Neighborhood Aided Implicit Active Contours

Transcription

1 Neighborhood Aided Implicit Active Contours Huafeng Liu 1,2, Yunmei Chen 3, Wufan Chen 4, and Pengcheng Shi 2 1 State Key Laboratory of Modern Optical Instrumentation Zhejiang University, Hangzhou, China 2 Department of Electrical and Electronic Engineering Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong 3 Department of Mathematics, University of Florida, Gainesville, Florida, USA 4 Key Laboratory of Medical Image Processing, Southern Medical University, Guangzhou, China Abstract We have developed a geometric deformable model that employs neighborhood influence to achieve robust segmentation for noisy and broken edges. The fundamental power of this strategy rests with the explicitly combination of regional inter-point constraints, image forces, and a priori boundary information for each geometric contour point within its adaptively determined local influence domain. This formulation thus naturally unifies the essences of the geometric and parametric snakes through automatic local scale selection, and exhibits their respective fundamental strengths of allowing stable boundary detection when the edge information is weak and possibly discontinuous, while maintaining the abilities to handle topological changes during front evolution. In particular, this paper presents an implementation of the method through local integration of the level set function and the image/prior-driven evolution forces, where the resulting partial differential equation is solved numerically using standard finite difference method. Experimental results on synthetic and real images demonstrate its superior performance. 1. Introduction One of the primary goals in computer vision has been to robustly recover the shape of objects from images of various qualities. Although there exist a large amount of dedicated methods more or less suited for particular applications, a strong and persistent interest has been the general paradigm based on the parametric and geometric deformable models [1, 5, 6, 8]. The mathematical foundations of these deformable models consist of two integral components, the geometry for shape representation and the mathematics/physics for constructing energy functional. Popular explicit shape representations include parametric and meshed curves/surfaces/volumes. For example, the classical snakes are explicitly represented as parameterized curves in Lagrangian formulations [5]. The evolution is then obtained by minimizing an energy measure on the entire contour, including both internal and external energies. The principal disadvantage of these parametric active contours (PACs) is that they cannot easily deal with topological changes. That is, unless specifically designed [10], the topology of the final object has to be the same as the initial one, even if it is in conflict with the image information. A different class of approaches, where active contours are implicitly represented as a level set of a higher-dimensional scalar function [11], has been introduced to overcome this problem. These geometric active contours (GACs) have received a large amount of attention in recent years because their ability to naturally handle topological complexity and variability without any additional machinery [1, 8]. Most explicit representations use basis functions with finite support, and thus, give the segmentation algorithms an easy way to perform local control of the shape. In contrast, implicit representations often do not have obvious influence on the shape position and thus the noise in the shape representations is difficult to deal with. A desired procedure should be capable of handling complicated topology and geometry as well as robust to noise. Recently, a hybrid shape modeling scheme based on the notion of a pedal curve has been proposed. In this work, a global prior is introduced using a parameterized model, and the local properties are fine-tuned using a geometric flow [17]. In addition, in order to achieve robust segmentation against varying imaging conditions, an important issue when using active contour algorithms is the appropriate, and often task-specific, choice and design of an image/prior driven external energy functional. Otherwise, they will face 1

