The fast wavelet X-ray transform

The wavelet X-ray transform computes one-dimensional wavelet transforms along lines in Euclidian space in order to perform a directional time-scale analysis of functions in several variables. A fast algorithm is proposed which executes this transformation starting with values given on a cartesian grid that represent the underlying function. The algorithm involves a rotation step and wavelet analysis/synthesis steps. The number of computations required is of the same order as the number of data involved. The analysis/synthesis steps are executed by the pyramid algorithm which is known to have this computational advantage. The rotation step makes use of a wavelet interpolation scheme. The order of computations is limited here due to the localization of the wavelets. The rotation step is executed in an optimal way by means of quasi-interpolation methods using (bi-)orthogonal wavelets.