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## The Discrete Multiple Wavelet Transform and Thresholding Methods (1998)

Venue: | IEEE Transactions in Signal Processing |

Citations: | 37 - 3 self |

### Citations

3534 | A theory of multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ... Under these conditions it can be shown that the filter coefficients, h k and g k , also satisfy the following condition, g k = (\Gamma1) k h 1\Gammak : (2) The discrete wavelet transform DWT (Mallat =-=[10]-=-) takes observations from a function f at 2 J equally spaced points, and recursively computes the wavelet coefficients using the following algorithm. Assign c 0;k = f(x k ); k = 0; \Delta \Delta \Delt... |

2970 |
Robust statistics
- Huber
- 2011
(Show Context)
Citation Context ...ture of the wavelet coefficients at each level, in terms of oe 2 . We obtained better results when V j is estimated directly from the observed coefficients using a robust covariance estimation method =-=[7]-=- than by estimating oe 2 and then recursively computing V j . 5 Simulation A series of simulations were carried out to assess the efficacy of bivariate thresholding using the Geronimo multiple wavelet... |

2600 |
Lectures on Wavelets
- Daubechies, Ten
- 2012
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Citation Context ...coefficients and keep the large `signal' coefficients. A brief introduction to wavelet methods is given in Nason & Silverman [10]. For a more comprehensive and theoretical account, consult Daubechies =-=[2]-=-. Multiple wavelets [6] have recently been formulated, using translates and dilates of more than one mother wavelet function. This paper describes a method of thresholding such multiple wavelets, by c... |

2205 | Biorthogonal bases of compactly supported wavelets,
- Cohen, Daubechies, et al.
- 1992
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Citation Context ...ssion is possible when the wavelet function is symmetric, has short support, and has a large number of vanishing moments (the latter determines the smoothness of the thresholded estimate). Daubechies =-=[2]-=- shows that the support, symmetry and number of vanishing moments are linked and that for orthogonal wavelets, an improvement in say vanishing moments would be offset by longer support or by worse sym... |

1277 | De-Noising by Soft-Thresholding
- Donoho
- 1995
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Citation Context ...not as critical as the frequency response and degree. We suggest that interpolation prefilter should be used if it is not clear which prefilter to use. 4 Thresholding Multiple Wavelets References [4] =-=[3]-=- [5] [9] and [1] discuss several different thresholding techniques for single wavelet decompositions. These aim to reduce the noise in an observed signal. Strela et al [13] have applied a thresholding... |

728 |
Ideal spatial adaptation via wavelet shrinkage
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ... is not as critical as the frequency response and degree. We suggest that interpolation prefilter should be used if it is not clear which prefilter to use. 4 Thresholding Multiple Wavelets References =-=[4]-=- [3] [5] [9] and [1] discuss several different thresholding techniques for single wavelet decompositions. These aim to reduce the noise in an observed signal. Strela et al [13] have applied a threshol... |

295 | Wavelet shrinkage: asymptopia? - Donoho, Johnstone, et al. - 1995 |

239 | Wavelet threshold estimators for data with correlated noise,”
- Johnstone, Silverman
- 1997
(Show Context)
Citation Context ...ntains the covariances between each coefficient element) will be block symmetric and circulant (assuming periodic boundary conditions). By a simple extension of the results in Johnstone and Silverman =-=[9]-=-, a vectorwise thresholding routine will be asymptotically optimal. Using a repeated signal of length one, each C 0;k will be i.i.d. with Var(C 0;k ) = oe 2 flfl ? This is a singular L \Theta L matrix... |

189 |
Fractal functions and wavelet expansions based on several scaling functions
- Geronimo, Hardin, et al.
- 1994
(Show Context)
Citation Context ...he large `signal' coefficients. A brief introduction to wavelet methods is given in Nason & Silverman [10]. For a more comprehensive and theoretical account, consult Daubechies [2]. Multiple wavelets =-=[6]-=- have recently been formulated, using translates and dilates of more than one mother wavelet function. This paper describes a method of thresholding such multiple wavelets, by considering the magnitud... |

132 | An overview of wavelet based multiresolution analyses.
- Jawerth, Sweldens
- 1994
(Show Context)
Citation Context ...efficients in the wavelet expansion of f(x) (Equation 1) can be obtained from the coefficients C 0;k by applying the DMWT. When L = 1 the starting coefficients are C 0;k = R f(x)OE(x \Gamma k)dx; see =-=[8]-=-, [2]. An approximation to the inner product could be obtained using c 0;k = P n f n OE(n \Gamma k). This corresponds to a prefilter with the coefficients Q n = OE(n). In practice, this prefilter is r... |

96 | Wavelet shrinkage using cross-validation.
- NASON, P
- 1996
(Show Context)
Citation Context ...ritical as the frequency response and degree. We suggest that interpolation prefilter should be used if it is not clear which prefilter to use. 4 Thresholding Multiple Wavelets References [4] [3] [5] =-=[9]-=- and [1] discuss several different thresholding techniques for single wavelet decompositions. These aim to reduce the noise in an observed signal. Strela et al [13] have applied a thresholding method ... |

