Interpolating implicit surfaces from scattered surface data using compactly supported radial basis functions

Abstract

We describe algebraic methods for creating implicit surfaces using linear combinations of radial basis interpolants

Incomplete Cholesky Preconditioners with Band‐Diagonalization for Sparse Symmetric Systems

“…In this section, we consider a surface reconstruction problem of 3D objects [1,2,6,7,9,10]. The surface can be reconstructed as an implicit function f (p) = 0, which is determined by solving a linear system derived from input scattered data.…”

“…where φ is a radial basis function [10], λ i are the weights, p i = (x i , y i , z i ) T ∈ R 3 are the locations of the constraints, and q(p) is a degree one polynomial : q(p) = q 0 + q 1 x + q 2 y + q 3 z. These unknowns, λ i and q i are determined by solving a symmetric linear system [2,7,10]:…”

“…On the other hand, there is one typical problem having the property that the process of the band-diagonalization can be done at a low computational cost, that is, implicit surface reconstruction problems interpolating 3D scattered data [1,2,6,7,9,10]. The band-diagonalization algorithm proposed in [6] is an efficient method for that : Non-zero element : Zero element :…”

Implicit surface reconstruction of 3D objects: Preconditioned iterative methods for the CSRBF‐type linear systems

“…The method of representing the object by an implicit surface reconstruction has drawn attention as a way of dealing with such problems [3][4][5][6][7][8]. This method has the following advantages.…”

“…In one type, all of the given data are treated systematically to determine the function f(p) from all data [1,[4][5][6]. In this method, the function f(p) is expressed as a linear combination of the basis and a polynomial.…”