2 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. Considerable progress has already been made in the designing of new external force (or energy) terms for geometric active contours. An extra stopping term for pulling back the contour if it passes the boundary has been investigated in [1]. A weighted area functional force has been introduced to help the snakes be more robust with respect to small gap problem [15]. Diffused external data forces, such as the gradient vector flow (GVF), have also been adopted [19]. And boundary and region information has been integrated under a curve-based minimization framework [12]. These later approaches own the benefits provided by external force potential field in achieving a larger capture range and robustness against the boundary leakage problem [12, 19]. More recently, region-aided geometric snake, which integrates gradient flow forces with region vector flow forces obtained through the diffusion of the region segmentation map, has been developed and implemented within level set platform [18]. These integrated forces give another way to be more robust toward weak edges. Similar region-based strategies have been explored by other works as well [2, 20]. However, real-world images are so different from each other that it is difficult to expect one of the aforementioned external energies to be able to segment efficiently all of them. We realize that, for geometric active contours, more robust results can be achieved if the behavior of any individual contour point is constrained by both local edge information of itself and that of its neighboring points. These inter-point relationships provide an expanded local view, and the global view in the extreme case when all the contour points are considered, of the object boundaries and thus aid in the delineation of noisy, diffused, and broken edges. In this paper, we present a geometric deformable model with external energy that makes uses of the image and prior information at the adaptively determined support domain around each point of interest, thus effectively enlarges the capture range of each point to have a better regional understanding of the edge information within its local neighborhood. It provides adaptive smoothing/averaging regularity through local integration of the level set function, as well as incorporates inter-point relationships through local integration of the external image/prior forces. As a result, we effectively obtain a hybrid active contour which combines the geometric deformable models with the internal energy term usually appeared in the parametric snakes. Experimental results on synthetic and real images 1. 1 We appreciate the anonymous reviewer for pointing out that, following the similar spirit, the Sobolev active contour (SAC) has attempted to minimize the energies from image and the cost of perturbing the curve [16]. Our work differs in the sense that it does local weak form averaging on both the level set function and the flow, while the SAC does so only for the flow itself 2. Neighborhood Aided Implicit Active Contours 2.1. Locally Regularized Implicit Shape Representation In practice, the geometry and topology of the objects to be segmented can be very complicated. Furthermore, due to the imperfections during the contour evolution process, the shape data are often corrupted with noise. A good active contour algorithm should be robust to noise, while owning the capability of dealing with complicated geometry and topology. There has been very active research aimed at deriving proper and efficient object shape representations, and implicit representations have attracted a lot of attention since they allows topological flexibility in segmentation algorithms. Let contour C be implicitly represented as the zero level set of a higher dimensional hypersurface φ (the level set function) [14]: C = {x φ(x,t)=0}, where x is a point in space and t is the time. The higher dimensionality of the representation provides one of major advantages: the flexible handling of changes in the topological genus. This implies that it can easily represent complicated shapes, split to form multiple objects, or merge with other objects to form a single structure. This is an important property when segmenting complex models with an unknown topological genus. Because the parameter domain of the implicit function could be the whole space, it usually needs to be restricted to some bounding box, and the most basic representation is a uniform scalar grid of sampled values of φ. Nevertheless, the use of a purely implicit representation is often insufficient, where one of the principal challenges is how to deal with the inherent noise in the data once the level set front has evolved. It means that the contour representations should be robust to noise and guarantee global consistency of the final segmentation results. Towards this direction, it is natural to smooth or regularize the level set values within a local neighborhood of any zero set point, i.e. instead of using the level set value φ of a point itself directly, we replace it with φ(x,t)= I N I (x)φ I = NΦ (1) with N I the interpolating shape function and φ I the level set value for sample point I in the neighborhood influence domain 2. This will guarantee to yield a smooth approximation φ for any front point, on the basis of available neighboring level set values within its influence domain. 2 Centered at each point, there is a adaptively determined surrounding region which is called the influence domain. See more discussion later.

