We present a new class of wavelet bases—Fresnelets—which is obtained by applying the Fresnel transform operator to a wavelet basis of L2. The thus constructed wavelet family exhibits properties that are particularly useful for analyzing and processing optically generated holograms recorded on CCD-arrays. We first investigate the multiresolution properties (translation, dilation) of the Fresnel ...

Orthonormal M-band wavelet bases have been constructed and applied by several authors. This paper makes three main contributions. First, it generalizes the minimal length Kregular 2-band wavelets of Daubechies to the M-band case by deriving explicit formulas for K-regular M-band scaling filters. Several equivalent characterizations of K-regularity are given and their significance explained. Sec...

Existing work on the representation of operators in one-dimensional, compactly-supported, orthonormal wavelet bases is extended to two dimensions. The non-standard form of the representation of operators is given in separable two-dimensional, periodic, orthonormal wavelet bases. The matrix representation of the partial differential operators ∂x and ∂y are constructed and a closed form formula f...

Biorthogonal bases of compactly-supported wavelets are characterized by the FIR perfect-reconstruction filterbanks to which they correspond. In this paper we develop explicit representations of all such filterbanks, allowing us to generate every possible biorthogonal compactly-supported wavelet basis. For these filterbanks, the product H(z) = H(z) e H(z) of the two lowpass filters must have N 2...

The main contribution of this work is a new paradigm for image compression. We describe a new multi-layered representation technique for images. An image is encoded as the superposition of one main approximation, and a sequence of residuals. The strength of the multi-layered method comes from the fact that we use different bases to encode the main approximation and the residuals. For instance, ...

Previously, the use of non-separable wavelets in image processing has been hindered by the lack of a fast algorithm to perform a non-separable wavelet transform. We present two such algorithms in this paper. The rst algorithm implements a periodic wavelet transform for any valid wavelet lter sequence and dilation matrices satisfying a trace condition. We discuss some of the complicating issues ...

This paper deals with continuous frames and continuous Riesz bases. We introduce continuous Riesz bases and give some equivalent conditions for a continuous frame to be a continuous Riesz basis. It is certainly possible for a continuous frame to have only one dual. Such a continuous frame is called a Riesz-type frame [13]. We show that a continuous frame is Riesz-type if and only if it is a con...

This paper describes a new algorithm to compute radiance in a synthetic environment. Motivated by the success of wavelet methods for radiosity computations we have applied multi wavelet bases to the computation of radiance in the presence of glossy reflectors. We have implemented this algorithm and report on some experiments performed with it. In particular we show that the convergence properti...

We obtain wavelet characterizations of Besov spaces and the Triebel–Lizorkin spaces associated with Aloc ∞ -weights. These characterizations are used to show that our wavelet bases are also greedy. c © 2009 Elsevier Inc. All rights reserved.

We present applications of variational – wavelet approach to different forms of nonlinear (rational) rms envelope equations. We have the representation for beam bunch oscillations as a multiresolution (multiscales) expansion in the base of compactly supported wavelet bases.