Distance Regularized Level Set Evolution and Its Application to Image Segmentation Chunming Li, Chenyang Xu, Senior Member, IEEE, Changfeng Gui, and Martin D

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010 3243 Distance Regularized Level Set Evolution and Its Application to Image Segmentation Chunming Li, Chenyang Xu, Senior Member, IEEE, Changfeng Gui, and Martin D. Fox, Member, IEEE

Abstract—Level set methods have been widely used in image contour as the zero level set of a higher dimensional function, processing and computer vision. In conventional level set formula- called a level set function (LSF), and formulate the motion of tions, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually de- the contour as the evolution of the level set function. Some key stroy the stability of the evolution. Therefore, a numerical remedy, ideas in the level set method were proposed earlier by Dervieux called reinitialization, is typically applied to periodically replace and Thomasset [2], [3] in the late 1970s, but their work did the degraded level set function with a signed distance function. not draw much attention. It was only after the work by Osher However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects and Sethian in [1], the level set method became well known numerical accuracy in an undesirable way. This paper proposes and since then has had far-reaching impact in various applica- a new variational level set formulation in which the regularity of tions, such as computational geometry, fluid dynamics, image the level set function is intrinsically maintained during the level processing, and computer vision. set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization In image processing and computer vision applications, the term and an external energy that drives the motion of the zero level set method was introduced independently by Caselles et al. level set toward desired locations. The distance regularization [4] and Malladi et al. [5] in the context of active contour (or term is defined with a potential function such that the derived snake) models [6] for image segmentation. Early active contour level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the models are formulated in terms of a dynamic parametric contour level set function, particularly a signed distance profile near the with a spatial parameter in [0,1], zero level set. This yields a new type of level set evolution called which parameterizes the points in the contour, and a temporal distance regularized level set evolution (DRLSE). The distance variable . The curve evolution can be expressed as regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to com- plicated implementations of conventional level set formulations, a (1) simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of where is the speed function that controls the motion of the more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can contour, and is the inward normal vector to the curve . be used in the finite difference scheme to reduce the number The curve evolution in (1) in terms of a parameterized con- of iterations, while ensuring sufficient numerical accuracy. To tour can be converted to a level set formulation by embedding demonstrate the effectiveness of the DRLSE formulation, we apply the dynamic contour as the zero level set of a time de- it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly pendent LSF . Assuming that the embedding LSF reduce computational cost. takes negative values inside the zero level contour and positive Index Terms—Forward and backward diffusion, image segmen- values outside, the inward normal vector can be expressed as tation, level set method, narrowband, reinitialization. , where is the gradient operator. Then, the curve evolution equation (1) is converted to the following partial differential equation (PDE): I. INTRODUCTION (2)

HE LEVEL set method for capturing dynamic interfaces which is referred to as a level set evolution equation. The ac- T and shapes was introduced by Osher and Sethian [1] in tive contour model given in a level set formulation is called an 1987. The basic idea of the level set method is to represent a implicit active contour or geometric active contour model. For specific active contours in parametric and their corresponding Manuscript received September 06, 2007; revised October 12, 2009; accepted level set formulations, the readers are referred to [7]. October 12, 2009. Date of publication August 26, 2010; date of current version A desirable advantage of level set methods is that they can November 17, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Scott T. Acton. represent contours of complex topology and are able to handle C. Li is with Institute of Imaging Science, Vanderbilt University, Nashville, topological changes, such as splitting and merging, in a natural TN 37232 USA (e-mail: [email protected]). and efficient way, which is not allowed in parametric active con- C. Xu is with Technology to Business Center, Siemens Corporate Research, Berkeley, CA 94704 USA (e-mail: [email protected]). tour models [6], [8], [9] unless extra indirect procedures are in- C. Gui is with Department of Mathematics, University of Connecticut, Storrs, troduced in the implementations. Another desirable feature of CT 06269 USA (e-mail: [email protected]). level set methods is that numerical computations can be per- M. D. Fox is with Department of Electrical and Computer Engineering, Uni- versity of Connecticut, Storrs, CT 06269 USA (e-mail: [email protected]). formed on a fixed Cartesian grid without having to parameterize Digital Object Identifier 10.1109/TIP.2010.2069690 the points on a contour as in parametric active contour models.

