Exponential splines and minimal-support bases for curve representation

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 B-splines. 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 minimal-support bases that reproduce ellipses and higher-order harmonic curves.

► Compact-support bases of exponential polynomials have an exponential B-spline factor.
► We found the bases of minimal support capable of reproducing exponential polynomials.
► We can reproduce ellipses and higher-order harmonic curves.
► We designed multiresolution and subdivision schemes to represent closed curves.
► We applied our findings to build efficient active contours.