Publications by Michaël Unser

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The fractional Laplacian (-Delta)(gamma/2) commutes with the primary coordination transformations in the Euclidean space Rd: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < gamma < d, its inver ...

The Matérn class is a parametric family of autocorrelation functions that is commonly used in geostatistics. We argue that a generalized, anisotropic version of this model is suitable for capturing the correlation structure of a variety of natural images. ...

A filterbank decomposition can be seen as a series of projections onto several discrete wavelet subspaces. In this presentation, we analyze the projection onto one of them—the low-pass one, since many signals tend to be low-pass. We prove a general but simple formula that allows the computation of the

l _{ 2 }

-error made by approximating the signal by its projection. This result provides a norm for evaluating the accuracy of a complete decimation/interpolation branch for arbitrary analysis and synthesi

We propose a new statistical dictionary learning algorithm for sparse signals that is based on an α-stable innovation model. The parameters of the underlying model—that is, the atoms of the dictionary, the sparsity index α and the dispersion of the transfo ...

Observing cellular responses to perturbations is central to generating and testing hypotheses in biology. We developed a massively parallel microchemostat array capable of growing and observing 1,152 yeast-GFP strains on the single-cell level with 20 min t ...

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 problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the least-squares sense. We then consider the reverse problem and introduce a new spline-convo

We present a new, robust algorithm for tracking fluorescent particles in dynamic image sequences obtained by brightfield or confocal microscopy. Specifically, we consider the problem of extracting the movement of chromosomal telomeres within the nucleus of a budding yeast cell. Our method has three components. The first is an alignment module that compensates for the movement of the biological structure under investigation. In our application, the images are aligned to the center of gravity of the nucleus w

In this paper we introduce a new reconstruction algorithm for X-ray differential phase-contrast Imaging (DPCI). Our approach is based on 1) a variational formulation with a weighted data term and 2) a variable-splitting scheme that allows for fast converge ...

In this study, we investigate the possibility of applying a continuous-time ARMA (CARMA) model to radio-frequency ultrasound signals. We consider the effect of the discretization process on the parameters of the continuous system, and we take into account ...

Localization microscopy relies on computationally efficient Gaussian approximations of the point spread function for the calculation of fluorophore positions. Theoretical predictions show that under specific experimental conditions, localization accuracy i ...

Our goal is to detect and localize areas of activation in the brain from sequences of fMRI images. The standard approach for reducing the noise contained in the fMRI images is to apply a spatial Gaussian filter which entails some loss of details. Here instead, we consider a wavelet solution to the problem, which has the advantage of retaining high-frequency information. We use fractional-spline orthogonal wavelets with a continuously-varying order parameter alpha; by adjusting alpha, we can balance spatial

In applications that involve interactive curve and surface modeling, the intuitive manipulation of shapes is crucial. For instance, user interaction is facilitated if a geometrical object can be manipulated through control points that interpolate the shape ...

We propose a reconstruction scheme adapted to MRI that takes advantage of a sparsity constraint in the wavelet domain. We show that, artifacts are significantly reduced compared to conventional reconstruction methods. Our approach is also competitive with ...

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 ...

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among ...

Interpolation is a vital tool in biomedical signal processing. Although there exists a substantial literature dedicated to noise-free conditions, much less is known in the presence of noise. Here, we document the breakdown of standard interpolation for noi ...

Growth and form in biology are often associated with some level of fractality. Fractal characteristics have also been noted in a number of imaging modalities. These observations make fractal modelling relevant in the context of bio-imaging. ...

This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finite-support ones are the square

Our goal is to detect and group different kinds of local symmetries in images in a scale- and rotation-invariant way. We propose an efficient wavelet-based method to determine the order of local symmetry at each location. Our algorithm relies on circular h ...

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 schemes including wavelets, splines, and bandlimited signal expansions. The formula takes the simple form of a Parseval's-like relation, where the Fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation op

Michaël Unser

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Left-inverses of fractional Laplacian and sparse stochastic processes

Non-Ideal Sampling and Adapted Reconstruction Using the Stochastic Matérn Model

A Theoretical Analysis of the Projection Error onto Discrete Wavelet Subspaces

Dictionary Learning Based on Sparse Distribution Tomography

A chemostat array enables the spatio-temporal analysis of the yeast proteome

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

Automatic Tracking of Particles in Dynamic Fluorescence Microscopy

Constrained regularized reconstruction of X-ray-DPCI tomograms with weighted-norm

Continuous-Domain ARMA Modeling for Ultrasound Tissue Characterization

3-D PSF fitting for fluorescence microscopy: implementation and localization application

Analysis of fMRI Data Using Spline Wavelets

Compactly-Supported Smooth Interpolators for Shape Modeling with Varying Resolution

Wavelet-Regularized Reconstruction For Rapid MRI

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

Support and approximation properties of Hermite splines

Regularized interpolation for noisy data

Innovation modelling and wavelet analysis of fractal processes in bio-imaging

Image Interpolation and Resampling

Template-Free Wavelet-Based Detection of Local Symmetries

Sampling of Periodic Signals: A Quantitative Error Analysis