Improved geometric constraints on deformable surface ... - CiteSeerX

Improved Geometric Constraints on Deformable Surface Model for Volumetric Segmentation

1

Jiuxiang Hu 1 ∗, Anshuman Razdan 1 , Gregory M. Nielson 2 , and Gerald E. Farin 2 Partnership for Research in Spatial Modeling (PRISM), 2 Department of Computer Science Arizona State University (ASU), Tempe AZ 85287, USA.

Abstract In this paper, we present a deformable surface geometric model for segmenting targets from volumetric data. This model deforms under external forces only, and changes its geometry and topology by using improved geometric constraints. External forces are calculated by using the sum of inflation forces, whose contributions take place on internal regions of objects, and the gradient forces, which play a key role when the surface is near the boundary of objects. A new set of geometric constraints is proposed which includes constraints on vertices, edges and faces. Once a constraint is broken, the corresponding topological transformation will occur to keep the geometric and topological integrity of the surface unaltered. We demonstrate that our model can efficiently segment complex anatomic structures from medical 3D images, and achieve the requirements of accuracy and geometry for image segmentation.

1. Introduction Volumetric segmentation is to extract a meaningful object or region from a 3D image or volume data, and has a wide application in biomedical image processing. There are many techniques that have been developed for segmentation of volumetric data. A ”good” segmentation method should satisfy the requirements on [1][6]: 1) accuracy, i.e., the segmented result reaches the proper boundary of objects; 2) well-behaved geometry, such as smoothness, and robustness against noise from the image. Image segmentation methods can generally be classified into two categories: 1) image-based only [1][6][7], which merely depends on image density and/or its differential derivatives, and includes the methods such as thresholding, region growing, edge detecting, k-mean and fuzzy clustering; 2) deformable surface model (DSM), which considers both image features and topological and geometric properties of objects without ∗ Corresponding

author, e-mail:[email protected]

Proceedings of the Geometric Modeling and Processing 2004 (GMP’04) 0-7695-2078-2/04 $ 20.00 © 2004 IEEE

a priori knowledge of its shape. DSMs include two types : parametric models [3][4][9][14] and geometric models [10][11]. In this paper, we focus on the deformable surface geometric model (DSGM). A geometric deformable surface is defined by a mesh (in this paper we use triangle meshes) within an image domain that can evolve by the internal forces coming from the surface itself and external forces derived from image data. This model starts from an initial mesh and deforms until it reaches an equilibrium in which the total force acting on each vertex is zero [11] [14]: Fvint + Fvext |v∈V = 0,

(1)

where V is the set of vertices of the deformable surface. The external forces are based on the image data and computed by some combinations of pressure forces, or potential force [10] or gradient vector flow [14]. They pull the surface toward desired object boundaries, and perform the shape discovery. The internal forces are obtained by some combinations of mean curvature motion [4], elastic forces, and Laplacian smoothing [3]. They are designed to hold the surface together, keep it from bending too much, make the surface smooth, and to perform the shape regularization. There are two key shortcomings of standard DSMs. First, it is difficult to balance the internal and external forces for a given application, because strong internal forces make it difficult for a deformable surface to move into boundary concavities [2][14]. In general, coefficients are assigned to the different forces applied, and the contributions of these forces to the equilibrium (1) are determined by mannul selection of proper parameters, and adjustment of the processing of deformation by visual inspection [3][14][15]. The basic idea is to increase the external force fields and to guide the surface toward the desired boundary. It is easier for a user to select a set of proper coefficients for 2D case, but will likely fail for 3D segmentations. The second problem is that the estimation of the internal force at a vertex is sensitive to the sampling of vertices around it. Several methods have been proposed to address this problem, including local model curvature computation

with least-squares error approximation of the Dupin indicatrix [4], and force estimation using the average distance between neighboring vertices [3] [11]. One additional difficulty is that the computation of internal force is often time consuming. In this paper, we apply a new equilibrium equation: Fvext |v∈V

= 0.

OP1 e2 t3 e3 t4

v0 e4 e5 t4

e1

v1

n

t1 t1 e6 t5

e2

e3

e t2

v2

e1

v3

OP2 (a)

(2)

(b)

(c)

Figure 1. Triangle mesh data structure. (a) Vertex structure. (b) Edge structure. (c) Face structure.