3 2.2. Neighbor-Influenced External Energy Based on the implicit shape representation, we can arrive at the following minimization problem [1, 6]: min φ δ ε (φ)g( I(x) ) φ(x) dx (2) where δ ε (φ) is the Dirac measure, denotes the gradient operator, and g(.) is a monotonically decreasing function for a given image I such that g(0) = 1 and lim x > g(x) =0. In other words, we try to find the minimal length geodesic curve that best takes into account the desired image characteristics. The minimization of the objective function is done using a gradient descent method: φ t = ( where κ = div φ φ ( ) φ gκ +( g. φ ) φ (3) ) is the curvature, and φ represents the norm of the gradient φ. Writing it in a brief form, we have φ + F φ =0 (4) t ( 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 edge has gaps or when regions are homogeneous but very noisy. The traditional way to solve this problem is to develop more complicated, and most likely task-specific, external energy terms. However, we argue from prior experiences with parametric active contour [5] that, more reliable and robust results can also be achieved if image/prior information from both the contour point itself and its neighboring points are properly included. It is clear that neighboring image force information provides useful constraints and gives better regional understanding of the local boundaries. Hence, in our effort, each front point moves under the influence of two forces: the data force provided by image information (and prior information if available) such as GVF, and the neighborhood force due to the interaction of the point with other points in the influence domain. With proper formulation of the neighboring interactions, front points at the weak edges or gaps will be dominated by the neighborhood force such that the front would be discouraged from leaking through the boundary. Because the neighborhood size, i.e. the influence domain size, is adaptively determined, for front points with good data force, their movement is still mostly controlled by image/prior information (with very small influence domain) and thus would stick to the 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-point relationship. In the same fashion as we compute the regularized level set values, one simple way of enforcing neighborhood influence is to replace the edge evolution function F at a given front point by F = N I F I (5) where F I is the edge evolution function value for sample point I in the neighborhood influence domain. This F function can now be regarded as a smoothed version of the original edge evolution function, influenced by its neighboring points through the shape functions N I by assigning proper weights to each point within the influence domain Neighborhood Aided Implicit Active Contours Since minimizing Equation (2) is equivalent to solve the PDE Equation (4), let us modify this original formulation to incorporate the neighborhood influence we have just discussed. Referring to Equation (4) as the strong form level set equation, we wish to convert this strong form formulation to a corresponding local weak form formulation. The basic premise is that by doing so, the original PDE is no longer exactly existed for every point of the problem domain, but rather in a local average sense within its influence domain for each point. Multiplying Equation (4) by the weighting shape function N and then integrate over the influence domain for each data point: N T φ t d+ N T F φ d =0 (6) The first term, the local integration of level set function (i.e. locally regularized implicit shape representation), focuses on holding the most relevant geometric information of the evaluation contours under the premise that the smoothness of the final curve can be well captured. The second term, the local integration of front evolution forces (i.e. neighborinfluenced external energy), aims to provide a joint potential force from the point data/prior force and its interactions

4 Ground Truth Image Corrupted with Gaussian noise (SNR:30dB) Corrupted with Gaussian noise (SNR:10dB) Corrupted with Gaussian noise (SNR:1dB) Figure 2. Segmentation results comparison between traditional level set (top) and neighborhood-influenced GAC (bottom) on weak edges. with its neighboring points. Due to the combined effect of these two local integrations on level set values and evolution forces, as well as the adaptive selection of the local influence domain, the local weak form level set equation (Equation (6)) posses more robust ability in segmenting object boundaries with poor contrast, high noise, and discontinuous edges. A major challenge in using numerical methods for the solutions of Equation (6) has been how to eliminate or reduce potential front oscillations. One way is to add leastsquares forms of the residuals (or the so called Galerkin least square approximation (GLS)) to the left side of the weak form statement [4]. It plays the same role as the essentially non-oscillatory (ENO) schemes. Hence, the following overall local weak form (variational) formulation is used to update the level set field: N T φ t d+ N T F φ d+ F ( φ NT φ)τ( φ + F φ )d =0 (7) t where τ is a positive stabilization parameter. As the solution tends to be exact, the residuals go to zero. If we substitute Equation (1) into Equation (6), we have M φ φ + fφ =0 (8) with φ = φ t. More generally, considering the GLS stabilizing terms at in Equation (7), we obtain: with (M φ + M GLS ) φ + f φ + f GLS =0 (9) M φ = M GLS = f φ = f GLS = N T Nd (10) ( N T φ F )τnd (11) φ N T F φ d (12) ( N T φ F )τf φ d (13) φ Error: Error: Error: Figure 3. Segmentation results under different noise levels. Top row: original images. Bottom row: neighborhood-influenced GAC segmentation results on the three noisy images. M GLS and f GLS terms