These desirable features, among others, have greatly promoted of this formulation still causes errors that may destroy the further development of level set methods and their applications signed distance property and eventually destabilize the level in image segmentation [10]–[12], [7], [13]–[16], tracking [14], set evolution, which renders it necessary to introduce separate [17], and stereo reconstruction [18], etc. reinitialization procedures, as reported in [26]. Weber et al. [26] Although level set methods have been used to solve a wide proposed an implementation strategy to avoid separate reinitial- range of scientific and engineering problems, their applications ization procedures in their implementation of the well-known have been plagued with the irregularities of the LSF that are geodesic active contour (GAC) model in [11]. The updating of developed during the level set evolution. In conventional level the LSF at each time step is performed by a complex procedure set methods, the LSF typically develops irregularities during its of solving an optimization problem, instead of an iteration evolution (see the lower row of Fig. 3 for an example), which scheme derived from the underlying evolution equation, which cause numerical errors and eventually destroy the stability of is a disagreement between the theory and its implementation. the level set evolution. To overcome this difficulty, a numerical In our preliminary work [27], we proposed a variational level remedy, commonly known as reinitialization [19], [20], was in- set formulation with an intrinsic mechanism of maintaining troduced to restore the regularity of the LSF and maintain stable the signed distance property of the LSF. This mechanism is level set evolution. Reinitialization is performed by periodically associated with a penalty term in the variational formulation stopping the evolution and reshaping the degraded LSF as a that penalizes the deviation of the LSF from a signed distance signed distance function [21], [20]. function. The penalty term not only eliminates the need for A standard method for reinitialization is to solve the fol- reinitialization, but also allows the use of a simpler and more lowing evolution equation to steady state: efficient numerical scheme in the implementation than those used for conventional level set formulations. However, this (3) penalty term may cause an undesirable side effect on the LSF in some circumstances, which may affect the numerical accuracy. where is the LSF to be reinitialized, and is the sign This paper proposes a more general variational level set for- function. Ideally, the steady state solution of this equation is a mulation with a distance regularization term and an external en- signed distance function. This reinitialization method has been ergy term that drives the motion of the zero level contour toward widely used in level set methods [21], [22]. Another method for desired locations. The distance regularization term is defined reinitialization is the fast marching algorithm [19]. Although with a potential function such that it forces the gradient mag- reinitialization as a numerical remedy is able to maintain the nitude of the level set function to one of its minimum points, regularity of the LSF, it may incorrectly move the zero level thereby maintaining a desired shape of the level set function, set away from the expected position [20], [23], [19]. There- particularly a signed distance profile near its zero level set. In fore, reinitialization should be avoided as much as possible particular, we provide a double-well potential for the distance [19, p. 140]. regularization term. The level set evolution is derived as a gra- Moreover, there are serious theoretical and practical problems dient flow that minimizes this energy functional. In the level in conventional level set formulations regarding the practice of set evolution, the regularity of the LSF is maintained by a for- reinitialization. Level set evolution equations in conventional ward-and-backward (FAB) diffusion derived from the distance level set formulations can be written in the form of (2) with a regularization term. As a result, the distance regularization com- speed function . The speed function , which is defined to pletely eliminates the need for reinitialization in a principled guide the motion of the zero level set, does not have a compo- way, and avoids the undesirable side effect introduced by the nent for preserving the LSF as a signed distance function. In penalty term in our preliminary work [27]. We call the level theory, it has been proven by Barles et al. [24] that the solutions set evolution in our formulation a distance regularized level to such Hamilton-Jacobi equations is not a signed distance func- set evolution (DRLSE). To demonstrate the effectiveness of the tion. However, the LSF is forced to be a signed distance func- DRLSE formulation, we apply it to an edge-based active con- tion as a result of reinitialization. This is obviously a disagree- tour model for image segmentation. We provide a simple and ef- ment between theory and its implementation, as pointed out by ficient narrowband implementation to further improve the com- Gomes and Faugeras [25]. From a practical viewpoint, the use of putational efficiency. Due to the distance regularization term, reinitialization introduces some fundamental problems yet to be the DRLSE can be implemented with a simpler and more effi- solved, such as when and how to apply the reinitialization [25]. cient numerical scheme in both full domain and narrowband im- There are no general answers to these problems so far [23] and, plementations than conventional level set formulations. More- therefore, reinitialization is often applied in an ad hoc manner. over, relatively large time steps can be used to significantly re- Due to the previously mentioned theoretical and practical duce the number of iterations and computation time, while en- problems associated with reinitialization, it is necessary to suring sufficient numerical accuracy. pursue a level set method that does not require reinitialization. This paper is organized as follows. In Section II, we first pro- Gomes and Faugeras [25] proposed a level set formulation that pose a general variational level set formulation with a distance consists of three PDEs, one of which is introduced to restrict regularization term. In Section III, we apply the proposed gen- the LSF to be a signed distance function, and the other two of eral formulation to an edge-based model for image segmenta- which describe the motion of the zero level contour. However, tion, and describe its implementations in full domain and nar- it is not clear whether there exists a solution to this system of rowband. Experimental results are shown in Section IV. Our three PDEs in theory. Moreover, the numerical implementation work is summarized in Section V. LI et al.: DRLSE AND ITS APPLICATION TO IMAGE SEGMENTATION 3245

