scholarly journals WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

2010 ◽  
Vol 08 (01) ◽  
pp. 109-132 ◽  
Author(s):  
PETER BALAZS ◽  
JEAN-PIERRE ANTOINE ◽  
ANNA GRYBOŚ
Keyword(s):  
Iterative Algorithms ◽  
Gabor Frames ◽  
Dual Frame ◽  
Mutual Relationship ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Frame Condition ◽  
Frame Bound ◽  
Special Case ◽  
Numerical Advantage

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

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

2013 ◽  
Vol 21 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Asghar Rahimi ◽  
Abolhassan Fereydooni
Keyword(s):  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Multiplier Operator ◽  
Frame Operator ◽  
Fixed Sequence ◽  
Numerical Efficiency ◽  
Bessel Sequences ◽  
Almost All

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.

scholarly journals Controlled K−g−Fusion Frames in Hilbert C∗−Modules

2022 ◽  
Vol 20 ◽  
pp. 1
Author(s):  
Mohamed Rossafi ◽  
Fakhr-dine Nhari
Keyword(s):  
Iterative Algorithms ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Fusion Frames ◽  
Fusion Frame ◽  
Perturbation Problem ◽  
The Subject

Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g−fusion frame and controlled K−g−fusion frame in Hilbert C∗−modules and we give some properties. Also, we study the perturbation problem of controlled K−g−fusion frame. Moreover, an illustrative example is presented to support the obtained results.

scholarly journals Inversion Free Algorithms for Computing the Principal Square Root of a Matrix

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Nicholas Assimakis ◽  
Maria Adam
Keyword(s):  
Riccati Equation ◽  
Newton Method ◽  
Continuous Time ◽  
Iterative Algorithms ◽  
Algebraic Method ◽  
Matrix Inversion ◽  
Square Root ◽  
Special Case ◽  
New Algorithms

New algorithms are presented about the principal square root of ann×nmatrixA. In particular, all the classical iterative algorithms require matrix inversion at every iteration. The proposed inversion free iterative algorithms are based on the Schulz iteration or the Bernoulli substitution as a special case of the continuous time Riccati equation. It is certified that the proposed algorithms are equivalent to the classical Newton method. An inversion free algebraic method, which is based on applying the Bernoulli substitution to a special case of the continuous time Riccati equation, is also proposed.

scholarly journals Stable Sparse Model with Non-Tight Frame

2020 ◽  
Vol 10 (5) ◽  
pp. 1771 ◽  
Author(s):  
Min Zhang ◽  
Yunhui Shi ◽  
Na Qi ◽  
Baocai Yin
Keyword(s):  
Image Restoration ◽  
Sparse Representation ◽  
Tight Frame ◽  
Closed Form Expression ◽  
Dual Frame ◽  
Two Phase ◽  
Sparse Models ◽  
Frame Operator ◽  
Sparse Model ◽  
Sparse System

Overcomplete representation is attracting interest in image restoration due to its potential to generate sparse representations of signals. However, the problem of seeking sparse representation must be unstable in the presence of noise. Restricted Isometry Property (RIP), playing a crucial role in providing stable sparse representation, has been ignored in the existing sparse models as it is hard to integrate into the conventional sparse models as a regularizer. In this paper, we propose a stable sparse model with non-tight frame (SSM-NTF) via applying the corresponding frame condition to approximate RIP. Our SSM-NTF model takes into account the advantage of the traditional sparse model, and meanwhile contains RIP and closed-form expression of sparse coefficients which ensure stable recovery. Moreover, benefitting from the pair-wise of the non-tight frame (the original frame and its dual frame), our SSM-NTF model combines a synthesis sparse system and an analysis sparse system. By enforcing the frame bounds and applying a second-order truncated series to approximate the inverse frame operator, we formulate a dictionary pair (frame pair) learning model along with a two-phase iterative algorithm. Extensive experimental results on image restoration tasks such as denoising, super resolution and inpainting show that our proposed SSM-NTF achieves superior recovery performance in terms of both subjective and objective quality.

Controlled g-fusion frame in Hilbert space

2021 ◽  
pp. 2150009
Author(s):  
Hanbing Liu ◽  
Yongdong Huang ◽  
Fengjuan Zhu
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Iterative Algorithms ◽  
Bessel Sequence ◽  
Numerical Efficiency ◽  
Fusion Frames ◽  
Controlled Fusion ◽  
Fusion Frame

Fusion frame is a generalization of frame, which can analyze signals by projecting them onto multidimensional subspaces. Controlled fusion frame as generalization of fusion frame, it can improve the numerical efficiency of iterative algorithms for inverting the fusion frame operators. In this paper, we first introduce the notion of controlled g-fusion frame, discuss several properties of controlled g-fusion Bessel sequence. Then, we present some sufficient conditions and some characterizations of controlled g-fusion frames. Finally, we study the sum of controlled g-fusion frames.

scholarly journals Approximation of the Inverse Frame Operator and Applications to Gabor Frames

2000 ◽  
Vol 103 (2) ◽  
pp. 338-356 ◽  
Author(s):  
Peter G. Casazza ◽  
Ole Christensen
Keyword(s):  
Gabor Frames ◽  
Frame Operator

Partial Gabor frames and dual frames

2021 ◽  
pp. 2150035
Author(s):  
Yu Tian ◽  
Hui-Fang Jia ◽  
Guo-Liang He
Keyword(s):  
Density Theorem ◽  
Gabor Frame ◽  
Gabor Frames ◽  
Dual Frame ◽  
Gabor System ◽  
Dual Frames ◽  
Gabor Systems

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

scholarly journals On Discrete Time Wilson Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Teena Kohli ◽  
Suman Panwar ◽  
S. K. Kaushik
Keyword(s):  
Discrete Time ◽  
Dual Pair ◽  
Gabor Frames ◽  
Sufficient Condition ◽  
Zak Transform ◽  
Frame Operator

In this paper, we define the discrete time Wilson frame (DTW frame) for l 2 ℤ and discuss some properties of discrete time Wilson frames. Also, we give an interplay between DTW frames and discrete time Gabor frames. Furthermore, a necessary and a sufficient condition for the DTW frame in terms of Zak transform are given. Moreover, the frame operator for the DTW frame is obtained. Finally, we discuss dual pair of frames for discrete time Wilson systems and give a sufficient condition for their existence.

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