are used to enhance the stability of Equation (6) without degrading accuracy. M φ, acting as diffused local internal energy, is responsible for the smoothness of the contour. If the influence domain reduced to a line segment, M φ becomes the length of the contour segment. This can be considered equivalent as a particular class of internal energy in the parametric models where the coefficient β is set to be zero. The f φ term suggests that each front point move under the influence of two forces: the typical data force provided by image/prior information such as gradient vector flow (GVF), and the neighborhood force due to the interaction of the point with other points in the influence domain. Due to the combined effects of internal energy and data force interactions with the neighboring points, our method exhibits robustness against boundary leakage while maintains the desired geometrical characteristics of traditional geometrical active contours. By taking finite differences in time domain to solve Equation (9), with time step Δt, we integrate through time using an explicit forward-euler updating procedure: φ n+1 = φ n Δt(M φ + M GLS ) 1 (f n φ + f n GLS) (14) 2.4. Discussions An attractive property of the neighborhood aided implicit active contour is that the local influence domain controls the behavior trade-off of a front point between a PAC or GAC point. Consider the case where the size of the local influence domain is approaching zero, that is, the level set value of the point is mainly determined by itself. With being very small, M φ will approach to be a unit matrix, while f φ will become F φ, and Equation (8) goes back to the strong form φ + F φ =0. On the other hand, if we

5 Figure 4. The ability of the neighborhood-influenced GAC to handle topological changes. enlarge the influence domain to cover the entire curve, the integration is taken over the whole contour. Perceptually, the behavior of all the points are now inter-related, and we are effectively having a parametric deformable model instead. Our framework thus depends crucially upon whether the size of influence domain has been properly determined. Influence domains of different sizes generate different M φ and f φ matrices, which in turn are suitable for different situations. For example, large influence domains are effective in robust segmentation of noisy images or object with broken edges. On the other hand, small influence domains are needed for object boundaries with many fine details. In this paper, the local neighborhood is adaptively selected based on image gradient estimation and front geometry, and will be discussed in the following section. The shape functions N also play an important role in our algorithm. There are several type methods that can be used to serve as interpolation functions. The polynomial shape function expressed as a linear combination of basis functions, which is commonly used in FEM methods, can be accommodated. Moving least squares, radial basis functions and partition of unity, which all use the paradigm of local approximations, provide a wide range of function choices. However, the different choices for the shape functions do not change the fundamental underlying concepts of our framework, although we realize that more sophisticated shape functions probably will produce more accurate results. The neighborhood force concept can be assigned with physics meanings. For example, each point can be viewed as a charged particle with some electric charges computed from gradient-magnitude image. Thus these charges will move under the influence of two forces: Lorenz force, related to the electric field generated by its fixed charges, and Coulomb force, due to the interaction of the particle with other particles in its influence domain. The neighborhood force in our framework can be viewed as the Coulomb force from this physics viewpoint. We should further point out that the adequate form for the evolution force/energy need to be researched further, including possible probabilistic formulation such that the internal energy term can be fit to the data by finding the term parameters that maximize the posterior probability. Figure 5. Segmentation of illusory contours. 3. Numerical Implementations The front evolution Equations (8) or (9) can be solved by a variety of numerical schemes, including the classical finite difference methods, the moving grids method, the finite element method, and the meshfree point cloud method [3]. Here, we present the procedures to implement the neighborhood influenced geometric active contours with standard finite difference method on regular grids 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 the point x, take positive sign if x is outside and negative if x is inside. Regardless of the specific numerical schemes used, the general level set updating procedures are: 1. Initialization: Initialize φ(, 0) to be the signed distance function of the initial contour. 2. Domain Representation: The domain is represented by uniform grids. Then, find all the points in the userspecified narrow band of the current zero level set. 3. Influence Domain Generation: Generate a proper influence domain for each node within the narrow band. (more details later) 4. Shape Function Construction: Choose the shape function to interpolate the level set function over influence domain. (more details later) 5. Evaluation of Integrals: Calculate M φ and M GLS, and construct f φ and f GLS based on image data and/or prior model constraints. 6. Updating Procedure: Update level set function φ using Equation (14). 7. Reinitialization: Re-initialize φ(,t +1)to be the signed distance function of its zero level set. 3 3 To maintain numerical accuracy, reinitialization process may be performed every two or three time steps.

