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The impetus for the local-clustered appearance model (Stough et al., 2007) is that more local regions will better specify the exterior than the common single homogeneous region approach. The question is what constitutes a region. For my purposes, consider that an organ or other volume whose local intensity distributions are distinguishable from those of neighboring volumes constitutes such a region. Examples of such volumes are neighboring organs and fat deposits. The presence of these regions, to the extent that they are distinct from one another, is a cause of the intensity inhomogeneity in the organ exterior. I avoid modeling the 3D shapes of these regions and for my appearance model consider them only as they affect the local distributions near the boundary of the object of interest. I apply the same general notion as in Chapter 3, where I determined profile types through clustering. Here I determine region types, corresponding to these different organs and volumes, through clustering on the RIQFs of smaller regions.

Over all training images Ii, I compute RIQFs qip for many local exterior regions

anchored to individual geometrically defined points on the object boundary indexed by

p. These points are in fact the spoke ends of the m-rep model representing the object (Section 2.4). The interior of the object is treated as one region. As in the global case,

0 0.2 0.4 0.6 0.8 1 800 850 900 950 1000 1050 1100 1150 1200 Quantile Intensity (Hounsfield)

Bladder RIQFs with Clusters

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Bladder RIQF Cluster 1

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Bladder RIQF Cluster 2

Figure 4.7: Clustering on local RIQFs for the bladder exterior. Left shows the pooled data with cluster centers overlaid (two clusters). The middle and right images show the two cluster populations with mean and±2 standard deviations overlaid. The reasonable separation into lighter and darker distributions is evidenced in Figure 4.6.

the contribution of a voxel toqip is weighted by its distance to the surface but further is

equal to zero if its closest point on the surface is not close enough to point p– ensuring locality. Thus each local exterior region has its own projection onto the surface, called a patch. This patch breadth parameter is the maximum distance from the anchor point

p to the projection of a voxel onto the surface for that voxel to still contribute to the RIQF of the local region anchored to p. For this experiment the patch breadth was set to 1.5 cm, which ensured that all near boundary voxels are sampled while the local regions overlap by a small amount. Parameters for this model are the density of points on the surface and the patch breadth, in addition to the distance weighting Gaussian of the global region model.

The patch breadth can also be a parameter to a continuous weighting rather than a step function. In the global model I described the distance weighting Gaussian as leading to a weight at each sample along a profile. These weights are precomputed since the profile is normal to surface and thus distance along the profile is distance from the surface. An additional patch breadth Gaussian can weight the profile itself in the following way. Each profile intersects the surface at some point, and that point has a distance to the anchor point of each local region. All of the voxels sampled along a profile can then be weighted according to which region is being sampled. The contribution of voxels sampled along this profile is then the product of the profile weight and the distance weighting Gaussian. Computing these weights involves minimal additional computation through my constructive sampling scheme.

I then cluster the pooled set of RIQFs for all boundary points and images,{qip,∀i, p},

using fuzzy c-means clustering (Bezdec, 1981). The Euclidean feature space distances inherent in this method hold for RIQFs (see Section 2.3.1). I specify the number of clusters. The results are the set of RIQF cluster centers {µk_{} and cluster membership}

scores{uk

distance in the M2 metric used: {µk}= min {µ˜1,...,µ˜K∈<n} X ∀i,p K X k=1 uk_ip||qip−µ˜k||2, (4.5) withuk ip∈[0,1] and P ku k

ip= 1 for giveniandp. More details of the iterative procedure

are presented below in Section 4.6.1.

The last two steps for training the local-clustered model are to partition the object boundary according to representative region type and to separately characterize the variability in the cluster populations. To partition the object model boundary, I use the explicit correspondence across training cases required of the shape model (see Section 4.5.1). Each pointpis assigned the region type/cluster center k that maximizes P

iu k ip,

the sum of cluster membership scores for that point over all images. Finally, PCA on the RIQFs making up the individual cluster populations results in Gaussian models of the RIQF variability per region type. Each positionpis then assigned the PCA model of its cluster. For the interior patches, PCA is applied to the pooled set of RIQFs sampled from all points over all training images, equivalent to finding one cluster on the interior. During target segmentation the image is sampled relative to a prospective model to populate the interior and exterior sets of RIQFs {qint

p ,∀p} and {qextp ,∀p}. For each

point p on the exterior, I compute the Mahalanobis distance of qext

p with respect to

the principal modes and variances of that point’s region type. For each point p on the interior, I compute the Mahalanobis distance ofqint

p with respect to the principal modes

and variances found for the interior. The image match is then the sum of these terms over all points, weighted by the points’ comparative importance (related to how many voxels contributed to each local RIQF).

Confirming evidence for this approach is found in the spatial distribution of region- type on the boundary that I observe in training. When I look for two clusters on the bladder data (Figure 4.7), the resultant boundary partition corresponds to lighter and

darker local distribution areas (Figure 4.6). This is anatomically justified because the bladder is surrounded mostly by lower intensity bowel and fat, with much brighter tissue from the pubic bone area and prostate inferior to it. There is similar evidence for the prostate, which has brighter tissue exterior to it in the pubic bones areas and bladder, with darker tissue elsewhere. As a result of the clustering scheme, searching for more than two clusters did not lead to such intuitive regions. Section 4.6.1 describes a new clustering on this data which does lead to cluster populations that accurately reflect the neighboring organs and volumes.

In the local-clustered scheme described in this section, through clustering on local RIQFs I determine region types which correspond to neighboring organs and volumes. The RIQFs of a cluster can be thought of as samples from one of these larger-scale volumes. The PCA model of a cluster describes the variability in these samples, which make up the set of local RIQFs belonging to that cluster. This PCA model does not, however, describe the variability in the RIQFs over training of the larger-scale volume associated with the cluster. In Section 4.6 below, I present an extension to the local- clustered approach that does account for the variability of these larger-scale RIQFs, which are constructed from the local RIQFs.