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Abstract

Hierarchical grids appear in various applications in computer graphics such as subdivision and multiresolution surfaces, and terrain models. Since the different grid types perform better at different tasks, it is desired to switch between regular grids to take advantages of these grids. Based on a 2D domain obtained from the connectivity information of a mesh, we can define simple conversions to switch between regular grids. In this paper, we introduce a general framework that can be used to convert a given grid to another and we discuss the properties of these refinements such as their transformations. This framework is hierarchical meaning that it provides conversions between meshes at different level of refinement. To describe the use of this framework, we define new regular and near-regular refinements with good properties such as small factors. We also describe how grid conversion enables us to use patch-based data structures for hexagonal cells and near-regular refinements. To do so, meshes are converted to a set of quadrilateral patches that can be stored in simple structures. Near-regular refinements are also supported by defining two sets of neighborhood vectors that connect a vertex to its neighbors and are useful to address connectivity queries.

7. Conclusion and future work

In this paper, we present hierarchical grid conversions and use it to systematically define refinements. We extend an existing patch-based hierarchical data structure called ACM for handling connectivity queries of semiregular models, to support hexagonal semiregular models and some additional refinements. From this enhanced support and newly defined conversions, we can best apply a given grid-type to the application-specific challenge. Based on the conversions between regular grids, we can extend ACM to support hexagonal semiregular models as well as a wider range of refinements such as 4–8 and 1-to-7 refinements. We have proposed new types of refinements that may be used to generate smooth subdivision schemes. Finding smoothing masks of these new refinements and their multiresolution filters can be a future work. ACM is designed for semiregular models and models obtained from adaptively subdividing patches. Extending ACM to support meshes that have a combination of regular patches and irregular connectivity is also a future work. One possibility is to combine ACM with a known data structure such as half-edges that can perform well for irregular patches of the mesh.