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## Complex wavelets for shift invariant analysis and filtering of signals (2001)

Venue: | J. Applied and Computational Harmonic Analysis |

Citations: | 384 - 40 self |

### Citations

3533 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...mplex wavelet transform as a useful tool for overcoming some of these problems. The discrete wavelet transform (DWT) is most commonly used in its maximally decimated form (Mallat’s dyadic filter tree =-=[1, 2]-=-). This works well for compression but its use for other signal analysis and reconstruction tasks has been hampered by two main disadvantages: • Lack of shift invariance, which means that small shifts... |

3202 |
A Wavelet Tour of Signal Processing
- Mallat
- 1999
(Show Context)
Citation Context ...mplex wavelet transform as a useful tool for overcoming some of these problems. The discrete wavelet transform (DWT) is most commonly used in its maximally decimated form (Mallat’s dyadic filter tree =-=[1, 2]-=-). This works well for compression but its use for other signal analysis and reconstruction tasks has been hampered by two main disadvantages: • Lack of shift invariance, which means that small shifts... |

1388 | The laplacian pyramid as a compact image code
- Burt, Adelson
- 1983
(Show Context)
Citation Context ...lectivity was proposed by Simoncelli et al. [4] and was based on Laplacian pyramids and steerable filters, designed in the frequency domain. Laplacian pyramids were first proposed by Burt and Adelson =-=[3]-=- and, in that form, also produced limited redundancy of 2 : 1 for 1-D and 1.33 : 1 for 2-D signals. In addition, they could provide moderate levels of shift invariance if the lowpass filters of the py... |

1277 | De-noising by soft-thresholding
- Donoho
- 1995
(Show Context)
Citation Context ...hieve subpixel motion compensation, although spline methods may be equally effective in this case. Denoising and deconvolution. It is well known that wavelet denoising methods such as those of Donoho =-=[9]-=- perform better when implemented with a shift-invariant transform such as the undecimated DWT (UWT). We have shown [11] that the DT CWT performs at least as well as the UWT in this context, and with s... |

560 | Shiftable multi-scale transforms
- Simoncelli, Freeman, et al.
- 1992
(Show Context)
Citation Context ... m-D); • Efficient order-N computation—only twice the simple DWT for 1-D (2m times for m-D). Another approach both to shift invariance and to directional selectivity was proposed by Simoncelli et al. =-=[4]-=- and was based on Laplacian pyramids and steerable filters, designed in the frequency domain. Laplacian pyramids were first proposed by Burt and Adelson [3] and, in that form, also produced limited re... |

189 | The Dual-Tree Complex Wavelet Transform: A New Efficient Tool Computer-Aided Palaeography 335 for Image Restoration and Enhancement.”
- Kingsbury
- 1998
(Show Context)
Citation Context ...s from substantially increased computation requirements compared to the fully decimated DWT and also exhibits high redundancy in the output information, making subsequent processing expensive too. In =-=[10, 11]-=-, we introduced a more computationally efficient approach to shift invariance, the dual-tree complex wavelet transform (DT CWT). Furthermore the DT CWT also gives much better directional selectivity w... |

174 | A filter bank for the directional decomposition of images: Theory and design,”
- Bamberger, Smith
- 1992
(Show Context)
Citation Context ...s steerable filters was directional selectivity achieved in 2-D, and then the redundancy increased to 5.33 : 1. An alternative approach to directional selectivity was pioneered by Bamberger and Smith =-=[5]-=-, using a combination of 2-D resampling methods and separable polyphase filter bank techniques. This produced fan-shaped directional decompositions of the input signal with maximal decimation (no redu... |

84 |
Lattice Structures for Optimal Design and Robust Implementation of Two-Channel PerfectReconstruction
- Vaidyanathan, Hoang
- 1988
(Show Context)
Citation Context ...rsion of the DT CWT lies in finding a good even-length lowpass filter with a delay of 14 sample which also satisfies the standard orthonormal perfect reconstruction condition of two-band filter banks =-=[6]-=-. To achieve a lowpass filter HL(z) of length 2n with a delay that approximates 14 sample, we find that the simplest approach is to design a linear-phase lowpass FIR filter HL2(z) of length 4n with ha... |

65 | Motion estimation using a complex-valued wavelet transform,
- Magarey, Kingsbury
- 1998
(Show Context)
Citation Context ...ng the shift invariant properties of the DT CWT and here we expand on these basic ideas and develop them somewhat further. 2. THE DUAL FILTER TREE Our work with complex wavelets for motion estimation =-=[7]-=- showed that complex wavelets could provide approximate shift invariance. Unfortunately we were unable to obtain PR and good frequency characteristics using short support complex FIR filters in a sing... |

52 |
Flexible design of multidimensional perfect reconstruction FIR 2-band filters using transformations of variables
- Tay, Kingsbury
- 1993
(Show Context)
Citation Context ...gonal filter sets which meet the above conditions quite well and are also nearly orthogonal. For the odd-length set, we designed (13,19)-tap filters using the (1-D) transformation of variables method =-=[8]-=-. Then for the even-length set, we designed a (12,16)-tap even-length set to minimize the error in Eq. (6). The analysis filter coefficients are listed in Table 1 (the reconstruction filters are obtai... |

