A connection between partial symmetry and inverse procedural modeling

In this paper, we address the problem of inverse procedural modeling: Given a piece of exemplar 3D geometry, we would like to find a set of rules that describe objects that are similar to the exemplar. We consider local similarity, i.e., each local neighborhood of the newly created object must match some local neighborhood of the exemplar. We show that we can find explicit shape modification rules that guarantee strict local similarity by looking at the structure of the partial symmetries of the object. By cutting the object into pieces along curves within symmetric areas, we can build shape operations that maintain local similarity by construction. We systematically collect such editing operations and analyze their dependency to build a shape grammar. We discuss how to extract general rewriting systems, context free hierarchical rules, and grid-based rules. All of this information is derived directly from the model, without user interaction. The extracted rules are then used to implement tools for semi-automatic shape modeling by example, which are demonstrated on a number of different example data sets. Overall, our paper provides a concise theoretical and practical framework for inverse procedural modeling of 3D objects.