Deformable models are curves or surfaces defined within an image domain that can move under the influence of internal forces coming within the model itself and external forces computed from the image data. The internal and external forces are defined so that the model will conform to an object boundary or other desired features within an image. Deformable models are widely used in many applications, including edge detection [5,10], shape modeling [15,18], segmentation [8,12], and motion tracking [12,19].

There are two general types of deformable models in the literature today: parametric deformable models [4,10,15,18] and geometric deformable models [2,3,14]. In this chapter, we focus on parametric deformable models, which synthesize parametric curves or surfaces within an image domain and allow them to move toward desired features, usually edges. Typically, the models are drawn toward the edges by potential forces, which are defined to be the negative gradient of potential functions. Additional forces, such as pressure forces [4], together with the potential forces make up the external forces. There are also internal forces designed to hold the model together (elasticity forces) and to keep it from bending too much (bending forces).

There have been two key difficulties with parametric deformable models. First, the initial model must, in general, be close to the true boundary, or else it will likely converge to the wrong result. Several methods have been proposed to address this problem, including multiresolution methods [11], pressure forces [4], and distance potentials [5]. The basic idea is to increase the capture range of the external force fields and to guide the model toward the desired boundary. The second problem is that deformable models have difficulties progressing into boundary concavities [1,7]. There has been no satisfactory solution to this problem, although pressure forces [4], control points [7], domain-adaptivity [6], directional attractions [1], and the use of solenoidal fields [16] have been proposed. Most of the methods proposed to address these problems, however, solve only one problem while creating new difficulties. For example, multiresolution methods have addressed the issue of capture range, but specifying how the deformable model should move across different resolutions remains problematic. Another example is that of pressure forces, which can push a deformable model into boundary concavities, but cannot be too strong or "weak" edges will be overwhelmed [17]. Pressure forces must also be initialized to push out or push in, a condition that mandates careful initialization.

In this chapter, we present a class of external force fields for deformable models that addresses both problems just listed. These fields, which we call gradient vector flow (GVF) fields, are dense vector fields derived from images by solving a vector diffusion equation which diffuses the gradient vectors of a gray-level or binary edge map computed from the image. GVF was first introduced in [23] and a generalization to GVF was then proposed in [22]. In this chapter, we present the GVF in the context of its generalized framework. We call the deformable model that uses the GVF field as its external force a GVF

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de/ormaWe modeZ. The GVF deformable model is distinguished from nearly all previous deformable model formulations in that its external forces cannot be written as the negative gradient of a potential function. Because of this, it cannot be formulated using the standard energy minimization framework; instead, it is specified directly from a dynamic force equation.

Particular advantages of the GVF deformable model over a traditional deformable model are its insensitivity to initialization and its ability to move into boundary concavities. As we show in this chapter, its initializations can be inside, outside, or across the object's boundary. Unlike deformable models that use pressure forces, a GVF deformable model does not need prior knowledge about whether to shrink or expand toward the boundary. The GVF deformable model also has a large capture range, which means that, barring interference from other objects, it can be initialized far away from the boundary. This increased capture range is achieved through a spatially varying diffusion process that does not blur the edges themselves, so multiresolution methods are not needed. The external force model that is closest in spirit to GVF is the distance potential forces of Cohen and Cohen [5]. Like GVF, these forces originate from an edge map of the image and can provide a large capture range. We show, however, that unlike GVF, distance potential forces cannot move a deformable model into boundary concavities. We believe that this is a property of all conservative forces that characterize nearly all deformable model external forces, and that exploring nonconservative external forces, such as GVF, is an important direction for future research in deformable models.

This chapter is organized as follows. We focus our most attention on the 2D case and introduce the formulation for traditional 2D parametric deformable models in Section 2. We next describe the 2D GVF formulation in Section 3 and demonstrate its performance on both simulated and real images in Section 4. We then briefly present the formulation for 3D GVF deformable models and their results in two examples in Section 5. Finally, in Section 6, we conclude this chapter and point out future research directions.

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