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## A physical interpretation of tight frames (2006)

Venue: | In: Harmonic Analysis and Applications, Applied and Numerical Harmonic Analysis |

Citations: | 18 - 6 self |

### Citations

2596 |
lectures on wavelets
- Daubechies, Ten
- 1992
(Show Context)
Citation Context ...ons, which are very attractive from the applied perspective. In particular, frame decompositions are resilient against noise and quantization, and provide numerically stable reconstruction algorithms =-=[6, 8, 15]-=-. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively [4, 7, 18, 19]. Frame decompositions may also reveal hidden ... |

1367 |
Theory of Communication
- Gabor
- 1946
(Show Context)
Citation Context ...e tight frames. The theory of frames was first introduced by Duffin and Schaeffer [10] in the 1950’s, furthering the study of nonharmonic Fourier series and the time-frequency decompositions of Gabor =-=[13]-=-. Decades later, the subject was reinvigorated following a publication of Daubechies, Grossman and Meyer [9]. Frames have subsequently become a state-of-the-art signal processing tool. A Physical Inte... |

465 |
A class of nonharmonic Fourier series
- Duffin, Schaeffer
- 1952
(Show Context)
Citation Context ... theory and in applications. In recent years, several inquiries have been made into some of the deeper issues of finite tight frames. The theory of frames was first introduced by Duffin and Schaeffer =-=[10]-=- in the 1950’s, furthering the study of nonharmonic Fourier series and the time-frequency decompositions of Gabor [13]. Decades later, the subject was reinvigorated following a publication of Daubechi... |

337 |
Texture classification and segmentation using wavelet frames
- Unser
- 1995
(Show Context)
Citation Context ...oversampled filter banks, have been studied extensively [4, 7, 18, 19]. Frame decompositions may also reveal hidden signal characteristics, and have therefore been used to solve problems of detection =-=[1, 3, 21]-=-. Frames have also been used to design unitary space-time constellations for multiple-antenna wireless systems [16]. Specific types of finite tight frames have been studied to solve problems in commun... |

274 |
Painless non-orthogonal expansions
- Daubechies, Grossmann, et al.
- 1986
(Show Context)
Citation Context ...ing the study of nonharmonic Fourier series and the time-frequency decompositions of Gabor [13]. Decades later, the subject was reinvigorated following a publication of Daubechies, Grossman and Meyer =-=[9]-=-. Frames have subsequently become a state-of-the-art signal processing tool. A Physical Interpretation of Tight Frames 3 Frames provide redundant vector space decompositions, which are very attractive... |

226 | Jr., “Grassmannian frames with applications to coding and communication
- Strohmer, Heath
- 2003
(Show Context)
Citation Context ...have also been used to design unitary space-time constellations for multiple-antenna wireless systems [16]. Specific types of finite tight frames have been studied to solve problems in communications =-=[14, 17, 20, 24]-=-. In addition, many techniques of constructing finite tight frames have been discovered, several of which involve group theory [5, 22]. Researchers have also been interested in tight frames whose elem... |

201 | Systematic design of unitary space-time constellations
- Hochwald, Marzetta, et al.
- 2000
(Show Context)
Citation Context ...l characteristics, and have therefore been used to solve problems of detection [1, 3, 21]. Frames have also been used to design unitary space-time constellations for multiple-antenna wireless systems =-=[16]-=-. Specific types of finite tight frames have been studied to solve problems in communications [14, 17, 20, 24]. In addition, many techniques of constructing finite tight frames have been discovered, s... |

199 | Quantized frame expansions with erasures
- Goyal, Kovačević, et al.
- 2001
(Show Context)
Citation Context ...have also been used to design unitary space-time constellations for multiple-antenna wireless systems [16]. Specific types of finite tight frames have been studied to solve problems in communications =-=[14, 17, 20, 24]-=-. In addition, many techniques of constructing finite tight frames have been discovered, several of which involve group theory [5, 22]. Researchers have also been interested in tight frames whose elem... |

140 | Optimal sequences and sum capacity of synchronous CDMA systems
- Viswanath, Anantharam
- 1984
(Show Context)
Citation Context ...etto and Fickus’s characterization of unit-norm tight frames [2]. Subsequent to the completion of our work, we learned that some of our results were independently obtained by Anantharam and Viswanath =-=[23]-=- in the context of wireless communications. At the conclusion of this chapter, we compare and contrast the two approaches. In the following section, we motivate our main results by introducing a physi... |

127 | Oversampled filter banks
- Cvetković, Vetterli
- 1998
(Show Context)
Citation Context ...vide numerically stable reconstruction algorithms [6, 8, 15]. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively =-=[4, 7, 18, 19]-=-. Frame decompositions may also reveal hidden signal characteristics, and have therefore been used to solve problems of detection [1, 3, 21]. Frames have also been used to design unitary space-time co... |

