In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and other nonpointlike structures, for which wavelets are poorly adapted. We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with linelike phenomena in dimension 2, planelike phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1+...+unxn) whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain. The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.

We develop an algorithm to reconstruct the wavelet coefficients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which significantly reduces the amount of exposure and computations in X-ray tomography. This property which distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet basis with sufficiently many vanishing moments, the ramp-filtered version of the scaling function as well as the wavelet function has extremely rapid decay. We show that the variance of the elements of the null-space is negligible in the locally reconstructed image. Also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm. To reconstruct a local region 20 pixels in radius in a 256 \Theta 256 image we require 12:5% of full e...

. We use the theory of the continuous wavelet transform to derive inversion formulas for the Radon transform on L 1 " L 2 (R d ). These inversion formulas turn out to be local in even dimensions in the following sense. In order to recover a function f from its Radon transform in a ball of radius R ? 0 about a point x to within error ffl, we can find ff(ffl) ? 0 such that this can be accomplished by knowing the projections of f only on lines passing through a ball of radius R+ ff(ffl) about x. We give explicit a priori estimates on the error in the L 2 and L 1 norms. in Proceedings of the Conference: 75 Years of the Radon Transform (Vienna, 1992), International Press Co., Ltd., 38--58 (1994). 0. Introduction. Given a function f defined on R d , its Radon transform, Rf , is defined by R ` f(s) = Z ` ? f(s` + y) dy; where ` 2 S d\Gamma1 and s 2 R. Rf(`; s) is the integral of f on the hyperplane in R d defined by fx: hx; `i = sg. The backprojection operator is given by...

Combined use of the X-ray (Radon) transform and the wavelet transform has proved to be useful in application areas such as diagnostic medicine and seismology. In the present paper, the wavelet X-ray transform is introduced. This transform performs one-dimensional wavelet transforms along lines in R n , which are parameterized in the same fashion as for the X-ray transform. It is shown that the transform has the same convenient inversion properties as the wavelet transform. The reconstruction formula receives further attention in order to obtain usable discretizations of the transform. Finally, a connection between the wavelet X-ray transform and the filtered backprojection formula is discussed. Keywords: wavelet transform, wavelet X-ray transform, windowed Radon transform, reconstruction, approximation, noise reduction, filtered backprojection 1. INTRODUCTION The X-ray transform (or Radon transform) and the wavelet transform play significant roles in a large range of application ar...

. Local or lambda tomography reconstructs f which has the same discontinuities as the searched-for density distribution f . Computing f , however, requires only local tomographic measurements. Local tomography is usually implemented by a filtered backprojection algorithm (FBA). In the present article we design reconstruction filters for the FBA such that 2m+1 f will be reconstructed for a given m 2 N0 . Moreover, we prove convergence and convergence rates for the FBA as the discretization step size goes to zero. To this end we express the FBA in the framework of approximate inverse. Based on our analysis we further propose a scheme which yields a proper scaling of the reconstruction filters. Numerical experiments illustrate the analytic results. Key words. Approximate inverse, local tomography, lambda tomography, filtered backprojection AMS subject classification. 65R20 1. Introduction. Local tomography recovers jump discontinuities of the searched for density distribution f in a...

In a special type of synthetic aperture radar (SAR), using low frequencies and a large relative bandwidth, one encounters the problem of determining the reAEectivity function of the ground from knowledge of its circular averages. This problem is closely related to computerized tomography and the Radon transform. In both cases an important step in the reconstruction consists of a ramp øltering of which a part is the Hilbert transform. These transforms are non-local and therefore very sensitive to noise. We show how it is possible to denoise and stabilize these transforms using wavelet based techniques. We also present a method for target detection based on a measure of the local oscillation. Introduction Background This licentiate thesis is written as the ønal part of the ECMI programme in applied mathematics. The European Consortium for Mathematics in Industry (ECMI) is a joint program between dioeerent mathematical institutes in Europe and aims at strengthening the collaboration be...