Cerebral Artery Segmentation with Level Set Methods

Transcription

1 H. Ho, P. Bier, G. Sands, P. Hunter, Cerebral Artery Segmentation with Level Set Methods, Proceedings of Image and Vision Computing New Zealand 2007, pp , Hamilton, New Zealand, December Cerebral Artery Segmentation with Level Set Methods H. Ho, P. Bier, G. Sands, and P. Hunter Bioengineering Institute, University of Auckland, Auckland, New Zealand Abstract In this paper we present the application of two level set methods (active contour without edges and fast marching) to the problem of segmenting arteries from medical images. The two methods fit in the same Hamilton-Jacobi type equation but their underlying mechanisms are quite different. The first method finds the objects of interest using an energy minimization equation without the information of image edges. The second method segments an image by evaluating its time-cross-map, which is calculated from the image gradient. We implement the first method from scratch, and use a medical imaging library (ITK) for the second. Their segmentation results and performances are compared. Each method s strengths and weaknesses are also analyzed. Keywords: Level set, cerebral artery, image segmentation, ITK 1 Introduction The level set paradigm was originally proposed by Osher and Sethian in 1988 to track the evolution of 2D contours or 3D surfaces [1]. Since its introduction, this technique has found applications in a wide range of fields, including computational geometry, fluid mechanics and visualization [2][3]. The application of the level set in image processing was initially explored by Caselles et al. [4] and Malladi et al. [5]. The basic idea is to embed an active contour in a higher dimensional surface as the zero level set. The active contour is then propagated along its normal direction according to a partial differential equation (PDE). The main advantages of this approach are its ability to trace topologically evolving surfaces by splitting and merging, and its ease of implementation in three or more space dimensions [6]. In recent years, level set methods have been utilized to segment vessel structures from 3D brain images [7][8]. The images are mainly CT angiography (CTA) or MR angiography (MRA) images, where blood vessels are enhanced in their intensities via injection of contrast agents. In this study we present our work in using level set methods to segment arteries from a 3D CTA image. We explore two level set methods: the active contour without edges method, and the fast marching method. We compare the performance of these methods and discuss each method s strengths and weaknesses. The structure of this paper is organized as follows: first we briefly introduce the mathematical formulation of the level set, and its varied forms for both 300 methods. Next we introduce some implementation details. The two methods are then applied to a 2D image for comparison. The fast marching method is also applied to a 3D image. Finally we discuss and contrast their segmentation results. 2 Level Set Method 2.1 Mathematical Formulation Let be a bounded open subset of R 2, and (x, y) be the discrete point set in. Consider an evolving curve ζ, which is embedded as the zero level set of a higher dimensional function φ. φ is defined φ(x, y, t = 0) = ±d (1) where d is the distance from every point in (x, y) to ζ at t = 0. The derivative of φ with respect to time is expressed in a PDE [1]: t = F, φ(x, y, 0) = φ 0(x, y) (2) where the set {(x, y) φ 0 (x, y) = 0} defines the initial contour, and F is the speed of propagation. Equation (2) is the most basic formulation of the level set equation. Although it looks simple, various level set methods can be constructed by designing different speed functions F, such as the geodesic level set [9], or the Canny-edge level set [10]. Next we will introduce a level set method where F has three terms.

