By Webber R.E.

During this thesis, numerous points of quadtree representations are analyzed. The quadtree is a hierarchical variable-resolution facts constitution compatible for representing the geometric items of special effects, the polygonal maps of laptop cartography, and the digitized photos of laptop vision.The research of quadtrees is gifted in 3 parts:A) a proper semantics for quadtree algorithms,B) greater algorithms for manipulating the normal quarter quadtree, andC) diversifications of the quadtree technique to the duty of representing polygonal maps.

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22 CHAPTER 2 Fisher [Fis58], Ward [War63], and others also represent an implicit quality criterion. 11) i=1 where K is the number of objects, wi is the weight of object i, ai is some numerical measure assigned to i – for instance the position in a metric space – and a¯i is the arithmetic mean of the numerical measures of all objects assigned to the same cluster as i. It is then of course desirable to minimize this measure. However, in spite of all these attempts, it should be noted that the naturalness criterion is still hard to grasp formally.

A similar idea from the area of conceptual clustering was developed earlier by Rowland and Vesonder [RV87]. If a cluster center and a new object are very similar, the cluster center is replaced by a generalization of itself such that it is also able to represent the new object. Aggarwal et al. [AHWY03] criticize that often single-pass algorithms are used for clustering data streams that ignore temporal trends. Instead, analyzing a long time span, historical data may prevail over current trends, which means that the most recent evolution in the stream is not registered by the user.

The requirements for a cluster hierarchy are homogeneity and monotonicity. The first means that intra-cluster densities should be homogeneous. The second implies that the density of a child cluster is always higher than that of its parents. In this way a new element is inserted into the cluster tree where it least disrupts the two criteria, either by appending the new object to the child list of a node already present in the tree if the density lies in a predefined interval or by opening an intermediate cluster if the density lies between the intervals of parent and child cluster.