Region-based segmentation approaches examine pixels in an image and form disjoint regions by merging neighborhood pixels with homogeneity properties based on a predefined similarity criterion. Suppose that I represents an image that is segmented into N regions, each of which is denoted as R where i = 1, 2,..., N, the regions must satisfy the following properties:

where £(•) is a logical predicate. The original image can be exactly assembled by putting all regions together (Eq. 3.9) and there should be no overlapping between any two regions R and Rj for i = j (Eq. 3.10). The logical predicate L( ) contains a set of rules (usually a set of homogeneity criteria) that must be satisfied by all pixels within a given region (Eq. 3.11), and it fails in the union of two regions since merging two distinct regions will result in an inhomogeneous region (Eq. 3.12).

The simplest region-based segmentation technique is the region growing, which is used to extract a connected region of similar pixels from an image [36]. The region growing algorithm requires a similarity measure that determines the inclusion of pixels in the region and a stopping criterion that terminates the growth of the region. Typically, it starts with a pixel (or a collection of pixels) called seed that belongs to the target ROI. The seed can be chosen by the operator or determined by an automatic seed finding algorithm. The neighborhood of each seed is then inspected and pixels similar enough to the seed are added to the corresponding region where the seed is, and therefore, the region is growing and its shape is also changing. The growing process is repeated until no pixel

R n Rj = 0 Wi, j = 1, 2,..., N; i = j L(R) = TRUE for i = 1, 2,...,N L(R U Rj) = FALSE Wi, j = 1, 2,..., N; i = j

can be added to any region. It is possible that some pixels may remain unlabeled when the growing process stops.

Hebert et al. [37] investigated the use of region growing to automated delineation of the blood pool with computer simulations and applied the method to three gated SPECT studies using Tc-99m pertechnetate, and the results were promising. Kim et al. [38] also investigated an integrated approach of region growing and cluster analysis (to be described later) to segment a dynamic [18F]fluorodeoxyglucose (FDG) PET dataset. Although qualitatively reasonable segmentation results were obtained, much more work is needed to overcome the difficulties in the formation of odd segments possibly due to spillover region boundaries, and evaluate the quantitative accuracy of the segmentation results using kinetic parameter estimation.

Region splitting methods take an opposite strategy to the region growing. These methods start from the entire image and examine the homogeneity criteria. If the criteria do not meet, the image (or subimage) is split into two or more subimages. The region splitting process continues until all subimages meet the homogeneity criteria. Region splitting can be implemented by quadtree partitioning. The image is partitioned into four subimages that are represented by nodes in a quadtree, which is a data structure used for efficient storage and representation. The partition procedure is applied recursively on each subimage until each and all of the subimages meet the homogeneity criteria.

The major drawback of region splitting is that the final image may contain adjacent regions R and Rj, which are homogeneous, i.e. L(R U Rj) = TRUE, and ideally this region should be merged. This leads to another technique called split-and-merge, which includes a merging step in the splitting stage, where an inhomogeneous region is split until homogeneous regions are formed. A newly created homogeneous region is checked against its neighboring regions and merged with one or more of these regions if they possess identical properties. However, this strategy does not necessarily produce quadtree partitioning of the image. If quadtree partitioning is used, an additional step may be added to merge adjacent regions (nodes) that meet the homogeneity criterion.

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