Image Segmentation Using Deformable Models

Chenyang Xu

Imaging and Visualization Department

Siemens Corporate Research, Inc.

Princeton, NJ, 08540, USA.

In the past four decades, computerized image segmentation has played an increasingly important role in medical imaging. Segmented images are now used routinely in a multitude of different applications, such as the quantification of tissue volumes [1], diagnosis [2], localization of pathology [3], study of anatomical structure [4, 5], treatment planning [6], partial volume correction of functional imaging data [7], and computer-integrated surgery [8, 9]. Image segmentation remains a difficult task, however, due to both the tremendous variability of object shapes and the variation in image quality (see Fig. 1). In particular, medical images are often corrupted by noise and sampling artifacts, which can cause considerable difficulties when applying classical segmentation techniques such as edge detection and thresholding. As a result, these techniques either fail completely or require some kind of postprocessing step to remove invalid object boundaries in the segmentation results.

To address these difficulties, deformable models have been extensively studied and widely used in medical image segmentation, with promising results. Deformable models are curves or surfaces defined within an image domain that can move under the influence of internal forces, which are defined within the curve or surface itself, and external forces, which are computed from the image data. The internal forces are designed to keep the model smooth during deformation. The external forces are defined to move the model toward an object boundary or other desired features within an image. By constraining extracted boundaries to be smooth and incorporating other prior information about the object shape, deformable models offer robustness to both image noise and boundary gaps and allow integrating boundary elements into a coherent and consistent mathematical description. Such a boundary description can then be readily used by subsequent applications. Moreover, since deformable models are implemented on the continuum, the resulting boundary representation can achieve subpixel accuracy, a highly desirable property for medical imaging applications. Figure 2 shows two examples of using deformable models to extract object boundaries from medical images. The result is a parametric curve in Fig. 2(a) and a parametric surface in Fig. 2(b). Although the term deformable models first appeared in the work by Terzopoulos and his collaborators in the late eighties [12–15], the idea of deforming a template for extracting image features dates back much farther, to the work of Fischler and Elschlager’s spring-loaded templates [16] and Widrow’s rubber mask technique [17]. Similar ideas have also been used in the work by Blake and Zisserman [18], Grenander et al. [19], and Miller et al. [20]. The popularity of deformable models is largely due to the seminal paper “Snakes: Active Contours” by Kass, Witkin, and Terzopoulos [13]. Since its publication, deformable models have grown to be one of the most active and successful research areas in image segmentation. Various names, such as snakes, active contours or surfaces, balloons, and deformable contours or surfaces, have been used in the literature to refer to deformable models.

There are basically two types of deformable models: parametric deformable models (cf. [13, 21–23]) and geometric deformable models (cf. [24–27]). Parametric deformable models represent curves and surfaces explicitly in their parametric forms during deformation. This representation allows direct interaction with the model and can lead to a compact representation for fast real-time implementation. Adaptation of the model topology, however, such as splitting or merging parts during the deformation, can be difficult using parametric models. Geometric deformable models, on the other hand, can handle topological changes naturally. These models, based on the theory of curve evolution [28–31] and the level set method [32, 33], represent curves and surfaces implicitly as a level set of a higher-dimensional scalar function. Their parameterizations are computed only after complete deformation, thereby allowing topological adaptivity to be easily accommodated. Despite this fundamental difference, the underlying principles of both methods are very similar.

In this talk, we first introduce parametric deformable models, and then describe geometric deformable models. Next, an explicit mathematical relationship between parametric deformable models and geometric deformable models is presented. Finally, we provide an overview of several extensions to these deformable models and point out future research directions.