II. DRLSE minimum point at (it may have other minimum points). In level set methods, a contour (or more generally a hyper- We will use such a potential in the proposed variational level surface) of interest is embedded as the zero level set of an LSF. set formulation (4). The corresponding level set regularization Although the final result of a level set method is the zero level term is referred to as a distance regularization term for set of the LSF, it is necessary to maintain the LSF in a good con- its role of maintaining the signed distance property of the LSF. dition, so that the level set evolution is stable and the numerical A simple and straightforward definition of the potential for computation is accurate. This requires that the LSF is smooth distance regularization is and not too steep or too flat (at least in a vicinity of its zero level (6) set) during the level set evolution. This condition is well satisfied by signed distance functions for their unique property , which has as the unique minimum point. With this poten- which is referred to as the signed distance property. For the tial , the level set regularization term can be 2-D case as an example, we consider a signed distance func- explicitly expressed as tion as a surface. Then, its tangent plane makes an equal angle of 45 with both the -plane and the -axis, which (7) can be easily verified by the signed distance property . For this desirable property, signed distance functions have been which characterizes the deviation of from a signed distance widely used as level set functions in level set methods. In con- function. ventional level set formulations, the LSF is typically initialized The energy functional was proposed as a penalty term and periodically reinitialized as a signed distance function. In in our preliminary work [27] in an attempt to maintain the signed this section, we propose a level set formulation that has an in- distance property in the entire domain. However, the derived trinsic mechanism of maintaining this desirable property of the level set evolution for energy minimization has an undesirable LSF. side effect on the LSF in some circumstances, which will be A. Energy Formulation With Distance Regularization described in Section II-D. To avoid this side effect, we introduce a new potential function in the distance regularization term Let be a LSF defined on a domain . We define . This new potential function is aimed to maintain the signed an energy functional by distance property only in a vicinity of the zero level set, while keeping the LSF as a constant, with , at lo- (4) cations far away from the zero level set. To maintain such a pro- where is the level set regularization term defined in the file of the LSF, the potential function must have minimum following, is a constant, and is the external en- points at and . Such a potential is a double-well ergy that depends upon the data of interest (e.g., an image for potential as it has two minimum points (wells). A double-well image segmentation applications). The level set regularization potential will be explicitly defined in Section II-C. Using term is defined by this double-well potential not only avoids the side effect that occurs in the case of , but also offers some appealing (5) theoretical and numerical properties of the level set evolution, as will be seen in the rest of this paper. where is a potential (or energy density) function B. Gradient Flow for Energy Minimization . The energy is designed such that it achieves a minimum when the zero level set of the LSF is located at desired In calculus of variations [28], a standard method to minimize position (e.g., an object boundary for image segmentation ap- an energy functional is to find the steady state solution of plications). An external energy will be defined in Section III in the gradient flow equation an application of the general formulation (4) to image segmen- (8) tation. The minimization of the energy can be achieved by solving a level set evolution equation, which will be given in Section II-B. where is the Gâteaux derivative of the functional . A naive choice of the potential function is for the This is an evolution equation of a time-dependent function regularization term , which forces to be zero. Such a with a spatial variable in the domain and a temporal level set regularization term has a strong smoothing effect, but variable , and the evolution starts with a given initial it tends to flatten the LSF and finally make the zero level con- function . The evolution of the time-dependent tour disappear. In fact, the purpose of imposing the level set function is in the opposite direction of the Gâteaux regularization term is not only to smooth the LSF , but also derivative, i.e., , which is the steepest descent direc- to maintain the signed distance property , at least in a tion of the functional . Therefore, the gradient flow is also vicinity of the zero level set, in order to ensure accurate compu- called steepest descent flow or gradient descent flow. tation for curve evolution. This goal can be achieved by using The Gâteaux derivative of the functional in (5) is a potential function with a minimum point , such that the level set regularization term is minimized when . Therefore, the potential function should have a (9) 3246 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010 where is the divergence operator and is a function where is the Laplace operator. Note that the term defined by computes the mean curvature of the level contours of the function . (10) The sign of the function indicates the property of the FAB diffusion term in the following two cases: From (4) and the linearity of Gâteaux derivative, we have • for , the diffusion rate is positive, and the diffusion (14) is forward, which decreases ; • for , the diffusion rate is negative, and (11) the diffusion becomes backward, which increases . Therefore, the FAB diffusion with potential forces where is the Gâteaux derivative of the external energy to 1 to maintain the signed distance property. However, this FAB functional with respect to . Then, the gradient flow of the diffusion has an unbounded diffusion rate energy is , which goes to negative infinity as approaches 0. This may cause an undesirable side effect on the LSF when (12) is close to 0, as will be described in Section II-D. This side effect can be avoided by using a double-well potential which, combined with (9), can be further expressed as defined next, for which the diffusion rate is bounded by a constant. (13) C. Double-Well Potential for Distance Regularization

This PDE is the level set evolution equation derived from the As mentioned in Section II-A, a preferable potential function proposed variational formulation (4). We solve this PDE with for the distance regularization term is a double-well poten- the Neumann boundary condition [29] and a given initial func- tial. Here, we provide a specific construction of the double-well tion . We call the level set evolution (13) a DRLSE for its potential as intrinsic capability of preserving the signed distance property if of the LSF, which is associated with the distance regularization (16) term in (4). Reinitialization is not needed in the implemen- if tation of DRLSE due to the intrinsic distance regularization effect embedded in the level set evolution, as described in the fol- This potential has two minimum points at and lowing. The level set evolution without reinitialization in our . preliminary work [27] is a special case of the DRLSE in (13). It is easy to verify that is twice differentiable in , The distance regularization effect in DRLSE can be seen from with the first and second derivatives given by the gradient flow of the energy if if (14) and if This flow can be expressed in standard form of a diffusion if equation The function , its derivatives and , and the corresponding function , are plotted in Fig. 1. It is easy to verify that the function satisfies with diffusion rate . Therefore, the flows in (13) and (14) have a diffusion effect on the level set function . This for all (17) diffusion is not a usual diffusion, as the diffusion rate and can be positive or negative for the potential used in DRLSE. (18) When is positive, the diffusion is forward diffusion, which decreases . When is negative, the diffusion is backward diffusion, which increases . Such diffusion is Therefore, we have called a forward-and-backward (FAB) diffusion. This FAB diffusion adaptively increases or decreases to force it to be close to one of the minimum points of the potential function , thereby maintaining the desired shape of the function . For the potential defined in (6), we have which verifies the boundedness of the diffusion rate for the po- . In this case, the PDE in (13) can be expressed as tential . The sign of the function for , as plotted in Fig. 1(d), indicates the property of the FAB diffusion in the fol- (15) lowing three cases: LI et al.: DRLSE AND ITS APPLICATION TO IMAGE SEGMENTATION 3247

Fig. 2. Distance regularization effect on binary step function with the double- well potential p a p @sA. (a) Initial LSF 0 . (b) Final LSF 0 after the evolu- Fig. 1. Double-well potential p a p @sA and its ﬁrst derivative p and second tion. (c) and (d) Show a cross section of 0 and jr0j for the ﬁnal function 0, derivative p , and corresponding function d @sA are shown in (a), (b), (c), and respectively. (d), respectively.

