Reconstruction and Representation of 3D Objects with Radial Basis Functions

1. Briefly summarize the paper’s contributions. Does it address a new problem? Does it present a new approach? Does it show new types of results?  [AS] This paper presents a fast approach to reconstruct smooth, manifold surfaces from point clouds and to repair incomplete meshes using polyharmonic Radial Basis Functions (RBFs), which were previously thought to be too slow to compute to be scalable.  [DS] The paper presents a way to use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data. RBFs are fitted and evaluated using fast methods without which this was computationally infeasible to do on large point sets and complex surfaces with arbitrary topology. The fast method used for RBF evaluation is the Fast Multipole Method (FMM). The RBF center reduction method is used to fit the RBF function to a point set. This has smaller memory requirements and faster evaluation times without a loss of accuracy. The resulting implicit function can be easily manipulated through Boolean operations. It also allows for arbitrary accuracy. RBFs slo allow for mesh repair and surface extrapolation.  [FP] Surface is implicitly defined from a global function given by the sum of biharmonics at sample positions and a linear shift. The nature of the radial basis functions used by the authors, |x−xi| , contrast with the type of functions used in previous approaches (e.g., Othake) : instead of being monotonic decreasing these are monotonic increasing functions. According to the authors, this selection of basis functions provides the smoothest interpolation. This characteristic makes the method robust to undersampling and useful for mesh repair.  [JD]  [LF]  [MK] The paper presents a new approach for surface reconstruction that fits RBF to the point, seeking the coefficients that evaluate to zero at the input samples. To avoid the trivial solution, the authors also add off-surface constraints, adding constraint points at a distance of d from the samples along the normal, and constraining the function to be at these points (depending on whether they are inside or outside the surface.)