Reconstruction Method

Following the above discussion, we proposed in [22,63] a segmentation/surface reconstruction method that is based on the following steps: (1) extract region-based statistics; (2) coarser image resolution; (3) define the object characteristic function; (4) PL manifold extraction by the tetra-cubes; (5) if needed, increase the resolution, return to step (3); and (6) apply T-surfaces model.

It is important to highlight that T-surfaces model can deal naturally with the self-intersections that may happen during the evolution of the surfaces obtained by step (4). This is an important advantage of T-surfaces.

Among the surfaces extracted in step (4), there may be open surfaces which start and end in the image frontiers and small surfaces corresponding to artifacts or noise in the background. The former is discarded by a simple automatic inspection. To discard the latter, we need a set of predefined features (volume, surface area, etc.) and corresponding lower bounds. For instance, we can set the volume lower bound as 8(r)3, where r is the dimension of the grid cells.

Besides, some polygonal surfaces may contain more than one object of interest (see Fig. 7.9). Now, we can use upper bounds for the features. These upper bounds are application dependent (anatomical elements can be used).

Figure 7.9: (a) PL manifolds for resolution 3 x 3. (b) Result with the highest (image) resolution.

Figure 7.10: Representation of the multiresolution scheme.

The surfaces whose interior have volumes larger than the upper bound will be processed in a finer resolution. By doing this, we adopted the basic philosophy of some nonparametric multiresolution methods used in image segmentation based on pyramid and quadtree approaches [3,8,41]. The basic idea of these approaches is that as the resolution is decreasing, small background artifacts become less significant relative to the object(s) of interest. So, it can be easier to detect the objects in the lowest level and then propagate them back down the structure. In this process, it is possible to delineate the boundaries in a coarser resolution (step (4)) and to re-estimate them after increasing the resolution in step (5).

It is important to stress that the upper bound(s) is not an essential point for the method. Its role is only to avoid expending time computation in regions where the boundaries enclose only one object.

When the grid resolution of T-surfaces is increased, we just reparameterize the model over the finer grid and evolve the corresponding T-surfaces.

For uniform meshes, such as the one in Fig. 7.10, this multiresolution scheme can be implemented through adaptive mesh refinement data structures [5]. In these structures each node in the refinement level l splits into nn nodes in level l + 1, where n is the refinement factor and n is the space dimension (n = 2 and n = 3 in our case). Such a scheme has also been explored in the context of level sets methods [61].

As an example, let us consider Fig. 7.9. In this image, the outer scale corresponding to the separation between the objects is finer than the object scales. Hence, the coarsest resolution could not separate all the objects. This happens for the bottom-left cells in Fig. 7.9(a). To correct this result, we increase the resolution only inside the extracted region to account for more details (Figure 7.9(b)).

We shall observe that T-surfaces makes use of only the data information along the surface when evolving the model toward the object boundary. Thus, we can save memory space by reading to main memory only smaller chunks of the data set, instead of the whole volume, as is usually done by the implementations of deformable surface models. Such point is inside the context of out-of-core methods which are discussed next.

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