Growing Least Squares for the Analysis of Manifolds in Scale‐Space

We present a novel approach to the multi‐scale analysis of point‐sampled manifolds of co‐dimension 1. It is based on a variant of Moving Least Squares, whereby the evolution of a geometric descriptor at increasing scales is used to locate pertinent locations in scale‐space, hence the name “Growing Least Squares”. Compared to existing scale‐space analysis methods, our approach is the first to provide a continuous solution in space and scale dimensions, without requiring any parametrization, connectivity or uniform sampling. An important implication is that we identify multiple pertinent scales for any point on a manifold, a property that had not yet been demonstrated in the literature. In practice, our approach exhibits an improved robustness to change of input, and is easily implemented in a parallel fashion on the GPU. We compare our method to state‐of‐the‐art scale‐space analysis techniques and illustrate its practical relevance in a few application scenarios.