2 2.2 Active Contour without Edges The active contour without edges method was introduced by Chan and Vese[11]. Figure 1 shows an image u(x, y). Let c 1 and c 2 denote the average image intensities inside and outside a random closed curve ζ respectively, and ζ 0 denotes the bounding contour of an object region u 0. A fitting term F is defined F (ζ) = inside(ζ) u 0 (x, y) c 1 2 dxdy + u 0 (x, y) c 2 2 dxdy (3) outside(ζ) 2.3 Fast Marching If the PDE governing the motion of contour φ has a simple form, e.g. it only has a single term, then a fast marching level set method can be constructed [10]. This method ensures that the contour front propagates quickly in homogeneous areas, yet moves slowly through high gradient regions such as edges and corners. To achieve this effect, the speed term F can be formulated as a negative exponential (e x ) or as a reciprocal function (1/(1 + x)) of the gradient image. Alternatively, F can be constructed as a Sigmoid function of the gradient image, for which we will use: t = 1 I β 1 + e ( α ) (6) where I is the magnitude of the gradient image, and α(< 0), and β(> 0) defines the region of the image we are interested in. This formulation guarantees that the speed term gets close to zero in high gradient regions, yet approaches one in homogeneous regions [10]. Since the time taken for a contour to reach each pixel (from one or several seed points) varies across the whole image, we can extract useful information by analyzing this time-map. Figure 1: Diagram of level set contour. F is also called the energy function. The minimum of F is achieved only when ζ is fitted onto ζ 0 enclosing the object region. Chan and Vese further regulated F [11] F (c 1, c 2, φ) = µ δ(φ(x, y)) (x, y) dxdy +λ 1 u(x, y) c 1 2 H(φ(x, y))dxdy +λ 2 u(x, y) c 2 2 (1 H(φ(x, y)))dxdy (4) where µ 0, λ 1, λ 2 > 0 are fixed parameters. H is the Heaviside function, and δ is the one-dimensional Dirac delta function. The first term at the right hand side (RHS) is a contour length term, the second and third are energy variance terms. Further manipulation of Equation (4) by minimizing F with respect to φ gives: [ ( ) t = δ(φ) µ div λ 1 (u c 1 ) 2 ] +λ 2 (u c 2 ) 2 (5) The first term inside the square brackets now forms a curvature term, which is used to smooth the contour. The second and third terms are used to minimize overall image energy. 3 Methodology 3.1 Numerical Method To solve Equation (5), the ( Dirac delta ) function δ(φ) and curvature term div need to be evaluated. Following the analysis of Chan and Vese, the Heaviside function H can approximated H(z) = 1 (1 + 2π ) 2 arctan(z) (7) therefore the Dirac delta function δ can be written δ(φ) = H (φ) = 1 π (1 + φ2 ) (8) The curvature term at time step n is approximated ( div = x +φ n ) x i,j ( x +φ n i,j )2 + (φ n i,j+1 φn i,j 1 )2 /4 ( + y y +φ n ) i,j (φ n i+1,j φn i 1,j )2 /4 + ( y +φ n i,j )2 where: x φ i,j = φ i,j φ i 1,j, x +φ i,j = φ i+1,j φ i,j y φ i,j = φ i,j φ i,j 1, y +φ i,j = φ i,j+1 φ i,j (10) (9) 301

3 Now, every term at the RHS of equation (5) can be evaluated. The value of φ can then be solved by a finite forward difference scheme: φ n+1 i,j = φ n i,j + t RHS (11) The above scheme is implemented in C/C++ code, with OpenGL for graphics. 3.2 Insight Toolkit (ITK) Instead of implementing the fast marching method from scratch, we decided to utilize the implementation available in the medical imaging package Insight Segmentation & Registration Toolkit (ITK) 1. ITK itself is an open-source, object-oriented software system for image processing, segmentation and registration [10]. Image processing algorithms in ITK are implemented as filters. These filters can be arranged in a pipeline to achieve required results. For example, a normal segmentation procedure for a medical image is: image denoising feature extraction segmentation. The corresponding ITK pipeline can be arranged anisotropic diffusion filter gradient magnitude filter Sigmoid filter fast marching level set filter. For more details of ITK infrastructure, data structure, algorithms and applications we refer the reader to references [3] and [10]. 