Bicriteria Shapes: Hierarchical Grouping and Aggregation of Polygons with an Efficient Graph-Cut Approach

An important task of pattern recognition and map generalization is to partition a set of polygons into groups and to aggregate the polygons within each group to a representative output polygon. We introduce a new method for this task called bicritria shapes. Following a classical approach, we define the output polygons by merging the input polygons with a set of triangles that we select from a conforming Delaunay triangulation of the input polygons’ exterior. The innovation is that we control the selection of triangles with a bicriteria optimization model that is efficiently solved via graph cuts. In particular, we minimize a weighted sum that combines the total area of the output polygons and their total perimeter. In a basic problem, we ask for a single solution that is optimal for a preset parameter value. In a second problem, we ask for a set containing an optimal solution for every possible value of the parameter. We discuss how this set can be approximated with few solutions and show that it is hierarchically nested. Hence, the output is a hierarchical clustering that can be used to obtain multiple levels of detail. An evaluation with building footprints as input concludes the article.