Using segmental active contour model, the contour is already close to the object boundary, therefore fine match scheme only need to search the neighborhood around the contour to exactly localize the object. At each iteration, the control points along the contour of model seek its search area and move to new place to minimize the energy of model.
The search area of control point is the region specified according to the position of and its neighboring points, in which can move at arbitrary direction to minimize the model energy. Both the width and depth of search area should be considered. Connect and its neighboring control points and define the middle point of these two line segments as and and compute their normal vectors respectively by :
Thus, the left and right edge of search area are confirmed by the normal vectors of and If choice appropriate depth of search area for a closed contour,
134 X. Ran and F. Qi
the width definition approach will form a ring-shaped search band around the contour of model, which not only avoids the self-intersection but also guarantees the integrality of search area for whole contour.
A fast adaptive algorithm is proposed for determining the depth of search area. Conforming to the direction of normal vector of the control point, The depth of search area extends outside and inside the model so that the model can expand or contract. Because of the pre-processing by segmental active contour model at previous stage, the initial depth of search area can be set relatively small. If the control point can move to a new place to make the similarity between and smaller at this iteration, its depth of search area remains unchanged at next iteration. Otherwise, the depth of search area may be reduced.
This adaptive method makes minimizing process focus on the deformed part of model and results in faster convergence.
In case of searching boundary concavity, edge-based segmentation methods can not perform well unless using some special process like the method proposed in [2] to diffuse the external force. The fine match scheme presented in this paper is based on statistical information of search areas and more effective to pull the contour into boundary concavity, which is illustrated in Fig.2. The search area of may be divided into two and as shown with shadow in Fig.2) by the line segments connected with its neighboring control points. Suppose that is moved to the search area is then divided into two new part by three line segments connected and their neighboring control points. That is to say, the area that belonged to before belongs to now. Accordingly, the MSE of intensity in either or is reduced and the similarity between and becomes smaller. In this way the contour may move into boundary concavity increasingly within finite iteration steps.
The algorithm of fine match scheme using statistical information is summa- rized as follows:
1. Determine the width of search area for each control point using (9) and (10), and specify the initial depth of search area. Compute the similarity between
and in each search area and the initial energy of model.
2. Seek the appropriate position for each control point in its search area to minimizing the model’s energy. If the similarity between and in search area of is reduced less than the threshold then do not move and reduce its depth of search area.
3. If the energy difference of model between this iteration and the last itera- tion is less than the threshold the algorithm is finished, otherwise, determine the search area of each control point, then go to step 2.
4 Experimental Results
To evaluate the proposed model and algorithm, two sets of experiments are presented. All the experiments are designed for extracting the boundary of ven- tricles from the brain MR images. The size of the image is 256*256, and the gray scales of the image are 256. In all experiments, the initialization of the model is
Segmental Active Contour Model 135
Fig. 3. Segmentation result of MRI brain image by our model. (a) initialization, (b) the result achieved at the fist stage using hierarchical deformation mecha- nism, (c) the final result by fine match algorithm.
provided by user. The first set of experiments demonstrates the performance of the segmental active contour model and its results at two processing stages. The second set of experiments compares our model with the classical snakes model[1]
and GVF-Snakes[2].
The initialization of the model is shown in Fig.3(a). Figure 3(b) shows the result of the segmental active contour model at the pre-processing stage using hierarchical deformation mechanism, which is achieved generally within several iterations. The contour of model is close to the actual boundary and describes the approximate shape of brain ventricle. The accurate result is achieved using fine match scheme at the following stage, as illustrated in Fig.3(c). The model exactly conforms to the ventricle’s boundary concavity.
Figure 4 gives the performance comparisons of the proposed model with the classical snakes and GVF-Snakes. All models use the same initial contour for segmenting the brain MR image, as shown in Fig.4(a). Using the classical snakes, the result is shown in Fig.4(b). In addition to being trapped by erroneous edges, this model has poor ability to move into the boundary concavity. Both GVF-Snakes and the proposed model have more accurate results, as shown in Fig.4(c) and Fig.4(d) respectively. However, the model presented in this paper utilizes the simpler and more efficient methods for computing internal energy, the convergence time is reduced. In case of segmenting images with a lot of fake edges around the actual boundary, our model may achieve better results, as it deforms depending on region-based method like region growing techniques instead of those edge-based methods.
5 Conclusion
In this paper, a segmental active contour model is presented for medical image segmentation. A key feature of our model is that it utilizes not only the edge-
136 X. Ran and F. Qi
Fig. 4. Comparisons of our model with classical snakes and GVF-snakes. (a) initialization for all models, (b) the result of classical snakes, (c) the result of GVF-snakes, (d) the result of our model.
based method but also the region-based method. The edge-based method is used to find the approximate shape of object in a short time. The region-based fine match scheme is then adopted to localize the contour on object’s boundary exactly. The experiments show that our new model is a competent approach for medical image segmentation.
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A Level Set Algorithm for Contour Tracking in Medical Images
Yue Li and Qingyu Tang
Department of Biomedical Engineering, Tsinghua University, Beijing, China liyue98@mails.tsinghua.edu.cn
Abstract. In this paper, a novel curve evolution strategy driven by boundary statistics for the segmentation of medical images is proposed and realized under the Level Set framework. It has a speed term similar to that of the Chan-Vese’s method [1] for bimodal pictures, but is driven by boundary statistics (the statistics of intensity in an observing window) instead of the global statistics. In the case of multimodal pictures, the target’s shape can, therefore, be more easily recovered. Here, we present methods for shape prediction based on the signed distance functions and extension field constructed from the boundary statistics. Employing the above techniques, our algorithm can adaptively handle both the sharp and smooth edges of the target, and its efficiency is demonstrated in the contour tracking of medical images.
Key words: curve evolution, level set, boundary statistics, contour tracking, shape prediction
1 Introduction
Algorithms based on curve evolution are classified as active contour. With the advantage of noise-insensitivity and its preservation of closed edge, it has brought much success to computer vision, especially in the challenging task of medical image segmentation. The first curve evolution model for image segmentation is the snake, proposed by Kass et al [2], a parametric contour. Sethian first proposed the geometric contour using level set. To deal with the problems of curve evolution, the curvature-dependent level set method, proposed by Osher and Sethian [3], has been used extensively. It allows for cusps and corners, is unchanged in higher dimensions and can naturally handle the topological changes in the evolving curve.
In the past decade, segmentation algorithms using geometric contour have focused on the evolution speed of the front or interface. Earlier level set tech- niques such as [4,5,6] only utilized the local strategy, they try to reduce the propagation speed where the image gradient is high. Compared with the local- based methods, regional statistics in curve evolution have prove advantageous, as they employ both the local and global information for pulling or pushing the front to the desired boundary. For example, the level set techniques based on Bayesian theory or clustering ([7,8]) are more robust than those mentioned pre- viously. Most of them, however, need prior knowledge of the distribution of the
G.-Z. Yang and T. Jiang (Eds.): MIAR 2004, LNCS 3150, pp. 137–144, 2004.
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138 Y. Li and Q. Tang
image’s intensity, and the estimations directly influence the propagating force and finally the outcome of the segmentation.
Chan et al [1] and Tsai et al [9] both proposed the curve evolution model based on the optimization of the Mumford-Shah function. By solving the region competition-based Mumford-Shah problem, one need not know the exact values of the regional statistics in advance and the initial curve could be arbitrary. The global bimodal method is simple but not suitable for most images, while the multimodal method is impractical. In addition, it becomes necessary to decide the number of modals in advance, which is a major drawback.
Another problem many level set methods face is the choice of the initial curve. Due to the CFL restriction, the front cannot move further than a single space step during each time interval. The position and the shape of the initial curve are therefore crucial to the computational period. If the initial curve is far from the target edge, a large number of iterations would be required in order to propagate the curve toward the target edge.
In this paper we will pay special attention to solving the following problems:
1.
2.
To avoid looking for a global minimum in the multimodal problem, we in- troduce boundary statistics to take the place of regional statistics for prop- agating the front.
In the contour tracking of 3D medical images, information of detected edges from previous slices can be used to predict edges in the current slice. The number of iterations needed to converge to the target edge can be reduced, and efficiency increases.
2 Method