Exponential splines and minimal-support bases for curve representation

Abstract

Our interest is to characterize the spline-like integer-shift-invariant bases capable of reproducing exponential polynomial curves. We prove that any compact-support function that reproduces a subspace of the exponential polynomials can be expressed as the convolution of an exponential B-spline with a compact-support distribution. As a direct consequence of this factorization theorem, we show that the minimal-support basis functions of that subspace are linear combinations of derivatives of exponential Bsplines. These minimal-support basis functions form a natural multiscale hierarchy, which we utilize to design fast multiresolution algorithms and subdivision schemes for the representation of closed geometric curves. This makes them attractive from a computational point of view. Finally, we illustrate our scheme by constructing minimalsupport bases that reproduce ellipses and higher-order harmonic curves.

A family of smooth and interpolatory basis functions for parametric curve and surface representation

“…The shifted exponential B-splines in (9) also have the same reproduction property. By combining (11) and (12) and considering an arbitrary shift m, we see that (13) which is a linear combination of polynomials in t of degree up to n that are multiplied by e αt . Thus, we can collect all the factors multiplying t k and rewrite them as b k to express (13) as (14) for n = 0, .…”

“…Such shapes can be constructed independently of the number of control points, which makes them particularly useful for deformable models where, when starting from an initial configuration, it is desirable to approximate shapes with arbitrary precision [8,9,13,27]. We construct symmetric interpolators that have the smallest support given α n 0 as described in Section 3.3.…”

“…Again, using the same basis function of our working example (18) . The basis functions that were used to construct the noninterpolatory surfaces correspond to the ones presented in [13]. For the non-interpolatory surfaces it is difficult and non-intuitive to associate a given control point to the specific surface patch that is modified when moving the control point through user-interaction.…”

“…We show that they are able to reproduce exponential polynomials which include the pure polynomials as a subset. This convenient property is particularly relevant for the exact rendering of conic sections such as circles, ellipses, or parabolas, as well as other trigonometric and hyperbolic curves and surfaces [13]. In its absence, one must resort to subdivision to tackle this aspect [14][15][16][17][18].…”

Spline-based framework for interactive segmentation in biomedical imaging

“…We show in Fig. 2 the resulting curve when sampling an epitrochoid with different M. The particular choice of ϕ determines the properties of the parametric model such as smoothness, reproduction of interesting shapes, or computational efficiency [5].…”