The Hermite radial basis functions (HRBFs) implicits have been used to reconstruct surfaces from scattered Hermite data points. In this work, we propose a closed-form formulation to construct HRBF-based implicits by a quasi-solution approximating the exact solution. A scheme is developed to automatically adjust the support sizes of basis functions to hold the error bound of a quasi-solution. Our method can generate an implicit function from positions and normals of scattered points without taking any global operation. Working together with an adaptive sampling algorithm, the HRBF-based implicits can also reconstruct surfaces from point clouds with non-uniformity and noises. Robust and efficient reconstruction has been observed in our experimental tests on real data captured from a variety of scenes.
@cite_30: A new implicit surface fitting method for surface reconstruction from scattered point data is proposed. The method combines an adaptive partition of unity approximation with least-squares RBF fitting and is capable of generating a high quality surface reconstruction. Given a set of points scattered over a smooth surface, first a sparse set of overlapped local approximations is constructed. The partition of unity generated from these local approximants already gives a faithful surface reconstruction. The final reconstruction is obtained by adding compactly supported RBFs. The main feature of the developed approach consists of using various regularization schemes which lead to economical, yet accurate surface reconstruction.
@cite_19: We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data.
@cite_22: We consider the problem of reconstructing a surface from scattered points sampled on a physical shape. The sampled shape is approximated as the zero level set of a function. This function is defined as a linear combination of compactly supported radial basis functions. We depart from previous work by using as centers of basis functions a set of points located on an estimate of the medial axis, instead of the input data points. Those centers are selected among the vertices of the Voronoi diagram of the sample data points. Being a Voronoi vertex, each center is associated with a maximal empty ball. We use the radius of this ball to adapt the support of each radial basis function. Our method can fit a user-defined budget of centers: The selected subset of Voronoi vertices is filtered using the notion of lambda medial axis, then clustered to fit the allocated budget.
There are schemes for finding a subset of optimal' centers from the point set in order to obtain fast reconstructions. @cite_19 proposed a greedy algorithm that iteratively appends centers which are corresponding to the maximal residual of the current RBF fitting until a desired accuracy is reached. Samozino presented the reconstruction with voronoi centered RBFs @cite_22 . @cite_30 proposed a reconstruction method which combines an adaptive POU approximation with least-squares RBF fitting. Different from @cite_30 , we adopt a quadric error function based on positions and normals of a point and its neighbors instead of local quadratic approximation. Therefore, our center selection step can generate a good spherical cover in a non-iterative way. The selected centers of spheres describe the input shape with a small error.