At first glance, this new model is just the omission of the internal force and is a simple form of the equilibrium equation (1). However, the new model can solve the problems mentioned above. The basic idea is that we divide the implementation of the shape discovery and shape regularization of DSGM into two steps instead of one step as in equilibrium equation (1). As we know, the main contribution of the internal force of a DSGM to the deformation of the surface is to smooth the surface while the external force pulls the surface toward the desired boundary of objects [3][10][11]. However, at the desired boundary where external forces are near zero, internal forces will play significant roles. They will deviate the surface from the real boundary, because internal forces are computed by the surface itself and can not be near to zero uniformly unless the surface is flat. The reason we do not use the internal force in our proposed DSGM is that it is difficult to utilize shape discovery in one step. To compensate the loss of shape realization gained by the internal force, we present a new set of geometric constraints on the deformable surface. We locally regulate the surface mesh based on these constraints. This model bridges the gap between local topological and geometric features, and global image information. Some new properties include:

A. Mesh Structure. A triangle mesh is a boundary representation of the original object. The topology of the triangle mesh is represented by a data structure which describes the connectivity information between vertices, edges and faces, while its geometry is represented by a geometric realization of the topological elements: points, lines and triangles. DSGM is constructed and maintained to avoid computationally expensive searches operated on the triangle mesh at each deformation. Geometric algorithms may be more efficient if more adjacency information is stored in the DSGM, but it requires the more memory as well as the effort to construct and maintain the data structure. Appropriate trade-offs between the efficiency of algorithms and the complexity of the data structure depends on specific applications. Various triangle mesh structures have been proposed for boundary representation in solid modeling, including winged-edge structure, vertexedge structure, and face-edge structure [8][13]. In this paper, a mesh structure that explicitly represents vertex-edge-face topology is designed to facilitate mesh deformation and maintenance. The deformable surface S(t) over time t ∈ R+ consists of a set of vertices denoted by V(t), and edges E(t) that form a closed surface defined by connected, oriented triangle facets T (t). Each vertex has a set of parameters(coordinates, normal, speed), and a list of its neighbors (edges and triangles), as shown in Figure 1(a). The position of the ith vertex, vi ∈ V, is represented by a 3D vector xi . Since every non-boundary edge is shared by two adjacent faces, each edge is associated with two triangles, as shown in Figure 1(b). Each face is composed of three edges, ordered in counterclockwise direction with respect to its outer normal as shown in Figure 1(c). The first layer of vertex v in the triangle mesh refers to all the vertices incident to vertex v, denoted by Nv1 . The layer can be efficiently extracted from searching the edge list of vertex (see Figure 1(a)). The second layer of vertex v, denoted by Nv2 is defined by

• Local and global topological transformation with ensured integrity and soundness are defined to incrementally construct a 2-manifold surface with arbitrary topology; • The evolution of this model is determined by external forces only, we do not consider the influence by internal forces needed by standard DSM. Therefore, unlike traditional DSGM, the surface does not oscillate when the surface is near the boundary of object.

2. Methods We will briefly review primary steps of our model.

2.1. Deformable Surface In this paper, a deformable surface is represented by a triangle mesh which is driven by a dynamic equation to capture the target topology with the associated 2-manifold structure.

Nv2 =

u∈Nv1

2 Proceedings of the Geometric Modeling and Processing 2004 (GMP’04) 0-7695-2078-2/04 $ 20.00 © 2004 IEEE

t2

Nu1 \ (Nv1 ∪ {v}).

(3)

where ni (t) is the unit normal vector to the surface at node vi , and T is the image intensity threshold of the target object, and Imax and Imin are the maximum and minimum values of the image I(x). This force is used to search for the iso-intensity surface of value T . Its principal is to inflate or deflate the model locally as long as it does not lie on the desired iso-intensity surface. This force contributes to the evolution of a model that is insensitive to noise far from the edges where the intensity is much lower/higher than the intensity of objects. However, when the surface is near the boundary of objects desired, this force is sensitive to noise, and oscillation of the surface will occur [12]. To stop the surface at significant edges, we include the external force from the discrete gradient of I:

B. Dynamic Equation. To find the solution of the equilibrium (2), the deformable surface starts from a given mesh as its initial state. Let xi (t) = (xi (t), yi (t), zi (t)) denotes the coordinates of a vertex vi ∈ V(t) in deformable surface S(t), and x˙ i its velocity and ¨ xi its acceleration. The behavior of deformable surface S(t) is governed by a Newtonian law of motion[3][4][10][12]: m¨xi + γ x˙ i = F(xi ),

(4)

where m is the mass of the vertex that determines the inertia in the model and γ is the damping factor which controls the rate of dissipation of the kinetic energy of the nodes, and F(xi ) is the external force applied on vi . To escape oscillations, γ must not be too low, and to avoid slow displacements, it must not be too high [3][10]. We applied the fourth-order Runge-Kutta method to solve the ordinary differential equation (4) instead of the Euler method. Although the Runge-Kutta method is slower than a Euler method, the Runge-Kutta method is more accurate and more stable, therefore, presents better behavior and robustness to the geometric and topological transformation which is unpredictable. Unlike standard GDSMs [11][12][14], we do not perform any energy minimization which is often performed with the associated Euler-Lagrange equation. Because our model does not apply any internal force which is often defined by spatial derivatives of the surface itself, it can not be derived from the type of Euler-Lagrange equation which involves spatial derivatives. Although Lachaud and Montanvert[10] did not perform any energy minimization, their approach shares several similarities with energy minimization since the two internal force used correspond to the regularization terms of first and second order spacial derivatives of DSGM.

f(2) (xi ) = b(Gσ (xi ) ∗ I(xi ))(∇Gσ (xi ) ∗ I(xi )),

where b is the factor to normalize this force and Gσ (x) is a 3D Gaussian function with standard deviation σ, ∇ is the gradient operator, and is the Laplacian operator, Gσ ∗ I denotes a Gaussian smoothing filter on the image I(x). The factor ∇Gσ (xi ) ∗ I(xi ) in right-hand side of (6) is the edge map [14], and Gσ (xi ) ∗ I(xi ) corresponds to the second derivative of image and is zero near the principal edges. The principal contribution of this force is near the edges, and is zero at the internal regions of objects and edges. The external force at each vertex vi is computed by f(vi ) = αf(1) (xi ) + βf(2) (xi ),

External forces push the DSM toward the stable position (2), which is defined from the image so that it takes on its smaller values at the feature of interest such as boundaries. Given a gray-level image I(x, y, z), viewed as a function of continuous spatial variables (x, y, z). There are two typical external forces designed to lead a deformable surface toward edges. One is inflation/delation force [10][12] which is derived from the difference between iso-value (determined by the intensity at the edges desired) and the intensity at each vertex. This force pushes the surface toward an iso-value surface. The typical inflation/deflation force defined by [10][12] is: I(xi ) − T ni , Imax − Imin

F(xi ) =

f(vi ) 1 1 f(u) + f(w), + 1 2 3 3|Nvi | 3|Nvi | 1 2 u∈Nv

i

w∈Nv

i

(8) where Nvji (j = 1, 2) is a set of vertices in the j th layer centered node vi , and |Nvji | is the cardinality of Nvji .

2.3. Topological Transformations In order to approximate the surface of objects in image from a initial mesh, the S(t) must change at each step under some controls till the deformation stops. There are two

(5) 3


(7)

where the weight α and β scale the external force f(1) and f(2) defined above, respectively, and are usually chosen to be of the same order, with α slightly larger than β so that a significant edge will stop the inflation/deflation, but with β large enough so that the surface will pass weak regions in objects. In addition, if the normal of node vi points away from the surface, a positive coefficient α is expected when the intensity of the targets is higher than T , a negative one when its intensity is lower than T . In order to improve robustness against noise and the ability to fill holes on boundary, we consider the external force used in the right-hand side of equation (2) as the average over a local neighborhood centered at node vi :

2.2. External Forces

f(1) (xi ) =

(6)

(a)

(a)

creation e (b)

Inversion

Figure 3. Global topological transformation. (a) axial transformation, (b) annular transformation

(b)

e

melting

faces, two types of topological transformations, (namely, local, and global) are defined in this paper for construction and maintenance of the manifold structure of the surface during deformation. Local topological transformations include face subdivision, edge inversion, edge creation, edge melting shown in Figure 2(a), (b) and (c) respectively. These change local manifold structure while keeping the topological properties of the object surface unaltered. Global topological transformations of closed and oriented surface[10] include axial transformations, where two parts of the surface are colliding or already intersected, and annular transformations, which occur when a connected part of the surface homotopic to a circle is shrinking to a point, shown in Figure 3(a) and (b), respectively. These two topological transformations change the topological properties of the object surface. According to the classification theorem [5], every oriented and closed surface is topologically equivalent to the sphere, or the connected sum of n tori. By this theorem, it can be proved that the proposed two types of topological transformations are complete such that a manifold with arbitrary topology can be constructed by a finite number of these transformations.

(c)

Figure 2. Local topological transformation. (a) face subdivision, (b) edge creation and inversion, (c) edge melting.

types of information in S(t): topological and geometric. Topological information includes the relationships among vertices, edges and faces. Geometric information is usually quantization for the edges and faces. In this subsection, we first discuss topological properties and transformations on a deformable surface. A deformable surface to be constructed has global as well as local topologies that are invariant under homeomorphisms, so that the resulting object is well-behaved. The global topology is that the surface satisfies the so-called manifold condition. Local topology, referring to the adjacency relationship between vertices, edges, and faces, should be considered in order to avoid global searches when the surface is deformed. The reason is that deformation is performed by constraints of the model on its vertices [11]. Based on the mathematical properties of 2-manifold sur-

2.4. Improved Geometric Constraints Some deformable models offer dynamic topological modification with user interaction or validation [3], or by performing series of contractions [10]. In [12], McIner4


e1

e2

(a)

v1 v2

?

(a)

Figure 4. Examples of failures to perform GC1-3.

satisfy geometrical constraints (GC2), two inversion transformations (see [10] or below) on these two edges will result in a spiked triangle. For the case shown in Figure 4(b), axial operation does not work on two close vertices. It is obvious that more complete set of geometrical constraints should be introduced. Besides GC1-3 listed above, we also introduce additional geometric constraints on edges, vertices and faces.

mimimum closed path

Figure 5. A minimum closed path around a intersection region when GC6 is not satisfied.

GC4 angle(e) > a , ∀e ∈ E, and a is a given positive value; GC5 |Nv1 | ≥ 3 , ∀v ∈ V;

ney and Terzopoulos automatically modify topology of a deformable surface with regard to its variable geometry by using an affine cell image decomposition to segment an image. They use a set of tetrahedra to partition the image, and keep track of the ”burnt” vertices of the simplified grid, which the surface has already crossed. They then select a new subset of tetrahedra to present the segmentation result at each deformation, and finally recompute the ”surface” at each iteration with a method similar to a marching tetrahedra isosurface extraction algorithm. The method adapts the topology to the geometry of its vertices according to the volumetric segmentation, not to surface deformation. In [10], Lachaud and Montanvert have automatically changed the topology and geometry according to simple distance constraints on the surface, and directly modified the surface under three geometrical constraints: given an invariant δ > 0 which is associated with the mesh, then

GC6 t ∩ s = ∅ , ∀t,

GC2 d(e) < 2.5δ, ∀e ∈ E; ∀v1 , v2 ∈ V and e(v1 , v2 ) ∈ E,

where d(e) and d(v1 , v2 ) are the length of edge e and the distance of vertices v1 and v2 , respectively. However, these geometrical constraints will fail for the cases shown in Figure 4. If the edges e1 and e2 shown in Figure 4(a) do not 5 Proceedings of the Geometric Modeling and Processing 2004 (GMP’04) 0-7695-2078-2/04 $ 20.00 © 2004 IEEE

s∈T;

where, angle(e) is the angle between two oriented triangles neighboring edge e. The process of deformation may cause some geometric constraints to be broken. In order to correct this we apply topological transformations on the vertices, edges or faces until all constraints are satisfied. Here, we emphasize the case when GC6 is not satisfied as shown in Figure 4. We will use axial transformation to fix this problem. First, two minimum closed pathes around the intersection regions of two parts of the surfaces are identified. Second, we generate a local coordinate u − v − w in which the line between the middles of two pathes is assigned to be w axis. Third, we computed the angles between u axis and the project of the vector,which is from the local origin to each vertex in the two pathes, on plane u − v. Finally, we move one vertex of the triangle, which is in the intersection regions and share a edge of two pathes, to the smallest angle between the vertex and a vertex in the opposite path. Figure 7 shows the topology adaptivity and geometry flexibility of our DSGM to segment a torus in a simulated volume data shown in Figure 6(a). The deformation starts from a spherical mesh shown in Figure 7(a), and the topology of the deformable surface was changed at iteration 120.

