Regularization of inverse visual problems involving discontinuities
IEEE Transactions on Pattern Analysis and Machine Intelligence
On the multi-level splitting of finite element spaces
Numerische Mathematik
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fast Surface Interpolation Using Hierarchical Basis Functions
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computation of thin-plate splines
SIAM Journal on Scientific and Statistical Computing
Wavelet methods for fast resolution of elliptic problems
SIAM Journal on Numerical Analysis
Curve and surface fitting with splines
Curve and surface fitting with splines
Approximation by multiinteger translates of functions having global support
Journal of Approximation Theory
Box splines
Applied numerical linear algebra
Applied numerical linear algebra
Reconstruction and representation of 3D objects with radial basis functions
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods
SIAM Journal on Scientific Computing
Scattered date interpolation from principal shift-invariant spaces
Journal of Approximation Theory
Scattered Data Interpolation with Multilevel B-Splines
IEEE Transactions on Visualization and Computer Graphics
Radial Basis Functions
IEEE Transactions on Image Processing
Reconstruction of nonuniformly sampled images in spline spaces
IEEE Transactions on Image Processing
Hi-index | 0.00 |
The problem of fitting a nice curve or surface to scattered, possibly noisy, data arises in many applications in science and engineering. In this paper we solve the problem using a standard regularized least square framework in an approximation space spanned by the shifts and dilates of a single compactly supported function @f. We first provide an error analysis to our approach which, roughly speaking, states that the error between the exact (probably unknown) data function and the obtained fitting function is small whenever the scattered samples have a high sampling density and a low noise level. We then give a computational formulation in the univariate case when @f is a uniform B-spline and in the bivariate case when @f is the tensor product of uniform B-splines. Though sparse, the arising system of linear equations is ill-conditioned; however, when written in terms of a short support wavelet basis with a well-chosen normalization, the resulting system, which is symmetric positive definite, appears to be well-conditioned, as evidenced by the fast convergence of the conjugate gradient iteration. Finally, our method is compared with the classical cubic/thin-plate smoothing spline methods via numerical experiments, where it is seen that the quality of the obtained fitting function is very much equivalent to that of the classical methods, but our method offers advantages in terms of numerical efficiency. We expect that our method remains numerically feasible even when the number of samples in the given data is very large.