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- 1. Narrow band region-based active contours and surfaces for 2D and 3D segmentation Julien Mille Universit´ Fran¸ois Rabelais de Tours, Laboratoire Informatique (EA2101) e c 64 avenue Jean Portalis, 37200 Tours, FranceAbstractWe describe a narrow band region approach for deformable curves and surfaces in the perspective of 2D and 3D imagesegmentation. Basically, we develop a region energy involving a ﬁxed-width band around the curve or surface. Classicalregion-based methods, like the Chan-Vese model, often make strong assumptions on the intensity distributions of thesearched object and background. In order to be less restrictive, our energy achieves a trade-oﬀ between local featuresof gradient-like terms and global region features. Relying on the theory of parallel curves and surfaces, we perform amathematical derivation to express the region energy in a curvature-based form allowing eﬃcient computation on explicitmodels. We introduce two diﬀerent region terms, each one being dedicated to a particular conﬁguration of the targetobject. Evolution of deformable models is performed by means of energy minimization using gradient descent. Weprovide both explicit and implicit implementations. The explicit models are a parametric snake in 2D and a triangularmesh in 3D, whereas the implicit models are based on the level set framework, regardless of the dimension. Experimentsare carried out on MRI and CT medical images, in 2D and 3D, as well as 2D color photographs.Key words: Segmentation, narrow band region energy, deformable model, active contour, active surface, level sets1. Introduction evolution algorithm. Conversely, implicit implementa- tions, based on the level set framework [4], handle the Segmentation by means of deformable models has been evolving boundary as the zero level of a hypersurface,a widely studied aspect of computer vision over the last deﬁned on the same domain as the image. They aretwo decades. Since their introduction by Kass et al. [1], often chosen for their natural handling of topologicaldeformable models have found many applications in im- changes and intuitive extensibility to higher dimensions.age segmentation and tracking. From an initial location, Their algorithmic complexity is a function of the imagewhich may be manually or automatically provided, these resolution, making them time-consuming. Despite themodels deform according to an iterative evolution algo- development of accelerating methods, like the narrowrithm until they ﬁt one or more structures of interest. The band technique [4] or the fast marching method [5], theirevolution method is usually derived from the minimization computational cost remains higher than their explicitof some energy functional, including regularizing terms counterparts.for geometrical smoothness and external terms relatingthe model to the data. They are powerful tools thanks Deformable models, whether they are explicit orto their ability to adapt their geometry and incorporate implicit, are attached to the image by means of a localprior knowledge about the structure of interest. edge-based energy or force. Since they consider only local boundaries, classical snakes are relatively blind, in the Several implementations of these active models were sense they are unable to reach boundaries if their initialdeveloped. Explicit deformable models represent the location is far from them. The increasing use of regionevolving boundary as a set of interconnected control terms inspired by the Mumford-Shah functional [6, 7] haspoints or vertices. Among these, the original 2D paramet- proven to overcome the limitations of uniquely gradient-ric contour and the 3D triangular mesh [2, 3] are intuitive based models, especially when dealing with data setsimplementations, in which the boundary is deformed by suﬀering from noise and lack of contrast. Indeed, manydirect modiﬁcations of vertices coordinates. The main anatomical structures encountered in medical imagingdrawback is that polygon and meshes do not modify lend themselves to region-based segmentation. Globaltheir topology naturally, i.e. techniques for detection statistical data computed over the entire region of interestof topological changes must be implemented beside the is a well established technique to improve the behaviour of snakes. Early work, including the anticipating snake by Ronfard [8] and the active region model by Ivins and Por- Email address: julien.mille@univ-tours.fr (Julien Mille) rill [9], introduced the use of region terms in the evolutionPreprint submitted to Computer Vision and Image Understanding May 18, 2009
- 2. of parametric snakes. The region competition methodby Zhu and Yuille [10] was developed later, combiningaspects of snakes and region growing techniques. Manypapers have dealt with region-based approaches using thelevel set framework, including the Chan-Vese model [11], (a) (b) (c)the deformable regions by Jehan-Besson et al. [12] andthe geodesic active regions by Paragios and Deriche [13]. Figure 1: Diﬀerent object conﬁgurations for diﬀerent region energiesThese implementations have the advantage of adaptivetopology at the expense of computational cost. In thecontext of 3D segmentation, a deformable mesh endowed of the image in medical data, the background containswith a Chan-Vese region energy was presented in [14], various anatomical structures, which diﬀer in their overallwhereas Dufour et al. [15] used an implicit active surface intensities and textures. In this context, the use of localto perform segmentation and tracking of cells, where features was already addressed in the literature. Forcomputations are particularly time-expensive. other work dealing with local statistics in region-based segmentation, the reader may refer to [18, 19, 20, 21]. Most existing region-based deformable models segment For the same purpose, active contours embedded withimages according to statistical data computed over the en- both edge and region terms were studied in [22, 23, 24]tire regions, i.e. the object of interest and the background. and extended to textured region segmentation [25]. InThese approaches have an underlying notion of homogene- cases (b) and (c), the background, made up of the ﬂoority, in the sense that image partitions should be uniform in and the plate, is now piecewise uniform. Case (b) depictsterms of intensity, whether prior knowledge on the distri- a particular situation where the background is uniform inbution of pixel intensities is available [16] or not. Instead a small band around the cup boundaries. We believe thatof raw pixel intensity, higher level features like texture de- many objects can be discriminated from the backgroundscriptors may also be considered [17]. We now focus on according to intensity features only in the vicinity of theirthe region energy of the Chan-Vese model [11]. Let Rin be boundaries, which leads to the development of our ﬁrstthe region enclosed by deformable curve Γ, and Rout its narrow band region energy. Extending the work in [26],complement. The energy penalizes the curve splitting the we formulate our energy as the intensity variance over animage into heterogeneous regions, using