94 | The discrete wavelet transform in
- Nason, Silverman
- 1994
(Show Context)
Citation Context ...ontains noise then a thresholding method will throw away the small `noise' coefficients and keep the large `signal' coefficients. A brief introduction to wavelet methods is given in Nason & Silverman =-=[10]-=-. For a more comprehensive and theoretical account, consult Daubechies [2]. Multiple wavelets [6] have recently been formulated, using translates and dilates of more than one mother wavelet function. ... |

84 | Short wavelets and matrix dilation equation,
- Strang, Strela
- 1995
(Show Context)
Citation Context ...e wavelet bases was that it is possible to have wavelet functions which are orthogonal and symmetric, have short support and have a higher degree of accuracy than a single wavelet of the same support =-=[12]-=-. This suggests that multiple wavelets will give better noise reduction and signal compression ratios than single wavelets, as demonstrated by Strela et al [13] and the results in Section 5. The follo... |

83 | The Application of Multiwavelet Filterbanks to Image Processing,
- Strela, Heller, et al.
- 1999
(Show Context)
Citation Context ...an a single wavelet of the same support [12]. This suggests that multiple wavelets will give better noise reduction and signal compression ratios than single wavelets, as demonstrated by Strela et al =-=[13]-=- and the results in Section 5. The following methods apply to any general orthogonal multiple wavelet basis but examples will use the double wavelet basis derived by Geronimo, et al [6] using the high... |

69 | Adaptive thresholding of wavelet coefficients
- Abramovich, Benjamini
- 1996
(Show Context)
Citation Context ...tematic high frequency component after reconstruction (Figure 1). Using a suitable prefilter overcomes this problem [7] [12] [13]. II -- Thresholding Multiple Wavelets Wavelet thresholding techniques =-=[1]-=- [4] [3] [5] [10] reduce the noise in an observed signal. Strela et al [12] have applied the single wavelet (univariate) thresholding method [4] to a GHM multiwavelet decomposition with encouraging re... |

67 | Design of Prefilters for Discrete Multiwavelet Transforms,
- Xia, Geronimo, et al.
- 1996
(Show Context)
Citation Context ... : ; c j;k;L ) ? and D j;k = (d j;k;1 ; : : : ; d j;k;L ) ? . Given the vector starting coefficients C 0;k , the wavelet coefficients can be found using the discrete multiple wavelet transform (DMWT) =-=[13]-=-. To obtain these starting coefficients, a method of mapping a sequence f k of univariate data to bivariate vectors has to be adopted. A matrix prefilter does this by partitioning the data into a sequ... |

53 | Orthogonal multiwavelets with vanishing moments,
- Strang, Strela
- 1994
(Show Context)
Citation Context ... : : ; /L . In Section 3.3 we showed that a piecewise constant function did not give zero wavelet coefficients after applying the DMWT, even though the Geronimo multiwavelets have 2 vanishing moments =-=[11]-=-. We introduce the degree of a prefilter Q (or repeated signal \Gamma), as the lowest degree of polynomial which gives non-zero wavelet coefficients, after preprocessing by Q (or \Gamma) and transform... |

10 | multiwavelet transforms - Xia, Zhang - 1996 |

9 |
Wavelet methods in statistics
- Downie
- 1997
(Show Context)
Citation Context ... 0 and 1, where M n is the maximum of n i.i.d. �� 2 L random variables. An appropriate sequence, which we term the multivariate universal threshold, is given bys2 n = 2 log n + (L \Gamma 2) log lo=-=g n [6]-=-. As n increases, the expected number of coefficients exceeding the threshold without containing signal tends to a small finite limit (1 in the case L = 2). In this sense, nearly all the coefficients ... |

1 |
Adaptive methods of wavelet coefficients as multiple hypothesis testing proceedure
- Abramovich, Benjamini
- 1994
(Show Context)
Citation Context ...as the frequency response and degree. We suggest that interpolation prefilter should be used if it is not clear which prefilter to use. 4 Thresholding Multiple Wavelets References [4] [3] [5] [9] and =-=[1]-=- discuss several different thresholding techniques for single wavelet decompositions. These aim to reduce the noise in an observed signal. Strela et al [13] have applied a thresholding method by Donoh... |

1 | Signal preprocessing for multiwavelets
- Downie
- 1998
(Show Context)
Citation Context ... have nonzero elements. Thresholding the small nonzero coefficients introduces a systematic high frequency component after reconstruction (Figure 1). Using a suitable prefilter overcomes this problem =-=[7]-=- [12] [13]. II -- Thresholding Multiple Wavelets Wavelet thresholding techniques [1] [4] [3] [5] [10] reduce the noise in an observed signal. Strela et al [12] have applied the single wavelet (univari... |