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Extremal Surface Based Projections Converge and Reconstruct with Isotopy

Many real world applications need to model a smooth surface from a noisy point sample. In order to eliminate unnecessary undulations on the output surface, it is necessary to project the points on a nearby smooth surface and then approximate the smooth surface with a surface reconstruction algorithm. Moving Least Square (MLS) surfaces in the family of extremal surfaces have been shown to be useful as projection targets. However, it is unknown if the extremal surface based projection procedure converges and if the target extremal surface is isotopic to the original sampled surface. We prove these two facts. The success of the entire projection method depends on the quality of the estimated normals at the sample points. We suggest an algorithm to estimate the normals from the noisy samples which can be of independent interest for other applications. We also present some experimental results.