The initialization should position the model near the desired solution while retaining certain properties such as smoothness, connectivity, etc. Given a rough initial estimate, the surface deformation process moves the surface model to ward specific features in the data. One must choose those properties of the input data to which the model will be attracted and what role the shape of the model will have in the deformation process. Typically, the deformation process combines a data term with a smoothing term, which prevents the solution from fitting too closely to noise-corrupted data. There are a variety of surface-motion terms that can be used in succession or simultaneously, in a linear combination to form F(x) in Eq. (8.4).

Curvature: This is the smoothing term. For the work presented here we use the mean curvature of the isosurface H to produce

The mean curvature is also the normal variation of the surface area (i.e., minimal surface area). There are a variety of options for second-order smoothing terms [41], and the question of efficient, effective higher-order smoothing terms is the subject of ongoing research [7, 31,42]. For the work in this chapter, we combine mean curvature with one of the following three terms, weighting it by a factor i, which is tuned to each specific application.

Edges: Conventional edge detectors from the image processing literature produce sets of "edge" voxels that are associated with areas of high contrast. For this work we use a gradient magnitude threshold combined with nonmaximal suppression, which is a 3D generalization of the method of Canny [16]. The edge operator typically requires a scale parameter and a gradient threshold. For the scale, we use small, Gaussian kernels with standard deviation a = [0.5,1.0] voxel units. The threshold depends on the contrast of the volume. The distance transform on this edge map produces a volume that has minima at those edges. The gradient of this volume produces a field that attracts the model to these edges. The edges are limited to voxel resolution because of the mechanism by which they are detected. Although this fitting is not sub-voxel accurate, it has the advantage that it can pull models toward edges from significant distances, and thus inaccurate initial estimates can be brought into close alignment with high-contrast regions, i.e. edges, in the input data. If E is the set of edges, and DE (x) is the distance transform to those edges, then the movement of the surface model is given by

Grayscale features—gradient magnitude: Surface models can also be attracted to certain grayscale features in the input data. For instance, the gradient magnitude indicates areas of high contrast in volumes. By following the gradient of such grayscale features, surface models are drawn to minimum or maximum values of that feature. Typically, grayscale features, such as the gradient magnitude, are computed with a scale operator, e.g., a derivative-of-Gaussian kernel. If models are properly initialized, they can move according to the gradient of the gradient magnitude and settle onto the edges of an object at a resolution that is finer than the original volume.

If G(x) is some grayscale feature, for instance G(x) = |VI(x)|, where I(x) is the input data (appropriately filtered—we use Gaussian kernels with a « 0.5), then

where a positive sign moves surface toward maxima and the negative sign toward minima.

Isosurface: Surface models can also expand or contract to conform to isosurfaces in the input data. To a first order approximation, the distance from a point x e U to the k-level surface of I is given by (I(x) — k) / |V11. If we let g(a) be a fuzzy threshold, e.g., g(a) = a/\J 1 + a2, then

causes the surfaces of 4 to expand or contract to match the k isosurface of I. This term combined with curvature or one of the other fitting terms can create "quasi-isosurfaces" that also include other considerations, such as smoothness or edge strength.

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