Because level set models move using gradient descent, they seek local solutions, and therefore the results are strongly dependent on the initialization, i.e., the starting position of the surface. Thus, one controls the nature of the solution by specifying an initial model from which the surface deformation process proceeds. We have implemented both computational (i.e. "semi-automated") and manual/interactive initialization schemes that may be combined to produce reasonable initial estimates directly from the input data.

Linearfiltering: We can filter the input data with a low-pass filter (e.g. Gaussian kernel) to blur the data and thereby reduce noise. This tends to distort shapes, but the initialization need only be approximate.

Voxel classification: We can classify pixels based on the filtered values of the input data. For grayscale images, such as those used in this chapter, the classification is equivalent to high and low thresholding operations. These operations are usually accurate to only voxel resolution (see [12] for alternatives), but the deformation process will achieve subvoxel results.

Topological/logical operations: This is the set of basic voxel operations that takes into account position and connectivity. It includes unions or intersections of voxel sets to create better initializations. These logical operations can also incorporate user-defined primitives. Topological operations consist of connected-component analyses (e.g. flood fill) to remove small pieces or holes from objects.

Morphological filtering: This includes binary and grayscale morphological operators on the initial voxel set. For the results in the chapter we implement openings and closings using morphological propagators [38,39] implemented with level set surface models. This involves defining offset surfaces of 0 by expanding/contracting a surface according to the following PDE, dt = ±lV0l' (8.5)

up to a certain time t. The value of t controls the offset distance from the original surface of 0(t = 0). A dilation of size a, Da, corresponds to the solution of Eq. (8.5) at t = a using the positive sign, and likewise erosion, Ea, uses the negative sign. One can now define a morphological opening operator

Figure 8.3: (a) Interactively positioning a CSG model relative to a Marching Cubes mesh. (b) Isosurface of a binary scan conversion of the initialization CSG model. (c) Final internal embryo structures.

Oa by first applying an erosion followed by a dilation of <, i.e. Oa< = Da o Ea<, which removes small pieces or thin appendages. A closing is defined as Ca< = Ea o Da<, and closes small gaps or holes within objects. Both operations have the qualitative effect of low-pass filtering the isosurfaces in < —an opening by removing material and a closing by adding material. Both operations tend to distort the shapes of the surfaces on which they operate, which is acceptable for the initialization because it will be followed by a surface deformation.

User-specified: For some applications it is desirable and easier for the user to interactively specify the initial model. Here, the user creates a Constructive Solid Geometry (CSG) model which defines the shape of the initial surface. In Fig. 8.3(a) the CSG model in blue is interactively positioned relative to a Marching Cubes mesh extracted from the original dataset. The CSG model is scan-converted into a binary volume, with voxels simply marked as inside (1) or outside (0), using standard CSG evaluation techniques [40]. An isosurface of the initialization volume dataset generated from the torus and sphere is presented in Fig. 8.3(b). This volume dataset is then deformed to produce the final result seen in Fig. 8.3(c).

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