GC1 d(e) > δ , ∀e ∈ E; 2.5 √ δ, 3

(c)

Figure 6. A simulated data added Gaussian noise with σ = 1.2 has two objects: a ball and a torus. (a) volumetric rendering by ray casting; (b) segmented by thresholding; (c) extracted by DSM.

(b)

GC3 d(v1 , v2 ) >

(b)

3D Image

? Initialize mesh

(a)

(b)

(d)

(c)

? Move the active vertices ? Apply topological transformation on vertices, edges and faces not satisfying geometric constraints

(e)

? Deform the surface again after subdivision Figure 8. Deformation algorithm.

(f )

(g)

(a)

Figure 7. Topology Adaptivity of the deformable surface during the reconstruction of torus. (a) At initialization; (b) at iteration 50; (c) at iteration 100; (d) at iteration 150; (e) at iteration 200, during the evolution, topology of surface was changed; (f) the deformation of mesh with low resolution stops; (g) the final surface.

(c)

Figure 9. The effect of internal force on deformation. (a) Spherical mesh generated by subdividing a icosahedron, 10 times deformation results (b) driven by the curvature regulation, (c) by the spring force.

olution mesh obtained by subdividing the mesh to regulate the shape of object. This is useful to prevent the surface from leaking on the boundary due to noise in the image. The second way is that we classify vertices on the surface into two types: active vertex and stable vertex. If the speed is lower than a given threshold, we then call it a stable vertices, otherwise we call it an active vertex. We confine the validation of geometric constraints on the surface to the active vertices, which is usually small part of the mesh. This is particularly efficient in a multi-resolution approach, where many vertices are quickly near their final position.

Comparison the results as shown in Figure 6(b) and 6(c), the segmentation results using our model have better geometric behavior than that using thresholding methods.

2.5. Deformation Process As shown in Figure 8, the DSM is composed of three procedures: mesh initialization, mesh deformation, and mesh refinement. We initialized the triangle surface in three steps: (1) create an icosahedra (or octahedron) whose center c and radius r are specified by user; (2) subdivide the icosahedra (or octahedron); (3) move the vertices of the mesh to the sphere by r(v−c)/||v||+c. After that the surface is free to evolve according to its dynamic and geometrical rules. There are two ways used to improve the efficiency of the algorithm. First, we utilize a low-resolution mesh (e.g., the mean of length of edges of the mesh is relatively longer) to quickly discover the shape of object, and use a higher res-

3. Experimental Results We have implemented the discussed methods and applied them to multiple examples of simulated, MRI and CT data sets. All DSGM parameters are currently set manually by experimentation. This process is performed once for a specific image modality or for a specific anatomic structure and requires only a few minutes of experimentation. 6


(b)

(a)

(c)

(b)

Figure 10. Convergence of GDSMs. Segmentation results using (a) thresholding; (b) our model; (c) standard model with internal forces.

(b)

(a)

The time step and deformation step parameters are set to achieve maximum DSGM efficiency. For many segmentation scenarios, the time step is set to 0.05 and deformation step to 5, and DSGM works well. Since our model does not utilize the internal force, the force parameter settings do not involve the problem of the ratio of external forces to internal forces, and there exists a wide range. In general, the selection of a coefficient of inflation/deflation always keeps α(I(xi ) − T ) > 0.

3.1. Contribution of Internal Force

(c)

In this paper, internal forces consist of normal and tangent component [3][4][10]. The normal of internal forces is computed by curvature regulation fn fn (vi ) = µ(x¯i − xi −

1 (¯y − y)), 1 |Nvi | 1 u∈Nv

(d)

Figure 11. DSGM segmenting a bone of thigh from MRI data. (a) Part of a slice, (b)after 30 times deformation, (c)after 60 times deformation, (d) after 120 times deformations with subdivision.