119 |
Finite normalized tight frames
- Benedetto, Fickus
(Show Context)
Citation Context ... finite tight frames have been discovered, several of which involve group theory [5, 22]. Researchers have also been interested in tight frames whose elements are restricted to spheres and ellipsoids =-=[2, 11]-=-, as well as the manifold structures of spaces of all such frames [12]. Much of our work is inspired by Benedetto and Fickus’s characterization of unit-norm tight frames [2]. Subsequent to the complet... |

119 | Quantized Overcomplete Expansions in R: Analysis, Synthesis and Algorithms.
- Goyal, Vetterli, et al.
- 1998
(Show Context)
Citation Context ...ons, which are very attractive from the applied perspective. In particular, frame decompositions are resilient against noise and quantization, and provide numerically stable reconstruction algorithms =-=[6, 8, 15]-=-. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively [4, 7, 18, 19]. Frame decompositions may also reveal hidden ... |

114 |
Equal-norm tight frames with erasures.
- Casazza, Kovacevic
- 2003
(Show Context)
Citation Context ...es have been studied to solve problems in communications [14, 17, 20, 24]. In addition, many techniques of constructing finite tight frames have been discovered, several of which involve group theory =-=[5, 22]-=-. Researchers have also been interested in tight frames whose elements are restricted to spheres and ellipsoids [2, 11], as well as the manifold structures of spaces of all such frames [12]. Much of o... |

98 | Frame-theoretic analysis of oversampled filter banks,”
- Bölcskei, Hlawatsch, et al.
- 1998
(Show Context)
Citation Context ...vide numerically stable reconstruction algorithms [6, 8, 15]. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively =-=[4, 7, 18, 19]-=-. Frame decompositions may also reveal hidden signal characteristics, and have therefore been used to solve problems of detection [1, 3, 21]. Frames have also been used to design unitary space-time co... |

68 | Filter Bank Frame Expansions with Erasures.
- Kovacevic, Dragotti, et al.
- 2002
(Show Context)
Citation Context ...vide numerically stable reconstruction algorithms [6, 8, 15]. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively =-=[4, 7, 18, 19]-=-. Frame decompositions may also reveal hidden signal characteristics, and have therefore been used to solve problems of detection [1, 3, 21]. Frames have also been used to design unitary space-time co... |

67 | Optimal frames for erasures
- Holmes, Paulsen
- 2004
(Show Context)
Citation Context ...have also been used to design unitary space-time constellations for multiple-antenna wireless systems [16]. Specific types of finite tight frames have been studied to solve problems in communications =-=[14, 17, 20, 24]-=-. In addition, many techniques of constructing finite tight frames have been discovered, several of which involve group theory [5, 22]. Researchers have also been interested in tight frames whose elem... |

34 | Generalised Welch bound equality sequences are tight frames,
- Waldron
- 2003
(Show Context)
Citation Context |

33 |
Properties of redundant expansions under additive degradation and quantization
- Cvetković
- 2001
(Show Context)
Citation Context ...ons, which are very attractive from the applied perspective. In particular, frame decompositions are resilient against noise and quantization, and provide numerically stable reconstruction algorithms =-=[6, 8, 15]-=-. Translation-invariant frames in 2(Z), being equivalent to perfect reconstruction oversampled filter banks, have been studied extensively [4, 7, 18, 19]. Frame decompositions may also reveal hidden ... |

13 |
Wavelet periodicity detection algorithms
- Benedetto, Pfander
- 1998
(Show Context)
Citation Context ...nd “frame potential” may be extended to the nonuniform setting. Inspired by the Coulomb force, consider the frame force FF(fm, fm′) ∈ RN of fm′ ∈ RN upon fm ∈ RN , FF(fm, fm′) = 2〈fm, fm′〉(fm − fm′). =-=(3)-=- This is essentially the same definition that appears in [2], but without the restriction of fm and fm′ to the unit sphere. Compared to the forces of the natural world, the frame force is bizarre. For... |

6 |
Wavelet analysis of spectogram seizure chips
- Benedetto, Colella
- 1995
(Show Context)
Citation Context ...oversampled filter banks, have been studied extensively [4, 7, 18, 19]. Frame decompositions may also reveal hidden signal characteristics, and have therefore been used to solve problems of detection =-=[1, 3, 21]-=-. Frames have also been used to design unitary space-time constellations for multiple-antenna wireless systems [16]. Specific types of finite tight frames have been studied to solve problems in commun... |

5 |
Modern Sampling Theory: Mathematics and Applications
- Strohmer
- 2000
(Show Context)
Citation Context |

1 |
Tight frames and their symmetries. Const. Approx
- Vale, Waldron
(Show Context)
Citation Context ...es have been studied to solve problems in communications [14, 17, 20, 24]. In addition, many techniques of constructing finite tight frames have been discovered, several of which involve group theory =-=[5, 22]-=-. Researchers have also been interested in tight frames whose elements are restricted to spheres and ellipsoids [2, 11], as well as the manifold structures of spaces of all such frames [12]. Much of o... |