6 Figure 6. Segmentation of brain tumor from ultrasound image. Top row: traditional level set results. Bottom row: neighborhoodinfluenced GAC results. 8. Convergence Test: Set proper convergence criterion to test whether the zero level set reaches object boundary. If no, go back to step Influence Domain Determination In the level set formulation based on distance measure, there is an entire family of isocontours of different level set values (although only one of which is the zero level set). For each data point in the narrow band, or a node, its all important influence domain is determined by a data-driven local operation. And the geometry of the resulting influence domain adapts to the isocontour segment to which it belongs (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 values (the dotted line in Fig. 1). From xa and along this 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 the active node, and Tscale is a scaling factor. The resulting isocontour segment defines the tangential span of the influence subdomain, and the two span-defining lines are constructed 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 in high gradient location and/or it is a geometrically significant landmark such as a corner. On the other hand, the span is large when the active node is far away from edges or is in 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 in the normal directions, are each Nspan distance in level set space from φ(xa ), i.e. the two green curves are isocontour segments for φ(x) = φ(xa )±Nspan. As long as contains enough nodes in addition to the active node, the value of Nspan should be as small as possible. In our experiments, we typically set Nspan = 1. Through the above procedures of defining the tangential and normal spans for the influence domain, all the grid points falling within are selected as nodes to construct the local shape function. Figure 7. Segmentation of bone structures from CT image (earlier frames are omitted). Top row: traditional level set results. Bottom row: neighborhood-influenced GAC results Shape Interpolation Functions Theoretically, the shape functions N can be any functions as long they satisfy the positive and unity conditions, and decrease in magnitude as the distance d(xi ) = xi xa increases, to enforce proper local neighbor influence. In our work, we construct shape function from polynomial basis functions of order m with non-constant coefficients. 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) () where C (s) () is a set of functions that have continuous derivatives up to order s on ; 3. {p(j) }m j=1 is a set of linearly independent functions over a subset of m of the given n points in. Typically, the basis functions consist of monomials of the lowest orders to ensure minimum completeness: p(x) = {xα } with α = 0, 1, 2,..., m. The MLS approximation for the level set value φ of each point in the influence domain is defined by 4 φh (x) = m pj (x)aj (x) pt (x)a(x) where p(x) is the polynomial basis functions of order, and a(x) are their coefficients. These coefficients can be obtained by performing a weighted least squares fit: J= n w(x xi )[pt (xi )a(x) φi ]2 IEEE (17) I with w(x xi ) a weight function, and n the number of points xi in the influence domain of x. The minimization condition requires J a = 0, and leads to: a(x) = A 1 (x)b(x)φ 4 It (18) is exactly the same procedure for the front evolution function F /06 $ (16) j=1

7 where n A(x) = w(x x I )p(x I )p T (x I ) (19) I B(x) = [B 1, B 2,..., B n ] (20) with B I = w(x x I )p(x I ) and Φ T = {φ 1,φ 2,..., φ n }. Note that we require n>>mwhich prevents the singularity of the matrix A and ensures the existence of A 1. Substituting the results into Equation (16), the dependent variable φ h can be expressed as n φ h (x) = N I (x)φ I NΦ (21) I where the MLS-derived shape function N I (x) is m N I (x) = p j (x)(a 1 (x)b(x)) ji = p T A 1 B I (22) j and N(x) =[N 1 (x),n 2 (x),..., N n (x)] = p T A 1 B. Further, the derivatives of the shape functions that are necessary to compute the gradients of the approximations are: N (x) =(p T ) A 1 B + p T ((A 1 ) B + A 1 B ) (23) It can be shown that if the weight functions w(x x I ) is continuous up to its first k derivatives, then the shape function is also continuous up to its first k derivatives [7]. The weight functions w(x x I ) play important roles in constructing the MLS shape functions. Theoretically, the weighting functions can be any functions as long they are positive and continuous together with their derivatives up to the desired degree, and they satisfies the positivity, compactness, and unity conditions. Defining d I = x x I, and r = d I /d mi, where d mi is the size of influence domain of the I th node, the weight function can be written