Fast computation of triangular Shepard interpolants

An Efficient Trivariate Algorithm for Tetrahedral Shepard Interpolation

“…Then, once the data points are stored in such blocks, an optimized searching technique is applied to detect the nearest neighbor points, thus enabling us to carry out a suitable choice of tetrahedra for 3D interpolation. Similar techniques were also studied in [7,4,5] in the context of partition of unity methods combined with the use of local radial basis functions, and suitably adapted to 2D interpolation via triangular Shepard interpolants [6]. Note that in this work we present the tetrahedral Shepard method and the related theoretical results for a generic domain Ω ⊂ R 3 .…”

“…, m and i / ∈ {j 1 , j 2 , j 3 , j 4 }. Moreover, they form a partition of unity, that is m j=1 B µ,j (x) = 1 (6) and consequently, for each i = 1, . .…”

“…, x n } in a compact convex domain Ω ⊂ R 2 . If the data points are scattered, that is they have not any structure, the triangular Shepard method [14] can be applied efficiently [6] to approximate and interpolate the target function f : Ω → R. This scheme has been introduced by Little in 1983 in light of some drawbacks of the more known Shepard method (see [16] for the original paper or [8] to get acquainted with some strategies to overcome them) and uses the same idea of combining in a convex way some values obtainable from the data f (x i ).…”

“…These triangulations are determined by minimizing the bound of the error of the linear interpolant on the vertices of the triangle, chosen in a set of nearby nodes. For such kind of triangulations the block-based partitioning structure procedure introduced in [7] can be easily applied to make the method very fast [6].…”

Construction of Cubic Timmer Triangular Patches and its Application in Scattered Data Interpolation

“…Most researchers have investigated surface interpolation based on triangulations of scattered data and there are several scattered data fitting techniques, such as the Delaunay triangulation method [1], radial basis function (RBF) [2], and moving least square (MLS) [3]. Very recently, new techniques for interpolating scattered data have been developed [1,4,5], which can be implemented in fast algorithms [6].…”