Continuous wavelet transform on the hyperboloid

In this paper we build a continuous wavelet transform (CWT) on the upper sheet of the 2-hyperboloid H-+(2). First, we define a class of suitable dilations on the hyperboloid through conic projection. Then, incorporating hyperbolic motions belonging to SO0(1, 2), we define a family of axisymmetric hyperbolic wavelets. The continuous wavelet transform W-f(a, x) is obtained by convolution of the scaled axisymmetric wavelets with the signal. The wavelet transform is proved to be invertible whenever wavelets satisfy a particular admissibility condition, which turns out to be a zero-mean condition. We then provide some basic examples and discuss the limit at null curvature. (c) 2007 Elsevier Inc. All rights reserved.