Publications by Michaël Unser

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We introduce stochastic models for flow fields with parameters that dictate the scale-dependent (self-similar) character of the field and control the balance between its rotational vs compressive behaviour. The development of our models is motivated by the ...

Quadtree-like pyramids have the advantage of resulting in a multiresolution representation where each pyramid node has four unambiguous parents. Such a centered topology guarantees a clearly defined up-projection of labels. This concept has been successfully and extensively used in applications of contour detection, object recognition and segmentation. Unfortunately, the quadtree-like type of pyramid has poor approximation powers because of the employed piecewise-constant image model. This paper deals with

We propose a discretization method for continuous-domain linear inverse problems with multiple-order total-variation (TV) regularization. It is based on a recent result that proves that such inverse problems have sparse polynomial-spline solutions. Our met ...

The use of the wavelet transform is explored for the detection of differences between brain functional magnetic resonance images (fMRI's) acquired under two different experimental conditions. The method benefits from the fact that a smooth and spatially localized signal can be represented by a small set of localized wavelet coefficients, while the power of white noise is uniformly spread throughout the wavelet space. Hence, a statistical procedure is developed that uses the imposed decomposition orthogonali

We introduce an extended class of cardinal LL-splines, where L is a pseudo-differential operator satisfying some admissibility conditions. We show that the LL-spline signal interpolation problem is well posed and that its solution is the unique minimizer ...

By interpreting the Green-function reproduction property of exponential splines in signal processing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the ...

We propose a novel method for the clustering of point-cloud data that originate from single-molecule localization microscopy (SMLM). Our scheme has the ability to infer a hierarchical structure from the data. It takes a particular relevance when quantitati ...

Optical tomography has been widely investigated for biomedical imaging applications. In recent years optical tomography has been combined with digital holography and has been employed to produce high-quality images of phase objects such as cells. In this p ...

The most essential ingredient of interpolation is its basis function. We have shown in previous papers that this basis need not be necessarily interpolating to achieve good results. On the contrary, several recent studies have confirmed that non-interpolating bases, such as B-splines and O-moms, perform best. This opens up a much wider choice of basis functions. In this paper, we give to the designer the tools that will allow him to characterize this enlarged space of functions. In particular, he will be ab

We present a simple but generalized interpolation method for digital images that uses multiwavelet-like basis functions. Most of interpolation methods uses only one symmetric basis function; for example, standard and shifted piecewise-linear interpolations use the “hat” function only. The proposed method uses q different multiwavelet-like basis functions. The basis functions can be dissymmetric but should preserve the “partition of unity” property for high-quality signal interpolation. The scheme of decompo

This work aims at the precise and efficient computation of the x-ray projection of an image represented by a linear combination of general shifted basis functions that typically overlap. We achieve this with a suitable adaptation of ray tracing, which is one of the most efficient methods to compute line integrals. In our work, the cases in which the image is expressed as a spline are of particular relevance. The proposed implementation is applicable to any projection geometry as it computes the forward and

X-ray differential phase-contrast tomography is a recently-developed modality for the imaging of low-contrast biological samples. Its mathematical model is based on the first derivative of the Radon transform and the images, in practice, are reconstructed ...

Images in fluorescence microscopy are inherently blurred due to the limit of diffraction of light. The purpose of deconvolution microscopy is to compensate numerically for this degradation. Deconvolution is widely used to restore fine details of 3D biologi ...

A ridge is a function that is characterized by a one-dimensional profile (activation) and a multidimensional direction vector. Ridges appear in the theory of neural networks as functional descriptors of the effect of a neuron, with the direction vector bei ...

We present a new surface energy potential for the segmentation of cylindrical objects in 3D medical imaging using parametric spline active contours (a.k.a. spline-snakes). Our energy formulation is based on an optimal steerable surface detector. Thus, we c ...

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gr ...

We show that a multi-dimensional scaling function of order γ (possibly fractional) can always be represented as the convolution of a polyharmonic B-spline of order γ and a distribution with a bounded Fourier transform which has neither order nor smoothness. The presence of the B-spline convolution factor explains all key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multi-scale differentiation property, and smoothness of the basis functions. The B-spline factori

We propose a deep-learning-based reconstruction method for cryo-electron microscopy (Cryo-EM) that can model multiple conformations of a nonrigid biomolecule in a standalone manner. Cryo-EM produces many noisy projections from separate instances of the sam ...

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 ...

Three-dimensional (3D) electron microscopy (3DEM) aims at the determination of the spatial distribution of the Coulomb potential of macromolecular complexes. The 3D reconstruction of a macromolecule using single-particle techniques involves thousands of 2D projections. One of the key parameters required to perform such a 3D reconstruction is the orientation of each projection image as well as its in-plane orientation. This information is unknown experimentally and must be determined using image-processing t

Michaël Unser

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Fractal Modelling And Analysis Of Flow-Field Images

Centered Pyramids

Solving Continuous-Domain Problems Exactly with Multiresolution B-Splines

Statistical Analysis of Functional MRI Data in the Wavelet Domain

Generalized Smoothing Splines and the Optimal Discretization of the Wiener Filter

Cardinal Exponential Splines: Part II—Think Analog, Act Digital

Graphic: Graph-Based Hierarchical Clustering For Single-Molecule Localization Microscopy

Learning Approach to Optical Tomography

Complete Parametrization of Piecewise-Polynomial Interpolators According to Degree, Support, Regularity, and Order

Multiwavelet-Like Bases for High Quality Image Interpolation

Generalized Ray Tracing with Basis Functions for Tomographic Projections

Iterative FBP For Improved Reconstruction Of X-Ray Differential Phase-Contrast Tomograms

DeconvolutionLab2: An Open-Source Software for Deconvolution Microscopy

Ridges, Neural Networks, and the Radon Transform

General Surface Energy for Spinal Cord and Aorta Segmentation

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Fractional Wavelets, Derivatives, and Besov Spaces

Multi-CryoGAN: Reconstruction of Continuous Conformations in Cryo-EM Using Generative Adversarial Networks

Divergence-Free Wavelet Frames

A Multiresolution Approach to Orientation Assignment in 3D Electron Microscopy of Single Particles