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# The 2 x 2 / 2 Curve Pyramid

The basic idea is to represent linear structures within images by curve relation. A cell of the underlying square grid is considered as the observation window through which the curve is observed. Consequently, only the intersection of the observed curve with the observation window is stored.

A NE curve relation

Curve relations are denoted by AB, where AB in (N=north, E=east, S=south, W=west, F) and F is a special end code when a curve ends within a cell.

The basic steps for deriving a lower resolution representation of the curve pyramid:

• split - subdivision of the cell contents by introducing a diagonal
• transitive closure - the curve relations of the four son cells are merged by computing the transitive closure of all relations
• reduce - the curve relations of the four diagonal elements of the new cell are selected

Reduction in the 2 x 2 / 2 curve pyramid.

Properties of the 2 x 2 / 2 curve pyramid

• a curve remains connected until it is completely covered by one cell. Then it disappears.
• length reduction property - short curves will disappear after a few levels and only long curves will be represented in higher levels of the pyramid.  This property is used for structural noise filtering. First, the curve pyramid is built up to a certain level and afterwards all curves which are not present at this level are removed from the levels below.