intensity devia- inner and an outer band around the evolving boundary.tions. In addition to length and area terms, the Chan-Vese Case (c) represents an even more general case, wheremodel has the following global data term: the outer band around the target object is piecewise C-V constant. Indeed, the cup is surrounded by the ﬂoor in Eregion [Γ] = λin (I(x)−kin )2 dx the upper half and the plate in the bottom half. The Rin (1) role of our second narrow band region energy is to handle +λout (I(x)−kout )2 dx conﬁgurations in which the outer neighborhood of the Rout target object presents several distinct areas.where kin and kout are intensity descriptors inside and In the paper, we ﬁrst describe the theoretical frame-outside the curve, respectively. By gradient descent, these work of the narrow band energy. This includes mathe-descriptors are assigned to average intensity values [11]. matical derivations to yield a suitable form for the regionAt the end of the segmentation process, region Rin is term, i.e. a formulation enabling natural implementation.expected to coincide with the target object. Hence, Our mathematical development is based on the theory ofalthough constraints on intensity deviations can be parallel curves and surfaces [27, 28]. We endeavour to de-adjusted by tuning parameters λin and λout , the global velop a framework which is applicable both to 2D and 3Dregion term is by deﬁnition devoted to segment uniform segmentation. Indeed, after describing our region termsobjects and backgrounds. Let us consider the images on a planar curve, we extend them to a deformable sur-depicted in ﬁg. 1, in which the object of interest is the face model. Then, in order to allow gradient descent af-white cup. Ignoring the inﬂuence of illumination changes, terwards, we determine the variational derivatives of thecase (a) is the typical conﬁguration which the Chan-Vese region energies with respect to the curve, thanks to calcu-model aims at, since the cup and the ﬂoor areas are nearly lus of variations, and extend them to the surface model asconstant with respect to color. well. Then, we deal with numerical implementation issues, including model structure and energy minimization. We Uniformity of intensity over regions is a rather ﬁrst present the explicit implementation, which lies in a 2Dstrong assumption. However, strict homogeneity is polygonal contour and a 3D triangular mesh. These mod-not necessarily a desirable property, especially for the els are able to perform resampling, in order to overcomebackground. The ideal case (a) is rarely encountered the lack of geometrical ﬂexibility of traditional snakes andin most of computer vision applications. For instance, meshes. We also provide a level-set implementation, whichwhen one wants to isolate a single structure from the rest 2
- 3. oﬀers the advantage of a common mathematical descrip- Γ[−B]tion in 2D and 3D, in addition to the topological adapt- Γability. Finally, experiments are carried out on medicaldata and natural color images. For both explicit and im- Γ[B]plicit implementations, the tests discuss the advantages ofour narrow band terms over other data terms includingedge energies and global region energies.2. Active contour model Bin2.1. Energies Bout The continuous active contour model is represented asa parameterized curve Γ with position vector c: Figure 2: Inner and outer bands for narrow band region energy Γ: Ω −→ R2 (2) u −→ c(u) = [x(u) y(u)]T Let Bin be the inner band domain and Bout the outerwhere x and y are continuously diﬀerentiable with respect band domain (see ﬁg. 2), and B the band thickness, whichto parameter u. The parameter domain is normalized: is constant as we move along Γ. We propose two diﬀerentΩ = [0, 1]. We assume that the curve is simple, i.e. non- region energies. Thus, in eq. (3), the region term will beintersecting, and closed: c(0) = c(1). Segmentation of either Eregion 1 or Eregion 2 . To obtain Eregion 1 , we consideran object of interest is performed by ﬁnding the curve Γ eq. (1) and we replace Rin with Bin and Rout with Bout ,minimizing the following energy functional: which yields: E[Γ] = ωEsmooth [Γ] + (1 − ω)Eregion [Γ] (3) Eregion 1 [Γ]= (I(x)−kin )2 dx + (I(x)−kout )2 dx (5)where Esmooth and Eregion are respectively the smooth- Bin Boutness and region energies. The user-provided coeﬃcientω weights the signiﬁcance of the smoothness term. We Increased ﬂexibility is achieved thanks to the narrow bandexpress the smoothness energy in terms of ﬁrst-order principle, since it does not convey a strict homogeneityderivative, as it appears in the original snake model by condition like classical region-based approaches. The sec-Kass et al. [1]: ond energy is a generalization of the ﬁrst one. Its purpose is to handle cases where the background is locally homo- dc 2 geneous in the vicinity around the object (see ﬁg. 1c). For Esmooth [Γ] = du (4) now, we express it using a local outer descriptor hout de- Ω du pending on current position x:The ﬁrst-order regularization term usually prevents thecontour to undergo large variations of its area. In our Eregion 2 [Γ]= (I(x)−kin )2 dx + (I(x)−hout (x))2 dx (6)case, it is a non desirable property, since the contour will Bin Boutbe initialized as a small shape inside a target object andinﬂated afterwards. Once discretized as a polygon, the In eq. (1), one may note that the Chan-Vese regioncontour is periodically reparameterized to keep control term is asymmetric, as region integrals are independentlypoints practically equidistant and to allow inﬂation. In weighted in order to favour minimization of intensity devi-this context, resampling and remeshing techniques are ation inside or outside. However, we use symmetric termsdiscussed in section 5. in our approach, as it is the most common case with region- based active contours. In what follows, we show in what Curve Γ splits the image domain D into an inner extent the narrow band principle allows easier implemen-region Rin and an outer region Rout , over which the tation than classical region-based approaches.homogeneity criterion is usually expressed. The narrowband principle, which has proven its eﬃciency in the 2.2. Parallel curvesevolution of level sets [4], is used in our approach to The theoretical background of our narrow band frame-formulate a new region term. Instead of dealing with the work is based on parallel curves, also known as ”oﬀsetentire domains delineated by the evolving curve, we only curves” [27, 28]. The curve Γ[B] is called a parallel curveconsider an inner and outer band both sides apart from of Γ if its position vector c[B] veriﬁesthe curve, as depicted in ﬁg. 2. One may note that the c[B] (u) = c(u) + Bn(u) (7)bands are not limited to the snake’s initial location andare updated during curve evolution. where B is a real constant, corresponding to the amount of translation, and n in the inward unit normal of Γ. 3
