Statistical optimality of Hermite splines

Abstract

Abstract-Hermite splines are commonly used for interpolating data when samples of the derivative are available, in a scheme called Hermite interpolation. Assuming a suitable statistical model, we demonstrate that this method is actually optimal for reconstructing random signals in Papoulis' generalized sampling framework. We focus on second-order Lévy processes-the integrated version of Lévy processes-and rely on cubic Hermite splines to approximate the original continuous-time signal from its samples and its derivatives at integer values. We statistically justify the use of this reconstruction scheme by demonstrating the equivalence between cubic Hermite interpolation and the linear minimum mean-square error (LMMSE) estimation of a secondorder Lévy process. We finally illustrate the cubic Hermite reconstruction scheme on an example of a discrete sequence sampled from the realization of a stochastic process.

Support and approximation properties of Hermite splines

“…The matrix being invertible (with determinant α 3 β 3 (α − β) = 0), we deduce that D(ω 0 ) = B(ω 0 ) = 0. Similary, (14) with ω = ω 0 and (ω 0 + 2π) implies that A(ω 0 ) = C(ω 0 ) = 0. Injecting this in (11) with ω = ω 0 , we deduce that β 2 (ω 0 ) = α 3 = 0, which contradicts our initial assumption.…”

“…The matrix being invertible (with determinant α 3 β 3 (α − β) = 0), we deduce that D(ω 0 ) = B(ω 0 ) = 0. Similary, (14) with ω = ω 0 and (ω 0 + 2π) implies that A(ω 0 ) = C(ω 0 ) = 0.…”

“…This hands-on intuition can be translated to formal properties of Hermite splines and mathematically characterized. As examples, the joint interpolation properties of Hermite splines (see Section 1.2), which ensure that, at integer values, the interpolated function exactly matches the sequences of samples and derivatives samples that were used to build it; their smoothness properties [11], which guarantee low curvature of the interpolated curve under some mild conditions; and their statistical optimality (in terms of MMSE) for the reconstruction of second-order Brownian motion from direct and first derivative samples [14]. In that spirit, we investigate in this work the theoretical counterpart of two additional features that are observed to grant Hermite splines their practical usefulness.…”

Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

“…We are especially interested in innovation models, for which one assumes that the signal can be whitened (i.e., transformed into a white noise) by the application of a linear operator [36], [37]. Non-periodic models have been studied in many situations, including the random processes associated with differential [38], [39] or fractional operators [40]. Extensions to non-Gaussian models are extensively studied by Unser and Tafti [41].…”

Sampling and (sparse) stochastic processes: A tale of splines and innovation

“…For particular configurations of analysis functions, it is possible as well to obtain multi-spline extensions of such solutions for the generalized sampling problem; in particular, for the Hermite interpolation problem where the reconstruction is based on the samples of the function and its derivatives [35].…”