This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘de-noising.’

Abstract. We define a notion of Radon Transform for data in an n by n grid. It is based on summation along lines of absolute slope less than 1 (as a function either of x or of y), with values at non-Cartesian locations defined using trigonometric interpolation on a zero-padded grid. The definition is geometrically faithful: the lines exhibit no ‘wraparound effects’. For a special set of lines equispaced in slope (rather than angle), we describe an exact algorithm which uses O(N log N) flops, where N = n2 is the number of pixels. This relies on a discrete projection-slice theorem relating this Radon transform and what we call the Pseudopolar Fourier transform. The Pseudopolar FT evaluates the 2-D Fourier transform on a non-Cartesian pointset, which we call the pseudopolar grid. Fast Pseudopolar FT – the process of rapid exact evaluation of the 2-D Fourier transform at these non-Cartesian grid points – is possible using chirp-Z transforms. This Radon transform is one-to-one and hence invertible on its range; it is rapidly invertible to any degree of desired accuracy using a preconditioned conjugate gradient solver. Empirically, the numerical conditioning is superb; the singular value spread of the preconditioned Radon transform turns out numerically to be less than 10%, and three iterations of the conjugate gradient solver typically suffice for 6 digit accuracy. We also describe a 3-D version of the transform.

. This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the split-radix FFT trick, Good's trick, the SchonhageStrassen trick, Schonhage's trick, Nussbaumer's trick, the cyclic SchonhageStrassen trick, and the Cantor-Kaltofen theorem. It emphasizes the underlying ring homomorphisms. 1.

. The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a two-dimensional curvilinear duct. The problem is discretized with a second-order accurate finitedifference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. The preconditioner is based on fast transforms, and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. Compared with band Gaussian elimination the preconditioned iterative method shows a significant gain in both storage requirement and arithmetic complexity. This research was supported by the U. S. National Science Foundationunder grant ASC-8958544 and by the Swedish National Board for Industrial and Technical Development (NUTEK). y Department of Scientific Computi...

In a wide range of applied problems of 2D and 3D imaging a continuous formulation of the problem places great emphasis on obtaining and manipulating the Fourier transform in Polar coordinates. However, the translation of continuum ideas into practical work with data sampled on a Cartesian grid is problematic. In this article we develop a fast high accuracy Polar FFT. For a given two-dimensional signal of size N × N, the proposed algorithm’s complexity is O(N^2 log N), just like in a Cartesian 2D-FFT. A special feature of our approach is that it involves only 1D equispaced FFT’s and 1D interpolations. A central tool in our method is the pseudo-Polar FFT, an FFT where the evaluation frequencies lie in an oversampled set of nonangularly equispaced points. We describe the concept of pseudo-Polar domain, including fast forward and inverse transforms. For those interested primarily in Polar FFT’s, the pseudo-Polar FFT plays the role of a halfway point—a nearly-Polar system from which conversion to Polar coordinates uses processes relying purely on 1D FFT’s and interpolation operations. We describe the conversion process, and give an error analysis of it. We compare accuracy results obtained by a Cartesian-based unequally-sampled FFT method to ours, both algorithms using a small-support interpolation and no pre-compensating, and show marked advantage to the use of the pseudo-Polar initial grid.

We show that the polar as well as the pseudo-polar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo-)polar grid by means of the inverse nonequispaced FFT.

This paper is dedicated to the memory of Professor Moshe Israeli 1940-2007, who passed away on February 18. Computing the Fourier transform of a function in polar coordinates is an important building block in many scientific disciplines and numerical schemes. In this paper we present the pseudo-polar Fourier transform that samples the Fourier transform on the pseudo-polar grid, also known as the concentric squares grid. The pseudo-polar grid consists of equally spaced samples along rays, where different rays are equally spaced and not equally angled. The pseudo-polar Fourier transform Fourier transform is shown to be fast (the same complexity as the FFT), stable, invertible, requires only

Abstract—The estimation of large motions without prior knowledge is an important problem in image registration. In this paper, we present the angular difference function (ADF) and demonstrate its applicability to rotation estimation. The ADF of two functions is defined as the integral of their spectral difference along the radial direction. It is efficiently computed using the pseudopolar Fourier transform, which computes the discrete Fourier transform of an image on a near spherical grid. Unlike other Fourier-based registration schemes, the suggested approach does not require any interpolation. Thus, it is more accurate and significantly faster. Index Terms—Global motion estimation, Fourier domain, pseudopolar FFT, image alignment. æ 1

We present a source-to-source compiler that processes matrix formulae in the form of Kronecker product factorizations. The Kronecker product notation allows for simple expressions of algorithms such as Walsh-Hadamard, Haar, Slant, Hartley, and FFTs as well as transpositions and wavelet transforms. The compiler is based on a set of term rewriting rules that translate high level matrix descriptions into parallel and sequential loops and assignment statements. We provide back-end translators for FORTRAN, FORTRAN-90, C and Matlab. 1

Abstract. This paper shows how the recently developed fractional fft algorithm (frft) can be used to retrieve option prices from the corresponding characteristic functions. The frft algorithm has the advantage of using the characteristic function information in a more efficient way than the straight fft. Therefore less function evaluations are typically needed and substantial savings in computational time can be made. Two experiments, based on the stochastic volatility and the variancegamma models, illustrate the benefits of using the fractional version of the fft and show that option prices can be delivered up to forty-five times faster without substantial losses of result accuracy. 1.