Publications de John Paul Ward

The n-term Approximation of Periodic Generalized Levy Processes

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

In this paper, we study the compressibility of random processes and fields, called generalized Levy processes, that are solutions of stochastic differential equations driven by d-dimensional periodic Levy white noises. Our results are based on the estimati ...

2020

Angular Accuracy of Steerable Feature Detectors

Michaël Unser, Arash Amini, Zsuzsanna Püspöki, John Paul Ward, Julien René Pierre Fageot

The detection of landmarks or patterns is of interest for extracting features in biological images. Hence, algorithms for finding these keypoints have been extensively investigated in the literature, and their localization and detection properties are well ...

SIAM PUBLICATIONS2019

Beyond Wiener's Lemma: Nuclear Convolution Algebras and the Inversion of Digital Filters

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

A convolution algebra is a topological vector space X that is closed under the convolution operation. It is said to be inverse-closed if each element of X whose spectrum is bounded away from zero has a convolution inverse that is also part of the algebra. ...

SPRINGER BIRKHAUSER2019

Rotation Invariance and Directional Sensitivity: Spherical Harmonics versus Radiomics Features

Michaël Unser, Adrien Raphaël Depeursinge, John Paul Ward, Julien René Pierre Fageot

We define and investigate the Local Rotation Invariance (LRI) and Directional Sensitivity (DS) of radiomics features. Most of the classical features cannot combine the two properties, which are antagonist in simple designs. We propose texture operators bas ...

SPRINGER INTERNATIONAL PUBLISHING AG2018

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

Michaël Unser, John Paul Ward, Quy Ha Nguyen

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal f ...

2017

Splines Are Universal Solutions of Linear Inverse Problems with Generalized TV Regularization

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by con ...

2017

On the Besov regularity of periodic Levy noises

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

In this paper, we study the Besov regularity of Levy white noises on the d-dimensional torus. Due to their rough sample paths, the white noises that we consider are defined as generalized stochastic fields. We, initially, obtain regularity results for gene ...

Academic Press Inc Elsevier Science2017

On the Continuous Steering of the Scale of Tight Wavelet Frames

Michaël Unser, Daniel Sage, Zsuzsanna Püspöki, John Paul Ward

In analogy with steerable wavelets, we present a general construction of adaptable tight wavelet frames, with an emphasis on scaling operations. In particular, the derived wavelets can be "dilated" by a procedure comparable to the operation of steering ste ...

SIAM2016

SpotCaliper: Fast Wavelet-Based Spot Detection with Accurate Size Estimation

Michaël Unser, Daniel Sage, Zsuzsanna Püspöki, John Paul Ward

Motivation: SpotCaliper is a novel wavelet-based image-analysis software providing a fast automatic detection scheme for circular patterns (spots), combined with the precise estimation of their size. It is implemented as an ImageJ plugin with a friendly us ...

Oxford Journals2016

Compressibility of Symmetric-α-Stable Processes

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

Within a deterministic framework, it is well known that n-term wavelet approximation rates of functions can be deduced from their Besov regularity. We use this principle to determine approximation rates for symmetric-α-stable (SαS) stochastic processes. Fi ...

IEEE2015

Interpretation of Continuous-Time Autoregressive Processes as Random Exponential Splines

Michaël Unser, John Paul Ward, Julien René Pierre Fageot

We consider the class of continuous-time autoregressive (CAR) processes driven by (possibly non-Gaussian) Lévy white noises. When the excitation is an impulsive noise, also known as compound Poisson noise, the associated CAR process is a random non-uniform ...

IEEE2015

Fast Detection and Refined Scale Estimation Using Complex Isotropic Wavelets

Michaël Unser, Daniel Sage, Zsuzsanna Püspöki, John Paul Ward

The dyadic scaling in the discrete wavelet transform can lead to a loss of precision, in comparison to the computationally unrealistic continuous wavelet transform. To overcome this obstacle, we propose a novel method to locally scale wavelets between dyad ...

IEEE2015

Optimal Isotropic Wavelets for Localized Tight Frame Representations

Michaël Unser, John Paul Ward, Pedram Pad

In this letter, we aim to identify the optimal isotropic mother wavelet for a given spatial dimension based on a localization criterion. Within the framework of the calculus of variations, we specify an Euler-Lagrange equation for this problem, and we find ...

IEEE Institute of Electrical and Electronics Engineers2015

Divergence-Free Wavelet Frames

Michaël Unser, Emrah Bostan, John Paul Ward

We propose an efficient construction of wavelet frames in any number of dimensions that are divergence-free. Our approach is based on applying the Leray projector, which is scale-invariant, to a standard wavelet frame. We prove that the projected wavelets ...

IEEE Institute of Electrical and Electronics Engineers2015

Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

Michaël Unser, John Paul Ward

To address the classic interior tomography problem where projections at each view extend only to the shadow of a circular region completely interior to the subject being scanned, previously we showed that the exact recovery of two- and three-dimensional pi ...

Siam Publications2015

Optimized Kaiser-Bessel Window Functions for Computed Tomography

Michaël Unser, Cédric René Jean Vonesch, Masih Nilchian, John Paul Ward

Kaiser-Bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties: 1) their short support leads to a low computational cost and 2) their rotational symmetry makes their imaging transform indepe ...

Ieee-Inst Electrical Electronics Engineers Inc2015

Interior Tomography Using 1D Generalized Total Variation. Part I: Mathematical Foundation

Michaël Unser, John Paul Ward

Motivated by the interior tomography problem, we propose a method for exact reconstruction of a region of interest of a function from its local Radon transform in any number of dimensions. Our aim is to verify the feasibility of a one-dimensional reconstru ...

Siam Publications2015

Harmonic Singular Integrals and Steerable Wavelets in $ L _{ 2 } $ $ (\mathbb{R} ^{ d } $ )

Michaël Unser, John Paul Ward

Here we present a method of constructing steerable wavelet frames in

L _{ 2 }

(\mathbb{R} ^{ d }

) that generalizes and unifies previous approaches, including Simoncelli's pyramid and Riesz wavelets. The motivation for steerable wavelets is the nee ...

Elsevier2014

Multiscale Interior Tomography Using 1D Generalized Total Variation

Michaël Unser, John Paul Ward

We propose a method for accurate and fast reconstruction of the interior of a 2D or 3D tomographic image from its incomplete local Radon transform. Unlike the existing interior tomography work with 2D total variation, the proposed algorithm guarantees exac ...

The University of Utah2014

Approximation Properties Of Sobolev Splines And The Construction Of Compactly Supported Equivalents

Michaël Unser, John Paul Ward

In this paper, we construct compactly supported radial basis functions that satisfy optimal approximation properties. Error estimates are determined by relating these basis functions to the class of Sobolev splines. Furthermore, we derive new rates for app ...

Siam Publications2014