NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, for example, nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This article provides a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.

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Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

Hyperbolic Cross Approximation

We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competitors.

Shearlet-Based Total Variation Diffusion for Denoising

We consider divergence form elliptic operators in dimension n ge; 2 with L∞ coefficients. Although solutions of these operators are only Holder-continuous, we show that they are differentiable (C1, α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales. This new numerical homogenization method is based on the transfer of a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales. Error bounds can be given and this method can also be used as a compression tool for differential operators. © 2006 Wiley Periodicals, Inc.

Metric‐based upscaling

An irregularly spaced sampling raster formed from a sequence of low-resolution frames is the input to an image sequence superresolution algorithm whose output is the set of image intensity values at the desired high-resolution image grid. The method of moving least squares (MLS) in polynomial space has proved to be useful in filtering the noise and approximating scattered data by minimizing a weighted mean-square error norm, but introducing blur in the process. Starting with the continuous version of the MLS, an explicit expression for the filter bandwidth is obtained as a function of the polynomial order of approximation and the standard deviation (scale) of the Gaussian weight function. A discrete implementation of the MLS is performed on images and the effect of choice of the two dependent parameters, scale and order, on noise filtering and reduction of blur introduced during the MLS process is studied

Superresolution and noise filtering using moving least squares

https://www-user.tu-chemnitz.de/~potts/paper/polarfft.pdf

On the computation of the polar FFT

A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on thresholding complex ridgelet coefficients combined with total variation (TV) minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the nonequispaced fast Fourier transform (NFFT) and then applying a dual-tree complex wavelet transform (DT CWT). The NFFT-based approach of the Radon transform completely avoids linear interpolations of the Cartesian-to-polar grid and requires only O(n2 log n) arithmetic operations for n by n arrays, while its inverse preserves the good reconstruction quality of the filtered backprojection. The DT CWT in the second step of the ridgelet transform provides approximate shift invariance on the projections of the Radon transform. After hard thresholding the ridgelet coefficients, they are restored using TV minimization to eliminate the pseudo-Gibbs artifacts near the discontinuities. Numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches.

Combined Complex Ridgelet Shrinkage and Total Variation Minimization

In this paper, we modify the robust local image estimation method of R. van den Boomgaard and J. van de Weijer for the approximation of scattered data. The derivation of our knot and data dependent approximation method is based on the relation between the Gaussian facet model in image processing and the moving least square technique known from approximation theory. Numerical examples demonstrate the advantages of our robust scattered data approximation.

Robust local approximation of scattered data

This paper is concerned with the fast summation of radial functions by the fast Fourier transform for nonequispaced data. We enhance the fast summation algorithm proposed in [20] by introducing a new regularization procedure based on the two-point Taylor interpolation by algebraic polynomials and estimate the corresponding approximation error. Our error estimates are more sophisticated than those in [20]. Beyond the kernels Kβ(χ)

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Fast NFFT based summation of radial functions

We develop a new algorithm for the fast computation of matrix–vector products with special matrices. More precisely we develop a method for the fast computation of sums f(yj):= ∑αkK(yj−xk) at non-equispaced nodes xk and yj (j=1,…,M) which requires only (N log N + (M + N)) arithmetic operations. Our algorithm is based on a novel approach to fast discrete trigonometric transforms at non-equispaced nodes. Copyright © 2004 John Wiley & Sons, Ltd.

Markus Fenn

Papers

On the computation of the polar FFT

Combined Complex Ridgelet Shrinkage and Total Variation Minimization

Robust local approximation of scattered data

Fast NFFT based summation of radial functions

Fast summation based on fast trigonometric transforms at non‐equispaced nodes