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. Key words. wavelet X-ray transform, wavelet transform, X-ray transform, Radon transform, windowed Radon transform, local Radon transform, reconstruction formula, wavelet orthonormal basis, biorthogonal wavelet expansion, wavelet frames. AMS subject classifications. 42C15, 44A12, 86A15 1...