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## Factoring wavelet transforms into lifting steps (1998)

Venue: | J. FOURIER ANAL. APPL |

Citations: | 572 - 8 self |

### Citations

2469 | Ten Lectures on Wavelets - DAUBECHIES - 1992 |

2163 | Orthonormal bases of compactly supported wavelets
- DAUBECHIES
- 1988
(Show Context)
Citation Context ... construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets [1, 8, 10, 11, 49], fully biorthogonal compactly supported wavelets =-=[12, 56]-=-, and recursive filter banks [25]. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building blocks are known. One of these is lifting. The original moti... |

1013 |
An introduction to wavelets
- Chui
- 1992
(Show Context)
Citation Context ...ogonal (prewavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets =-=[1, 8, 10, 11, 49]-=-, fully biorthogonal compactly supported wavelets [12, 56], and recursive filter banks [25]. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building bl... |

615 |
Multiresolution approximation and wavelet orthonormal bases of L2
- Mallat
- 1989
(Show Context)
Citation Context ... [13, 18, 22, 34, 21]. In the mid eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets =-=[30, 31, 34]-=-; this led to new constructions, such as the smooth orthogonal, and compactly supported wavelets [16]. Later many generalizations to the biorthogonal or semiorthogonal (prewavelet) case were introduce... |

607 | Wavelets and Subband Coding
- Vetterli, Kovacevic
- 1995
(Show Context)
Citation Context ...ing community. The roots of critically sampled wavelet transforms are actually older than the word “wavelet” and go back to the context of subband filters, or more precisely quadrature mirror filters =-=[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]-=-. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions [13, 18, 22, 34, 21]. In the mid eightie... |

585 |
The lifting scheme: A custom-design construction of biorthogonal wavelets
- Sweldens
- 1996
(Show Context)
Citation Context ... any matrix This family was derived independently, but without the use of lifting, by several people: Reissell [38], Tian and Wells [47], and Strang [43]. The derivation using lifting can be found in =-=[44]-=-. 4swith polynomial entries and determinant one can be factored into such elementary matrices. For those familiar with the common notation in this field, this is written as . The proof relies on the 2... |

532 | The lifting scheme: A construction of second generation wavelets,” Jounnul on Murheimrical Analysis
- Sweldens
- 1998
(Show Context)
Citation Context ...varying update coefficients can be computed [46]. This thus immediately allows for a (2,2) type transform for irregular samples. These spatial lifting steps can also be used in higher dimensions (see =-=[45]-=-) and leads e.g., to wavelets on a sphere [40] or more complex manifolds. Note that the idea of using spatial wavelet constructions for building second generation wavelets has been proposed by several... |

410 |
Digital Coding of Waveforms
- Jayant, Noll
- 1984
(Show Context)
Citation Context ...nt is zero. The operation of computing a prediction and recording the detail we will call a lifting step. The idea of retaining rather than is well known and forms the basis of so-called DPCM methods =-=[26, 27]-=-. This idea connects naturally with wavelets as follows. The prediction steps can take care of some of the spatial correlation, but for wavelets we also want to get some separation in the frequency do... |

347 | Wavelet transforms that map integers to integers
- Calderbank, Daubechies, et al.
- 1998
(Show Context)
Citation Context ... from the factorization, only three extra steps are needed to avoid scaling. This is particularly important when building integer to integer wavelet transforms in which case scaling is not invertible =-=[6]-=-. 7.4. Interpolating filters. In case the low-pass filter is half band, or , the corresponding scaling function is interpolating. Since , the factorization can be done in two steps: The filters constr... |

335 | Multifrequency channel decompositions of images and wavelet models
- Mallat
- 1989
(Show Context)
Citation Context ... [13, 18, 22, 34, 21]. In the mid eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets =-=[30, 31, 34]-=-; this led to new constructions, such as the smooth orthogonal, and compactly supported wavelets [16]. Later many generalizations to the biorthogonal or semiorthogonal (prewavelet) case were introduce... |

327 |
Wavelets and filter banks: theory and design
- Vetterli, Herley
- 1992
(Show Context)
Citation Context ... construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets [1, 8, 10, 11, 49], fully biorthogonal compactly supported wavelets =-=[12, 56]-=-, and recursive filter banks [25]. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building blocks are known. One of these is lifting. The original moti... |

282 | Spherical wavelets: Efficiently representing functions on the sphere
- Schröder, Sweldens
(Show Context)
Citation Context ...6]. This thus immediately allows for a (2,2) type transform for irregular samples. These spatial lifting steps can also be used in higher dimensions (see [45]) and leads e.g., to wavelets on a sphere =-=[40]-=- or more complex manifolds. Note that the idea of using spatial wavelet constructions for building second generation wavelets has been proposed by several researchers: The lifting scheme is inspired b... |

275 |
Painless non-orthogonal expansions
- Daubechies, Grossmann, et al.
- 1986
(Show Context)
Citation Context ...[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions =-=[13, 18, 22, 34, 21]-=-. In the mid eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets [30, 31, 34]; this le... |

