. The experiment also demonstrates the attractivemulti-scale properties of WF-GDM. Each contour point ofthe WF-GDM at the blurred area will have large influencedomain, and thus detect the boundary properly, while it willhave very small influence domain and behave just like the

5Do note that for WF-GDM, the force term is then influenced by itsneighboring points in the formulation.


traditional GAC point elsewhere. We have also performedcomparative tests to examine WF-GDM’s tolerance to ad-ditive noises. In this experiment, a harmonic shape is gen-erated according to r = a + bcos(mθ + c),m = 6, andthen various levels of noises (SNR = 1−30dB) are addedto the original shape, as shown in Fig. 3 (top). The WF-GDM segmentation results are presented in Fig. 3 (bottom)with quantitative assessments of average errors (the erroris defined as the distance between the estimated boundarypoint and its corresponding true one). The ability of theWF-GDM to handle topological changes is demonstrated inFig. 4, where starting from a single front, the WF-GDMmanages to split and capture the boundaries of and all threeobjects. Similarly, it can be shown that WF-GDM is alsocapable to merge from multiple fronts. We have appliedWF-GDM to identify illusory contours, which are intrin-sic phenomena in human vision. As shown in Fig. 5, it iseasy for human to recognize the contour inside the images,even though the contour is spatially discontinuous. The tra-ditional boundary-based level set often fails because thereis no apparent boundary indication in the edge map for themissing part, while WF-GDM correctly detects the illusoryboundary when using the same force term. We have alsoshown the WF-GDM segmentation results on medical andnatural scene images. Fig. 6 and Fig. 7 compare the WF-GDM and TLS-FD segmentation of brain tumor from diffi-cult ultrasound images and bone structures from CT imagerespectively. While WF-GDM yields satisfactory solutionsin both cases, the TLS-FD suffers from sensitivity to localminima due to only use the local information. Further ex-amples of WF-GDM on several natural scene images areshown in Fig. 8 [9].

Acknowledgments

This work is supported by the 973 Program of China(2003CB716104), the Hong Kong Research Grants Council(CERG HKUST6252/04E), and the National Natural Sci-ence Foundation of China (60403040, 60021201).

References[1] V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active con-

tours. International Journal of Computer Vision, 22(1):61–79, 1997.

[2] T. Chan and L. Vese. Active contours without edges. IEEE

Transactions on Image Processing, 10(2):266–277, 2001.[3] H. Ho, Y. Chen, H. Liu, and P.Shi. Level set active con-

tours on unstructured point cloud. In Computer Vision and

Pattern Recognition, volume II, pages 690–697, San Diego,June 2005.

[4] T. Hughes, L. Franca, and G. Hulbert. A new finite el-ement formulation for computational fluid dynamics: Thegalerkin least-squares method for advective-diffuse equa-

tions. Computer Methods in Applied Mechanics and Engi-

neering, 73:173–189, 1989.[5] M. Kass, A. Witkin, and D. Terzopoulos. SNAKES: Active

contour models. International Journal of Computer Vision,1:321–332, January 1988.

[6] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, andA. Yezzi. Gradient flows and geometric active contour mod-els. In IEEE International Conference on Computer Vision,pages 810–815, Boston, USA, 1995.

[7] P. Lancaster and K. Salkauskas. Surface generated by mov-ing least squares methods. Mathematics of Computation,37(155):141–158, 1981.

[8] R. Malladi, J. Sethian, and B. Vemuri. Shape modeling withfront propagation: A level set approach. IEEE Transactions

on Pattern Analysis and Machine Intelligence, 17(2):158–175, 1995.

[9] D. Martin, C. Fowlkes, D. Tal, and J. Malik. A databaseof human segmented natural images and its application toevaluating segmentation algorithms and measuring ecologi-cal statistics. In Proc. 8th Int’l Conf. Computer Vision, vol-ume 2, pages 416–423, July 2001.

[10] T. McInerney and D. Terzopoulos. Topologically adaptablesnakes. In International Conference on Computer Vision,pages 840–845, 1995.

[11] S. Osher and J. Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi for-mulations. Journal of Computational Physics, 79:12–49,1988.

[12] N. Paragios and R. Deriche. Coupled geodesic active regionsfor image segmentation: a level set apporach. In Proceedings

of the Europe Conference on Computer Vision, pages 224–240, 2000.

[13] N. Paragios, O. Mellina-Gottardo, and V. Ramesh. Gradientvector flow fast geodesic active contours. In Proceedings

of the International Conference on Computer Vision, pages67–73, 2001.

[14] J. Sethian. Level Set Methods and Fast Matching Methods:

Evolving Interfaces in Computational Geometry, Fluid Me-

chanics, Computer Vision and Material Science. CambridgeUniv. Press, London, 1999.

[15] K. Siddiqi, Y. Lauziere, A. Tannenbaum, and S. Zucker. Areaand length-minimizing flows for shape segmentation. IEEE

Transactions on Image Processing, 7:433–443, 1998.[16] G. Sundaramoorthi, A. Yezzi, and A. Mennucci. Sobolev

active contours. In VLSM, pages 109–120, 2005.[17] B. Vemuri, Y. Guo, and Z. Wang. Deformable pedal curves

and surfaces: hybird geometric active models for shape re-covery. International Journal of Computer Vision, 44:137–155, 2001.

[18] X. Xie and M. Mirmehdi. RAGS: Region-aided geometricsnake. IEEE Trans. Imag. Process., 13(5):640–652, May2004.

[19] C. Xu and J. Prince. Generalized gradient vector flow exter-nal forces for active contours. Signal Process., 71(2):131–139, 1998.

[20] A. Yezzi, A. Tsai, and A. Willsky. A fully global approach toimage segmentation via coupled curve evoluation equations.J. Vis. Commun. Image. Represent., 13(1-2):195–216, 2002.


neighborhood aided implicit active contours

Documents