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Hierarchical Distance-based Aggregation

Approximation is one of the key techniques used in manipulating geometric objects. Aggregation is a form of approximation that joins objects that are likely to be grouped together as a single unit. This can improve efficiency, reduce complexity and generate levels of detail, and has application in rendering, cartography and molecular structure design. In this work, we present a general framework to aggregate nearby objects together. The proximity of the objects is determined based on distances between them. Grouped objects are approximated into shapes similar to alpha shapes. This reduces volume as compared to convex hull approximation. However, typical alpha shape approximation, which uses a constant alpha value, fails to adapt to non-uniform inputs. To address this, we use a function to adaptively determine the value of alpha based on the properties of the objects to be aggregated. Varying the threshold distance that determines the group of nearby objects, we can create a hierarchy of aggregation for any environment. We evaluate our method using two dimensional objects. The quality of the aggregated objects is evaluated using shape metrics including area, perimeter and circularity and is compared with convex hull and alpha shape approximations. Our method produces aggregates which are closer to the original objects than alpha shapes and convex hulls. We also demonstrate how the levels in aggregation can be used to solve motion planning problems more efficiently in complex environments. We have also extended our implementation to three dimensional objects with results similar to two dimensional environments.