Directional and time-scale wavelet analysis

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. The wavelet X-ray transform performs one-dimensional wavelet transforms along lines in R(n) which are parameterized in the same fashion as for the X-ray transform. The reconstruction formula for this transform gives rise to a continuous family of elementary projections. These projections provide the building blocks of a directional wavelet analysis of functions in several variables. Discrete wavelet X-ray transforms are described which make use of wavelet orthonormal bases and, more generally, of biorthogonal systems of wavelet Riesz bases. Some attention is given to approximation results which involve wavelet X-ray analysis in several directions.