Deformable Model Initialization 1451 Background

An essential prerequisite of interactive segmentation, which affects overall accuracy as well as efficiency of the method, is the sound initialization of the underlying model. On the one hand, the initial guess has a critical impact on the quality of the segmentation outcome. On the other hand, tedious and time-consuming manual initialization procedures forfeit possible time savings of the segmentation phase.

Although these are well-known facts, emphasis in the literature is usually placed on extensions of the deformable models, while an initial position relatively close to the desired solution is assumed. Nevertheless, the determination of such an initial guess with mouse-based interfaces, especially in 3D, poses a problem.

In the following, two approaches to aid a user in the fast generation of an initialization for a deformable model are described. In the first method, a priori shape knowledge is used for efficient initialization, thus reducing the amount of user interaction. In the second approach, the human-computer interface itself is enhanced by using multimodal interaction metaphors stemming from virtual reality techniques.

14.5.1.1 Ziplock Snakes

Ziplock Snakes emphasize on the improvement of the result based on the user's initialization [60]. They reduce the requirements on the initialization while

Figure 14.12: Segmentation process using Ziplock Snakes. It can be observed how the single segments are optimized from the user selected endpoints towards the center of the segments.

increasing the influence of this information. Traditional Snakes rely on the "closeness" of the initial Snake to the desired result. Depending on the underlying image, the term "closeness" transfers to an almost complete, manual delineation of the desired object's boundary. Ziplock Snakes in contrast require only the specification of the endpoints of the Snakes in the vicinity of clearly visible edge segments, which implies a well-defined edge direction. The system then optimizes the location of the user-supplied points to ensure that they are indeed good edge points, and extracts the associated edge directions. These anchor elements are used as boundary conditions and the edge information is then propagated along the Snake starting from them. The resulting behavior is visually similar to closing a zip, as can be observed in Fig. 14.12. The optimization of the energy term starts by defining the initial Snake as the solution of the corresponding homogeneous version of the system of differential Eq. (14.10). The selected endpoints provide the necessary boundary conditions v(0), v'(0), v(1), and v'(1) to solve this equation directly, i.e. Eq. (14.10) has a unique solution. At this stage the Snake "feels" absolutely no external image forces, as — Pv = 0 for the homogeneous case. Assuming that the user selects both endpoints near dominant edge fragments in the image, this initialization ensures that the Snake already lies close to its optimal position at both ends. During the ongoing iterative optimization process, the image potential P is turned on progressively for all the Snake vertices, starting from the extremities. Two types of Snake nodes are discerned, depending on whether the potential force field FP is turned on (active nodes) for that vertex or not (passive nodes).

The user interaction closely resembles the Life-Wire approach: start- and endpoints of single segments have to be specified and the complete contour is assembled from several segments. The potential discontinuities arising at the connecting vertices are compensated by the fact that these vertices were selected on salient edges with clear directions.

Ziplock Snakes improve the overall convergence properties of Snakes and the probability of getting trapped in an undesirable local minimum is considerably reduced in most cases. However, gaps in object boundaries, misleading edges, and object outlines with low contrast represent insuperable obstacles that are quite usual in medical imagery.

14.5.1.2 Velcro Surfaces

The 3-D analogs of Ziplock Snakes are called Velcro surfaces, as their behavior mimics a piece of Velcro that is progressively clamped onto the surface of interest.

Following a natural extension from 1- to 2-D manifolds, points become lines. In the case of the Snakes under scrutiny this observation states that the initialization of 3-D models requires the specification of lines as boundary conditions. This conclusion comprises the original goal of the Ziplock framework—to reduce the user interaction. From the end-user's perspective, the specification of point landmarks for the initialization of the surface models is more desirable as it can be provided faster and more reliably. Velcro surfaces aim at such a landmark based initialization.

Assuming a set of anchor points and surface normals are given, a solution for the homogeneous equation (thin plate problem without external forces, t = 0, see Eq. (14.22))

can be computed. Specifying boundary conditions for isolated points of de-formable surfaces in principle leads to the theory of weak solutions and the associated mathematical framework for the minimization problem. The solution of the set of Eq. (14.26) belongs to the Sobolev space and is, therefore, a weak solution. It is a smooth surface that is as close as possible to a sphere and interpolates the given points.

Given a total number of M user-supplied anchor points Pi(4 < i < M, non-coplanar) and the normal vector at their locations, the system of equations reduces to

where V* stands for either V(1)*, V(2)*, or V(3)*, the reduced vectors of the three coordinate functions, and K* for an (N — M) x (N — M) sparse matrix that is now invertible and can be solved using a sparse linear solver. Closed 3-D objects can be initialized by selecting at least four non-coplanar points. Of course, since F*, depends on the surface's current position, Eq. (14.27) cannot, in general, be solved in closed form.

The algorithm for the approximation of the underlying image data is analogous to the 2-D case. Starting from the initial shape that is approximately correct in the neighborhood of the selected anchor points, the image potential is taken into account progressively for all surface vertices.

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