• for , the diffusion rate is positive, and shown in Fig. 2(a). The final function of the diffusion is shown the diffusion (14) is forward, which decreases ; in Fig. 2(b), which exhibits the shape of a signed distance func- • for , the diffusion rate is tion in a band around the zero level set and a flat shape outside negative, and the diffusion becomes backward, which in- the band. We call this band a signed distance band (SDB). The creases ; width of the SDB is controlled by the constant . This is be- • for , the diffusion rate is positive, cause, when the function becomes a signed distance function and the diffusion (14) is forward, which further decrease in the SDB, its values vary from to across the band at down to zero. a rate of . This implies that the width of the SDB is The key differences between the FAB diffusions with potentials approximately . and are the boundedness of the corresponding diffusion The evolution from a binary step function to the function rate and the diffusion behavior for the case ,as of the previously described profile is described as follows. At can be seen from the previous descriptions for both cases. the zero crossing of the binary step function , the values of are larger than 1 due to a sharp jump from to . D. Distance Regularization Effect As a result, the diffusion rate is positive and, there- We demonstrate the distance regularization effect of DRLSE fore, the diffusion (14) is forward, which keeps decreasing the by simulating the FAB diffusion (14) with the initial function gradient magnitude until it approaches 1. The decreasing being a binary step function. The binary step function is of is stopped when it reaches 1, otherwise, the FAB dif- defined by fusion (14) would become backward and, therefore, increases back to 1, because the diffusion rate is negative if (19) for . As a result, the function converges to a otherwise signed distance in a band around its zero level set, i.e., the previ- where is a constant, and is a region in the domain ously mentioned SDB. On the two sides of the SDB, is ini- . Despite the irregularity of the binary step function, the FAB tially 0, which indicates that the diffusion rate . diffusion (14) is able to evolve the LSF into a function with de- Therefore, the diffusion (14) is forward, which keeps the func- sired regularity. It is worth noting that a binary step function tion flat on the two sides of the SDB. The signed distance can be generated extremely efficiently. Such initialization is de- property of the final function near the zero level set can be seen sirable in many practical applications for its computational effi- more clearly from 1-D cross sections of the and its gra- ciency and simplicity, as will be demonstrated in an application dient magnitude . Fig. 2(c) and (d) shows 1-D cross of DRLSE for image segmentation in Section III. sections of and , respectively, for .In The effect of the distance regularization with the double-well the SDB, the values of are sufficiently close to 1, as shown potential can be seen from the following numerical in Fig. 2(d). simulation of the FAB diffusion (14). We used a binary step The advantage of the double-well potential over the simple function on a 100 100 grid, as defined by (19), with the potential can be clearly seen when a binary step function is region being a rectangle. To test the regularization effect of used as the initial function. For the binary initial function, we the FAB diffusion in DRLSE, we used a large in (19) have in the interior of the regions and to create a particularly steep shape of at the zero crossing, as . Thus, the FAB diffusion with is backward 3248 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010 diffusion with an arbitrarily large diffusion rate. In this case, the the central difference scheme is more accurate and efficient strong backward diffusion drastically increases and cause than the first-order upwind scheme that is commonly used in oscillation in , which finally appears as periodic “peaks” and conventional level set formulations. “valleys” in the converged LSF. Although these “peaks” and “valleys” appear at a certain distance to the zero level set, they III. APPLICATION TO IMAGE SEGMENTATION may slightly distort the zero level contour. This undesirable side The general DRLSE formulation in (4) can be used in var- effect can be avoided by using the double-well potential , ious applications with different definitions of the external en- as described previously. ergy . For image segmentation applications, a variety of image information, including region-based or edge-based image E. Finite Difference Scheme formation, can be used to define the external energy. In this sec- The DRLSE in (13) can be implemented with a simple fi- tion, we only provide an application of DRLSE to an active con- nite difference scheme as follows. We consider the 2-D case tour model using edge-based information in the external energy, with a time dependent LSF . The spatial derivatives as a demonstration of the effectiveness of the general DRLSE and in our model are approximated by the cen- formulation. tral difference. We use fixed space steps for computing spatial derivatives. The temporal partial derivative A. Edge-Based Active Contour Model in Distance Regularized is approximated by the forward difference. The time de- Level Set Formulation pendent LSF is given in discretized form with Let be an image on a domain , we define an edge indicator spatial index and temporal index . Then, the level set evo- function by lution equation is discretized as the following finite difference equation , where is the ap- (23) proximation of the right hand side in the evolution equations. This equation can be expressed as where is a Gaussian kernel with a standard deviation . The convolution in (23) is used to smooth the image to reduce the (20) noise. This function usually takes smaller values at object boundaries than at other locations. which is an iteration process used in the numerical implemen- For an LSF , we define an energy functional tation of DRLSE. by Given spacial steps , the choice of the time step for this finite difference scheme must satisfy the Courant- (24) Friedrichs-Lewy (CFL) condition for numerical stability (see Appendix A). In practice, one can use a relatively where and are the coefficients of the energy large time step to speed up the curve evolution, in which functionals and , which are defined by case the parameter must be relatively small to meet the CFL condition. (25) It is worth noting that, although the FAB diffusion term in and DRLSE is derived from the proposed variational level set formulation in (4), we believe that it can be incorporated into more (26) general level set evolution equations that are not necessarily derived from a variational formulation. For example, consider where and are the Dirac delta function and the Heaviside a typical level set evolution equation in conventional level set function, respectively. formulation With the Dirac delta function , the energy computes the line integral of the function along the zero level contour (21) of . By parameterizing the zero level set of as a contour , the energy can be expressed as a line integral where is a scalar function and is a vector valued function. In . The energy is minimized when the standard numerical solution to this PDE, spatial derivatives are zero level contour of is located at the object boundaries. Note discretized by upwind scheme [20]. Central difference scheme that the line integral was first introduced is not stable for this PDE. by Caselles et al. as an energy of an parameterized contour in Adding the distance regularization term into the PDE, the their proposed geodesic active contour (GAC) model [11]. level set evolution in (21) becomes a DRLSE formulation The energy functional computes a weighted area of the region . For the special case , (22) this energy is exactly the area of the region . This energy is introduced to speed up the motion of the zero level Due to the added distance regularization term, all the spatial contour in the level set evolution process, which is necessary derivatives in (22) can be descretized by central difference when the initial contour is placed far away from the desired scheme, and the corresponding numerical scheme is stable object boundaries. In this paper, we use LSFs that take negative without the need for reinitialization. It is well known that values inside the zero level contour and positive values outside. LI et al.: DRLSE AND ITS APPLICATION TO IMAGE SEGMENTATION 3249

