A Physical Interpretation of Tight Frames

Casazza, Peter G.; Fickus, Matthew; Kovačević, Jelena; Leon, Manuel T.; Tremain, Janet C.

doi:10.1007/0-8176-4504-7_4

Cited by 79 publications

(80 citation statements)

References 26 publications

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“…The equivalence (4)⇔ (5) follows from the definition of A. (1)⇔ (4). Similarly, the definition of A shows that …”

Section: Cross-gramian and Gramian Matrices A B C By Their (I J)thmentioning

confidence: 86%

“…An important early insight [2] into tight frames provides a physical interpretation in terms of minimal energy configurations. As a part of this program the following theorem characterizes the possible norms e n of the elements in tight frames; see [4]. The inequality in condition (2) below is known as the fundamental inequality of tight frames.…”

Section: Introductionmentioning

confidence: 99%

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A note on finite dual frame pairs

Christensen

Powell

Xiao

2012

Proc. Amer. Math. Soc.

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Abstract. The purpose of this note is to extend certain key results in frame theory from the setting of tight frames to dual pairs of frames. We provide various characterizations of dual frame pairsBased on this we characterize those scalar sequences {α n } N n=1 for which there exist dual pairs of framesfor K d such that α n = e n , f n . This generalizes the well-known fundamental inequality of tight frames.

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“…The equivalence (4)⇔ (5) follows from the definition of A. (1)⇔ (4). Similarly, the definition of A shows that …”

Section: Cross-gramian and Gramian Matrices A B C By Their (I J)thmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A note on finite dual frame pairs

Christensen

Powell

Xiao

2012

Proc. Amer. Math. Soc.

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“…As such, they have been studied extensively, but only recently have Benedetto and Fickus [13] formally shown why tight frames and orthonormal bases indeed belong together. In their work, they characterized all unit-norm tight frames, while in [40], the authors did the same for nonequal norm tight frames.…”

Section: What Can Coulomb Teach Us?mentioning

confidence: 99%

“…The authors gave one solution to the problem using an explicit construction; characterization of all solutions to this problem using a physical interpretation of frame theory was given in Theorem 4.3 [40], Section 4.…”

Section: Cdma Systemsmentioning

confidence: 99%

An Introduction to Frames

Kovačević

Chebira

2007

FNT in Signal Processing

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“…We choose these parameters such that the set of frame functions form a Parseval tight frame [1]. For Parseval tight frames, we can establish bounds on the energy of the error when wavelet coefficients are thresholded.…”

Section: Characterization Of the Scaling And Wavelet Functionsmentioning

confidence: 99%

A Multiscale Data Representation for Distributed Sensor Networks

Wagner¹,

Sarvotham²,

Baraniuk³

Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005.

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Characterization of the scaling and wavelet functionsThe sensors are irregularly spaced on a 2D grid. We assume that the sensors sample a slowly varying field. To reconstruct the original field, we interpolate the sensor values over R 2 in the following manner. The approximate value of the field at any arbitrary location is equal to the value recorded by the sensor closest to it. Thus, the function space F spanned by the reconstructed field is the space of piecewise constants whose discontinuities lie on voronoi cell boundaries.Our goal is to first find a suitable set of functions that span the function space F . Motivated by the Haar wavelet transform, we pick one function that captures the average value of the sensor field, and N funtions that capture the deviation of each sensor value from the average. Since we use (N + 1) functions that span the N dimensional function space F , we refer to the functions as frame functions. Example of such frame functions in 1 − D are shown in Figure 1. The scaling function is constant over the entire region. The wavelet functions have discontinuities only at the boundaries of one voronoi cell. The wavelet functions have zero mean. Thus, the dot product of a wavelet function and the scaling function is zero.We need to pick the parameters of the frame functions (namely the heights of the wavelet functions) appropriately. We choose these parameters such that the set of frame functions form a Parseval tight frame [1]. For Parseval tight frames, we can establish bounds on the energy of the error when wavelet coefficients are thresholded.Let the support of the voronoi regions be ∆ i , for i = 1, 2, ..., N . LetDefine the scaling function as The i th wavelet function, W i , takes the value k i for all the voronoi cells except at the i th cell, where it takes value k i . Since the average value of the wavelet functions are zero, we haveTo satisfy the Parseval tight frame conditions, we look at the energy of the signal in the signal and wavelet domains. The energy of the signal is given byThe scaling coefficient s and the wavelet coefficient w are given by s =

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A Physical Interpretation of Tight Frames

Cited by 79 publications

References 26 publications

A note on finite dual frame pairs

A note on finite dual frame pairs

An Introduction to Frames

A Multiscale Data Representation for Distributed Sensor Networks

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