- 4. An important property resulting from the deﬁnition in eq. (7) is that the velocity vector of parallel curves de- pends on the curvature of Γ. The velocity vector of curve Γ[B] is expressed as a function of the velocity vector of Γ, as well as its curvature and normal. Using the identity nu = − κcu , we have: c[B] u = cu + Bnu = (1 − Bκ)cu (11) which yields, for the length element of inner parallel curve: Γ[B1 ] Γ[B2 ] Γ ℓ[B] = c[B] u = ℓ |1 − Bκ| The same development is valid for Γ[−B] , replacing B with −B. This is a known result in parallel curve the-Figure 3: Main curve (black) and two parallel curves. Small trans- ory [32, 33]. The expressions of ℓ[B] and ℓ[−B] suggest thelation B1 yields regular curve (blue) whereas large translation B2 smoothness condition of curves Γ[B] ans Γ[−B] . Indeed,yields a curve with singulaties (red). Corresponding points on par- their length elements should remain strictly positive. Thisallel curves are linked with dashed lines. implies a constraint on the maximal curvature of curve Γ, i.e. the band width should not exceed the radius of cur- vature. We should assume that Γ is smooth enough suchHereafter, we will use the index [B] to denote all quantities that:related to the parallel curve. The deﬁnition in eq. (7) 1 1 − < κ(u) < , ∀u ∈ Ω (12)is suitable to our narrow band formulation, in the sense B Bthat bands Bin and Bout are bounded by parallel curves If condition 12 is well veriﬁed, curves Γ[B] and Γ[−B] areof Γ, respectively Γ[B] and Γ[−B] . This implies that both simple and regular. The impact of this assumption oncurves are continuously diﬀerentiable and do not exhibit numerical implementation is discussed in section 5.singularities. Fig. 3 depicts a case where width B2 ,unlike B1 , is larger than the curve’s radius of curvature, 2.3. Transformation of area integralyielding singularities (also known as cusps). Afterwards, In this section, we show that the domain integrals ap-we refer to the eroded inner region by Rin[B] , bounded pearing in eq. (5) can be expressed in terms of c and B.by Γ[B] , and the dilated inner region by Rin [−B] bounded This conversion is mandatory for the calculation of theby Γ[−B] . variational derivative of Eregion with respect to c. More- over, it brings a formulation suitable for implementa- Before introducing our simpliﬁcation, let us recall the tion on explicit models. The proof is based on Green-notion of line integral. Given a real-valued function f de- Riemann theorem, stating that for every region R, ifﬁned over R2 and a domain D ⊂ R2 , we introduce the [P (x, y) Q(x, y)]T is a continuously diﬀerentiable R2 → R2general notation J(f, D) representing the integral of f over vector ﬁeld, then:domain D. If D is a region R, J(f, R) is an area integral ∂Q ∂Pwhereas if D is a curve Γ, J(f, Γ) is written as a line inte- − dxdy = P dx + Qdy ∂x ∂y ∂Rgral: R dc J(f, Γ) = f (c(u)) du (8) In order to apply the theorem on J(f, R), where f is a Ω du real-valued function deﬁned on the image domain D, onewhere the length element (or velocity) should determine vector ﬁeld [P Q] such that dc ∂Q ∂P ℓ= (9) − = kf (x, y) du ∂x ∂y where k is a real constant. By choosing P and Q as follows,makes J(f, Γ) intrinsic, i.e. independent of the param- the previous condition is satisﬁed:eterization. This idea was ﬁrst introduced in deformable xmodels with the geodesic active contour model [29, 30, 31]. 1 Q(x, y) = f (t, y)dtFrom now on, we will use indexed notations for derivatives: 2 −∞ 1 y (13) dc d2 c P (x, y) = − f (x, t)dt cu = , cuu = 2 ... (10) 2 −∞ du du Hereafter, we will rely on the following equation to trans-The curvature of Γ is: form region integrals: xu yuu − xuu yu xu yuu − xuu yu κ(u) = = J(f, R) = P dx + Qdy (14) (x2 u + 2 3 yu ) 2 ℓ3 ∂R 4
- 5. Γ1 Γ1 2.4. Region energies B Thanks to the previous result, we now express our two Γ2 Γ2 narrow band region energies in terms of contour, curvature and band thickness. The ﬁrst narrow band region energy, which aims at minimizing the intensity deviation in the two bands, is rewritten: B (a) (b) Eregion 1 [Γ] = (I(c+bn) − kin )2 ℓ(1−bκ)dbduFigure 4: Region enclosed by two simple closed curves Γ1 and Γ2 (a) B Ω 0 (18)split into inﬁnitesimal quadrilaterals (b) + (I(c−bn) − kout )2 ℓ(1+bκ)dbdu Ω 0 For the second narrow band region energy, intensity devi- Let us consider the more general case of a band B ation should be minimized in the outer band locally alongbounded by curves Γ1 and Γ2 , as depicted in ﬁg. 4a. Rely- the curve. Hence, we replace global descriptor kout ofing on eq. (14), the integral of f over B is expressed using eq. (18) by a local counterpart, which is now a function ofGreen’s theorem: the position on the curve: J(f, B) = J(f, R1 ) − J(f, R2 ) B Eregion 2 [Γ] = (I(c+bn) − kin )2 ℓ(1−bκ)dbdu = x1u P (c1 ) + y1 u Q(c1 )du (15) B Ω 0 (19) Ω + (I(c−bn) − hout (u))2 ℓ(1+bκ)dbdu − x2u P (c2 ) + y2 u Q(c2 )du Ω 0 Ω Up to now, we have used intensity descriptors without ex- ˜We introduce a family of curves {Γ(α)}α∈[0,1] interpolating plicitly providing their expressions. They may be consid- ˜from Γ1 to Γ2 . The position vector of Γ is ered as unknowns which will be determined during energy minimization. Their values will be determined by calculus c(α, u) = (1 − α)c2 (u) + αc1 (u) ˜ of variations of the energies, described in section 4.Relying on the following equality, 1 3. Active surface model d c1 − c2 = (1 − α)c2 + αc1 dα 0 dα 3.1. Energiesand using integration by parts, we transform eq. (15) and The active contour method approach naturally extendsshow that J(f, B) can be directly expressed as a function to a three dimensional segmentation problem. In a con-of f , c1 and c2 (a detailed derivation is provided in ap- tinuous space, a deformable model is represented by a pa-pendix A.1). rameterized surface Γ. 1 Γ : Ω2 −→ R3 J(f, B) = f (˜)(c1 − c2 ) × cu dαdu c ˜ (16) (u, v) −→ s(u, v) = [x(u, v) y(u, v) z(u, v)]T Ω 0This expression is intuitively understood since (1 − α)c2 + In all subsequent derivations, we will assume a closed sur-αc1 sweeps all curves between Γ1 and Γ2 as α varies from 0 face with a parameterization homeomorphic to a torus:to 1. The cross product corresponds to the area of the in-ﬁnitesimal quadrilaterals spanned by (c1− 2 ) and cu , as de- c ˜ s(0, v) = s(1, v) ∀v ∈ Ω (20)picted in ﬁg. 4b. Relying on parallel curves, the mathemat- s(u, 0) = s(u, 1) ∀u ∈ Ωical deﬁnition of bands Bin allows us to express J(f, Bin ) or a sphere:in a convenient form. We apply the general result ineq. (16) on inner band Bin , considering curves Γ and Γ[B] s(0, v) = s(1, v) ∀v ∈ Ωinstead of Γ1 and Γ2 . Using a variable thickness b (see s(u1 , 0) = s(u2 , 0) ∀(u1 , u2 ) ∈ Ω2 (21)appendix A.2 for more details), we ﬁnally obtain: s(u1 , 1) = s(u2 , 1) ∀(u1 , u2 ) ∈ Ω2 B J(f, Bin ) = f (c + bn)ℓ(1 − bκ)dbdu (17) Note that these parameterizations are given only for math- Ω 0 ematical transformation purpose and do not generate anyThe formulation for J(f, Bout ) is easily obtained by re- constraint on the topology of the surface once this lastplacing b with −b in eq. (17). This expression is especially one is discretized. Hence, they do not restrict numericaluseful when the curve is discretized as a polygonal line, as implementation. The surface is endowed with the energydescribed in section 5. 5