4 Results 4.1 Chan-Vese method A 2D image (size: ) of brain arteries is chosen as the test image. Figure 2 illustrates the results of the Chan-Vese method for different iteration times. Four images (labelled A, B, C and D) in the figure show the initial contour (a circle), the contour after 10, 30 and 50 iterations. The initial contour is driven by an energy minimization force until converging to a final contour, which yields the segmentation result. In this case convergence is reached after 50 iterations within 1 second on a PC (2.40 GHz Intel CPU). Note, that the zero level set is implicitly embedded in the level set formulation. Its physical location can be extracted by locating the pixels where φ = 0 in Equation (11). To elucidate this concept, the surface of function φ for the resulting image after 50 iterations (Image D, Figure 2) is plotted in Figure 3 using Matlab. In the image, the positive φ values appear in the bright area (ridges), whereas negative φ values present as the dark regions (plateau, valley). The zero level set, that is when φ = 0, is compromised 1 Figure 2: Segmentation using the Chan-Vese method. of the cross-border contours between which yield the segmentation result. Figure 3: 3D surface of φ. 4.2 Fast marching method As stated before, ITK is employed for fast marching segmentation. It was ported into CMGUI, the front end of CMISS 2, the Continuum Mechanics, Image analysis, Signal processing and System Identification software developed in our Bioengineering Institute. CMGUI implements image processing routines via computed fields, which in turn call the ITK library code. These routines can be used for both 2D and 3D segmentation. As a comparison, we adopt the same image as used for the Chan-Vese method. The parameters for the ITK image processing pipeline (original image gradient magnitude filter Sigmoid filter fast marching level set filter) are configured the α and β values are set to 0.3/255 and 2.0/255 for the Sigmoid function; A seed point is placed in the lower left corner. After applying the fast marching filter a time-cross-map is generated. The timecross-map is further segmented using a thresholding filter. The results are visualized via extract

4 ing isosurfaces in CMGUI. Figure 4 shows four snapshots from the pipeline: the original image and seed point (Image A), the gradient magnitude image (Image B), the time-cross-map image (Image C) and the final segmentation result (Image D). The total computation time is less then 1 second on the same PC. (size: ) of cerebral arteries are imported into CMGUI. The α, β values in the Sigmoid function are set to 1.0/255 and 2.0/255 respectively. As in the 2D case, the location of seed point(s) needs to be carefully selected. Figure 5 shows two segmentation results when one seed point is placed at different locations. In the left image, the seed point (A) is placed inside a vessel. The level set front stops at somewhere along the axial direction. In the right image, the seed point (B) is placed at the centre, where no large vessels exist. The fast marching method quickly segments out main vessels (the computation takes 10 seconds). Figure 4: Segmentation using fast marching method. The segmented image yields a similar result to the Chan-Vese method except near the lower right corner. This can be explained from the time-crossmap (Image C, Figure 4), which tells how long it takes for the level set front to reach every pixels in the image. The darker the colour, the less time it takes, and vice versa. When the contour front moves near the edges, its speed becomes close to zero so it takes a long time for the front to cross edges. As a consequence, the regions enclosed by edges have (very) high time-cross value (up to ). They are visualized in white colour after scaling. Observe that there is a vessel separating the lower right corner from the seed point, which means the contour front needs a long time to move into the corner. This artifact can be easily removed by placing another seed point in the lower right corner. In this experiment we placed the seed point in the homogeneously dark background. It is not difficult to imagine that if the seed point is placed inside a vessel, then only that single vessel and its connected vessels can be segmented (proof to be shown in the 3D case). Therefore to have sparse, separated vessels segmented we have to place multiple seed points into these vessels D image segmentation The extension of the fast marching method from 2D to 3D is straightforward. Since the difference between 2D and 3D image handling is implicitly handled by the generic ITK framework, the explicit ITK pipeline for both dimension is extractly the same. As an example, a stack of 52 images Figure 5: Fast marching method for 3D. In this example, we take advantage of the a-priori knowledge of CT angiography images, in which vessels are enhanced in brighter areas (as is bone), yet other soft tissues are left as dark regions (background). The segmentation therefore can be carried out in a semi-automatic way: manually place seed points, followed by automatic segmentation. 