Kluwer - Handbook of Biomedical Image Analysis Vol

scale, while keeping small details on a finer scale intact. Tamed Snakes combine the hierarchical modeling and Snake-like edge delineation. They adhere to the concept of hierarchical shape representations with several scales of resolution to provide the necessary interactive modeling capabilities while being suitable for numerical simulations.

Hierarchical modeling consists of (a) an iterative refinement of the geometry, which defines a hierarchy of representations and (b) a local detail encoding, which represents the details on a finer level with respect to the next coarser one. Subdivision curves are best suited for such hierarchical modeling, as their representation implicitly comprises a hierarchy of refined shapes. These curves are constructed using univariate subdivision schemes defined as the iterative application of an operator that maps a given polygon Pl = [vi(l)] to a refined polygon

Pl+1 = [vi(l+1)], where l denotes the level of the hierarchy. Such an operator is given by two rules for computing the new so-called even vertices Pl+1

and the new odd vertices Pl!+1

The Tamed Snakes as introduced by Hug [52] employ the DLG-subdivision scheme [53], given by

As the even vertices remain unchanged the subdivision operator has interpolating behavior. The free tension parameter ω has to be chosen inside the interval 0 < ω < 18 to obtain a limit curve that has a continuous tangent vector.

Local details, i.e. transformations of the vertices vi(l) from their given position, are encoded by establishing a local coordinate system fi(l) in each vertex vi(l) and by representing the details with respect to this frame.

The subdivision scheme suggests to start the segmentation process with a reasonably coarse model and to iteratively adjust and refine the control vertices of the resulting curve. Since the subdivision scheme is interpolating, only the odd vertices of the current level must be adjusted to align with a correct boundary position before proceeding to the next finer level. In doing so, the prediction of the refinement operator improves continuously with respect to the vertex position on the next finer level and converges to the correct boundary position.

The traditional Snake energy has to be modified to combine the described coarse-to-fine approach with the Snake-like edge tracking. Tamed Snakes replace the elastic rod model term (Eq. 14.9) by a spring energy similar to the

task. In case of clear boundaries though, the segmentation is not as fast and elegant as with traditional Snakes.

14.4.5 Tamed Surfaces

Analogous to the 2-D case, a subdivision scheme is employed to generate a hierarchy of triangles, which is subject to the governing equations of the physical model. The modified Butterfly subdivision scheme for triangulated surfaces has been suggested for this purpose [54]. It was originally introduced by the same authors [55] as the DLG subdivision scheme in two dimensions and exhibits similar properties: it has interpolating behavior and has a tangent-continuous limit surface, i.e., C1. As for most subdivision methods for triangulated surfaces, quaternary subdivision is used. To correct for degeneracies resulting from topologically irregular neighborhoods, i.e. for vertices with valence other then six, the extensions proposed by Zorin [56] have to be incorporated, hence the term “modified” in the name of the scheme.

Considering the extensions, the weights for the new vertices Vl! are computed as a function of the valence of the vertices vi. To solve Eq. (14.22) on triangular meshes, the Laplace operator has to be replaced by a discrete operator L. One example of such an operator is the so-called umbrella-operator U introduced by Kobbelt [57]:

where n is the valence of the vertex vi. This operator clearly does not consider the geometric constellation of the neighborhood of vi, but results in a simple computation of the Laplace operator with reasonable accuracy for regular meshes.

The approximation of differential operators on arbitrary, discrete 2-D manifolds poses a complex problem. In contrast to the 1-D situation, where the adjacent vertices are always the left and right neighbors of the current vertex, there exists no such fixed relationship for 2-D manifolds. Many different methods for the computation of discrete operators have been proposed in the past few years [58, 59].

At this point it has to be noted that practical implementations of 3-D snakes pose additional challenges that have to be considered. In general, the 3-D

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: startand 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, τ = 0, see Eq. (14.22))

can be computed. Specifying boundary conditions for isolated points of deformable 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, noncoplanar) 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) × (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

Fv" 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.

14.5.2 Model-Based Initialization

The previously described model-based approaches employing statistical encoding of large organ populations can also be successfully applied to efficient initialization of interactive methods [61]. The underlying idea is to apply statistical shape analysis for examining the remaining variability of shape due to interactive point-wise subtraction of variation. The key element is the optimal selection of principal landmarks that carry as much shape information as possible. The goal is to remove as much variation as possible by selecting points that have a maximal reduction potential. The overall process will be described below, considering the previously mentioned population of 71 hand segmented corpus callosi.

Similar to the automatic approach, the first step is the generation of a compact statistical shape description of all object instances in the database. First, we calculate the mean shape p and the instance specific difference vector

pi = pi − p.

To find the eigensystem of our data, the difference vectors are projected into a lower dimensional space whose basis M is constructed by the Gram–Schmidt

The principal components defining the eigenmodes in shape space are then given

˜

by back projecting the eigenvectors U :

14.5.2.1 Point-wise Subtraction of Variation

After the statistical analysis of the anatomical shape, this information can be used to progressively eliminate variation by point-wise fixation of control points. After defining the coordinate system with the AC–PC line, the initialization starts with the average model p (Fig. 14.13(a)). Additional boundary conditions are then introduced by moving control vertices to approximately correct positions

Figure 14.13: (a) Boundary conditions for an initial outline are established by prescribing a position for each coarse control vertex. (b) Shape variations caused by adding the basis vectors defining the x- and y-translation of one point to the average model. The various shapes are obtained by evaluating p + ωU rk with ω {−2, . . . , 2} and k {xj , yj }.

on the object border. In the next step, given the a priori shape knowledge and these constraints, the most natural initialization outline should be chosen. In the context of PCA, this means choosing the model with minimal Mahalanobis distance Dm.

The solution to this task is to find two vectors in variation space describing decoupled x- and y-translations of a given point j in object space with minimal overall variations. Once these vectors are found, all possible boundary conditions can be satisfied by adding these appropriately weighted vectors to the mean shape.

Let rxj and ryj denote the two unknown basis vectors causing unit x- and y- translation of the point j respectively. The Dm of these two vectors is then given by

Taking into account that xj and yj depend only on two rows of U , we define the submatrix U j according to the following expression:

◦

In order to minimize Dm subject to the constraint of a separate x- or y-translation by one unit, we establish the Lagrange function L:

The vectors lxj and lyj denote the required Lagrange multipliers. To find the optimum of L(rk, lk), we calculate the derivatives with respect to all elements

If the basis vectors and the Lagrange multipliers are combined according to

Rj = [rxj ryj ] and L j = [lxj lyj ], Eq. (14.34 ) can be rewritten as two linear matrix equations:

The two basis vectors rxj and ryj (resulting from simple algebraic operations on Eqs. (14.35) and (14.36)) are then given by

While rxj describes the translation of xj by one unit with constant yj and minimal shape variation, ryj alters yj correspondingly. The resulting effect caused by adding these shape-based basis vectors to the average model is illustrated in

Fig. 14.13(b). The most probable shape pˇ given the displacement [ xj , yj ]T

for the control vertex j is consequently determined by

xj

pˇ = p + U Rj . (14.38)

yj

After obtaining the most probable shape for a given control vertex, we now

have to ensure that subsequent modifications do not alter the adjusted vertex.

Therefore, we remove the components from the statistic that cause a displace-

ment of the point. The first step is to subtract the basis vectors Rj , weighted by the example specific displacement U j = [ xj , yj ]iT , from the parameter representation bi of each instance i: