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Wavelets, sparsity and biomedical image reconstruction

M. Unser

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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, image-denoising procedure that essentially amounts to discarding small wavelet coefficients (soft-thresholding). The crucial observation is that this type of “sparsity-promoting” algorithm is the solution of a l1-norm 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 wavelet-based algorithms for 3D deconvolution microscopy, and MRI reconstruction (with multiple coils and/or non-Cartesian k-space sampling). These methods were developed at the EPFL in collaboration with imaging scientists and are, for the most part, providing state-of-the-art performance.

Recent Advances in Biomedical Imaging and Signal Analysis

M. Unser

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Proceedings of the Eighteenth European Signal Processing Conference (EUSIPCO'10), Ålborg, Denmark, August 23-27, 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 ℓ1-type regularization in the wavelet domain, which favors sparse solutions. We discuss some corresponding iterative skrinkage-thresholding algorithms (ISTA) for sparse signal recovery and introduce a multi-level variant for greater computational efficiency. We illustrate the method with two concrete imaging examples: the deconvolution of 3-D fluorescence micrographs, and the reconstruction of magnetic resonance images from arbitrary (non-uniform) k-space 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 operator-based 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 self-reversible 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 (higher-order Riesz wavelets) extracts the partial derivatives of order N of the signal (e.g., gradient or Hessian). We also introduce a signal-adapted design, which yields a PCA-like 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

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Invited talk: Mathematics and Image Analysis (MIA'16), 18-20 January, 2016, Institut Henri Poincaré, Paris, France.

In recent years, significant progress has been achieved in the resolution of ill-posed linear inverse problems by imposing l1/TV regularization constraints on the solution. Such sparsity-promoting schemes are supported by the theory of compressed sensing, which is finite dimensional for the most part.

In this talk, we take an infinite-dimensional point of view by considering signals that are defined in the continuous domain. We claim that non-uniform 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 sparsity-promoting regularization, albeit in a discretized form. The latter corresponds to the log-likelihood of the projection of the stochastic model onto a finite-dimensional reconstruction space.

Steerable wavelet transforms and multiresolution monogenic image analysis

M. Unser

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

We introduce an Nth-order extension of the Riesz transform that has the remarkable property of mapping any primary wavelet frame (or basis) of L2(ℝ2) into another "steerable" wavelet frame, while preserving the frame bounds. Concretely, this means we can design reversible multi-scale 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 polyharmonic-spline wavelets that replicate the behavior of the Nth-order partial derivatives of an isotropic Gaussian kernel.
The case N=1 is of special interest because it provides a functional framework for performing wavelet-domain monogenic signal analyses. Specifically we introduce a monogenic wavelet representation of images where each wavelet index is associated with a local orientation, an amplitude and a phase. We present examples of applications to directional image analyses and coherent optical imaging.

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

M. Unser

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Plenary talk, proceedings of the Fourth French Biyearly Congress on Applied and Industrial Mathematics (SMAI'09), La Colle sur Loup, France, May 25-29, 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 real-valued 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 self-similarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L2(ℝ2) and lend themselves to the construction of wavelet bases. The other fundamental implication is that the corresponding wavelets behave like multi-scale 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 Laplace-gradient 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 Gaussian-like smoothing kernel that has been optimized for better steerability. We demonstrate that this multiresolution representation is well suited for a variety of image-processing 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, Laplacian-Like Wavelet Bases, and the Whitening of Fractal Processes," IEEE Transactions on Image Processing, vol. 18, no. 4, pp. 689-702, April 2009. [2] D. Van De Ville, M. Unser, "Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid," IEEE Transactions on Image Processing, vol. 17, no. 11, pp. 2063-2080, November 2008.

Wavelet Demystified

M. Unser

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

This 2-hour tutorial focuses on wavelet bases : it covers the concept of multi-resolution analysis, the construction of wavelets, filterbank algorithms, as well as an in-depth discussion of fundamental wavelet properties. The presentation is progressive starting with the example of the Haar transform and essentially self contained.
We emphasize the crucial role of splines in wavelet theory, presenting a non-standard point of view that simplifies the mathematical formulation. The key point is that any wavelet (or scaling function) can be expressed as the convolution of a (fractional) B-spline and a singular distribution, and that all fundamental spline properties (reproduction of polynomials, regularity, order of approximation, etc.) are preserved through the convolution operation. A direct implication is that the wavelets have vanishing moments and that they behave like multiscale differentiators. These latter two properties are the key for understanding why wavelets yield sparse representations of piecewise-smooth signals.

Affine Invariance, Splines, Wavelets and Fractional Brownian Fields

M. Unser

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Mathematical Image Processing Meeting (MIPM'07), Marseilles, France, September 3-7, 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 self-similarity of the fractional Brownian field, and (2) the fact that the polyharmonic splines specify a multiresolution analysis of L2(ℝd) 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

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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

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SIAM Review, vol. 42, no. 1, pp. 43-67, 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 one-sided power functions x+α = max(0, x)α, belong to L1 and are α-Hölder continuous for α > 0. We construct the corresponding B-splines 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 B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the B-splines which are not compactly supported for non-integral α's. Their most astonishing feature (in reference to the Strang-Fix 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 L2 (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuously-varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon's general theory of radial (m,s)-splines (including thin-plate 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. 626-638, 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 multi-scale 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 one-dimensional 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.

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ISBN 0-8493-9483-X, 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 non-experts 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:

  • Wavelet Transform: Theory and Implementation
  • Wavelets in Medical Imaging and Tomography
  • Wavelets and Biomedical Signal Processing
  • Wavelets and Mathematical Models in Biology
  • BibTeX reference
  • Full review of this book
    Akram Aldroubi and Michael Unser, Eds., Wavelets in Medicine and Biology, CRC Press, Boca Raton, FL, 1996.
    A. Bultheel
    Journal of Approximation Theory, vol. 90, no. 3, pp. 458-459, September 1997.
    Wavelets have built a strong reputation in the context of signal and image processing. The editors of this book have invited several specialists to contribute a chapter illustrating this in the (bio)medical and biological sciences.

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