- 6. functional E. Replacing c by s in eq. (3), we obtain the The gaussian curvature κG and mean curvature κM maysurface energy to be minimized. The smoothness term is: be expressed in terms of coeﬃcients of the fundamental 2 2 forms: ∂s ∂s LN − M 2 Esmooth [Γ] = + dudv (22) κG = ∂u ∂v EG − F 2 Ω2 GL − 2F M + EN κM =Considering now that image I is a R3 → R function, the 2(EG − F 2 )narrow band region energy is a function of the volume Normal derivatives nu and nv are orthogonal to n. In theintegrals over the two bands Bin and Bout : tangential plane at point s(u, v), they can be expressed as combinations of basis vectors su and sv according to the Eregion 1 [Γ] = (I(x) − kin )2 dx Weingarten equations [34]: Bin (23) + (I(x) − kout )2 dx F M − GL F L − EM nu = su + sv Bout EG − F 2 EG − F 2 (26) F N − GM F M − ENand similarly for Eregion 2 . As in the two dimensional case, nv = su + sv EG − F 2 EG − F 2terms deﬁned over bands are not computed as is. Theyshould undergo some mathematical transformation in or- which lead to the following combinations, holding meander to be diﬀerentiated and implemented. This is done and gaussian curvatures:through the framework of parallel surfaces described in nu ×sv + su ×nv = −2κM su ×svthe next section. (27) nu ×nv = κG su ×sv3.2. Parallel surfaces An important property, resulting from eq. (27), is that the In three dimensions, regions Bin and Bout are bounded normal vector of parallel surface Γ[B] is colinear to theby Γ and its parallel surfaces Γ[B] and Γ[−B] , respectively. normal vector of Γ. Its magnitude is a function of theAs an example, if Γ describes a sphere, Bin and Bout may mean and gaussian curvatures of Γ:be viewed as two empty balls with thickness B. s[B] u ×s[B] v = (su + Bnu ) × (sv + Bnv ) (28) s[B] (u, v) = s(u, v) + Bn(u, v) (24) = (1 − 2BκM + B 2 κG )su ×svand similarly for s[−B] . As previous, B is the constant Considering the magnitude of the previous vector, we ob-band thickness and n(u, v) is the unit inward normal: tain the area element of the parallel surface: su × sv a[B] = s[B] u ×s[B] v n(u, v) = (29) su × sv = a 1 − 2BκM + B 2 κGA surface integral of f over Γ is which will be useful for expressing the simpliﬁed form of ∂s ∂s the narrow band region energy described below. J(f, Γ) = f (s(u, v)) × dudv ∂u ∂v Ω2 3.3. Transformation of volume integral Volume integrals can be converted to surface integralswhere the area element thanks to the divergence theorem, also known as Green- ∂s ∂s Ostrogradski’s theorem. For every volumic region R, given a(u, v) = × (25) ∂u ∂v F(x) = [P (x) Q(x) R(x)]T a continuously diﬀerentiablemakes the surface integral J(f, Γ) independent of the pa- R3 → R3 vector ﬁeld, we have:rameterization. In accordance with our mathematicalderivations in the previous section, we demonstrate how div(F) dV = F, N dA (30)the normal vector of parallel surface can be expressed as R ∂Ra function of su ×sv . Moreover, we show how the various where dA and dV are the diﬀerential area and volumesurface curvatures intervene in this expression. To express elements, respectively. N is the surface outward normal.surface curvature, we introduce basic elements of diﬀeren- The divergence of vector ﬁeld F is:tial geometry [34, 33]. E, F and G are the coeﬃcients of ∂P ∂Q ∂Rthe ﬁrst fundamental form, whereas L, M and N are the div(F) = + + (31)coeﬃcients of the second fundamental form. At a given ∂x ∂y ∂zsurface point s(u, v), we have If the boundary ∂R is parameterized by s(u, v), the surface integral can be written: E = su , su F = su , sv G = sv , sv L = − nu , su = n, suu ∂s ∂s F, N dA = − F(s(u, v)), × dudv M = − nu , sv = − nv , su = n, suv ∂u ∂u ∂R Ω2 N = − nv , sv = n, svv (32) 6
- 7. where the negative sign appears since n(u, v) is the unit 4. Calculus of variationsinward normal. To convert the volume integral of f intoa surface integral, one should ﬁnd F such that div(F) = Image segmentation is performed through numericalf . This condition is veriﬁed by choosing P , Q and R as minimization of the energy functional using gradient de-follows: scent. The negative discretized variational derivative of 1 x the energy term is usually considered for the descent direc- P (x, y, z) = f (t, y, z)dt tion. In this section, we express the variational derivatives 3 0 1 y of the energies, especially focusing on the region terms, for Q(x, y, z) = f (x, t, z)dt (33) both contour and surface. 3 0 z 1 R(x, y, z) = f (x, y, t)dt 4.1. Active contour 3 0 Let us consider a general energy term E, depending onIn what follows, we demonstrate how we can express the the curve position c and its successive derivatives:3D region energy in terms of surface integrals. The deriva-tion is similar in philosophy to the 2D case, since our 3Dscheme is also based on curvature. As in the 2D section, E[Γ] = L(c, cu , cuu ) du Ωwe consider a general case of a volumic band B boundedby surfaces Γ1 and Γ2 . The variational derivative of the energy with respect to the curve can be computed thanks to calculus of variations [1]: J(f, B) = J(f, R1 ) − J(f, R2 ) ˜This theorem is based on a family of surfaces Γ(α) δE ∂L d ∂L d2 ∂L 0≤α≤1 = − + 2 ∂c (37) δΓ ∂c du ∂cu du uuwith position vector: ˜(α, u, v) = (1 − α)s1 (u, v) + αs2 (u, v) s According to the Euler-Lagrange equation, if curve Γ is a local minimizer of E, the previous variational derivativeUsing the divergence theorem in eq. (32), the volume in- vanishes. Curve evolution is achieved by iterative solvingtegral over the region bounded by two surfaces Γ1 and Γ2 of the Euler-Lagrange equation, by means of gradient de-can be expressed as follows (details of the proof are given scent. It is more convenient to calculate the variationalin appendix A.3): derivative of each energy. From eq. (3), we have: 1 J(f, B) = f (˜) s2 − s1 , ˜u ×˜v dαdudv s s s (34) δE δEsmooth δEregion 0 =ω + (1 − ω) Ω2 δΓ δΓ δΓIn the previous expression, the scalar triple product is the The derivative of the smoothness term is well known [1],volume of the parallelepiped spanned by vectors (s2 − s1 ), since eq. (37) is easily applicable on Esmooth :˜u and ˜v . We apply this general result in our case, wheres sΓ1 = Γ and Γ2 = Γ[B] . Given the area element of parallel δEsmooth d2 csurface in eq. (29), we write the ﬁnal approximation of the = −2 2 (38) δΓ duvolume integral: As regards the ﬁrst narrow band region energy, it is more J(f, Bin ) = conveniently diﬀerentiated when expressed with integrals B f (s+bn) su ×sv (1−2bκM +b2 κG )dbdudv (35) over Rin and its related regions, rather than over bands. 0 Therefore, the inner band term is split between Rin and the Ω2 eroded inner region Rin [B] , whereas the outer band termThe transformation from eq. (34) to eq. (35) is detailed is split between Rin and the dilated inner region Rin[−B] ,in appendix A.4. Again, the volume integral over outer which leads to the following variational derivative:band Bout is obtained by replacing b with −b. The ﬁrstnarrow band region energy is found by replacing adequates δEregion 1 =quantities in eq. (18): δΓ B δJ (I−kin )2 , Rin δJ (I−kin )2 , Rin [B] (39) Eregion 1 [Γ] = a[b] (I(s[b] )−kin )2 dbdudv + − δΓ δΓ Ω2 0 δJ (I−kout )2 , Rin[−B] δJ (I−kout )2 , Rin B (36) + − δΓ δΓ + a[−b] (I(s[−b] )−kout )2 dbdudv 0 In this way, region terms are transformed using Green’s Ω2 theorem and subsequently derived. From the appendixwhere area elements a[b] and a[−b] should be expanded ac- in [10], we have:cording to eq. (29). The explicit form of the second energymay be obtained by replacing kout with surface-dependent δJ(f, Rin )local descriptor hout (u, v). = −ℓf (c)n (40) δΓ 7
