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Michaël Unser

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Related publications (506)

Page 4 of 26

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

Michaël Unser, Kunal Narayan Chaudhury

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions-the B-spline factorization theorem. In particular, starting from well-localized sc ...

IEEE2009

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

Michaël Unser, Julien René Fageot, Arash Amini

The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specif ...

Springer Birkhauser2014

Controlled learning of pointwise nonlinearities in neural-network-like architectures

Michaël Unser, Alexis Marie Frederic Goujon, Stanislas Ducotterd

We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the second-order tota ...

2025

On the Shiftability of Dual-Tree Complex Wavelet Transforms

Michaël Unser, Kunal Narayan Chaudhury

The dual-tree complex wavelet transform (DT-ℂWT) is known to exhibit better shift-invariance than the conventional discrete wavelet transform. We propose an amplitude-phase representation of the DT-ℂWT which, among other things, offers a direct explanation ...

IEEE2010

Duality Mapping for Schatten Matrix Norms

Michaël Unser, Shayan Aziznejad

In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Holder inequality for Schatten norms. We prove in our main result that ...

2021

Bayesian Denoising Of Generalized Poisson Processes With Finite Rate Of Innovation

Michaël Unser, Arash Amini, Ulugbek Kamilov

We investigate the problem of the optimal reconstruction of a generalized Poisson process from its noisy samples. The process is known to have a finite rate of innovation since it is generated by a random stream of Diracs with a finite average number of im ...

Ieee2012

Sparsest piecewise-linear regression of one-dimensional data

Michaël Unser, Julien René Fageot, Thomas Jean Debarre, Quentin Alain Denoyelle

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient a ...

Elsevier2021

Performance Analysis of Reconstruction Techniques for Frequency-Domain Optical-Coherence Tomography

Michaël Unser, Chandra Sekhar Seelamantula

We address the issue of noise robustness of reconstruction techniques for frequency-domain optical-coherence tomography (FDOCT). We consider three reconstruction techniques: Fourier, iterative phase recovery, and cepstral techniques. We characterize the re ...

Institute of Electrical and Electronics Engineers2010

An Inner-Product Calculus for Periodic Functions and Curves

Michaël Unser, Daniel Andreas Schmitter, Anaïs Laure Marie-Thérèse Badoual

Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and

L _{ 2 }

distances between such curves. In partic ...

IEEE Institute of Electrical and Electronics Engineers2016

Super-resolution orientation estimation and localization of fluorescent dipoles using 3-D steerable filters

Michaël Unser, Theo Lasser, Iwan Märki, François Aguet, Stefan Geissbühler

Fluorophores that are fixed during image acquisition produce a diffraction pattern that is characteristic of the orientation of the fluorophore’s underlying dipole. Fluorescence localization microscopy techniques such as PALM and STORM achieve super-resolu ...

OSA2009

Sparse Stochastic Processes and Discretization of Linear Inverse Problems

Michaël Unser, Ulugbek Kamilov, Emrah Bostan, Masih Nilchian

We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies cont ...

Ieee-Inst Electrical Electronics Engineers Inc2013

Full Motion and Flow Field Recovery from Echo Doppler Data

Michaël Unser, Michael Sühling, Muthuvel Arigovindan

We present a new computational method for reconstructing a vector velocity field from scattered, pulsed-wave ultrasound Doppler data. The main difficulty is that the Doppler measurements are incomplete, for they do only capture the velocity component along ...

Institute of Electrical and Electronics Engineers2007

WSPM or How to Obtain Statistical Parametric Maps Using Shift-Invariant Wavelet Processing

Michaël Unser, Dimitri Nestor Alice Van De Ville, Thierry Blu

Recently, we have proposed a new framework for detecting brain activity from fMRI data, which is based on the spatial discrete wavelet transform. The standard wavelet-based approach performs a statistical test in the wavelet domain, and therefore fails to ...

IEEE2006

Efficient Shape Priors for Spline-Based Snakes

Michaël Unser, Daniel Andreas Schmitter, Ricard Delgado Gonzalo, Virginie Sophie Uhlmann

Parametric active contours are an attractive approach for image segmentation, thanks to their computational efficiency. They are driven by application-dependent energies that reflect the prior knowledge on the object to be segmented. We propose an energy i ...

2015

Learning Activation Functions in Deep (Spline) Neural Networks

Michaël Unser, Shayan Aziznejad, Pakshal Narendra Bohra, Harshit Gupta, Joaquim Gonçalves Garcia Barreto Campos

We develop an efficient computational solution to train deep neural networks (DNN) with free-form activation functions. To make the problem well-posed, we augment the cost functional of the DNN by adding an appropriate shape regularization: the sum of the ...

2020

Semi-Orthogonal Wavelets That Behave like Fractional Differentiators

Michaël Unser, Dimitri Nestor Alice Van De Ville, Thierry Blu

The approximate behavior of wavelets as differential operators is often considered as one of their most fundamental properties. In this paper, we investigate how we can further improve on the wavelet's behavior as differentiator. In particular, we propose ...

SPIE2005

Hessian Schatten-Norm Regularization for Linear Inverse Problems

Michaël Unser, John Paul Ward, Stamatios Lefkimmiatis

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which a ...

Ieee-Inst Electrical Electronics Engineers Inc2013

Representer Theorems for Sparsity-Promoting $ ℓ _{ 1 } $ Regularization

Michaël Unser, Julien René Fageot, Harshit Gupta

We present a theoretical analysis and comparison of the effect of

ℓ _{ 1 }

versus

ℓ _{ 2 }

regularization for the resolution of ill-posed linear inverse and/or compressed sensing problems. Our formulation covers the most general setting where the s ...

IEEE2016

Fast 3D reconstruction method for differential phase contrast X-ray CT

Michaël Unser, Michael Thompson McCann, Masih Nilchian

We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructio ...

2016

The n-term Approximation of Periodic Generalized Levy Processes

Michaël Unser, Julien René Fageot, John Paul Ward

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