Frame Operator Research Articles

Analytical Construction and Fast Computation of Frequency Warping Operators and Dual Frame

Frequency warping has been introduced as a unitary operator over a nonfinite domain. In practical applications, the need of a feasible implementation forces to redefine frequency warping over finite domains in the framework of redundant transformations called frames, where the unitary property is replaced by the tight frame property, i.e., the adjoint operator behaves as pseudo-inverse. In case a tight frame cannot be designed, inversion can be achieved by finding the dual frame. In this work, with reference to nonsmooth warping functions, we present a new approach to the construction and fast computation of the dual frame operator based on an operator power series. With respect to previous approaches, many limitations are overcome while maintaining important properties of frequency warping.

Patch-ordering-based wavelet frame and its use in inverse problems.

In our previous work [1] we have introduced a redundant tree-based wavelet transform (RTBWT), originally designed to represent functions defined on high dimensional data clouds and graphs. We have further shown that RTBWT can be used as a highly effective image-adaptive redundant transform that operates on an image using orderings of its overlapped patches. The resulting transform is robust to corruptions in the image, and thus able to efficiently represent the unknown target image even when it is calculated from its corrupted version. In this paper, we utilize this redundant transform as a powerful sparsity-promoting regularizer in inverse problems in image processing. We show that the image representation obtained with this transform is a frame expansion, and derive the analysis and synthesis operators associated with it. We explore the use of this frame operators to image denoising and deblurring, and demonstrate in both these cases state-of-the-art results.

Class of bounded operators associated with an atomic system

$K$-frames, more general than the ordinary frames, have been introduced by Laura G{\u{a}}vru{\c{t}}a in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Using the frame operator, we find a class of bounded linear operators in which a given Bessel sequence is an atomic system for every member in the class.

ON HILBERT TRANSFORM OF GABOR AND WILSON SYSTEMS

Given that the Gabor system {EmbTnag}m,n∈ℤ is a Gabor frame for L2(ℝ), a sufficient condition is obtained for the Gabor system {EmbTnaHg}m,n∈ℤ to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L2(ℝ). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {EmbTnag}m,n∈ℤ commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system [Formula: see text] to be a Wilson frame given that [Formula: see text] is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence [Formula: see text] commute with each other.

Perceptual frames in frequency estimation.

This study is an introductory investigation of cognitive frames, focused on perceptual frames divided into information and formal perceptual frames, which were studied based on sub-additivity of frequency estimations. It was postulated that different presentations of a response scale would result in different percentage estimates of time spent watching TV or using the Internet. The results supported the existence of perceptual frames that influence the perception process and indicated that information perceptual frames had a stronger effect than formal frames. The measures made possible the exploration of the operation of perceptual frames and also outlined the relations between heuristics and cognitive frames.

A frame based shrinkage procedure for fast oscillating functions

In non-parametric regression analysis the advantage of frames with respect to classical orthonormal bases is that they can furnish an efficient representation of a more broad class of functions. For example, fast oscillating functions as audio, speech, sonar, radar, EEG and stock market are much more well represented by a frame, with similar oscillating characteristic, than by a classical orthonormal basis. In this respect, a new frame based shrinkage estimator is derived as the Empirical Regularized version of the optimal Shrinkage estimator generalized to the frame operator. An analytic expression of it is furnished leading to an efficient implementation. Results on standard and real test functions are shown.

Structure of nonstationary Gabor frames and their dual systems

We investigate the structural properties of dual systems for nonstationary Gabor frames. In particular, we prove that some inverse nonstationary Gabor frame operators admit a Walnut-like representation, i.e. the operator acting on a function can be described by weighted translates of that function. In this case, which only occurs when compactly supported window functions are used, the canonical dual frame partially inherits the structure of the original frame, with differences that we describe in detail. Moreover, we determine a necessary and sufficient condition for a pair of nonstationary Gabor frames to form dual frames, valid under mild restrictions. This condition is then applied in a simple setup, to prove the existence of dual pairs of nonstationary Gabor systems with coarser frequency sampling than allowed by previous results [3]. We also explore a connection to recent work of Christensen, Kim and Kim on Gabor frames with compactly supported window function.

Construction of Fusion Frame Systems in Finite Dimensional Hilbert Spaces

We first investigate the construction of a fusion frame system in a finite-dimensional Hilbert space𝔽nwhen its fusion frame operator matrix is given and provides a corresponding algorithm. The matrix representations of its local frame operators and inverse frame operators are naturally obtained. We then study the related properties of the constructed fusion frame systems. Finally, we implement the construction of fusion frame systems which behave optimally for erasures in some special sense in signal transmission.

Frame properties of low autocorrelation random sequences

The goal is to construct random frames and study properties of such frames. Starting with the construction of unimodular random sequences whose expected autocorrelations can be made arbitrarily low outside the origin, these random sequences are used to construct frames for C. Using recent theory of non-asymptotic analysis of random matrices, the eigenvalue distribution of the corresponding frame operator is studied.

The Bijectivity of the Tight Frame Operators in Lebesgue Spaces

 Abstract—The motivation of this dissertation mainly is which affine frame wavelet systems span Lebesgue spaces have been investigated less. The technique we used is analogous to technique of Calderon-Zygmund operators, but we rely on Calderon-Zygmund decomposition theorem. We prove our main results without smoothness assumption on frame wavelets. We prove that affine tight frame wavelets span Lebesgue spaces under the condition, we also show that the affine tight frame operator extends from 2 () L  to a bounded, linear and bijective operator on () p L  , for 1. p    Under such condition, the

Tetris Tight Frames Construction via Hadamard Matrices

We present a new method to construct unit norm tight frames by applying altered Hadamard matrices. Also we determine an elementary construction which can be used to produce a unit norm frame with prescribed spectrum of frame operator.

Sum and direct sum of frame sequences

Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences (in the same Hilbert space) to be a frame sequence, and thereby show that the sum of two strongly disjoint frame sequences may not be a frame sequence. We also show that the closedness of the sum of the synthesis operators of two frame sequences and that of the sum of the frame operators of the same frame sequences are not related. Other observations are also included.

Weighted fusion frame construction via spectral tetris

Fusion frames consist of a sequence of subspaces from a Hilbert space and corresponding positive weights so that the sum of weighted orthogonal projections onto these subspaces is an invertible operator on the space. Given a spectrum for a desired fusion frame operator and dimensions for subspaces, one existing method for creating unit-weight fusion frames with these properties is the flexible and elementary procedure known as spectral tetris. Despite the extensive literature on fusion frames, until now there has been no construction of fusion frames with prescribed weights. In this paper we use spectral tetris to construct more general, arbitrarily weighted fusion frames. Moreover, we provide necessary and sufficient conditions for when a desired fusion frame can be constructed via spectral tetris.

Controlled G-Frames and Their G-Multipliers in Hilbert spaces

Abstract Multipliers have been recently introduced by P. Balazs as operators for Bessel sequences and frames in Hilbert spaces. These are opera- tors that combine (frame-like) analysis, a multiplication with a fixed sequence ( called the symbol) and synthesis. One of the last extensions of frames is weighted and controlled frames that introduced by P.Balazs, J-P. Antoine and A. Grybos to improve the numerical efficiency of iterative algorithms for inverting the frame operator. Also g-frames are the most popular generalization of frames that include almost all of the frame extensions. In this manuscript the concept of the controlled g- frames will be defined and we will show that controlled g-frames are equivalent to g-frames and so the controlled operators C and C' can be used as preconditions in applications. Also the multiplier operator for this family of operators will be introduced and some of its properties will be shown.

Constructing all self-adjoint matrices with prescribed spectrum and diagonal

The Schur---Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive proofs have subsequently been found. Most of these constructive proofs rely on Givens rotations, and none have been shown to be able to produce every example of such a matrix. We introduce a new construction method that is able to do so. This method is based on recent advances in finite frame theory which show how to construct frames whose frame operator has a given prescribed spectrum and whose vectors have given prescribed lengths. This frame construction requires one to find a sequence of eigensteps, that is, a sequence of interlacing spectra that satisfy certain trace considerations. In this paper, we show how to explicitly construct every such sequence of eigensteps. Here, the key idea is to visualize eigenstep construction as iteratively building a staircase. This visualization leads to an algorithm, dubbed Top Kill, which produces a valid sequence of eigensteps whenever it is possible to do so. We then build on Top Kill to explicitly parametrize the set of all valid eigensteps. This yields an explicit method for constructing all self-adjoint matrices with a given spectrum and diagonal, and moreover all frames whose frame operator has a given spectrum and whose elements have given lengths.

Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

We prove a sufficient condition for frame-type wavelet series in L p , the Hardy space H 1, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calderón–Zygmund constants of the frame kernel.

Survey of MapReduce frame operation in bioinformatics

Bioinformatics is challenged by the fact that traditional analysis tools have difficulty in processing large-scale data from high-throughput sequencing. The open source Apache Hadoop project, which adopts the MapReduce framework and a distributed file system, has recently given bioinformatics researchers an opportunity to achieve scalable, efficient and reliable computing performance on Linux clusters and on cloud computing services. In this article, we present MapReduce frame-based applications that can be employed in the next-generation sequencing and other biological domains. In addition, we discuss the challenges faced by this field as well as the future works on parallel computing in bioinformatics.

Bilinear Mappings and the Frame Operator

The theory of frames has been actively developed by many authors over the past two decades, both for its applications to signal processing, and for its deep connections to other areas of mathematics such as operator theory. Central to the study of frames is the frame operator. We initiate an investigation that extends the frame operator to the bilinear setting.

Hilbert-Schauder frame operators

We introduce a new concept of frame operators for Banach spaces we call a HilbertSchauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of p and Lp -spaces with 1 < p 2 . Finally, we give a new isomorphic characterization of Hilbert spaces. Mathematics subject classification (2010): Primary 46B15, 46B28, 46B45, 47B38; Secondary 47A20.

The Duals of Fusion Frames for Experimental Data Transmission Coding of High Energy Physics

The experimental data transmission is an important part of high energy physics experiment. In this paper, we connect fusion frames with the experimental data transmission implement of high energy physics. And we research the utilization of fusion frames for data transmission coding which can enhance the transmission efficiency, robust against erasures, and so forth. For this application, we first characterize a class of alternate fusion frames which are duals of a given fusion frame in a Hilbert space. Then, we obtain the matrix representation of the fusion frame operator of a given fusion frame system in a finite-dimensional Hilbert space. By using the matrix representation, we provide an algorithm for constructing the dual fusion frame system with its local dual frames which can be used as data transmission coder in the high energy physics experiments. Finally, we present a simulation example of data coding to show the practicability and validity of our results.