- 8. In appendix B, we develop the calculation of the varia- The local outer descriptor function hout is determined bytional derivative of the general term J(f, Rin[B] ), which solving another Euler-Lagrange equation:results in: δEregion 2 δJ(f, Rin[B] ) =0 = −ℓ(1 − Bκ)f (c[B] )n δhout δΓ which yields the average weighted intensity along the out-Its counterpart on the dilated region Rin [−B] is obtained ward normal line of length B, at a given contour point:by replacing B with −B in eq. (67). This eventually leads Bto: ℓ(1 + bκ)I(c − bn)db 0 δEregion 1 hout (u) = B = ℓ[ δΓ ℓ(1 + bκ)db −(I(c)−kin )2 + (1−Bκ)(I(c[B] )−kin )2 (41) 0 −(1+Bκ)(I(c[−B] )−kout )2 + (I(c)−kout )2 n According to the previous deﬁnition of hout , we assume that piecewise constancy over the outer band is veriﬁed ifThe energy should also be minimized with respect to inten- intensity is uniform along ﬁnite length lines in the direc-sity descriptors kin and kout . These are found by solving tion normal to the object boundary. For a given point on the contour, the length element is constant and may be ∂Eregion 1 ∂Eregion 1 omitted, which reduces the mean intensity to: =0 and =0 ∂kin ∂kout B 2which yield average intensities on the inner and outer hout (u) = (1 + bκ)I(c − bn)db (45) B(2 + Bκ) 0bands: B 4.2. Extension to the surface 1 kin = I(c+bn)ℓ(1−bκ)dbdu The general energy term depending on surface position |Bin | Ω 0 B (42) as well as its u and v-derivatives, 1 kout = I(c−bn)ℓ(1+bκ)dbdu |Bout | Ω 0 E[Γ] = L(s, su , sv , suu , suv , svv ) dudvBand areas |Bin | and |Bout | are expressed by considering Ω2eq. (17) with f (x) = 1: has the following variational derivative [35]: 2 B |Bin | = ℓ B− κ du δE ∂L d ∂L d ∂L Ω 2 = − − 2 (43) δΓ ∂s du ∂su dv ∂sv B |Bout | = ℓ B+ κ du d2 ∂L d2 ∂L d2 ∂L Ω 2 + 2 ∂s + + 2 du uu dudv ∂suv dv ∂svvThe derivative in eq. (41) holds the term The variation of the smoothness term is straightforward(I(c) − kout )2 − (I(c) − kin )2 , which is clearly in ac- to calculate and is a function of the laplacian:cordance with the region-based segmentation principle.Indeed, the sign of the above quantity depends on the δEsmooth ∂2s ∂2slikeness of the current point’s intensity with respect to kin = −2 + 2 (46) δΓ ∂u2 ∂vor kout . If I(c) is closer to kin than kout , the contourwill locally expand, as it would be the case with a region As in the 2D case, it is practical to diﬀerentiate the regiongrowing approach. Moreover, one may note that this term term when formulated in terms of integrals over Rin , Rin [B]is also found in the Chan-Vese region-based method. The and Rin[−B] . Hence, a similar derivation as in eq. (39) isderivative holds additional curvature-dependent terms performed. A detailed calculation of the variational deriva-which are addressed in section 5. tive of a general term J(f, Rin ) may be found in the ap- pendix of [36]. It gives: We now deal with the second region energy. Fromeq. (41), we extrapolate a consistent variational derivative δJ(f, Rin ) = −af (s)nof the second narrow band region term. We obtain: δΓ δEregion 2 The variation of the term expressed on the eroded inner ≈ ℓ[ region is extended from appendix B. In particular, the δΓ (44) ﬁnal result of eq. (67) gives: −(I(c)−kin )2 + (1−Bκ)(I(c[B] )−kin )2 δJ(f, Rin [B] ) −(1+Bκ)(I(c[−B] )−hout )2 + (I(c)−hout )2 n = −a(1 − 2BκM + BκG )f (s[B] )n δΓ 8
- 9. and similarly on the dilated inner region. Replacing the pj pjcurvature-dependent terms of eq. (41) and (44), this even- αijtually leads to: θij δEregion 1 βij = a −(I(s)−kin )2 + (I(s)−kout )2 pi pi δΓ +(1−2BκM +B 2 κG )(I(s[B] )−kin )2 (47) −(1+2BκM +B 2 κG )(I(s[−B] )−kout )2 nMinimizing Eregion 1 with respect to intensity descrip- Figure 5: Angles in the neighborhood of pi for discrete mean and gaussian curvature estimationtors kin and kout , we end up with average intensities: B 1 kin = a[b] I(s[b] ) dbdudv where ptk , k = 1, 2, 3 are the vertices of triangle t. In |Bin | 0 a given triangle, vertex indices should be sorted so that Ω2 1 B the normal vector points towards the inside of the sur- kout = a[−b] I(s[−b] ) dbdudv face. Since the iterative evolution algorithm described |Bout | 0 Ω2 below modiﬁes vertex coordinates, all normals should be updated after each iteration (when all vertices have beenThe derivative of Eregion 2 may be obtained from eq. (47), moved). For the contour, the discretized length element ℓreplacing kout with local descriptor hout (u, v). As in the 2D associated to pi iscase, minimizing Eregion 2 with respect to hout , we obtainthe average intensity along outer normal line segment at pi − pi−1 + pi − pi+1surface point s (u, v): ℓi = 2 hout (u, v) = For the mesh, to compute the area element associated to B pi , we use the sum of areas of its neighboring triangles: 3 (48) I(s[−b] )(1+2bκM +b2 κG )db 3B(1 + B) + B 3 0 |Ni | (pi − pNi [k] ) × (pi − pNi [k+1] )Once the ﬁrst variations of the smoothness and region Ai = 2terms are known, we are able to perform gradient descent k=1of the discretized energy over explicit implementations of The area element is simply ai = Ai /3. The sum of areathe curve and surface. elements equal to the sum of triangle areas, which is itself the total mesh area. To estimate the mean and gaussian5. Implementation on explicit models curvatures, we use the discrete operators described in [38] and [39].5.1. Polygon and mesh To describe the discrete forms of active 2D contour and 13D surface models simultaneously, we introduce a general κM i = (cot αij + cot βij )(pi − pj ) 4Aiframework. The contour is a discrete closed curve, whereas j∈Ni the surface model is a triangular mesh built by subdividing 1 an icosahedron [37]. The models have a constant global κG i = 2π − θij Aitopology, their initial shape being circular and spherical, j∈Nirespectively. Both are made up of a set of n vertices,denoted pi = [xi yi ]T in 2D and pi = [xi yi zi ]T in 3D. where αij , βij and θij are the angles formed by pi , pj andEach vertex pi has a set of neighboring vertices, denoted their common neighboring vertices, as shown in ﬁg. 5.Ni . In the 2D contour, index i is the discrete equivalentof the curve parameter, hence Ni = {i − 1, i + 1}. For 5.2. Reparameterizationthe mesh, Ni is sorted in such a way that the k th and To maintain a stable vertex distribution along the 2D(k + 1)th neighbors of pi are also neighbors between them. contour (or 3D surface), adaptive resampling (or remesh-While the computation of tangent and normal vectors is ing) is performed [3]. The contour is allowed to add orstraightforward on the polygon, computing normals on the delete vertices to keep the distance between neighboringmesh needs some explanation. The normal of vertex pi is vertices homogeneous. It insures that every couple ofthe mean computed over the normals of the neighboring neighbors (pi , pj ) satisﬁes the constraint:triangles [3]. The normal nt of a given triangle is thenormalized cross product between two of its edges. w ≤ pi − pj ≤ 2w (50) (pt2 − pt1 ) × (pt3 − pt1 ) where w is the sampling between consecutive vertices. nt = (49) Resampling the 2D contour is simple: when the distance (pt2 − pt1 ) × (pt3 − pt1 ) 9