255 |
Decomposition of Hardy functions into square integrable wavelets of constant shape
- GROSSMANN, MORLET
- 1984
(Show Context)
Citation Context ...[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions =-=[13, 18, 22, 34, 21]-=-. In the mid eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets [30, 31, 34]; this le... |

195 |
Subband Coding of Images
- Woods
- 1991
(Show Context)
Citation Context ...ing community. The roots of critically sampled wavelet transforms are actually older than the word “wavelet” and go back to the context of subband filters, or more precisely quadrature mirror filters =-=[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]-=-. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions [13, 18, 22, 34, 21]. In the mid eightie... |

155 |
and Björn Jawerth, Decomposition of Besov spaces
- Frazier
- 1985
(Show Context)
Citation Context ...[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions =-=[13, 18, 22, 34, 21]-=-. In the mid eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and Meyer provided the connection between subband filters and wavelets [30, 31, 34]; this le... |

150 | Building your own wavelets at home
- Sweldens, Schröder
- 1996
(Show Context)
Citation Context ...as above we could then see that a good predictor is of the form where the varies spatially and depends on the irregularity of the grid. Similarly spatially varying update coefficients can be computed =-=[46]-=-. This thus immediately allows for a (2,2) type transform for irregular samples. These spatial lifting steps can also be used in higher dimensions (see [45]) and leads e.g., to wavelets on a sphere [4... |

149 | Interpolating Wavelet Transforms
- Donoho
- 1992
(Show Context)
Citation Context ...folds. Note that the idea of using spatial wavelet constructions for building second generation wavelets has been proposed by several researchers: The lifting scheme is inspired by the work of Donoho =-=[19]-=- and Lounsbery et al. [29]. Donoho [19] shows how to build wavelets from interpolating scaling functions, while Lounsbery et al. build a multiresolution analysis of surfaces using a technique that is ... |

142 |
Exact reconstruction technique for tree structured subband coders
- Smith, Barnwell
(Show Context)
Citation Context ...ing community. The roots of critically sampled wavelet transforms are actually older than the word “wavelet” and go back to the context of subband filters, or more precisely quadrature mirror filters =-=[35, 36, 42, 50, 51, 52, 53, 57, 55, 59]-=-. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions [13, 18, 22, 34, 21]. In the mid eightie... |

128 | Fast algorithms for discrete and continuous wavelet transforms
- Rioul, Duhamel
- 1992
(Show Context)
Citation Context ...gorithm is not necessarily the best way to implement the wavelet transform. Lifting is only one idea in a whole tool bag of methods to improve the speed of a fast wavelet transform. Rioul and Duhamel =-=[39]-=- discuss several other schemes to improve the standard algorithm. In the case of long filters, they suggest an FFT based scheme known as the Vetterli-algorithm [56]. In the case of short filters, they... |

127 |
Fast Algorithms for Digital Signal Processing
- Blahut
- 1984
(Show Context)
Citation Context ...M The Euclidean algorithm was originally developed to find the greatest common divisor of two natural numbers, but it can be extended to find the greatest common divisor of two polynomials, see, e.g, =-=[4]-=-. Here we need it to find common factors of Laurent polynomials. The main difference with the polynomial case is again that the solution is not unique. Indeed the gcd of two Laurent polynomials is def... |

95 |
Filter banks allowing perfect reconstruction
- Vetterli
(Show Context)
Citation Context |

82 |
Lattice structures for optimal design and robust implementation of 2-channel PR-QMF banks
- Vaidyanathan, Hoang
- 1988
(Show Context)
Citation Context |

74 |
A family of polynomial spline wavelet transforms
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...ogonal (prewavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets =-=[1, 8, 10, 11, 49]-=-, fully biorthogonal compactly supported wavelets [12, 56], and recursive filter banks [25]. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building bl... |

70 |
Wavelets and recursive filter banks
- Herley, Vetterli
- 1993
(Show Context)
Citation Context ...nd thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets [1, 8, 10, 11, 49], fully biorthogonal compactly supported wavelets [12, 56], and recursive filter banks =-=[25]-=-. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building blocks are known. One of these is lifting. The original motivation for developing lifting was... |

70 |
Redundancy rate distortion analysis of multiple description coding using pairwise correlating transforms
- Orchard, Wang, et al.
- 1997
(Show Context)
Citation Context ... quantization cells which are optimal like in the orthogonal case. In a multiple description setting, it has been shown that this generalization to biorthogonality allows for substantial improvements =-=[58]-=-. 6. Lifting allows for adaptive wavelet transforms. This means one can start the analysis of a function from the coarsest levels and then build the finer levels by refining only in the areas of inter... |

68 |
Perfect reconstruction FIR filter banks: some properties and factorizations
- Vetterli, LeGall
- 1989
(Show Context)
Citation Context |

