Search results for "Orthonormal basis"
showing 10 items of 31 documents
More wavelet-like orthonormal bases for the lowest Landau level: Some considerations
1994
In a previous work, Antoine and I (1994) have discussed a general procedure which 'projects' arbitrary orthonormal bases of L2(R) into orthonormal bases of the lowest Landau level. In this paper, we apply this procedure to a certain number of examples, with particular attention to the spline bases. We also discuss Haar, Littlewood-Paley and Journe bases.
Block Based Deconvolution Algorithm Using Spline Wavelet Packets
2010
This paper presents robust algorithms to deconvolve discrete noised signals and images. The idea behind the algorithms is to solve the convolution equation separately in different frequency bands. This is achieved by using spline wavelet packets. The solutions are derived as linear combinations of the wavelet packets that minimize some parameterized quadratic functionals. Parameters choice, which is performed automatically, determines the trade-off between the solution regularity and the initial data approximation. This technique, which id called Spline Harmonic Analysis, provides a unified computational scheme for the design of orthonormal spline wavelet packets, fast implementation of the…
Generalized Riesz systems and orthonormal sequences in Krein spaces
2018
We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly non self-adjoint Hamiltonians.
Gabor-like systems in ${cal L}^2({bf R}^d)$ and extensions to wavelets
2008
In this paper we show how to construct a certain class of orthonormal bases in starting from one or more Gabor orthonormal bases in . Each such basis can be obtained acting on a single function with a set of unitary operators which operate as translation and modulation operators in suitable variables. The same procedure is also extended to frames and wavelets. Many examples are discussed.
Hamiltonians defined by biorthogonal sets
2017
In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mat…
Projector operators in clustering
2016
In a recent paper, the notion of quantum perceptron has been introduced in connection with projection operators. Here, we extend this idea, using these kind of operators to produce a clustering machine, that is, a framework that generates different clusters from a set of input data. Also, we consider what happens when the orthonormal bases first used in the definition of the projectors are replaced by frames and how these can be useful when trying to connect some noised signal to a given cluster. Copyright © 2016 John Wiley & Sons, Ltd.
Riesz-like bases in rigged Hilbert spaces
2015
The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{\xi_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.
Experimental studies on continuous speech recognition using neural architectures with “adaptive” hidden activation functions
2010
The choice of hidden non-linearity in a feed-forward multi-layer perceptron (MLP) architecture is crucial to obtain good generalization capability and better performance. Nonetheless, little attention has been paid to this aspect in the ASR field. In this work, we present some initial, yet promising, studies toward improving ASR performance by adopting hidden activation functions that can be automatically learned from the data and change shape during training. This adaptive capability is achieved through the use of orthonormal Hermite polynomials. The “adaptive” MLP is used in two neural architectures that generate phone posterior estimates, namely, a standalone configuration and a hierarch…
Polynomial Spline-Wavelets
2015
This chapter presents wavelets in the spaces of polynomial splines. The wavelets’ design is based on the Zak transform, which provides an integral representation of spline-wavelets. The exponential wavelets which participate in the integral representation are counterparts of the exponential splines that were introduced in Chap. 4. Fast algorithms for the wavelet transforms of splines are presented. Generators of spline-wavelet spaces are described, such as the B-wavelets and their duals and the Battle-Lemarie wavelets whose shifts form orthonormal bases of the spline-wavelet spaces.
Three-dimensional object detection under arbitrary lighting conditions
2006
A novel method of 3D object recognition independent of lighting conditions is presented. The recognition model is based on a vector space representation using an orthonormal basis generated by the Lambertian reflectance functions obtained with distant light sources. Changing the lighting conditions corresponds to multiplying the elementary images by a constant factor and because of that, all possible lighting views will be elements that belong to that vector space. The recognition method proposed is based on the calculation of the angle between the vector associated with a certain illuminated 3D object and that subspace. We define the angle in terms of linear correlations to get shift and i…