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Quantitative Fourier Analysis of Approximation Techniques: Part I—Interpolators and Projectors

Related publications (45)

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Sampling in Practice: Is the Best Reconstruction Space Bandlimited?

Michaël Unser, Dimitri Nestor Alice Van De Ville, Sathish Ramani

Shannon's sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some “shift-invariant” space, which may or may not be bandlimited. In this paper, we present some further justification for ...

IEEE2005

Hexagonal Versus Orthogonal Lattices: A New Comparison Using Approximation Theory

Dimitri Nestor Alice Van De Ville, Thierry Blu

We provide a new comparison between hexagonal and orthogonal lattices, based on approximation theory. For each of the lattices, we select the “natural” spline basis function as generator for a shift-invariant function space; i.e., the tensor-product B-spli ...

IEEE2005

Pade-Jacobi Approximants

Jan Sickmann Hesthaven

We discuss projection based Pad ́e-Jacobi approximants in general and present in particular an exact rational approximation to the Sign function. This serves as vehicle to analyze the behavior of Pad ́e-Jacobi approximants for discontinuous functions. The ...

2004

Sampling innovations

Pina Marziliano

Sampling theory has prospered extensively in the last century. The elegant mathematics and the vast number of applications are the reasons for its popularity. The applications involved in this thesis are in signal processing and communications and call out ...

EPFL2002

Discretization of the radon transform and of its inverse by spline convolutions

Michaël Unser, Michael Stefan Daniel Liebling, Stefan Horbelt

We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths hi and degrees ni. We apply our results to derive spline-convolution-based algorithms for two closely related prob ...

IEEE2002

Nondyadic and nonlinear multiresolution image approximations

Maria Arrate Munoz Barrutia

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics ...

EPFL2002

Sampling of Periodic Signals: A Quantitative Error Analysis

Michaël Unser, Thierry Blu, Mathews Jacob

We present an exact expression for the

L _{ 2 }

error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function. This formulation is applicable to a wide variety of linear approximation schem ...

IEEE2002

An Error Analysis for the Sampling of Periodic Signals

Michaël Unser, Thierry Blu, Mathews Jacob

We analyze the representation of periodic signals in a scaling function basis. This representation is sufficiently general to include the widely used approximation schemes like wavelets, splines and Fourier series representation. We derive a closed form ex ...

SampTA2001

Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets

Michaël Unser, Thierry Blu

In a previous paper, we proposed a general Fourier method which provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the u ...

IEEE1999

Wavelet Projections for Volume Rendering

Michaël Unser, Martin Vetterli

We extended Gross's method of volume wavelet rendering by computing splats via an orthogonal projection operator. The method decomposes the volume data into a wavelet pyramid representation in the spline domain. The splats of the basis functions are approx ...

EUROGRAPHICS1999

Sampling of Periodic Signals: A Quantitative Error Analysis

Michaël Unser, Thierry Blu, Mathews Jacob

We present an exact expression for the

L _{ 2 }

IEEE2002