as a function of normalized radius r. In our implementation, we use the cubic spline function: w(r) = 2 3 4r2 +4r 3 for r r +4r2 4 3 r3 for 1 2 <r 1 0 for r>1 (24) For two-dimensional case, tensor product concepts are employed to the construction of the cubic spline weighting functions. The tensor product weight function at any point is given by w(x x I )=w(r x ).w(r y )=w x.w y (25) where w(r x ) or w(r y ) is given by Equation (24) with r replace by r x or r y respectively; r x and r y are given by r x = x x I d mix, r y = y y I d miy (26) Figure 8. Neighborhood-influenced GAC segmentation of natural scene images. where d mix (or d miy ) is the size of influence domain in x (or y) direction. Integrating over the influence domain can be carried out through numerical techniques which approximate a continuous integral over into a discrete sum: n q f(ξ)d = N I f(ξ l ) (27) l=1 where n q is number of grid points in the influence domain. 4. Experimental Evaluations We have conducted several experiments on synthetic and real images with our neighborhood-aided implicit active contour (WF-GDM), and compared the segmentation results with those from the traditional geometric active contour (TLS-FD) with the same original force term. 5 The same force term is used for all experiments [13]: F = 1 βgκ (1 β)g(g. φ)/ φ, g = 1+ G I, with g the image gradient force, κ the contour curvature, β a constant weighting 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 image and then perform TLS-FD and WF-GDM segmentation on the noisy one. Clearly, the traditional level set curve keeps shrinking and leaks through the weak edges. The WF- GDM does not suffer from the leakage problem, and converges close to the true boundary since the neighborhood points information offers useful global view on the image boundaries. The experiment also demonstrates the attractive multi-scale properties of WF-GDM. Each contour point of the WF-GDM at the blurred area will have large influence domain, and thus detect the boundary properly, while it will have very small influence domain and behave just like the 5 Do note that for WF-GDM, the force term is then influenced by its neighboring points in the formulation.

8 traditional GAC point elsewhere. We have also performed comparative tests to examine WF-GDM s tolerance to additive noises. In this experiment, a harmonic shape is generated according to r = a + bcos(mθ + c),m =6, and then various levels of noises (SNR =1 30dB) are added to 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 error is defined as the distance between the estimated boundary point and its corresponding true one). The ability of the WF-GDM to handle topological changes is demonstrated in Fig. 4, where starting from a single front, the WF-GDM manages to split and capture the boundaries of and all three objects. Similarly, it can be shown that WF-GDM is also capable to merge from multiple fronts. We have applied WF-GDM to identify illusory contours, which are intrinsic phenomena in human vision. As shown in Fig. 5, it is easy for human to recognize the contour inside the images, even though the contour is spatially discontinuous. The traditional boundary-based level set often fails because there is no apparent boundary indication in the edge map for the missing part, while WF-GDM correctly detects the illusory boundary when using the same force term. We have also shown the WF-GDM segmentation results on medical and natural scene images. Fig. 6 and Fig. 7 compare the WF- GDM and TLS-FD segmentation of brain tumor from difficult ultrasound images and bone structures from CT image respectively. While WF-GDM yields satisfactory solutions in both cases, the TLS-FD suffers from sensitivity to local minima due to only use the local information. Further examples of WF-GDM on several natural scene images are shown 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 Science Foundation of China ( , ). References [1] V. Caselles, R. Kimmel, and G. Sapiro. Geodesic active contours. International Journal of Computer Vision, 22(1):61 79, [2] T. Chan and L. Vese. Active contours without edges. IEEE Transactions on Image Processing, 10(2): , [3] H. Ho, Y. Chen, H. Liu, and P.Shi. Level set active contours on unstructured point cloud. In Computer Vision and Pattern Recognition, volume II, pages , San Diego, June [4] T. Hughes, L. Franca, and G. Hulbert. A new finite element formulation for computational fluid dynamics: The galerkin least-squares method for advective-diffuse equations. Computer Methods in Applied Mechanics and Engineering, 73: , [5] M. Kass, A. Witkin, and D. Terzopoulos. SNAKES: Active contour models. International Journal of Computer Vision, 1: , January [6] S. 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