Applications of Gabor and Wavelet Expansions

We investigate the relationship between the Radon transform and certain phase space localization functions, namely the continuous Gabor and wavelet transforms. We derive inversion formulas for the Radon transform based on the Gabor and wavelet transform. Some of these formulas give a direct reconstruction of f or of 1=2 f from the Radon transform data. Others show how the Gabor and wavelet transforms of f or 1=2 f can be recovered directly from the Radon transform data. We suggest ways in which these formulas can lead to eecient reconstruction algorithms and can be applied to noise reduction in reconstructed images. The Radon transform is a mathematical tool which is used to describe an image (which may be thought of as a function of several, typically two, variables) in terms of intensity averages over lines or hyperplanes in several directions. Typically such averages can be easily measured while the function itself is inaccessible. In computerized tomogra-phy (CT) scanners, for example, one wishes to determine the tissue density function in a cross-section of the human body from non-invasive measurements. The basic problem is the accurate recovery of the unknown function or at least relevant features of the unknown function in a stable fashion and requiring the fewest possible measurements. In addition to medical applications, the Radon transform has also been used in astronomy, electron microscopy, optics, geophysics De]. The Radon transform has recently been proposed as the basis of a recovery instrument for space plasmas, and in determining the chemical composition of ames ZCMB]. Given a function f deened on R d , its Radon transform, Rf consists of the average of f over all hyperplanes in R d. For example, the Radon transform in the plane (d = 2) would consist of the integrals over all lines of a function deened in the plane. In planar imaging, these averages can be found by measuring the attenuation of a beam passing through a two dimensional slice of the body. In some applications, one also needs to consider integration 1