(9)

i

and the tangent component by a spring force ft ft (vi ) = ν

(d(vi , u) − d0 )

u∈Nv1

y − xi , d(u, vi )

on the sampling of nodes around each vertex, although this mesh is much more regular than the mesh after deformation. Figure 9(c) shows the result after 10 times deformation from the sphere mesh shown in Figure 9(a) driven by spring forces. The mean and standard deviation of edge length are 3.58 and 0.011 for the original mesh, respectively, whereas they are 3.63 and 0.0047 for the mesh shown in Figure 9(c). Although the spring forces improve the mesh regularity, the shape of the mesh is poor. These results have shown that internal forces are sensitive to the sampling of vertices in DSGM.

(10)

i

where µ is the ’rigidity’ coefficient, ν the ’stiffness’ coefficient, and d0 the expectancy of the edge length for the triangular mesh. In the following experiments, the d0 for every deformable mesh is automatically set to be the average of edge length. In the first example, we only apply internal forces to deform a regular spherical mesh. A regular mesh on a sphere is generated by subdividing a icosahedron and then moving the vertices to the sphere as shown in Figure 9(a). Except for the twelve vertices on which the first layer has 5 nodes, all vertices have 6 neighbors. Here, the ’rigidity’ and ’stiffness’ coefficient are set to be -0.45 and 1.2, respectively. The result after 10 times deformating is shown in Figure 9(b). It is easy to see that the internal forces along normal of vertex on the sphere mesh is not uniform and depends

3.2. Convergence to Boundary Concavity In order to compare the convergence to boundary concavity by using traditional geometric DSGM and by our model without internal forces, a controlled experiment is designed in following way: we first segment a binary vol7


(a)

(b)

(c)

(d)

(a)

(e)

(f )

(c)

(d)

Figure 13. Comparison of the segmentation results obtained by our DSGM with by standard DSGM. (a) CT data including a carotid artery bifurcation with three calcium plaques, rendered by ray casting method; (b) Both the carotid artery bifurcation and plaques segmented using the proposed DSGM; (c) Only carotid artery bifurcation segmented by region growing; (d) Only carotid artery bifurcation by standard DSGM.

(g)

Figure 12. Our DSGM segmenting a part of pelvis bone with one genus from CT image volume (different views).

ume data (see Figure 10(a)) and treat that segmentation as our reference, which includes a cube caved with a smaller cuboid in a 64 × 64 × 64 simulated volume data. Its volume size and surface area are 52.07 and 88.70. We then apply our DSGM without internal force to segment this object, where isovalue T = 0.5 (same as the threshold selected in the first step), α = 4.5 and β = 0. The deformation starts from a sphere triangle mesh, and the segmentation result is shown in Figure 10(b), whose volume size and surface are are 52.14 and 87.91. Third, we utilize internal forces including curvature regulations with µ = −0.35 and spring forces with ν = 1.2, and use the same external forces as that used in the second step, and set the ratio of internal forces and external forces between 0.3 and 0.5. Figure 10 shows the surface after deformation stopped. Its volume size and surface area are 59.23 and 92.34, respectively. It can be seen that the final deformable surface without internal forces more closely approximates the true boundary than that with internal forces.

the external forces were used to overcome the problems associated with noise, and a relatively high resolution of the mesh was applied to prevent the surface from leaking into neighboring structures. A deformable surface started from the sphere centered at (128, 256, 44) and radius of 6 pixels. The external force parameters were set to the following values: α = 7.8 and β = 0.2. The segmentation takes about 4.5 minutes on an Dell workstation PW350 with 1.7Gz CPU and 2G memory, and stopped after 120 steps of deformation at low resolution (see Figure 11(b) and (c)), and took another 10 deformations after subdivision. The final result are shown in Figure 11(d). Figure 12(b)-(g) show the deformation of the model to segment a part of pelvis bone with a genus from CT image to demonstrate the topological adaptability of our model. In this example, we set the model parameters: α = 4.5, and β = 0.3 and normalized isovalue T = 0.49. In the last example, we extract a carotid artery bifurcation with three calcium plaques from a CT image. The image dimensions are 57 × 63 × 99. To compute of external forces, we used model parameters: α = 30, β = 0 and T = 0.08 for segmenting blood vessel, and α = −20, β = 0, and T = 0.2 for extracting calcium plaques, respectively. We manually seed with a smell sphere mesh and segmentation then proceeds automatically. The final result is shown in Figure 13(b). Figure 13(c) shows the segmenta-

