We examine two different matrix representations of curves and surfaces based on or constructed by interpolation through points. Both are essentially implicit representations of objects given as parametric models or given as a point cloud, and both are quite powerful since they reduce geometric operations to linear algebra. First, we examine a representation by interpolation matrices, developed ...

The question of the physical significance of the Mueller matrix average is addressed by means of an analysis of interpolation processes. We draw a comparison between two interpolation processes. The first one is related to the classical Euclidean metrics and the second one is based on the log-Euclidean metrics. Both the associated interpolation procedures are depicted with their underlying phys...

In this paper, we build up a framework for sparse interpolation. We first investigate the theoretical limit of the number of unisolvent points for sparse interpolation under a general setting and try to answer some basic questions of this topic. We also explore the relation between classical interpolation and sparse interpolation. We second consider the design of the interpolation points for th...

This paper presents a method to reduce the complexity of the deterministic maximum likelihood (DML) estimator in the wideband direction-of-arrival (WDOA) problem, which is based on interpolating the array projection matrix in the temporal frequency variable. It is shown that an accurate interpolator like Chebyshev’s is able to produce DML cost functions comprising just a few narrowband-like sum...

In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients whenever a nodes is added to or removed from the interpolation. The second algorithm, wh...

Let Aρ denote the set of functions analytic in |z| < ρ but not on |z| ρ 1 < ρ < ∞ . Walsh proved that the difference of the Lagrange polynomial interpolant of f z ∈ Aρ and the partial sum of the Taylor polynomial of f converges to zero on a larger set than the domain of definition of f . In 1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation, and H...

An algorithm is described for extracting a polynomial matrix factor featuring any subset of the zeros of a given non-singular polynomial matrix. It is assumed that the zeros to be extracted are given as input data. Complex or repeated zeros are allowed. The algorithm is based on interpolation and relies upon numerically reliable subroutines only. It makes use of a procedure that computes the ge...

We study a subspace of bivariate trigonometric polynomials for interpolating functions on the sphere. We give an explicit construction for a system of interpolation nodes, and the corresponding basis for this space, that allows a (discrete) fast Fourier transform–type formula for the interpolant. We prove that the uniform norm of our interpolation operator is of the order (log M)2, where M is t...

In this paper we propose a 3D image reconstruction algorithm for a 3D Compton camera being developed at the University of Michigan. We present a mathematical model of the transition matrix of the camera which exploits symmetries by using an adapted spatial sampling pattern in the object domain. For each projection angle, the sampling pattern is uniform over a set of equispaced nested hemisphere...