The primary use of active contours in biomedical imaging is user-guided segmentation. Active contour models help improving accuracy and repeatability of the segmentation results.2 In addition, user-guided segmentation techniques such as snakes and active contours speed up the segmentation process, since the user specifies only a few points instead of tracing the entire contour.
After their inception, snakes were almost instantly used in medical image segmentation and registration. Examples include the contour extraction and segmentation of neuronal dendrites in electron micrographs,5 the extraction of brain structures from MR images,3,29 or the detection of blood-wall interfaces in CT, MR, and ultrasound images.26 In most cases, application of the active contour model is one step in a sequence of image processing steps. The live-wire technique is used in a similar manner. Examples of studies that use medical images include the determination of the thickness of femoral cartilage in MR images of osteoarthritic patients,28 the determination of cartilage volume in MR images,13 and the segmentation of the left heart ventricle in electron-beam CT images.31
One interesting extension is the inclusion of image texture in the energy function.18 In this example, texture is represented by the local variance. Therefore, the image becomes two-valued with an intensity and a local variance value assigned to each pixel. The rate of change, to be used in the potential energy function is then determined from both values. A boundary that attracts the snake may therefore be an intensity transition or a transition of the local variance. According to Lorigo et al.,18 the two-valued method was more stable and less prone to leaking the contour into adjacent areas than was a conventional intensity-based snake.
Another example of using image information beyond the intensity gradient is a study where coronary arteries were segmented in three-dimensional CT images.36 In this study, the crucial preprocessing step was the computation of a maximum a posteriori (MAP) estimation of the probability of each pixel belonging to a class of either blood, myocardium, or lung. The generation of a MAP image yielded a presegmented image where blood and myocardium areas already belonged to different classes. Active contours were used to separate the coronary arteries from other blood-filled regions, such as the aorta or heart chambers. The presegmentation step allowed the active contour algorithm to perform particularly robust final segmentation and subsequent three-dimensional reconstruction of the coronary arteries from two-dimensional active contours.
FIGURE 6.18 External forces to obtain a parametric representation of brain sulci. The initial active contour is pulled into the sulcus by a normal force F1 that acts on every point of the active contour (left). A second force, F2, is responsible for keeping the active contour centered within the sulcus as contour deformation into the sulcus takes place. Force F2 is computed by determining the center of mass of the brain surface inside a small circle around any point of the active contour. Two examples are shown (right): a case (a) where the contour is almost centered and F2 is close to zero, and a hypothetical case (b) where a point outside the sulcus is pulled back inside by a force caused by the mass of the outer cortical surface lying to the left of the point.
An area where active contours are widely used is segmentation and mapping of the brain and brain structures. A three-dimensional active contour model introduced by Vaillant and Davatzikos32 uses two forces to pull an elastic balloon model into the sulci (the valleys) of the brain surface. The forces are illustrated in Figure 6.18. A force F1, acting normal to the active contour, pulls the elastic surface into the sulcus, while a force F2, determined from the center of mass in a circle around each point of the contour, ensures that the active contour remains centered with respect to the sulcus. Such a model can be used for the registration of brain images10 and to generate projections of the sulcal depth and shape, because the force acting on each point can be projected back on the initial contour.
Fourier descriptors have also been useful in the segmentation of MRI brain fea-tures,29 in this case the segmentation of the corpus callosum. Through a training process, a mean model of the corpus callosum shape was obtained. Biological variation was analyzed by performing a principal component analysis of the covariance matrix of the Fourier coefficients, and it was found that the variance becomes very small after the twelfth eigenvector. Consequently, a flexible model was built from the first 12 eigenmodes. After training, two steps were applied to segment the corpus callosum: First, the Hough transform, together with a small set of dominant modes, was used to find the initial optimum placement of the mean model. Second, elastic deformation, restricted by the eigenmodes selected, was allowed to determine the ideal contour of the individual corpus callosum. In an extension, the two-dimensional model was extended into three-dimensional Fourier models.
An interesting application for two-dimensional active contours is the improvement in the three-dimensional surface representation of volumetric image features. Most frequently, biomedical images (primarily, CT and MRI) are acquired with an anisotropic voxel size; that is, the slice thickness in the z (axial)-direction is much larger than the in-plane resolution. A typical example would be a CT image with 0.2 x 0.2 mm pixel size in each slice with a slice thickness of 2 mm. Three-dimensional reconstructions from such a data set do not look smooth, but jagged, showing the steps in the z-direction. An example is shown in Figure 6.19, where a segmented lung obtained from a three-dimensional CT image was rendered. Resolution in the axial direction is about sixfold lower than in-plane resolution. Accordingly, the three-dimensional surface rendered shows steps and jagged regions. After interpolation, the resulting surface has a smoother appearance and allows better identification of image details. To obtain an interpolated surface from two-dimensional snakes, the object is first segmented in each slice by using a conventional snake algorithm. It is possible to progress from slice to slice by using the snake from the previous slice as initialization shape, because differences between slices are generally small. In the next step, new contours are created by interpolating between corresponding vertices of the existing snakes in the axial direction (Figure 6.20). With very low computational effort, a smooth surface can be obtained.
FIGURE 6.19 Demonstration of interpolation with active contours. Three-dimensional renderings such as the segmented lung in this example show jaggedness and steps because of the anisotropic voxel size of most CT and MRI images. Once a parametrized contour is obtained, it is straightforward to create new vertices by interpolation between slices. The resulting surface rendering (B) is smoother and allows better focus on the details.
FIGURE 6.20 Creating new contours through interpolation between slices. In each slice, a snake (dark lines) is defined by its vertices (black circles). New vertices can be created by interpolating in the axial direction (gray circles) and by using these vertices to generate new contours (gray lines).
A similar method has been proposed by Raya and Udupa,22 where an additional step is introduced: The segmented contour is converted into a gray-scale distance map with positive values inside the contour and negative values outside the contour. These gray-value slices are then interpolated and converted to binary by detection of zero crossings. A combination of the live-wire technique with this type of gray-scale interpolation has been used by Schenk et al.25 for the semiautomatic segmentation and three-dimensional reconstruction of the liver in CT images. A different approach was pursued by Leventon and Gibson.17 In this approach, two orthogonal scans are segmented by using surface nets, and the nets are linked and relaxed. The resulting surface is considerably smoother than the surface of one single data set.
Because of its closeness to physical models, it is possible to use active contours and active surfaces for the simulation of material elasticity. For example, Bro-Nielsen4 used elastic active cubes to simulate soft tissue deformation caused by movement of the jaw. Roth et al.23 used a finite-element approach to simulate changes of facial features under planned cosmetic surgery to be used as a tool for preoperative planning. One step further ahead, deformable models can be integrated into a neurosurgery simulation system33 capable of simulating the tissue reaction to cutting, pulling, and prodding. A further enhancement of the experiment was achieved through three-dimensional binocular vision and force feedback.
The basic mathematical foundation and computer implementation of deformable models are well established and in widespread use. Nonetheless, the field of de-formable models is very extensive and under active research. Three-dimensional active surfaces are of particular interest in the biomedical community for the shape analysis of three-dimensional objects with practical applications in image registration and the relationship between shape and disease.
