Local spline approximation methods

Approximation Methods for Thin Plate Spline Mappings and Principal Warps

The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p × p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. In practical applications, however, a surprisingly good a...

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to Σ∆ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is hr−1. This rate is optimal in the sense that a function can be...

Best Quadratic Spline Approximation

We present a method for hierarchical data approximation using quadratic simplicial elements for domain decomposition and field approximation. Higher-order simplicial elements can approximate data better than linear elements. Thus, fewer quadratic elements are required to achieve similar approximation quality. We use quadratic basis functions and compute best quadratic simplicial spline approxim...

Parallel Spline Approximation

Three-monotone spline approximation

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x0, x1, x2, x3] f are nonnegative for all choices of distinct points x0, . . . , x3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s ∈ Cr−1[a, b]) with n −1 equidistant knots in (a, b), which is also 3-monotone and satisfies ‖ f − s‖L∞[a,b] ≤ cω...