Directional and time-scale wavelet analysis

Combined use of the X-ray (Radon) transform and the wavelet transform has p roved to be useful in application areas such as diagnostic medicine and sei smology. The wavelet X-ray transform performs one-dimensional wavelet trans forms 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 r ise to a continuous family of elementary projections. These projections pro vide the building blocks of a directional wavelet analysis of functions in several variables. Discrete wavelet X-ray transforms are described which ma ke use of wavelet orthonormal bases and, more generally, of biorthogonal sy stems of wavelet Riesz bases. Some attention is given to approximation resu lts which involve wavelet X-ray analysis in several directions.