5 Discussion In this paper we demonstrated two level set methods for segmenting arterial structures from 2D and 3D images. The active contour without edges method (Chan-Vese method) was implemented from scratch, whilst we utilized a well-developed medical imaging library (ITK) for the fast marching method. Both methods employed the same Hamilton-Jacobi style equation [1], however the theories underpinning each method are quite different. The Chan- Vese model does not need image gradient information or seed points. Its active contour is driven by overall energy minimization forces. The fast marching method, on the other hand, depends on image gradients to evaluate the time-cross-map, and is highly sensitive to the placement of seed points. In general, it is straightforward to extend level set algorithms from 2D to 3D, which can be seen in the fast marching example. The price, however, is the increased computational cost due to higher dimension. This can be problematic for 303

5 some algorithms. For example, in the case of Chan- Vese model, extending the algorithm from 2D to 3D implies evaluating the energy variance for all 3D voxels in evolving surfaces for every iteration, instead of all the 2D pixels in a closed curve. This can be computationally prohibitive. A solution is the so called narrow band method, which only calculates the region close to the level set surface [2, 6]. The drawback is the extra step required to construct the narrow band, and extra consideration necessary to avoid local minima. The fast marching method, on the other hand, does not need to trace the level set propagation for every iteration. Instead, it evaluates the time-cross-map from the gradient image only once. The time-crossmap is then thresholded to yield the segmentation result. Therefore, fast marching method is more efficient in 3D segmentation, given the seed points are positioned properly. Acknowledgements We would like to thank Mr. S. Blackett for his work on porting ITK routines into CMGUI, and Dr. A. Frangi for providing the CTA image. References [1] S. Osher and J. Sethian, Fronts propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulations, J. of Comp. Physics, pp , [2] S. Osher and R. Fedkiw, Level set methods and dynamic implicit surfaces. Springer-Verlag New York, [3] T. S. Yoo, Insight into Images: Principles and Practice for Segmentation, Registration, and Image Analysis. A K Peters, Ltd., [4] V. Caselles, F. Catte, T. Coll, and F. Dibos, A geometric model for active contours in image processing, Num. Math., vol. 22, pp , [5] R. Malladi, J. A. Sethian, and B. C. Vemuri, Shape modeling with front propagation, IEEE Trans. Patt. Anal. Mach. Intell., vol. 17, pp , [6] J. A. Sethian, Adaptive fast marching and level set methods for propagating interfaces, in Acta Math. Univ. Comenian. (N.S.), vol. 67, 1998, pp Figure 6: Arterial tree from CTA image. At last, we should stress the fact that there is no single perfect segmentation method for all medical images. Sometimes a simple intensity thresholding will generate satisfactory results, yet in some other cases we have to employ highly complicated algorithms, or use the hybrid form of them. Still, there are cases when all algorithms could fail and we have to resort to manual segmentation. This is the case for the 3D unmasked CTA image shown in Figure 6. We can use fully automatic Chan- Vese method for intracranial arteries, and use semiautomatic fast marching method for carotid arteries at neck, but have to manually extract the transcranial arteries, which are mingled with skull structure. The extracted vasculature provides us with the necessary geometry information for further blood flow modelling. [7] R. Manniesing, B. Velthuis, M. van Leeuwen, I. van der Schaaf, P. van Laar, and W. Niessen, Leve set based cerebral vasculature segmentation and diameter quantification in CT angiography, Medical Image Analysis, pp , [8] H. Scherl, J. Hornegger, M. Prummer, and M. Lell, Semi-automatic level-set baed segmentation and stenosis quantification of the internal carotid artery in 3D CTA data sets, Medical Image Analysis, pp , [9] V. Caselles, F. Catte, T. Coll, and F. Dibos, Geodesic active contours, Int l J. of Comp. Vis., vol. 66, pp. 1 31, [10] L. Ibanez, W. Schroeder, L. Ng, J. Cates, and I. S. Consortium, The ITK Software Guide. Kitware, Inc., [11] T. F. Chan and L. A. Vese, Active contours without edges, IEEE Transactions on Image Processing, vol. 10, pp ,