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Wavelets 
Swiss Mathematical Society, Abel Prize Day 2020, September 25, University of Bern 

M. Unser 


The work of Yves Meyer (Abel Prize 2017) Six talks on the work of the 20172020 Abel Prize Laureates. 





Wavelets, sparsity and biomedical image reconstruction 

M. Unser 

Imaging Seminar, University of Bern, Inselspital November 13, 2012. 

Our purpose in this talk is to advocate the use of wavelets for advanced biomedical imaging. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a sparse representation of images. We then discuss a simple, but remarkably effective, imagedenoising procedure that essentially amounts to discarding small wavelet coefficients (softthresholding). The crucial observation is that this type of “sparsitypromoting” algorithm is the solution of a l1norm minimization problem. The underlying principle of wavelet regularization is a powerful concept that has been used advantageously for compressed sensing and for reconstructing images from limited and/or noisy measurements. We illustrate the point by presenting waveletbased algorithms for 3D deconvolution microscopy, and MRI reconstruction (with multiple coils and/or nonCartesian kspace sampling). These methods were developed at the EPFL in collaboration with imaging scientists and are, for the most part, providing stateoftheart performance. 



Recent Advances in Biomedical Imaging and Signal Analysis 

M. Unser 

Proceedings of the Eighteenth European Signal Processing Conference (EUSIPCO'10), Ålborg, Denmark, August 2327, 2010, EURASIP Fellow inaugural lecture. 

Wavelets have the remarkable property of providing sparse representations of a wide variety of "natural" images. They have been applied successfully to biomedical image analysis and processing since the early 1990s. In the first part of this talk, we explain how one can exploit the sparsifying property of wavelets to design more effective algorithms for image denoising and reconstruction, both in terms of quality and computational performance. This is achieved within a variational framework by imposing some ℓ_{1}type regularization in the wavelet domain, which favors sparse solutions. We discuss some corresponding iterative skrinkagethresholding algorithms (ISTA) for sparse signal recovery and introduce a multilevel variant for greater computational efficiency. We illustrate the method with two concrete imaging examples: the deconvolution of 3D fluorescence micrographs, and the reconstruction of magnetic resonance images from arbitrary (nonuniform) kspace trajectories. In the second part, we show how to design new wavelet bases that are better matched to the directional characteristics of images. We introduce a general operatorbased framework for the construction of steerable wavelets in any number of dimensions. This approach gives access to a broad class of steerable wavelets that are selfreversible and linearly parameterized by a matrix of shaping coefficients; it extends upon Simoncelli's steerable pyramid by providing much greater wavelet diversity. The basic version of the transform (higherorder Riesz wavelets) extracts the partial derivatives of order N of the signal (e.g., gradient or Hessian). We also introduce a signaladapted design, which yields a PCAlike tight wavelet frame. We illustrate the capabilities of these new steerable wavelets for image analysis and processing (denoising). 



Sparsity and the optimality of splines for inverse problems: Deterministic vs. statistical justifications 

M. Unser 


Invited talk: Mathematics and Image Analysis (MIA'16), 1820 January, 2016, Institut Henri Poincaré, Paris, France. 

In recent years, significant progress has been achieved in the resolution of illposed linear inverse problems by imposing l1/TV regularization constraints on the solution. Such sparsitypromoting schemes are supported by the theory of compressed sensing, which is finite dimensional for the most part. In this talk, we take an infinitedimensional point of view by considering signals that are defined in the continuous domain. We claim that nonuniform splines whose type is matched to the regularization operator are optimal candidate solutions. We show that such functions are global minimizers of a broad family of convex variational problems where the measurements are linear and the regularization is a generalized form of total variation associated with some operator L. We then discuss the link with sparse stochastic processes that are solutions of the same type of differential equations.The pleasing outcome is that the statistical formulation yields maximum a posteriori (MAP) signal estimators that involve the same type of sparsitypromoting regularization, albeit in a discretized form. The latter corresponds to the loglikelihood of the projection of the stochastic model onto a finitedimensional reconstruction space.




Steerable wavelet transforms and multiresolution monogenic image analysis 

M. Unser 


Engineering Science Seminar, University of Oxford, UK, January 15, 2010. 

We introduce an Nthorder extension of the Riesz transform that has the remarkable property of mapping
any primary wavelet frame (or basis) of L2(ℝ^{2}) into another "steerable" wavelet frame, while preserving
the frame bounds. Concretely, this means we can design reversible multiscale decompositions in
which the analysis wavelets (feature detectors) can be spatially rotated in any direction
via a suitable linear combination of wavelet coefficients. The concept provides a rigorous
functional counterpart to Simoncelli's steerable pyramid whose construction was entirely based
on digital filter design. It allows for the specification of wavelets with any order of steerability
in any number of dimensions. We illustrate the method with the design of new steerable polyharmonicspline
wavelets that replicate the behavior of the Nthorder partial derivatives of an isotropic Gaussian kernel.




Wavelets and Differential Operators: From Fractals to Marr's Primal Sketch 

M. Unser 

Plenary talk, proceedings of the Fourth French Biyearly Congress on Applied and Industrial Mathematics (SMAI'09), La Colle sur Loup, France, May 2529, 2009. 