- 10. step ∆t: (t+1) (t) pj pj pi = pi + ∆tf (pi ) (52) where f (pi ) is the force vector, expressed in terms of thepa pb pa pn+1 pb pa pb discretization of the energy derivative at a given vertex pi : pi pi pi δE f (pi ) = − δΓ c=pi = ωfsmooth (pi ) + (1 − ω)fregion (pi )Figure 6: Remeshing operations on the triangular mesh: vertex in-serting (middle) and deleting (left) We ﬁrst consider the region force fregion (the smoothness force is studied in the next section). To compute band areas and means on the polygonal contour, we apply the discretization templates in eq. (51) on expressions of areasbetween pi − pi+1 exceeds 2w, the line segment is split in eq. (43) and intensity means in eq. (42). Similarly, quan-by creating a new vertex at coordinates (pi + pi+1 )/2. tities in the 3D region energy in eq. (36) are implemented.When pi − pi+1 gets below w, vertex pi+1 is deleted For the ﬁrst narrow band region energy, the correspondingand pi is connected to pi+2 . force on the contour is: Active surface remeshing is described in [3] and [14]. fregion1 (pi ) = (I(pi )−kin )2 − (I(pi )−kout )2 ni (53)While resampling the 2D contour is rather straightforward,remeshing the 3D active surface is carefully performed. In addition to the squared diﬀerences between I(c) andAdding or deleting vertices modiﬁes local topology, thus the average band intensities, the variational derivative intopological constraints should be veriﬁed. Let us consider eq. (41) also contains curvature-based terms depending onthe couple of neighbors (pi ,pj ). To perform vertex adding the intensity at points c[B] and c[−B] . Actually, theseor deleting, pi and pj should share exactly two common terms turn out to go against the region growing or shrink-neighbors, denoted pa and pb . When pi − pj > 2w, a ing principle, as they oppose the other terms dependingnew vertex is created at the middle of line segment pi pj on I(c). As stated in [40], the usual energy gradient mayand connected to pa and pb (see middle part of ﬁg. 6). not be consistently the best direction to take, which jus-When pi − pj < w, pj is deleted and pi is translated to tiﬁes our choice to remove side eﬀect terms. Moreover,the middle location (see right part of ﬁg. 6). Vertex merg- by doing so, we keep the same evolution principle as theing prevents neighboring vertices from getting too close, Chan-Vese region term. In a similar way, the force result-which might result in vertex overlapping and intersections ing from the second narrow band region energy is:between triangles. Adding vertices allows the contour and fregion2 (pi ) = (I(pi )−kin )2 − (I(pi )−µNL (pi ))2 nisurface to inﬂate signiﬁcantly while keeping a suﬃcient (54)vertex density. with5.3. Energy minimization κi (B + 1) b=B We give the discrete forms of quantities used in the µNL (pi ) = B 1 + (1 + bκi )I(pi − bni ) 2region energies. Over Bin , area integrals are computed b=1according to the following template formula, which is a 5.4. Gaussian ﬁlteringdiscrete implementation of eq. (17): Minimizing the regularization term boils down to apply n B−1 laplacian smoothing of the contour - see eqs. (38) and (46) J(f, Bin ) ≈ f (pi + bni )ℓi (1 − bκi ) (51) - which is formalized by the following PDE: i=1 b=0 ∂c ∂2cwhere ℓi , ni and κi are the discretized length element, nor- =α 2 ∂t ∂umal and curvature at vertex pi , using ﬁnite diﬀerences.A similar computation is performed over the outer band, where α ∈ [0, 1/2]. When discretized, laplacian smoothingin which b varies from −B to −1. There are two com- only intervenes in the direct neighborhood of vertices andplementary techniques to address the regularity condition is consequently limited in space. However, highly noise-in eq. (12). The ﬁrst one is to prevent each vertex from corrupted data require very strong regularity constraintmaking a sharp angle with its neighbors, so that its cur- on the contour. Should the regularization be insuﬃcient,vature κi is well bounded. Moreover, the case of a neg- the contour is exposed to unstable behaviour. Hence, toative length element can be handled. Hence, in eq. (51), achieve more diﬀuse regularization, the contour is con-ℓi (1−bκi ) is actually computed as max(0, ℓi (1−bκi )) and volved with a gaussian kernel of zero mean and standardsimilarly on the outer band. Vertex coordinates are itera- deviation σ. Regularization often comes with curve shrink-tively modiﬁed using gradient descent of eq. (3) with time age, as studied in [41]. To avoid this unwanted eﬀect, we 10
- 11. use the two-pass method of Taubin [42]. This consists in 5.6. Implementation of Green’s and divergence theoremsperforming the gaussian smoothing twice, ﬁrstly with a Our experiments, described in section 7, include apositive weight and secondly with a negative one. The comparison between segmentation results obtained withforce, resulting from the ﬁrst pass, applied on a given ver- our narrow band region terms and the ones obtainedtex is with the Chan-Vese region energy in eq. (1). The imple- k=η mentation of the latter on the explicit polygon and mesh 1 k2 raises the diﬃculty of computing region integrals. The fsmooth (pi ) = √ exp − 2 pi+k − pi σ 2π k=−η 2σ direct solution consists in using region ﬁlling algorithms (55) to determine inner pixels, as in [9] and [14], which wouldwhere η is the rank of neighborhood. One usually admits be computationally expensive if performed after eachthat the value of the gaussian distribution is nearly zero deformation step. Another solution, which we chose,beyond 3σ, we choose η = 3σ. This force is used instead is based on an discretization of Green-Riemann andof a discretization of the variational derivative in eq. (38). Green-Ostrogradski theorems in 2D and 3D, respectively.A similar smoothing is performed on the deformable meshwhen working with 3D data. In eq. (55), the k th neigh- In the 2D case, we consider Green’s theorem as statedbors of pi should then be replaced with successive rings of in eqs. (13) and (14). For instance, a brute-force imple-neighbors - the ﬁrst one being Ni - around pi . The choice mentation of the integral J(I, Rin ) on the polygon wouldof standard deviation σ has a substantial impact on the yield:ﬁnal segmentation result and is discussed in section 7. n xi 1 yi+1 − yi−1 J(I, Rin ) ≈ I(k, yi )5.5. Bias force 2 i=1 2 k=0 yi In a particular case, the formulation of fregion presents xi+1 − xi−1 − I(xi , l)a shortcoming. Indeed, the magnitude of fregion is low 2 l=0when kin and kout are similar, since in eq. (53), the term(I(pi )−kout )2 − (I(pi )−kin )2 tends to 0. This situation Computing this term in this way may be time-consuming,arise when the contour, including the bands, is initialized since intensities should be summed horizontally andinside a uniform area. However, we expect the contour vertically at each vertex position. Nevertheless, it isto grow if the intensity at the current vertex matches the possible to compute and store the summed intensitiesinner band features, whatever the value of kout . Thus, we only once, before polygon deformation is performed. Thisintroduce a bias fbias expanding the boundary if I(pi ) is reduces the algorithmic complexity to O(n), whereasnear kin : narrow band region energies induce a O(nB) complexity. Note that the additonal memory cost imputed to the fbias (pi ) = − 1 − (I(pi ) − kin )2 ni 2D arrays, storing summed intensities in the x and y directions for each pixel, is insigniﬁcant.This bias acts like the balloon force described in [35]. Itsinﬂuence should decrease as kin gets far from kout . To do On the other hand, for volumetric images, the extraso, we weight fbias with a negative exponential-like coeﬃ- memory burden caused by a similar implementation of thecient γ. divergence theorem would be problematic. In this case, in addition to the initial image, we store a unique 3D array S 1 − (kin − kout )2 γ = holding summed intensities in the x, y and z dimensions. 