Publications by Michaël Unser

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We give a methodology-oriented perspective on directional image analysis and rotation-invariant processing. We review the state of the art in the field and make connections with recent mathematical developments in functional analysis and wavelet theory. We unify our perspective within a common framework using operators. The intent is to provide image-processing methods that can be deployed in algorithms that analyze biomedical images with improved rotation invariance and high directional sensitivity. We sta

We introduce an integrated framework for detecting brain activity from fMRI data, which is based on a spatial discrete wavelet transform. Unlike the standard wavelet-based approach for fMRI analysis, we apply the suitable statistical test procedure in the spatial domain. For a desired significance level, this scheme has one remaining degree of freedom, characterizing the wavelet processing, which is optimized according to the principle of minimal approximation error. This allows us to determine the threshol

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 ...

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 ...

We present a new framework to estimate and visualize heart motion from echocardiograms. For velocity estimation, we have developed a novel multiresolution optical flow algorithm. In order to account for typical heart motions like contraction/expansion and shear, we use a local affine model for the velocity in space and time. The motion parameters are estimated in the least-squares sense inside a sliding spatio-temporal window. The estimated velocity field is used to track a region of interest which is repre

Generalized sampling provides a general mechanism for recovering an unknown input function

f(x) \in H

from the samples of the responses of m linear shift-invariant systems sampled at 1 ⁄ mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace

V(\varphi) = span \{\varphi(x - k)\} _{ k \in Z }

; for example, the family of bandlimited signals with

\varphi(x) = sinc(x)

. This implies that the reconstruction will be perfect when the input sig

Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. We show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, we call their use generalized interpolation; they involve a prefiltering step when correctly applied. We explain why the approximation order inherent in any basis function is important to limit

Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together …through fractals. First, we identify and characterize the whole class of self-similar radial basis functions that can be localized to yield conventional

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 ...

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 ...

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 ...

We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of

L _{ 2 }

via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is suffic

We propose a general methodology (PURE-LET) to design and optimize a wide class of transform-domain thresholding algorithms for denoising images corrupted by mixed Poisson-Gaussian noise. We express the denoising process as a linear expansion of thresholds ...

The projection of sample measurements onto a reconstruction space represented by a basis on a regular grid is a powerful and simple approach to estimate a probability density function. In this paper, we focus on Riesz bases and propose a projection operato ...

The discrete cosine transform (DCT) is known to be asymptotically equivalent to the Karhunen-Loève transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes. Since being uncorrelated under the Gaussian hypothesis is synonymous with independ ...

Our aim is to optimize wavelet-based feature extraction for differentiating between the classical versus atypical pattern of usual interstitial pneumonia (UIP) in volumetric CT. Our proposal is to act on the bandwidth of steerable wavelets while maintainin ...

We show the feasibility and the potential of a new signal processing algorithm for the high-resolution deconvolution of OCT signals. Our technique relies on the description of the measures in a parametric form, each set of four parameters describing the optical characteristics of a physical interface (e.g., complex refractive index, depth). Under the hypothesis of a Gaussian source light, we show that it is possible to recover the 4K parameters corresponding to K interfaces using as few as 4K uniform sample

Parametric active contour models are one of the preferred approaches for image segmentation because of their computational efficiency and simplicity. However, they have a few drawbacks which limit their performance. In this paper, we identify some of these problems and propose efficient solutions to get around them. The widely-used gradient magnitude-based energy is parameter dependent; its use will negatively affect the parametrization of the curve and, consequently, its stiffness. Hence, we introduce a ne

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 ...

We develop a novel 2D functional learning framework that employs a sparsity-promoting regularization based on second-order derivatives. Motivated by the nature of the regularizer, we restrict the search space to the span of piecewise-linear box splines shi ...

Michaël Unser

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Transforms and Operators for Directional Bioimage Analysis: A Survey

Integrated Wavelet Processing and Spatial Statistical Testing of fMRI Data

Learning Activation Functions in Deep (Spline) Neural Networks

Semi-Orthogonal Wavelets That Behave like Fractional Differentiators

Myocardial Motion Analysis and Visualization from Echocardiograms

Generalized Sampling: Stability and Performance Analysis

Interpolation Revisited

Wavelets, Fractals, and Radial Basis Functions

Hessian Schatten-Norm Regularization for Linear Inverse Problems

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

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

Approximation Error for Quasi-Interpolators and (Multi-) Wavelet Expansions

Image Denoising in Mixed Poisson-Gaussian Noise

Bona Fide Riesz Projections For Density Estimation

Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

Optimized Steerable Wavelets for Texture Analysis of Lung Tissue In 3-D CT: Classification of Usual Interstitial Pneumonia

A New High-Resolution Processing Method for the Deconvolution of Optical Coherence Tomography Signals

Efficient Energies and Algorithms for Parametric Snakes

The n-term Approximation of Periodic Generalized Levy Processes

Learning of Continuous and Piecewise-Linear Functions With Hessian Total-Variation Regularization