In this case, if the initial contour is placed outside the object, the the DRLSE formulation, neither reinitialization nor velocity coefficient in the weighted area term should be positive, so extension procedures are needed in the narrowband implemen- that the zero level contour can shrink in the level set evolution. tation due to the intrinsic distance regularization effect in the If the initial contour is placed inside the object, the coefficient level set evolution. The narrowband implementation of DRLSE should take negative value to expand the contour. From the level is simple and straightforward, in which the iteration process set evolution given in (30), we can see that the role of in this only consists of updating the LSF according to the difference energy term is to slow down the shrinking or expanding of equation (20) and constructing the narrowband, as described the zero level contour when it arrives at object boundaries where in the following. Moreover, the narrowband implementation takes smaller values.1 of the DRLSE model (30) allows the use of a large time step In practice, the Dirac delta function and Heaviside function in the finite difference scheme to significantly reduce the in the functionals and are approximated by the fol- number of iterations and computation time, as in its full domain lowing smooth functions and as in many level set methods implementation. [20], [30], defined by We denote by an LSF defined on a grid. A grid point is called a zero crossing point, if either and are of (27) opposite signs or and are of opposite signs. The set of all the zero crossing points of the LSF is denoted by . and Then, we construct the narrowband as (28) (31)

Note that is the derivative of , i.e., . The parameter where is a square block centered at the is usually set to 1.5. point . We can set to be the smallest value , in which With the Dirac delta function and Heaviside function in case the narrowband is the union of the 3 3 neighborhoods (25) and (26) being replaced by and , the energy functional of the zero crossing points. is then approximated by The narrowband implementation of the DRLSE consists of the following steps: Step 1) Initialization. Initialize an LSF to a function . Then, construct the initial narrowband (29) , where is the set of the zero crossing points of . This energy functional (29) can be minimized by solving the Step 2) Update the LSF. Update following gradient flow: on the narrowband as in (20). Step 3) Update the narrowband. Determine the set of all the zero crossing pixels of on , denoted by . Then, update the narrowband by setting (30) . Step 4) Assign values to new pixels on the narrowband. given an initial LSF . The first term on the right For every point in but not in , set hand side in (30) is associated with the distance regularization to if , or else set to , where energy , while the second and third terms are associated is a constant, which can be set to as a default with the energy terms and , respectively. Equation value. (30) is an edge-based geometric active contour model, which Step 5) Determine the termination of iteration. If either is an application of the general DRLSE formulation (13). For the zero crossing points stop varying for consec- simplicity, we refer to this active contour model as a DRLSE utive iterations or exceeds a prescribed maximum model in this section and Section IV. number of iterations, then stop the iteration, otherwise, go to Step 2. B. Narrowband Implementation The readers are referred to our conference paper [33] for more details in the narrowband implementation of the level set for- The computational cost of a level set method can be greatly mulation in our preliminary work [27]. reduced by confining the computation to a narrowband around the zero level set. For conventional level set formulations, C. Initialization of Level Set Function the narrowband implementation requires even more frequent The DRLSE not only eliminates the need for reinitialization, reinitialization [31] or an additional step of velocity extension but also allows the use of more general functions as the initial [32], adding further sophistication in the implementation. For LSFs. We propose to use a binary step function in (19) as the ini- 1The function g in the energy e can be replaced by a different function f tial LSF, as it can be generated extremely efficiently. Moreover, defined by other types of image information, such as region-based information, the region in (19) can sometimes be obtained by a simple and which can take positive and negative values, thereby allowing bidirectional motion (i.e., shrinking or expanding at different locations) of the contour in a single efficient preliminary segmentation step, such as thresholding, process of curve evolution. such that is close to the region to be segmented. Thus, only 3250 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010

Fig. 3. Behavior of level set evolution in the DRLSE model (upper row) and GAC model (lower row). The LSF in the DRLSE model evolves from a binary step function 0 to a function 0 with desirable regularity, as shown in (b); the LSF in the GAC model evolves from a signed distance function 0 to an LSF 0 with irregularities, as shown in (f). (a) Isocontours of the initial LSF 0 (a binary step function). (b) Converged LSF 0 of the DRLSE model. (c) Isocontours of 0 at levels PY I, 0, 1, and 2. (d) Zero level contour of the LSF 0 overlaid on the image. (e) Isocontours of the initial LSF 0 (a signed distance function). (f) LSF 0 after 50 iterations in the GAC model. (g) Isocontours of 0 at levels PY I, 0, 1, and 2. (h) Zero level contour of 0 overlaid on the image. a small number of iterations are needed to move the zero level aries, the value of should be chosen relatively small to avoid set from the boundary of to the desired object boundary. boundary leakage. As explained in Section II-D, the LSF evolves from a binary We first demonstrate the level set regularization effect of the step function to an approximate signed distance function on an distance regularization term by applying the DRLSE model (30) SDB around the zero level set, and the width of the SDB is to a synthetic image shown in Fig. 3. To show the advantage of about . In practice, the image domain is a discrete grid, and the DRLSE formulation, we also applied the well-known GAC the SDB should have at least one grid point on each side of the model [11] to this image, and compared the behaviors of the zero level contour. Therefore, we suggest that be chosen from level set evolution in the two formulations. The level set evolu- the range . In this paper, we usually set for the tion in the GAC model is definition of the binary step function in (19) as the initial LSF, unless otherwise specified. (32)