1 + ρ(kin − kout )2 (56) Its corresponding continuous expression is: fregion+bias (pi ) = γfbias (pi ) + (1 − γ)fregion (pi ) z y xwhere ρ should be high enough to ensure a quickly de- S(x, y, z) = I(x′ , y ′ , z ′ )dx′ dy ′ dz ′ −∞ −∞ −∞creasing slope (any value above 50 turns out to be suit-able). Hence, fbias is predominant when mean intensities which is actually computed according to the following re-are close. Its inﬂuence decreases to the advantage of fregion cursive scheme:as inner and outer mean intensities become signiﬁcantly S(x, y, z) = S(x−1, y, z) + S(x, y−1, z)diﬀerent. One may note that the bias force is also ap-plied when using the second region force in eq. (54). Con- + S(x, y, z−1) − S(x−1, y−1, z)sequently, our region terms maintain the same ability to − S(x−1, y, z−1) − S(x, y−1, z−1)grow or retract than classical region-based active contours. + S(x−1, y−1, z−1) + I(x, y, z)The region bias guarantees the contour has a similar cap-ture range as other region-based models. As in 2D, S needs to be computed only once at the be- ginning of the segmentation process. Then, during surface 11
- 12. evolution, image primitives P , Q and R are determined by If it is assumed that ψ remains a signed euclidean distanceﬁnite diﬀerences. We provide the details for P : function, the narrow band region energy may be written as: 1 ∂2S P (x, y, z) = Eregion 1 [ψ] = 3 ∂y∂z 1 ≈ (S(x, y, z) − S(x, y−1, z) H(ψ(x)+B)(1−H(ψ(x)))(I(x)−kin )2 dx 3 −S(x, y, z−1) + S(x, y−1, z−1)) D + H(ψ(x))(1−H(ψ(x)−B))(I(x)−kout )2 dxFor an implementation of the divergence theorem in the Dcontext of 3D segmentation, the reader may refer to [43]. where the Heaviside step function H is used in a similar6. Level set implementation manner as in [45] or [46]. Unlike in the explicit case, the regularity condition in eq. (12) has no impact on the im- We provide an implicit implementation of our narrow plementation of band integrals in the implicit case. Dueband energies as well. In addition to topological ﬂexibility, to the geometric nature of level sets, there is no need tothe level set formulation presents the advantage of a com- compute any length element and the curvature does notmon formulation for both 2D and 3D models. We consider intervene in the computation of band integrals. Indeed,the level set function ψ : Rd → R, where d is the image considering the sign of ψ, pixels belonging to Bin or Boutdimension. The contour or surface is the zero level set of are easily determined by dilating the front with the cir-ψ. We deﬁne the region enclosed by the contour or sur- cular window WB . Regarding the average intensity alongface by Rin = {x|ψ(x) ≤ 0}. Instead of forces applied outward normal lines, we rely on the curvature-based for-on vertices, we now deal with speeds applied to function mulation of the explicit curve:samples. Function ψ deforms according to the evolutionequation: hout (x) = B 2 ∂ψ I(x + bnψ (x))(1 + bκψ (x))db = F (x) ∇ψ(x) ∀x ∈ Rd (57) B(2 + Bκψ (x)) 0 ∂t where the unit outward normal to the front at x is:where speed function F is to some extent the level set-equivalent of the explicit energy in eq. (3), i.e. a weighted ∇ψ(x) nψ (x) =sum of smoothness and region terms: ∇ψ(x) F (x) = ωFsmooth (x) + (1 − ω)Fregion (x) This last expression is only adequate when x is located on the zero-level front. Computed as is, to maintain nψIn the level set framework, regularization is usually per- normal to the front, ψ should remain a distance function.formed with a curvature-dependent term. With this tech- This implies to update ψ as a signed euclidean distancenique, for the same reasons as explained in section 5.4, the in the neighborhood of the front before estimating normaleﬀect is limited to the direct neighborhood of pixels. In vectors. Regardless of the dimension of ψ, the curvatureorder to achieve a regularization as diﬀuse as in eq. (55), is expressed in terms of divergence:we replace the usual curvature term with a gaussian con-volution, as in [44]: ∇ψ(x) κψ (x) = div ∇ψ(x) 2 1 x′ −x Fsmooth (x)= √ exp − ψ(x′ ) −ψ(x) which allows to write the level-set formulation of the sec- σ 2πx′ ∈W (x) 2σ 2 ond region term. From eq. (19), it follows: 3σwhere W is a circular window of a given radius around x: Eregion 2 [ψ] = Wη (x) = {x′ | x′ −x ≤ η} H(ψ(x)+B)(1−H(ψ(x)))(I(x)−kin )2 dx + DParameters ω and σ play the same role as in the explicit B 2implementation described in section 5. Areas, volumes and δ(ψ(x)) (I(x+bnψ (x))−hout (x)) (1+bκψ (x))dbdxaverage intensities upon inner and outer bands are easily 0 Dcomputed on the level set implementation, since a circularwindow of radius B may be considered around each pixel For a point x located on the front, the speeds correspond-located on the front. ing to the narrow band region terms are: Bin = {x|ψ(x) ≤ 0 and ∃x′ ∈ WB (x) s.t. ψ(x′ ) = 0} Fregion 1 (x) = (I(x)−kout )2 − (I(x)−kin )2 Bout = {x|ψ(x) ≥ 0 and ∃x′ ∈ WB (x) s.t. ψ(x′ ) = 0} Fregion 2 (x) = (I(x)−hout (x))2 − (I(x)−kin )2 12
- 13. On the implicit 2D contour, the curvature may be ex- to the data term of the Chan-Vese model [11] described inpanded as eq. (1): 2 2 ψxx ψy − 2ψx ψy ψxy + ψx ψyy κψ = Eglobal [Γ] = λ (I(x)−kin )2 dx 2 (ψx + ψy )3/2 2 Rin +(2 − λ) (I(x)−kout )2 dxAccording to [47], the mean and gaussian curvatures of theimplicit surface are: Rout 2 2 2 2 2 2 ψxx (ψy + ψz ) + ψyy (ψx + ψz ) + ψzz (ψx + ψy ) Weights on inner and outer integrals are expressed in terms κM ψ = of a single parameter λ, since the full data term is already (ψx + ψy + ψz )3/2 2 2 2 weighted by (1−ω). In subsequent experiments, the asym- ψxy ψx ψy + ψxz ψx ψz + ψyz ψy ψz metric conﬁguration (λ = 1) will be explicitly indicated. −2 (ψx + ψy + ψz )3/2 2 2 2 Otherwise, the symmetric conﬁguration is used. The data 2 2 term of the combined model by Kimmel [23] holds a robust ψi (ψjj ψkk − ψjk ) + 2ψ (i,j,k)∈C alignment term, encouraging intensity variations normal to κGψ = the curve, and a symmetric global region term: (ψx + ψy + ψz )2 2 2 2where C = {(x, y, z), (y, z, x), (z, x, y)} is the set of circular Ecombined [Γ] = − | ∇I(c), n | dushifts of (x, y, z). Eventually, the reader may note that the Ω bias technique used in the explicit implementation is also +β (I(x)−kin )2 dx + (I(x)−kout )2 dxapplied in the level set model. The level set function ψevolves according to the narrow band technique [4], so that Rin Routonly pixels located in the neighborhood of the front are For the edge