w)I2 + [G(w + :)I' = 1. Its periodic extension then satisfies +*(w) 2 1 and hence a sampling function with q w) = +(w)/G*(w) belongs to L2(Z2). 5) Daubechies Wavelets: The scaling function for the simplest class of Daubechies wavelets, those with support on [0, 31, are defined by the dilation equations REFERENCES [ 11 I. Daubechies, " Orthonormal bases of compactly supported wavelets, " Comm. c...

Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such tha...

In the first part of this paper we construct an algorithm for implementing the discrete wavelet transform by means of matrices in SO2(R) for orthonormal compactly supported wavelets and matrices in SLm(R),m ≥ 2, for compactly supported biorthogonal wavelets. We show that in 1 dimension the total operation count using this algorithm can be reduced to about 50% of the conventional convolution and...

There has recently been interest in the use of orthonor-mal bases for the purposes of SISO system identiication. Concurrently, but separately, there has also been vigorous work on estimation of MIMO systems by computa-tionally cheap and reliable schemes. These latter ideas have collectively become known as`4SID' methods. This paper is a contribution overlapping these two schools of thought by s...

The purpose of this paper is to generalize a result byDonoho, Huo, Elad and Bruckstein on sparse representations of signals/images in a union of two orthonormal bases. We consider general (redundant) dictionaries in finite dimension, and derive sufficient conditions on a signal/image for having a unique sparse representation in such a dictionary. In particular, it is proved that the result of D...

We introduce a sparse image representation that takes advantage of the geometrical regularity of edges in images. A new class of one-dimensional wavelet orthonormal bases, called foveal wavelets, are introduced to detect and reconstruct singularities. Foveal wavelets are extended in two dimensions, to follow the geometry of arbitrary curves. The resulting two dimensional “bandelets” define orth...

We consider sparse representations of signals from redundant dictionaries which are unions several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in representations. However, numerical computations sparks generally combinatorial. For bases, two lower bounds on the via mutual coherence were established previous work. constructively prove that both them tight. ...

In many areas of mathematics, orthonormal bases are a standard tool for representing vectors or operators. For certain problems, however, it is preferable to use an overcomplete, non-orthogonal set of vectors instead of an orthonormal basis, and thereby incorporate redundancy in the representation. Frames are overcomplete sets which are associated with redundant linear representations that have...

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec’ optimal 1/4 re...