Fredrik Andersson

A multi-scale approach to hyperbolic evolution equations with limited smoothness

We discuss how techniques from multiresolution analysis and phase space transforms can be exploited in solving a general class of evolution equations with limited smoothness. We have wave propagation in media of limited smoothness in mind. The frame that appears naturally in this context belongs to the family of frames of curvelets. The construction considered here implies a full-wave description on the one hand but reveals the geometrical properties derived from the propagation of singularities on the other hand. The approach and analysis we present (i) aids in the understanding of the notion of scale in the wavefield and how this interacts with the configuration or medium, (ii) admits media of limited smoothness, viz. with Hölder regularity s ≥ 2, and (iii) suggests a novel computational algorithm that requires solving for the mentioned geometry on the one hand and solving a matrix Volterra integral equation of the second kind on the other hand. The Volterra equation can be solved by recursion—as in the computation of certain multiple scattering series—revealing a curvelet–curvelet interaction. We give precise estimates expressing the degree of concentration of curvelets following the propagation of singularities.

Unique structure and optics of the lesser eyes of the box jellyfish Tripedalia cystophora.

The visual system of box jellyfish comprises a total of 24 eyes. These are of four types and each probably has a special function. To investigate this hypothesis the morphology and optics of the lesser eyes, the pit and slit eyes, were examined. The pit eyes hold one cell type only and are probably mere light meters. The slit eyes, comprising four cell types, are complex and highly asymmetric. They also hold a lens-like structure, but its optical power is minute. Optical modeling suggests spatial resolution, but only in one plane. These unique and intriguing traits support strong peripheral filtering.

A New Frequency Estimation Method for Equally and Unequally Spaced Data

Spectral estimation is an important classical problem that has received considerable attention in the signal processing literature. In this contribution, we propose a novel method for estimating the parameters of sums of complex exponentials embedded in additive noise from regularly or irregularly spaced samples. The method relies on Kroneckers theorem for Hankel operators, which enables us to formulate the nonlinear least squares problem associated with the spectral estimation problem in terms of a rank constraint on an appropriate Hankel matrix. This matrix is generated by sequences approximating the underlying sum of complex exponentials. Unequally spaced sampling is accounted for through a proper choice of interpolation matrices. The resulting optimization problem is then cast in a form that is suitable for using the alternating direction method of multipliers (ADMM). The method can easily include either a nuclear norm or a finite rank constraint for limiting the number of complex exponentials. The usage of a finite rank constraint makes, in contrast to the nuclear norm constraint, the method heuristic in the sense that the problem is non-convex and convergence to a global minimum can not be guaranteed. However, we provide a large set of numerical experiments that indicate that usage of the finite rank constraint nevertheless makes the method converge to minima close to the global minimum for reasonably high signal to noise ratios, hence essentially yielding maximum-likelihood parameter estimates. Moreover, the method does not seem to be particularly sensitive to initialization and performs substantially better than standard subspace-based methods.

Discrete Almost-Symmetric Wave Packets and Multiscale Geometrical Representation of (Seismic) Waves

We discuss a multiscale geometrical representation of (seismic) waves through a decomposition into wave packets. Wave packets can be thought of as certain localized “fat” plane waves. Here, we construct discrete almost-symmetric 3-D wave packets by using the unequally spaced fast Fourier transform. The resulting discrete transform is unitary, implying that the reconstruction operator is simply the adjoint of the decomposition operator. Another relevant aspect of the discretization scheme is the appearance of parameters that control the tiling of the phase space that corresponds with the dyadic parabolic decomposition, preserving the relative parabolic scaling while adapting to the physical problem at hand. We consider applications in exploration and global seismology, in particular for higher dimensional data regularization, seismic map migration, denoising, directional regularity analysis, and feature extraction.

Fast Fourier Methods for Synthetic Aperture Radar Imaging

In synthetic aperture radar (SAR) one wishes to reconstruct the reflectivity function of a region on the ground from a set of radar measurements taken at several angles. The ground reflectivity is found by interpolating measured samples, which typically lie on a polar grid in frequency space, to an equally spaced rectangular grid in frequency space, then computing an inverse Fourier transform. The classical polar format algorithm (PFA) is often used to perform this interpolation. We describe two other methods for performing the interpolation and imaging efficiently and accurately. The first is the gridding method, which is widely used in the medical imaging community. The second method uses unequally- spaced fast Fourier transforms (USFFTs), a generic tool for arbitrary sampling geometries. We present numerical and computational comparisons of these three methods using both point scattering data and synthetic X-band radar reflectivity predictions of a construction backhoe.

Fast inversion of the Radon transform using log-polar coordinates and partial back-projections

In this paper a novel filtered back-projection algorithm for inversion of a discretized Radon transform is presented. It makes use of invariance properties possessed by both the Radon transform and its dual. By switching to log-polar coordinates, both operators can be expressed in a displacement invariant manner. Explicit expressions for the corresponding transfer functions are calculated. Furthermore, by dividing the back-projection into several partial back-projections, inversion can be performed by means of finite convolutions and hence implemented by an FFT-algorithm. In this way, a fast and accurate reconstruction method is obtained.

Sparse approximation of functions using sums of exponentials and AAK theory

We consider the problem of approximating functions by sums of few exponentials functions, either on an interval or on the positive half-axis. We study both continuous and discrete cases, i.e. when the function is replaced by a number of equidistant samples. Recently, an algorithm has been constructed by Beylkin and Monzon for the discrete case. We provide a theoretical framework for understanding how this algorithm relates to the continuous case.

Data analysis tools for uncertainty quantification of inverse problems

We present exploratory data analysis methods to assess inversion estimates using examples based on l2- and l1-regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation and construction of training sets of functions with controlled local regularity.

Extended structure tensors for multiple directionality estimation

Standard structure tensors provide a robust way of directionality estimation of waves (or edges) but only for the case when they do not intersect. In this work, a structure tensor extension using a one-way wave equation is proposed as a tool for estimating directionality in seismic data and images in the presence of conflicting dips. Detection of two intersecting waves is possible in a two-dimensional case. In three dimensions both two and three intersecting waves can be detected. Moreover, a method for directionality filtering using the estimated directions is proposed. This method makes use of the ideas of a one-way wave equation but can be applied to generic images not related to wave propagation.

Operator-Lipschitz estimates for the singular value functional calculus

We consider a functional calculus for compact operators, acting on the singular values rather than the spectrum, which appears frequently in applied mathematics. Necessary and sufficient conditions for this singular value functional calculus to be Lipschitz-continuous with respect to the Hilbert-Schmidt norm are given. We also provide sharp constants.