and combined data terms, image gradient ∇Itreated. is computed on data convolved with ﬁrst-order derivative of gaussian, where scale s is empirically chosen to yield the7. Results and discussion most signiﬁcant edges. The choice of s is a tradeoﬀ be- tween noise removal and edge sharpness. The variational Regarding the results, we should ﬁrst point out that the derivatives of the previous terms are:goal of our experiments is not to compare explicit and im-plicit implementations, since it is well accepted that both δEedge = −∇ ∇I(c) − αnexhibit their own advantages. These ones are typically δΓtopological and geometrical freedom for level sets. On the δEglobal = [−λ(I(c)−kin )2 + (2−λ)(I(x)−kout )2 ]nother hand, explicit approaches with polygonal snakes and δΓ δEcombinedtriangular meshes yield less computational cost than level = −sign( ∇I(c), n )∇2 I(c)nset and allow more control. The purpose of our tests is δΓ +β[−(I(c)−kin )2 + (I(c)−kout )2 ]nto compare the behavior of active contours and surfacesendowed with diﬀerent data terms, on both explicit and We perform segmentation of visually uniform structures.implicit implementations. We intend to show the interest Since we are looking for perceptually homogeneous ob-of the narrow band approach regardless of the implemen- jects, segmentation quality can be assessed visually. Onetation. To do so, the narrow band region energies are can reasonably admit that the target object correspondscompared with an edge term, the global region term of to the area containing the major part of the initial region.the Chan-Vese model [11] as well as the combined term For all 2D datasets, the model, whether it is our explicitby Kimmel [23]. For every data term involved, we used contour or the level set, is initialized as a small circlethe same smoothness term. Hence, the full energy is ob- fully or partially inside the area of interest, far fromtained by replacing Eregion with the following data terms the target boundaries. Similarly, for 3D images, ourin eq. (3). The edge term is based on the image gradient active surface and the 3D level set are initialized as smallmagnitude: spheres. On a given image, a common initialization is used for all models. The initial curve is depicted for some Eedge [Γ] = − ∇I(c) du + α dx experiments. In all subsequent ﬁgures, explicit contours Ω and surfaces are drawn in red whereas implicit ones Rin appear in blue.where α weights an additional balloon force [35] increasingthe capture range. It allows the contour to be initialized For all experiments, the regularization weight ω is setfar from the target boundaries, similarly to a region-based to 0.5. The standard deviation of the gaussian smooth σcontour. The following global region energy is equivalent takes its values between 1 and 2, which achieves the best regularization for all tested images. On noisy data, we 13
- 14. Global Narrow band 1 Figure 8: Left ventricle in MRI with level setfound that contours and surfaces with lower σ are prone toboundary leaking. In addition, insuﬃcient regularizationmakes level set implementations leave spurious isolatedpixels inside and outside the inner region. Conversely,values above 2 might prevent the surface from propagating Global Narrow band 1into narrow structures, like thin blood vessels. Moreover,since the width of the gaussian mask is proportional to σ, Figure 9: Kidney in abdominal CT with parametric contour (top)large standard deviations lead to signiﬁcant increase of and level set (bottom)computational cost, especially for 3D segmentation. Fig. 7 shows segmentation results of the brain ventriclein a 2D axial MRI (Magnetic Resonance Imaging) slice. only the implicit method was tested on this dataset. InAs it is the case with many medical datasets, the partially order to keep a critical eye on our approach, we draw theblurred boundaries between ventricle and gray matter attention on its equivalence with the Chan-Vese model onprevent the extraction of reliable edges in places. For the this particular image. The background is not uniform butedge-based model, we could not ﬁnd a suitable balloon still signiﬁcantly darker than the target object. Thus, theweight α and gradient scale s preventing the contour from classical region speed manages to make the front stabilizebeing trapped in spurious noisy edges inside the shape on the actual boundaries. Fig. 8 depicts the typical casewhile stopping on the actual boundaries. As regards in which there is no particular beneﬁt in using the narrowcombined and symmetric global approaches (see columns band region energy.2 and 4), the contour did not manage to grow, as inner andouter average intensities were not suﬃciently diﬀerent. In ﬁg. 9 and 10, we illustrate the recovery of the kidneySetting λ to 0.9, we decrease the signiﬁcance of inner and aorta inner walls, respectively, in 2D CT (Computeddeviation to the beneﬁt of outer deviation, consequently Tomography) data. The global region energy makes theallowing the contour to grow. As depicted in column 3, contour leak outside the target object, into neighboringthe contour undesirably ﬂows into the gray matter part structures of rather light gray intensity. Indeed, sinceas soon as this area is reached, which is inconsistent with the black background occupies a large area of the image,the initialization in our context. In this column, one may every bright grayscale is considered as a part of innernote that the drawn curves are not the ﬁnal ones, but statistics. Conversely, the narrow band energy, whichintermediate states to illustrate the leaking eﬀect. Indeed, ignores the major part of the background, has a moreon this particular dataset, gray/white separation is the local view and manages to keep the contour inside themost probable partition with respect to a global two-class kidney and the aorta. One may note that all segmentedsegmentation. objects of interest are homogeneous. It has no incidence if we consider the inner band or the whole inner region We also tested the snake on slices of the human for the homogeneity criterion in Eregion 1 . Indeed, in theheart in short-axis view, where the shape of the endo- region force of eq. (53), descriptor kin may be equally thecardium, i.e. the inner wall of the left ventricle, should average intensity on Rin or Bin .be recovered. The slices were extracted from 3D+TMRI sequences. Finding the endocardium boundary is The band thickness B is an important parameter ofmade more complex by the presence of papillary muscles, our method and should be discussed. Apart from itsappearing as small dark areas inside the blood pool, as impact on the algorithmic complexity - computing averageshown in ﬁg. 8. This requires topological changes, hence intensities µ(Bin ) and µ(Bout ) takes at least O(nB) oper- 14
- 15. Edge Global Global (asymmetric) Combined Narrow band 1Figure 7: Brain ventricle in 2D axial MRI, with parametric contour (top) and level set (bottom). The initial curve is drawn in black dashedline Global Narrow band 1 Figure 11: Succesive gaussian smoothings of an axial slice of 3D CTFigure 10: Aorta in abdominal CT with parametric contour (top) dataand level set (bottom) 15

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