Publications by Michaël Unser

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Due to magnetic field inhomogeneities, EPI images are geometrically distorted, predominantly along the phase-encoding direction. Currently, the distortion is either ignored or compensated manually using a warping function defined through a set of landmarks. We propose an automatic method to unwarp the geometric distortion of EPI images by registering them with corresponding undistorted anatomical MRI images. We show that cubic splines are optimal interpolating functions for both landmark interpolation and a

We investigate the distributional extension of the k-plane transform in Rd and of related operators. We parameterize the k-plane domain as the Cartesian product of the Stiefel manifold of orthonormal k-frames in Rd with Rd-k. This parameterization imposes ...

We consider the problem of the reconstruction of a continuous-time function f(x) ∈ H from the samples of the responses of m linear shift-invariant systems sampled at 1 ⁄ m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H =

L _{ 2 }

) is considerably larger than the subspace of bandlimited functions. Second, we use a more general specification of the reconstruction subspace V(φ), so that the output of

We propose a new technique for two-dimensional phase unwrapping. The unwrapped phase is found as the solution of an inverse problem that consists in the minimization of an energy functional. The latter includes a weighted data fidelity term that favors spa ...

Phosphorylation is involved in numerous neurodegenerative diseases. In particular, alpha-synuclein is extensively phosphorylated in aggregates in patients suffering from synucleinopathies. However, the share of this modification in the events that lead to ...

The problem of estimating continuous-domain autoregressive moving-average processes from sampled data is considered. The proposed approach incorporates the sampling process into the problem formulation while introducing exponential models for both the cont ...

We have developed an algorithm for the rigid-body registration of a CT volume to a set of C-arm images. The algorithm uses a gradient-based iterative minimization of a least-squares measure of dissimilarity between the C-arm images and projections of the C ...

We present a new formulation of a family of proximity operators that generalize the projector step for phase retrieval. These proximity operators for noisy intensity measurements can replace the classical "noise-free" projection in any projection-based alg ...

We propose to design the reduction operator of an image pyramid so as to minimize the approximation error in the

l _{ p }

-sense (not restricted to the usual p=2), where p can take non-integer values. The underlying image model is specified using shift-invariant basis functions, such as B-splines. The solution is well-defined and determined by an iterative optimization algorithm based on digital filtering. Its convergence is accelerated by the use of first and second order derivatives. For p close to 1,

We aim at the solution of inverse problems in imaging, by combining a penalized sparse representation of image patches with an unconstrained smooth one. This allows for a straightforward interpretation of the reconstruction. We formulate the optimization as a bilevel problem. The inner problem deploys classical algorithms while the outer problem optimizes the dictionary and the regularizer parameters through supervised learning. The process is carried out via implicit differentiation and gradient-based opti

This paper proposes a new family of bivariate, non-separable splines, called hex-splines, especially designed for hexagonal lattices. The starting point of the construction is the indicator function of the Voronoi cell, which is used to define in a natural way the first-order hex-spline. Higher order hex-splines are obtained by successive convolutions. A mathematical analysis of this new bivariate spline family is presented. In particular, we derive a closed form for a hex-spline of arbitrary order. We also

We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function φ(x). We first give the expression of the φ's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support functions (MOMS), is made of linear combinations of the B-spline of same order and of its derivatives. We provide the explicit form of the MOMS that maximize the approximatio

Shift-orthogonal wavelets are a new type of multiresolution wavelet bases that are orthogonal with respect to translation (or shifts) within one level, but not with respect to dilations across scales. In this paper, we characterize these wavelets and investigate their main properties by considering two general construction methods. In the first approach, we start by specifying the analysis and synthesis function spaces, and obtain the corresponding shift-orthogonal basis functions by suitable orthogonalizat

Phase retrieval, a nonlinear problem prevalent in imaging applications, has been extensively studied using random models, some of which with i.i.d. sensing matrix components. While these models offer robust reconstruction guarantees, they are computational ...

Understanding the biological underpinnings of movement and action requires the development of tools for quantitative measurements of animal behavior. Drosophila melanogaster provides an ideal model for developing such tools: the fly has unparalleled geneti ...

We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preser ...

We designed two efficient and user-friendly tools for the segmentation and analysis of images containing chromosomes or, more generally, rod-shaped elements that are spread on microscopic slides. The segmentation tool allows to automatically extract the pr ...

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 approximated on a multiresolution grid. Using least-squares approximation ensures the smallest possible error for a given sampling step size. The approximation error on the grid is derived as a function of the sampling step h. The choice of the appropriate wavel

It is well known that spatial averaging can be realized (in space or frequency domain) using algorithms whose complexity does not scale with the size or shape of the filter. These fast algorithms are generally referred to as constant-time or O(1) algorithm ...

An attractive formulation of the sampling problem is based on the principle of a consistent signal reconstruction. The requirement is that the reconstructed signal is indistinguishable from the input in the sense that it yields the exact same measurements. ...

Michaël Unser

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Compensation of Unidirectional Geometric Distortion in EPI Using Spline Warping

DISTRIBUTIONAL EXTENSION AND INVERTIBILITY OF THE k-PLANE TRANSFORM AND ITS DUAL

A Generalized Sampling Theory without Band-Limiting Constraints

Isotropic Inverse-Problem Approach for Two-Dimensional Phase Unwrapping

Phosphorylation does not prompt, nor prevent, the formation of alpha-synuclein toxic species in a rat model of Parkinson's disease

A Sampling Theory Approach for Continuous ARMA Identification

An Optimized Spline-Based Registration of a 3D CT to a Set of C-Arm Images

Proximity Operators for Phase Retrieval

$ l _{ p } $ -Multiresolution Analysis: How to Reduce Ringing and Sparsify the Error

Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction

Hex-Splines: A Novel Spline Family for Hexagonal Lattices

MOMS: Maximal-Order Interpolation of Minimal Support

Shift-Orthogonal Wavelet Bases

Structured Random Model for Fast and Robust Phase Retrieval

FlyLimbTracker: An Active Contour Based Approach for Leg Segment Tracking in Unmarked, Freely Behaving Drosophila

Fast Interscale Wavelet Denoising of Poisson-Corrupted Images

User-Friendly Image-Based Segmentation and Analysis of Chromosomes

Wavelet Projections for Volume Rendering

Fast O(1) Bilateral Filtering Using Trigonometric Range Kernels

Consistent Sampling and Signal Recovery