Hierarchically Partitioned Implicit Surfaces For Interpolating Large Point Set Models

We present a novel hierarchical spatial partitioning method for creating interpolating implicit surfaces using compactly supported radial basis functions (RBFs) from scattered surface data. From this hierarchy of functions we can create a range of models from coarse to fine, where a coarse model approximates and a fine model interpolates. Furthermore, our method elegantly handles irregularly sampled data and hole filling because of its multiresolutional approach. Like related methods, we combine neighboring patches without surface discontinuities by overlapping their embedding functions. However, unlike partition-of-unity approaches we do not require an additional explicit blending function to combine patches. Rather, we take advantage of the compact extent of the basis functions to directly solve for each patch’s embedding function in a way that does not cause error in neighboring patches. Avoiding overlap error is accomplished by adding phantom constraints to each patch at locations where a neighboring patch has regular constraints within the area of overlap (the function’s radius of support). Phantom constraints are also used to ensure the correct results between different levels of the hierarchy. This approach leads to efficient evaluation because we can combine the relevant embedding functions at each point through simple summation. We demonstrate our method on a several very large models including the Thai statue from the Stanford 3D Scanning Repository. Using hierarchical compactly supported RBFs we interpolate all 5 million vertices of the model.