Invariance is an attractive principle for specifying image processing algorithms. In this presentation, we promote affine invariance—more precisely, invariance with respect to translation, scaling and rotation. As starting point, we identify the corresponding class of invariant 2D operators: these are combinations of the (fractional) Laplacian and the complex gradient (or Wirtinger operator). We then specify some corresponding differential equation and show that the solution in the realvalued case is either a fractional Brownian field or a polyharmonic spline, depending on the nature of the system input (driving term): stochastic (white noise) or deterministic (stream of Dirac impulses). The affine invariance of the operator has two important consequences: (1) the statistical selfsimilarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L_{2}(ℝ^{2}) and lend themselves to the construction of wavelet bases. The other fundamental implication is that the corresponding wavelets behave like multiscale versions of the operator from which they are derived; this makes them ideally suited for the analysis of multidimensional signals with fractal characteristics (whitening property of the fractional Laplacian) [1]. The complex extension of the approach yields a new complex wavelet basis that replicates the behavior of the Laplacegradient operator and is therefore adapted to edge detection [2]. We introduce the Marr wavelet pyramid which corresponds to a slightly redundant version of this transform with a Gaussianlike smoothing kernel that has been optimized for better steerability. We demonstrate that this multiresolution representation is well suited for a variety of imageprocessing tasks. In particular, we use it to derive a primal wavelet sketch—a compact description of the image by a multiscale, subsampled edge map—and provide a corresponding iterative reconstruction algorithm. References: [1] P.D. Tafti, D. Van De Ville, M. Unser, "Invariances, LaplacianLike Wavelet Bases, and the Whitening of Fractal Processes," IEEE Transactions on Image Processing, vol. 18, no. 4, pp. 689702, April 2009. [2] D. Van De Ville, M. Unser, "Complex Wavelet Bases, Steerability, and the MarrLike Pyramid," IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 20632080, November 2008. 



Wavelet Demystified 

M. Unser 

Invited presentation, Technical University of Eindhoven, The Netherlands, May 31, 2006. 

This 2hour tutorial focuses on wavelet bases : it covers the concept of multiresolution analysis, the construction of wavelets,
filterbank algorithms, as well as an indepth discussion of fundamental wavelet properties. The presentation is progressive starting with the example
of the Haar transform and essentially self contained. 



Affine Invariance, Splines, Wavelets and Fractional Brownian Fields 

M. Unser 

Mathematical Image Processing Meeting (MIPM'07), Marseilles, France, September 37, 2007. 

Invariance is an attractive principle for specifying image processing algorithms. In this work, we concentrate on affine—more precisely, shift, scale and rotation—invariance and identify the corresponding class of operators, which are fractional Laplacians. We then specify some corresponding differential equation and show that the solution (in the distributional sense) is either a fractional Brownian field (Mandelbrot and Van Ness, 1968) or a polyharmonic spline (Duchon, 1976), depending on the nature of the system input (driving term): stochastic (white noise) or deterministic (stream of Dirac impulses). The affine invariance of the operator has two remarkable consequences: (1) the statistical selfsimilarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L_{2}(ℝ^{d}) and lend themselves to the construction of wavelet bases. We prove that these wavelets essentially behave like the operator from which they are derived, and that they are ideally suited for the analysis of multidimensional signals with fractal characteristics (isotopic differentiation, and whitening property). This is joint work with Pouya Tafti and Dimitri Van De Ville. 



Wavelet Games 

M. Unser, M. Unser 

The Wavelet Digest, vol. 11, Issue 4, April 1, 2003. 

Summary: A dialogue between father and son about wavelets, Legos, and other existential questions. 

Fractional Splines and Wavelets 

M. Unser, T. Blu 

SIAM Review, vol. 42, no. 1, pp. 4367, March 2000. 

We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α > 1. These splines, which involve linear combinations of the onesided power functions x_{+}^{α} = max(0, x)^{α}, belong to L^{1} and are αHölder continuous for α > 0. We construct the corresponding Bsplines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional Bsplines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the Bsplines which are not compactly supported for nonintegral α's. Their most astonishing feature (in reference to the StrangFix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree [α]. For α > 1/2, they satisfy all the requirements for a multiresolution analysis of L^{2} (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuouslyvarying order parameter. Our construction also yields symmetrized fractional Bsplines which provide the connection with Duchon's general theory of radial (m,s)splines (including thinplate splines). In particular, we show that the symmetric version of our splines can be obtained as solution of a variational problem involving the norm of a fractional derivative. 



A Review of Wavelets in Biomedical Applications 

M. Unser, A. Aldroubi 

Proceedings of the IEEE, vol. 84, no. 4, pp. 626638, April 1996. 

In this paper, we present an overview of the various uses of the wavelet transform (WT) in medicine and biology. We start by describing the wavelet properties that are the most important for biomedical applications. In particular, we provide an interpretation of the continuous WT as a prewhitening multiscale matched filter. We also briefly indicate the analogy between the WT and some of the biological processing that occurs in the early components of the auditory and visual system. We then review the uses of the WT for the analysis of onedimensional physiological signals obtained by phonocardiography, electrocardiography (ECG), and electroencephalography (EEG), including evoked response potentials. Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms); image reconstruction and acquisition schemes (tomography, and magnetic resonance imaging (MRI)); and multiresolution methods for the registration and statistical analysis of functional images of the brain (positron emission tomography (PET), and functional MRI). In each case, we provide the reader with some general background information and a brief explanation of how the methods work. The paper also includes an extensive bibliography. 

Wavelets in Medicine and Biology 

A. Aldroubi, M.A. Unser, Eds. 

ISBN 084939483X, CRC Press, Boca Raton FL, USA, 1996, 616 p. 

For the first time, the field's leading international experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. This book provides guidelines for all those interested in learning about waveletes and their applications to biomedical problems. The introductory material is written for nonexperts and includes basic discussions of the theoretical and practical foundations of wavelet methods. This is followed by contributions from the most prominent researchers in the field, giving the reader a complete survey of the use of wavelets in biomedical engineering. The book consists of four main sections:



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