52 | Hidden Markov tree modeling of complex wavelet transforms,” in
- Romberg, Choi, et al.
- 2001
(Show Context)
Citation Context ...y long filters (13 to 19 taps), but there is then a computation penalty. The first problem is more fundamental and has implications for subsequent hierarchical algorithms, such as hidden Markov trees =-=[16, 18]-=-. To overcome all of the above (and with no significant penalties), we now propose a Q-shift dual tree, as in Fig. 3, in which all the filters beyond level 1 are even length, 238 NICK KINGSBURY FIG. 3... |

41 | Shift Invariant Properties of the Dual-Tree Complex Wavelet Transform.
- Kingsbury
- 1999
(Show Context)
Citation Context ...to offer a useful combination of properties not available from these earlier approaches—principally perfect reconstruction, greater directional selectivity, and a natural multiscale decomposition. In =-=[12]-=- we proposed a way of analyzing the shift invariant properties of the DT CWT and here we expand on these basic ideas and develop them somewhat further. 2. THE DUAL FILTER TREE Our work with complex wa... |

40 |
Image processing with complex wavelets”, philos.
- Kingsbury
- 1999
(Show Context)
Citation Context ...illustrated at level 4 of the transforms. The complex wavelets provide 6 directionally selective filters, while real wavelets provide 3 filters, only two of which have a dominant direction. detail in =-=[13]-=-. It is important to note that these redundancy values are not affected either by the number of levels of the transform or by whether row and column processing is done at each level before proceeding ... |

30 |
Complex wavelet features for fast texture image retrieval
- Rivaz, Kingsbury
- 1999
(Show Context)
Citation Context ...DT CWT at levels 1 and 2 to give improved high frequency selectivity. Texture analysis and synthesis. The multi-scale and directional properties of the DT CWT make it well suited for texture analysis =-=[14, 15]-=-. In this context it behaves similarly to a multi-scale Gabor filter bank, but is more efficiently implemented. Shift invariance is important too, as it makes the texture feature vectors independent o... |

18 | Satellite image deconvolution using complex wavelet packets
- Jalobeanu, Blanc-Féraud, et al.
- 2000
(Show Context)
Citation Context ...parent-child structure), whereas this is much more difficult for the UWT in which each child has multiple parents. Finally it is straightforward to combine deconvolution with denoising, and Jalobeanu =-=[19]-=- has shown one way of doing this, which involves the use of a packet DT CWT at levels 1 and 2 to give improved high frequency selectivity. Texture analysis and synthesis. The multi-scale and direction... |

17 |
Digital watermarking using complex wavelets,” in
- Loo, Kingsburry
- 2000
(Show Context)
Citation Context ...ocal spectral characteristics of the image that is being marked, in order to make the mark as invisible as possible (due to perceptual masking) and as robust as possible against denoising attacks. In =-=[22]-=-, we propose a system which achieves this by watermarking in the complex wavelet domain, since it is very easy, using the DT CWT, to provide the spatially varying spectral characteristics that are req... |

16 | Multiscale classification using complex wavelets and hidden Markov tree models
- Romberg, Choi, et al.
(Show Context)
Citation Context ...y long filters (13 to 19 taps), but there is then a computation penalty. The first problem is more fundamental and has implications for subsequent hierarchical algorithms, such as hidden Markov trees =-=[16, 18]-=-. To overcome all of the above (and with no significant penalties), we now propose a Q-shift dual tree, as in Fig. 3, in which all the filters beyond level 1 are even length, 238 NICK KINGSBURY FIG. 3... |

15 |
Content based image retrieval through object extraction and querying
- Kam, Ng, et al.
- 2000
(Show Context)
Citation Context ... CWT to give combined texture and colour feature vectors at multiple scales allows the development of effective hierarchical segmentation algorithms, which can then be applied to image classification =-=[17, 21]-=-. In [18] Romberg et al. have shown how the HMT may be used effectively with the DT CWT in this context, as well as for denoising. Watermarking. Effective image watermarks must be approximately matche... |

14 |
Fast segmentation using level set curves of complex wavelet surfaces
- Rivaz, Kingsbury
- 1998
(Show Context)
Citation Context ... CWT to give combined texture and colour feature vectors at multiple scales allows the development of effective hierarchical segmentation algorithms, which can then be applied to image classification =-=[17, 21]-=-. In [18] Romberg et al. have shown how the HMT may be used effectively with the DT CWT in this context, as well as for denoising. Watermarking. Effective image watermarks must be approximately matche... |

11 |
Rotationally invariant texture features using the dual-tree complex wavelet transform
- Hill, Bull, et al.
(Show Context)
Citation Context ...y to a multi-scale Gabor filter bank, but is more efficiently implemented. Shift invariance is important too, as it makes the texture feature vectors independent of precise texture location, and Hill =-=[20]-=- has shown how to make the texture features rotationally invariant. Finally the perfect reconstruction features of the DT CWT allow iterative texture synthesis techniques to be used [10]. Segmentation... |

10 |
Texture classification using dualtree complex wavelet transform
- Hatipoglu, Mitra, et al.
- 1999
(Show Context)
Citation Context ...DT CWT at levels 1 and 2 to give improved high frequency selectivity. Texture analysis and synthesis. The multi-scale and directional properties of the DT CWT make it well suited for texture analysis =-=[14, 15]-=-. In this context it behaves similarly to a multi-scale Gabor filter bank, but is more efficiently implemented. Shift invariance is important too, as it makes the texture feature vectors independent o... |