An Introduction to Frames

Kovačević, Jelena; Chebira, Amina

doi:10.1561/2000000006

Cited by 78 publications

(35 citation statements)

References 141 publications

(231 reference statements)

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“…In particular, a tight frame represents a matrix whose coherence matrix is as close as possible to an orthogonal matrix according the Frobenius norm. Frames have been widely used in many applications such as denoising, CDMA systems and multiantenna code design [12].…”

Section: B Tight Framesmentioning

confidence: 99%

On the benefit of using tight frames for robust data transmission and compressive data gathering in wireless sensor networks

Chen

Rodrigues

Wassell

2012

2012 IEEE International Conference on Communications (ICC)

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Abstract-Compressive sensing (CS), a new sampling paradigm, has recently found several applications in wireless sensor networks (WSNs). In this paper, we investigate the design of novel sensing matrices which lead to good expected-case performance -a typical performance indicator in practice -rather than the conventional worst-case performance that is usually employed when assessing CS applications. In particular, we show that tight frames perform much better than the common CS Gaussian matrices in terms of the reconstruction average mean squared error (MSE). We also showcase the benefits of tight frames in two WSN applications, which involve: i) robustness to data sample losses; and ii) reduction of the communication cost.

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Section: B Tight Framesmentioning

confidence: 99%

On the benefit of using tight frames for robust data transmission and compressive data gathering in wireless sensor networks

Chen

Rodrigues

Wassell

2012

2012 IEEE International Conference on Communications (ICC)

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“…Frames were defined by Duffin and Schaeffer [16] to address some deep questions in non-harmonic Fourier series. Traditionally, frames were most popular in signal processing [20], but today, frame theory has an abundance of applications in pure mathematics, applied mathematics, engineering, medicine and even quantum communication [8,15,20,26,1,5].…”

Section: Introductionmentioning

confidence: 99%

The road to equal-norm Parseval frames

Bodmann

Casazza

2010

Journal of Functional Analysis

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The construction of equal-norm Parseval frames is fundamental for many applications of frame theory. We present a construction method based on a system of ordinary differential equations, which generates a flow on the set of Parseval frames that converges to equal-norm Parseval frames. We developed this method to address a question posed by Vern Paulsen: How close is a nearly equal-norm, nearly Parseval frame to an equal-norm Parseval frame? The distance estimate derived here can be used to substantiate numerically found, approximate constructions of equal-norm Parseval frames. The estimate is valid for a fairly general class of frames -requiring that the dimension of the Hilbert space and the number of frame vectors is relatively prime. In addition, we re-phrase our distance estimate to show that certain projection matrices which are nearly constant on the diagonal are close in Hilbert-Schmidt norm to ones which have a constant diagonal.

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“…2). It decomposes a signal into localized space-frequency subbands using multiresolution transforms, both bases and frames [24]- [29]. In each subband, multiresolution classification extracts features, classifies them, and produces a local classification decision.…”

Section: B Multiresolution Classificationmentioning

confidence: 99%

Semi-Supervised Multiresolution Classification Using Adaptive Graph Filtering With Application to Indirect Bridge Structural Health Monitoring

Chen

Cerda

Rizzo

et al. 2014

IEEE Trans. Signal Process.

119

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We present a multiresolution classification framework with semi-supervised learning on graphs with application to the indirect bridge structural health monitoring. Classification in real-world applications faces two main challenges: reliable features can be hard to extract and few labeled signals are available for training. We propose a novel classification framework to address these problems: we use a multiresolution framework to deal with nonstationarities in the signals and extract features in each localized time-frequency region and semi-supervised learning to train on both labeled and unlabeled signals. We further propose an adaptive graph filter for semi-supervised classification that allows for classifying unlabeled as well as unseen signals and for correcting mislabeled signals. We validate the proposed framework on indirect bridge structural health monitoring and show that it performs significantly better than previous approaches.Index Terms-Multiresolution classification, semi-supervised learning, discrete signal processing on graphs, adaptive graph filter, indirect bridge structural health monitoring.

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An Introduction to Frames

Cited by 78 publications

References 141 publications

On the benefit of using tight frames for robust data transmission and compressive data gathering in wireless sensor networks

On the benefit of using tight frames for robust data transmission and compressive data gathering in wireless sensor networks

The road to equal-norm Parseval frames

Semi-Supervised Multiresolution Classification Using Adaptive Graph Filtering With Application to Indirect Bridge Structural Health Monitoring

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