where is a constant, which plays a similar role as the parameter IV. EXPERIMENTAL RESULTS in the DRLSE model (30), and is the edge indicator function This section shows the results of the DRLSE model (30) for defined by (23). both synthetic and real images. There are parameters , and In this experiment, we set , and in this model, and the time step for the implementation. for the DRLSE model, and the time step The model is not sensitive to the choice of and , which can and for the GAC model. We generated a signed be fixed for most of applications. The choice of the parameter distance function as the initial LSF for the GAC model. is subject to the CFL condition described in Section II-E. Unless To see the behavior of the level set evolution (32) in the GAC otherwise specified, these parameters are fixed as model, we applied it without using reinitialization in this exper- , and in this paper. iment. To demonstrate the regularization effect of the DRLSE The parameter needs to be tuned for different images. A formulation, we used a binary step function defined by nonzero gives additional external force to drive the motion of , with , as the initial LSF for the DRLSE the contour, but the resulting final contour may slightly deviate model, which has discontinuity at the zero crossing. The isocon- from the true object boundary due to the shrinking or expanding tours of these two initial LSFs and at levels , effect of the weighted area term. To avoid such deviation, one 0, 1, and 2 are overlaid on the image, as shown in Fig. 3(a) and can refine the final contour by further evolving the contour for (e). a few iterations with the parameter . For images with Despite the discontinuity of the binary step function as weak object boundaries, a large value of may cause boundary the initial LSF, the DRLSE model is able to regularize the LSF, leakage, i.e., the active contour may easily pass through the ob- and thereby maintain stable level set evolution and ensure accu- ject boundary. Therefore, for images with weak object bound- rate computation. Fig. 3(b) shows the LSF of the DRLSE model LI et al.: DRLSE AND ITS APPLICATION TO IMAGE SEGMENTATION 3251 after 240 iterations. This LSF exhibits the shape of a signed distance function in a vicinity of the zero level set, and a flat shape outside the vicinity. The regularity of the LSF can be also seen from its isocontours. Fig. 3(c) shows five isocontours at levels , 0, 1, and 2 of the LSF given by the DRLSE model, with the white contour being the zero level contour and the black contours being the other four isocontours. The signed distance property of this LSF can be seen from the uniform distances between isocontours, as shown in Fig. 3(c). With an intrinsic capability of distance regularization, the DRLSE model is able to evolve Fig. 4. Comparison of the DRLSE model and the GAC model on three syn- the LSF stably, while moving the zero level set toward the de- thetic images in (a), (b), and (c). For each images, the mean errors of the DRLSE " aHXHR HXHHR sired object boundary, as shown in Fig. 3(d). model with and are shown as the two lighter bars in (d), and the GAC model with reinitialization at every five iterations and at every ten it- By contrast, the level set evolution in the GAC model con- erations are shown as the two darker bars in (d). stantly degrades the LSF, from a nice signed distance function to a function with undesirable irregularities. Fig. 3(f) shows the LSF after 50 iterations of the GAC model, denoted by , different values of are quite close to each other, with very which exhibits very steep shape in some regions and very flat similar mean errors. All the mean errors for the three images shape in other regions. The irregularities of the LSF can are below 0.5 pixels, as shown in Fig. 4(d). be clearly seen from its isocontours at levels of ,0,1, For the GAC model, we use the time step . The co- and 2, as shown in Fig. 3(g). The isocontours are densely dis- efficients of the balloon force in the GAC model (32) is set to tributed in regions where the LSF is too steep, and are sparsely . We obtained two results for each image by applying distributed in regions where the LSF is too flat. Such irregu- the GAC model: one is obtained with reinitializations at every larities of the LSF causes numerical errors in the computation five iterations, and the other with reinitializations at every ten involving the derivatives of the LSF, and further iteration of the iterations. In this experiment, the reinitialization for the GAC LSF will further degrade the LSF, which would finally destroy model is performed by solving the PDE (3) with 20 iterations the stability of the level set evolution. The degraded LSF has (more iterations may cause larger displacement of the zero level to be “repaired” by replacing it with a signed distance function contour). The mean errors of the corresponding results for the for further computation. This is why reinitialization is required three images are shown as the darker bars in Fig. 4(d). For each in conventional level set formulations as a numerical remedy to image, the two result of the GAC model with the two frequencies maintain the regularity of the LSF. In all the experiments in the of applying reinitialization are somewhat different, with the cor- following, reinitialization is applied periodically when the GAC responding mean errors larger than those of the DRLSE model. model is used for comparison with the DRLSE model. From this experiment, we can see that the errors of the GAC While reinitialization can be used as a numerical remedy model become larger when reinitialization is applied more fre- to maintain the regularity of the LSF in conventional level quently. The mean error for the image in Fig. 4(c) is larger than set formulations, it inevitably introduces additional numerical one pixel when reinitialization is applied at every five iterations. errors, which can be quite considerable if it is used inappro- This reveals a difficulty in applying reinitialization in conven- priately. This is shown in the following experiment. We apply tional level set formulations: while reinitialization is required to the DRLSE model in (30) and the GAC model on synthetic reshape the degraded LSF periodically, too frequently applying images and compare the accuracy of the segmentation results. it may cause considerable error. There is no general answer to The true object boundaries are known for the synthetic images, the problem of choosing an appropriate frequency of applying which can be used to quantitatively evaluate the accuracy reinitialization. Therefore, reinitialization is often applied in an of the segmentation results. A metric to evaluate the accu- ad-hoc manner in the implementation of conventional level set racy of a segmentation result is the mean error defined by methods. There is no such issue in the implementation of the , where is a contour as the proposed DRLSE formulation, and the numerical accuracy is segmentation result, is the true object contour, ensured by the intrinsic distance regularization on the LSF. are the points on the contour , and is the distance The DRLSE model (30) has been applied to real images. For from the point to the contour . examples, Fig. 5 shows the results of the DRLSE model for five We applied the DRLSE model and the GAC model to the real images: a CT image with a tumor (the dark area) in human synthetic images in Fig. 4(a)–(c), each with a star-shaped object, liver, an image of a pot, a microscope image of cells, an image whose boundary is known and used as the ground truth. We used of a T-shaped object, and an MR image of a human bladder. For the same initial contour for both models. In this experiment, we this experiment, we used the narrowband implementation of the use and a large time step in our model DRLSE model as described in Section III-B, with the smallest (30). It is worth noting that our method is not sensitive to the parameter in the construction of the narrowband as in choice of the parameter . This is demonstrated by applying (31). We recorded the CPU times consumed for the five images. the DRLSE model with two different values of and In this experiment, the narrowband algorithm is implemented . The mean errors of the corresponding results for by a C program in MEX format, which was compiled and run in the three images are shown as the lighter bars in Fig. 4(d). For Matlab. The CPU times were obtained by running the program each image, the two results of the DRLSE model with the two on a Lenovo ThinkPad notebook with Intel (R) Core (TM) 2 3252 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 19, NO. 12, DECEMBER 2010

