Multiresolution (MR) provides a useful framework for hierarchical representation and manipulation of geometric objects. It consists of two main operations: decomposition (finding coarse points and details) and reconstruction (subdivision plus error correction). One way to construct MR is to use a reverse subdivision (RS) approach by minimizing the error between fine points and subdivided coarse points. However, after a few levels of decomposition using current RS techniques, the coarse points are usually high energy and do not preserve the overall structure of the fine points. This limits the reverse subdivision-based editing and synthesis applications to produce accurate results.
This thesis introduces a new reverse subdivision framework, entitled "Smooth Reverse Subdivision", that considers the smoothness of the coarse points as a factor in the decomposition. Using a weighted least-squares approach, a trial set of reverse subdivision operations are optimized globally to have a balance between providing a good approximation of the fine points and reducing the energy of the coarse points. Then, by finding a local representation for the resulting operations, the work is extended to subdivision schemes for general topology surfaces. The resulting decomposition and reconstruction operations feature linear processing time. To achieve a compact representation for these operations, a novel fairing operation with local inverse is presented. The new decomposition operations resulting from this new fairing technique can be reversed locally without over-representation. In addition, the new fairing operation provides a well-defined structure for constructing new subdivision and reverse subdivision schemes.
Finally, smooth reverse subdivision is used in example MR editing and synthesis applications as well as image compression. Smooth coarse models resulting from this work, preserve the overall structure of the fine models. This improves the results of current MR editing and synthesis applications.