transforms to compress digitally scanned fingerprints. NASA’s Mars rovers use them to compress images acquired by their 18 cameras. The new JPEG 2000 format is based on wavelet transforms; the smaller files allow one to store images using less memory and to transmit those images faster and more reliably. Researchers also use wavelet transforms to clean signals and images—that is, to reduce unwa...

[1] Rainfall variability occurs over a wide range of time scales owing to processes initiated by cloud microphysics and sustained by atmospheric circulation. A central topic in rainfall research is to determine whether rainfall variability at a given scale is caused by dynamics acting at some other scales. Random multiplicative cascades (RMCs) are standard approaches for describing rainfall var...

Fast trigonometric transforms and periodic orthogonal wavelet transforms are essential tools for numerous practical applications. It is very important that fast algorithms work stable in a floating point arithmetic. This survey paper presents recent results on the worst case analysis of roundoff errors occurring in floating point computation of fast Fourier transforms, fast cosine transforms, a...

The continuous curvelet and shearlet transforms have recently been shown to be much more effective than the traditional wavelet transform in dealing with the set of discontinuities of functions and distributions. In particular, the continuous shearlet transform has the ability to provide a very precise geometrical characterization of general discontinuity curves occurring in images. In this pap...

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. The wavelet transform of a signal is an L2-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves ...

New classes of diadic and triadic biorthogonal wavelets and wavelet-type frames in signal space are presented. The construction employs interpolatory filters with rational transfer functions that originate from continuous and discrete splines. These filters have linear phase. They are amenable either to fast cascading or parallel recursive implementation. The wavelets are applied to compression...

We show that translations and dilations of a p–adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p–adic wavelets. In this sense the continuous p–adic wavelet transform coincides with the discrete p–adic wavelet transform. The p–adic multiresolution approximation is introduced and relation with the real multiresolution approximat...

Complex discrete wavelet transforms have significant advantages over real wavelet transforms for certain signal processing problems. Two approaches to the implementation of complex wavelet transforms have been proposed earlier. Both approaches require discrete-time allpass systems having approximately linear-phase and (fractional) delay. This paper compares the results when different allpass sy...

The main goal of the paper is to show that wavelet transforms and packets have the multiparametric representation in the form of a product of the rotation Jacobi matrices. These representations we call the third and the fourth canonical multiparametric form. Each multiparametric wavelet transform (MPWT) depends on several free Jacobi parameters. When parameters are changed multiparametric trans...