A simple construction of an orthonormal basis starting with a so called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. In this paper, we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.

We present a method to constrain galaxy parameters directly from three-dimensional data cubes. The algorithm compares directly the data with a parametric model mapped in x, y, λ coordinates. It uses the spectral line-spread function (LSF) and the spatial point-spread function (PSF) to generate a 3-dimensional kernel whose characteristics are instrument specific or user generated. The algorithm returns the intrinsic modeled properties along with both an 'intrinsic' model data cube and the modeled galaxy convolved with the 3D-kernel. The algorithm uses a Markov Chain Monte Carlo (MCMC) approach with a nontraditional proposal distribution in order to efficiently probe the parameter space. We demonstrate the robustness of the algorithm using 1728 mock galaxies and galaxies generated from hydrodynamical simulations in various seeing conditions from 0. 6 to 1. 2. We find that the algorithm can recover the morphological parameters (inclination, position angle) to within 10% and the kinematic parameters (maximum rotation velocity) to within 20%, irrespectively of the PSF in seeing (up to 1. 2) provided that the maximum signal-to-noise ratio (SNR) is greater than ∼ 3 pixel −1 and that ratio of galaxy half-light radius to seeing radius is greater than about 1.5. One can use such an algorithm to constrain simultaneously the kinematics and morphological parameters of (nonmerging) galaxies observed in nonoptimal seeing conditions. The algorithm can also be used on adaptive-optics (AO) data or on high-quality, high-S/N data to look for nonaxisymmetric structures in the residuals.

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