Fig. 5. Results of narrowband implementation of the DRLSE model for ﬁve real images, with CPU times 0.12, 0.36, 0.09, 0.92, and 0.23 s consumed for the images from left to right. Upper row: Input images and initial contours. Lower row: segmentation results.

Fig. 6. Curve evolution in the narrowband implementation of the DRLSE model for an MR image of bladder. The initial contour, and the contours at iterations 50, 140, and 220 are shown from left to right.

Duo CPU, 2.40 GHz, 2 GB RAM, with Matlab 7.4 on Windows Fig. 7. Results of the DRLSE model in narrowband implementation using ini- Vista. The upper row of Fig. 5 shows the initial contours overlaid tialization from thresholding. Column 1: the input images. Column 2: the initial on the five images. We used the same time step and contours obtained by thresholding on the input image. Column 3: intermediate chose the parameter as 2.5, , 1.5, for these contours, after ten iterations for the upper image and one iteration for the lower image. Column 4: the final contours, after 110 iterations for the upper image images (from left to right). The corresponding results of the and five iterations for the lower image. DRLSE model are shown in the lower row. The sizes of the five images, from left to right, are 110 112, 148 202, 65 83, 192 254, and 107 180 pixels, and the corresponding CPU ically require even more frequent reinitializations or additional times consumed by our algorithm are 0.12, 0.36, 0.09, 0.92, and numerical tricks, such as velocity extension. 0.23 s, respectively. To show the curve evolution process of the As mentioned in Section III-C, the binary initial LSF can be DRLSE model, we display the zero level contours at the itera- obtained efficiently by applying thresholding or other simple tions 50, 140, and 220 for the MR image in Fig. 6. and efficient methods as preliminary segmentation on the input To compare the computation speeds of the DRLSE model and image. Fig. 7 shows the results of the DRLSE model in narrow- the GAC model, we applied them to the same images, with the band implementation for an MR image of a human brain (the same initial contours in Fig. 5. For simplicity and fair compar- corpus callosum is the object to be segmented) in the upper row ison, we use the full domain implementation of both models in and a synthetic noisy image in the lower row. We apply thresh- pure Matlab. The results of the full domain implementation of olding on the input image to obtain the region for the defi- the DRLSE model are sufficiently close to those of the narrow- nition of the initial LSF in (19). Column 2 of Fig. 7 shows band implementation shown in Fig. 5. By choosing appropriate the zero level contours of the initial level set function , and parameters for the GAC model and applying fast marching al- Columns 3 and 4 show intermediate contours and the final con- gorithm for reinitialization, we obtained similar results as those tours, respectively. In this experiment, the parameter is set to of the DELSE model shown in Fig. 5. The CPU times consumed and 0 for the images in the upper and lower rows, respec- by the DRLSE model for the five images, from left to right: 3.41, tively. It is worth noting that binary initial LSF for the syn- 15.25, 1.48, 32.67, and 8.54 s, and the corresponding CPU time thetic noisy image is highly irregular, and so are the zero level consumed by the GAC model are 22.98, 173.71, 11.34, 215.42, contours, as shown in Column 2. Despite the initial severe irreg- and 80.22 s, respectively. Obviously, the DRLSE model is sig- ularity, the LSF is quickly regularized in the level set evolution nificantly faster than the GAC model. This is mainly due to the due to the distance regularization term in the DRLSE model. fact that the DRLSE model does not need reinitialization and The regularity of the LSF is significantly improved after only allows the use of a larger time step, which significantly reduces one iteration, with more regular zero level contours as shown in the number of iterations and computation time. It is worth noting Column 3. After five iterations, the LSF exhibits desirable reg- that computational efficiency advantage of the DRLSE formu- ularity and zero level contour converges to the desired object lation over conventional level set formulations would be even boundary, as shown in Column 4. The whole iteration process more obvious in narrowband implementations, as the latter typ- only took 0.02 s for this image with 128 128 pixels. LI et al.: DRLSE AND ITS APPLICATION TO IMAGE SEGMENTATION 3253

V. C ONCLUSION The boundedness of the coefficients and al- We have presented a new level set formulation, which we lows the use of a fixed and relatively large time step in call DRLSE. The proposed DRLSE formulation has an intrinsic an finite difference scheme. This can be seen from the capability of maintaining regularity of the level set function, Courant–Friedrichs–Lewy (CFL) condition for the finite differ- particularly the desirable signed distance property in a vicinity ence schemes. From the decomposition (33), we have the CFL of the zero level set, which ensures accurate computation and condition as for fixed space stable level set evolution. DRLSE can be implemented by a steps . Note that this CFL condition appeared in simpler and more efficient numerical scheme than conventional [34] in the context of image denoising using a FAB diffusion. level set methods. DRLSE also allows more flexible and effi- This CFL condition and the boundedness of and in (34) cient initialization than generating a signed distance function imply that a fixed time step can be used in the finite difference as the initial LSF. As an application example, we have applied scheme for the diffusion equation (14) with , as long DRLSE to an edge-based active contour model for image as . If the coefficients and are not bounded, segmentation, and provided a simple and efficient narrowband then the time step (inversely proportional to and ) has to implementation of this model. This active contour model in be sufficiently small to satisfy the CFL condition and, therefore, DRLSE formulation allows the use of relatively large time steps more iterations are needed to obtain converged result. This will to significantly reduce iteration numbers and computation time, make the algorithm more time-consuming. while maintaining sufficient numerical accuracy in both full domain and narrowband implementations, due to the intrinsic ACKNOWLEDGMENT distance regularization embedded in the level set evolution. The authors would like to thank S. Zhou for discussion on Given its efficiency and accuracy, we expect that the proposed nonlinear diffusion, K. Konwar for his contribution in devel- distance regularized level set evolution will find its utility in oping the narrowband implementation of the proposed algo- more applications in the area of image segmentation, as well rithm, and J. Macione, L. He, X. Han, and M. Rousson for their as other areas where level set method has been and could be helpful comments and proofreading. applied.