3.3. MRI and CT Data Sets We applied our DSGM to segment and reconstruct the bone thigh from a T2 MRI image. The data consisted of a stack of 72 slices each with 5122 pixels, 2 bytes per pixel (voxel size 0.70mm × 0.70mm × 2.0mm). Figure 11(a) shows part of a slice of the volume data. Low contrast and signal-to-noise, especially vicinity of other structures (see Figure 11(a) make the task of segmentation more difficult. In this example, relatively large values of coefficients for 8 Proceedings of the Geometric Modeling and Processing 2004 (GMP’04) 0-7695-2078-2/04 $ 20.00 © 2004 IEEE

(b)

tion result of only carotid artery bifurcation by using region growing. However, standard DSGM will fail to segment the small branch of the carotid artery bifurcation (see Figure 13(d), even if we select smaller parameters: µ = −0.04 and ν = 0.2 to compute internal forces. The reason is that the two plaques squeeze the entrance of the small branch and create a bottle neck (see Figure 13(a)), while internal forces, which keep a mesh smooth, will prevent the mesh from moving into the narrow entrance.

[4] A. Ghanei and H. Soltanian-Zadeh, A Discrete Curvature-Based Deformable Surface Model With Application to Segmentation of Volumetric Image. IEEE Trans. Information Technology in Biomedicine, 6(4):285-295, 2002. [5] H. B. Griffiths, Surfaces. Cambridge University Press 1976. [6] J. Hu , A. Razdan , G. Nielson , G. Farin , D. P. Baluch , D. G. Capco, Volumetric Segmentation Using Weibull E-SD Fields. IEEE Trans. Vis. and Computer Graphics ,9(3):320- 328, 2003.

4. Conclusions We developed a deformable surface model without internal forces but with improved geometrical constraints, which can be used for the segmentation of 3D structures with low contrast and low homogeneity, and vicinity of other structures. By evaluating simulated data, it was concluded that accuracy of the model is higher than that of standard deformable surface model. By comparing with some applications to real MRI data, it can be seen that geometry of object segmented by the model is smoother than that by traditional image segmentation methods such as region growing. Future work can be directed toward finding a complete and necessary set of geometric constraints for the model, making the model more independent of the parameters, and automatically initializing the mesh in the objects of interest.

[7] J. Hu , A. Razdan , G. Nielson , G. Farin , Segementing Linear Parts using Layered Region Growing. The 8th internationals Conference for 3D Digitisation and Modeling Professionals: 3D Modeling , 2003. [8] J. Huang and C.H. Menq , Combinatorial Manifold Mesh Reconstruction and Optimization from Unorganized Points with Arbitray topology. CAD, 34:149-165, 2002. [9] M. Kass, A. Witkin, and D. Terzopoulos, Snakes: Active Contour Models. Internat. J. Computer Vision ,1:321-331, 1988.

Acknowledgements

[10] J. O. Lachaud and A. Montanvert, Deformable Meshes with Automated Topology Changes for Coarse-to-Fine Three-Dimensional Surface Extraction. Medical Image Analysis, 3(2):187-207, 1998.

This work was supported by the Defense Advanced Research Projects Agency (MDA 972-00-1-0027) and the National Science Foundation (IIS-998016 and ACI-0083609) and the National Institutes of Health (HD 32621) and the Office of Naval Research (N00014-02-1-0287). We would like to thank PRISM at ASU for providing the data and computing resources.

[11] J. Miller, D. Breen,W. Lorenson, R. O’Bara, and R. M. Wozny, Geometrically Deformed Models: A Method for Extracting Closed Geometric Models from Volume Data. Computer Graphics (SIGGRAPH’91 Proc)., 25(4):217-226, 1991.

References

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