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[16] C. Li, C. Kao, J. C. Gore, and Z. Ding, “Minimization of region-scal- Chenyang Xu (S’94–M’01–SM’06) received the able fitting energy for image segmentation,” IEEE Trans. Image B.S. degree in computer science and engineering Process., vol. 17, no. 10, pp. 1940–1949, Oct. 2008. from the University of Science and Technology of [17] D. Cremers, “A multiphase levelset framework for variational motion China, Hefei, in 1993 and the M.S.E. and Ph.D. segmentation,” Scale Space Meth. Comput. Vis., pp. 599–614, June degrees in electrical and computer engineering from 2003. the Johns Hopkins University, Baltimore, MD, in [18] H. Jin, S. Soatto, and A. Yezzi, “Multi-view stereo reconstruction of 1995 and 1999, respectively. dense shape and complex appearance,” Int. J. Comput. Vis., vol. 63, In 2000, he joined the Siemens Corporate Re- no. 3, pp. 175–189, July 2005. search, Princeton, NJ, as a member of technical staff. [19] J. Sethian, Level Set Methods and Fast Marching Methods. 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Control Optim., vol. 31, no. 2, pp. 439–469, Mar. 1993. [25] J. Gomes and O. Faugeras, “Reconciling distance functions and level sets,” J. Vis. Commun. Image Represent., vol. 11, no. 2, pp. 209–223, Changfeng Gui received the B.S. and M.S. degrees Jun. 2000. in mathematics from Peking University, Beijing, [26] M. Weber, A. Blake, and R. Cipolla, “Sparse finite elements for China, and the Ph.D. degree in mathematics at the geodesic contours with level-sets,” in Proc. Eur. Conf. Comput. Vis., University of Minnesota, Minneapolis, in 1991. 2004, pp. 391–404. He is currently a Professor in the Department of [27] C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without Mathematics, the University of Connecticut, Storrs. re-initialization: A new variational formulation,” in Proc. IEEE Conf. His research interests include nonlinear partial dif- Comput. Vis. Pattern Recognit., 2005, vol. 1, pp. 430–436. ferential equations, applied mathematics, and image [28] G. Aubert and P. Kornprobst, Mathematical Problems in Image processing. Processing: Partial Differential Equations and the Calculus of Varia- Dr. Gui was a recipient of Andrew Aisens- tions. New York: Springer-Verlag, 2002. dadt Prize, Centre de Recherches Mathematiques, [29] L. Evans, Partial Differential Equations. Providence, RI: Amer. Canada, in 1999, and PIMS Research Prize, Pacific Institute of Mathematical Math. Soc., 1998. Sciences, in 2002. [30] H. Zhao, T. Chan, B. Merriman, and S. Osher, “A variational level set approach to multiphase motion,” J. Comput. Phys., vol. 127, no. 1, pp. 179–195, Aug. 1996. [31] D. Adalsteinsson and J. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys., vol. 118, no. 2, pp. 269–277, May Martin D. Fox (S’69–M’73) received the B.E.E. de- 1995. gree from Cornell University, Ithaca, NY,in 1969, the [32] D. Adalsteinsson and J. Sethian, “The fast construction of extension Ph.D. Degree from Duke University, Durham, NC, in velocities in level set methods,” J. Comput. Phys., vol. 148, no. 1, pp. 1972, and the M.D. degree from University of Miami, 2–22, Jan. 1999. Coral Gables, in 1983. [33] C. Li, C. Xu, K. Konwar, and M. D. Fox, “Fast distance preserving He has published extensively in the areas of level set evolution for medical image segmentation,” in Proc. 9th Int. Medical Imaging and Biomedical Engineering. He Conf. Control Autom. Robot. Vis., 2006, pp. 1–7. has taught at the University of Connecticut, Storrs, [34] G. Gilboa, N. Sochen, and Y. Zeevi, “Forward-and-backward diffu- since 1972 where he is presently a Professor of sion processes for adaptive image enhancement and denoising,” IEEE Electrical and Computer Engineering. His research Trans. Image Process., vol. 11, no. 7, pp. 689–703, Jul. 2002. interests include ultrasound imaging and Doppler, microcontroller based devices, biomedical instrumentation, and multidimen- Chunming Li received the B.S. degree in math- sional image processing. ematics from Fujian Normal University, Fujian, China, the M.S. degree in mathematics from Fudan University, Shanghai, China, and the Ph.D. degree in electrical engineering from University of Con- necticut, Storrs, CT, in 2005. He was a Research Fellow at the Vanderbilt University Institute of Imaging Science, Nashville, TN, from 2005 to 2009. He is currently a Researcher in medical image analysis at the University of Penn- sylvania, Philadelphia. His research interests include image processing, computer vision, and medical imaging, with expertise in image segmentation, MRI bias correction, active contour models, variational and PDE methods, and level set methods. He has served as referee and com- mittee member for a number of international conferences and journals in image processing, computer vision, medical imaging, and applied mathematics.