67 |
Multiscale methods for pseudo-differential equations on smooth manifolds
- Dahmen, Prößdorf, et al.
- 1994
(Show Context)
Citation Context ...that the wavelet coefficients and are zero.) Dahmen and collaborators, independently of lifting, worked on stable completions of multiscale transforms, a setting similar to second generation wavelets =-=[7, 15]-=-. Again independently, both of Dahmen and of lifting, Harten developed a general multiresolution approximation framework based on spatial prediction [23]. In [14], Dahmen and Micchelli propose a const... |

56 |
A general framework for compactly supported splines and wavelets
- Chui, Wang
- 1992
(Show Context)
Citation Context ...ogonal (prewavelet) case were introduced. Biorthogonality allows the construction of symmetric wavelets and thus linear phase filters. Examples are: the construction of semiorthogonal spline wavelets =-=[1, 8, 10, 11, 49]-=-, fully biorthogonal compactly supported wavelets [12, 56], and recursive filter banks [25]. Various techniques to construct wavelet bases, or to factor existing wavelet filters into basic building bl... |

51 |
Two-channel perfect reconstruction FIR QMF structures which yield linear phase analysis and synthesis
- Nguyen, Vaidyanathan
- 1989
(Show Context)
Citation Context |

49 |
Families of multiresolution and wavelet spaces with optimal properties
- Aldroubi, Unser
- 1993
(Show Context)
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49 |
Multiresolution representation of data: A general framework
- Harten
- 1996
(Show Context)
Citation Context ...tting similar to second generation wavelets [7, 15]. Again independently, both of Dahmen and of lifting, Harten developed a general multiresolution approximation framework based on spatial prediction =-=[23]-=-. In [14], Dahmen and Micchelli propose a construction of compactly supported wavelets that generates complementary spaces in a multiresolution analysis of univariate irregular knot splines. The const... |

47 |
A cardinal spline approach to wavelets
- Chui, Wang
- 1991
(Show Context)
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47 |
Theory and design of M-channel maximally decimated quadrature mirror with arbitrary M having perfect reconstruction property
- Vaidyanathan
- 1987
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46 |
den Enden. New networks for perfect inversion and perfect reconstruction
- Bruekers, van
- 1992
(Show Context)
Citation Context ...n as . The proof relies on the 2000 year old Euclidean algorithm. In the filter bank literature subband transform built using elementary matrices are known as ladder structures and were introduced in =-=[5]-=-. Later several constructions concerning factoring into ladder steps were given [28, 41, 48, 32, 33]. Vetterli and Herley [56] also use the Euclidean algorithm and the connection to diophantine equati... |

45 |
Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices
- Vaidyanathan
- 1989
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41 |
Filters for distortion-free two-band multirate filter banks
- Mintzer
- 1985
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36 |
Local decomposition of refinable spaces
- Carnicer, Dahmen, et al.
- 1996
(Show Context)
Citation Context ...that the wavelet coefficients and are zero.) Dahmen and collaborators, independently of lifting, worked on stable completions of multiscale transforms, a setting similar to second generation wavelets =-=[7, 15]-=-. Again independently, both of Dahmen and of lifting, Harten developed a general multiresolution approximation framework based on spatial prediction [23]. In [14], Dahmen and Micchelli propose a const... |

32 |
Ondelettes et Operateurs; I: Ondelettes, II: Operateurs de Calder6n-Zygmund, III: (with R. Coifman) Operateurs multilineaires
- Meyer
- 1990
(Show Context)
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31 |
Multiresolution surfaces of arbitrary topological type
- Lounsbery, DeRose, et al.
- 1997
(Show Context)
Citation Context ...of using spatial wavelet constructions for building second generation wavelets has been proposed by several researchers: The lifting scheme is inspired by the work of Donoho [19] and Lounsbery et al. =-=[29]-=-. Donoho [19] shows how to build wavelets from interpolating scaling functions, while Lounsbery et al. build a multiresolution analysis of surfaces using a technique that is algebraically the same as ... |

30 |
A Computational Theory of Laurent Polynomial Rings and Multidimensional FIR Systems. Coadv. with Tsit-Yuen
- Park
- 1995
(Show Context)
Citation Context ...ls and applications. 7. Finally, under certain conditions it is possible to construct ladder like structures in higher dimensions using factoring of multivariate polynomials. For details, we refer to =-=[37]-=-. Acknowledgments. The authors would like to thank Peter Schröder and Boon-Lock Yeo for many stimulating discussions and for their help in computing the factorizations in the example section, Jelena K... |

24 |
TDM-FDM transmultiplexer: digital polyphase and FFT
- Bellanger, Daguet
- 1974
(Show Context)
Citation Context ...nverse transform: first apply the polyphase matrix and then join even and odd. The polyphase representation is a particularly convenient tool to express the special structure of the modulation matrix =-=[3]-=-. The polyphase representation of a filter is given by where contains the even coefficients, and contains the odd coefficients: or We assemble the polyphase matrix as so that and and We define similar... |

22 |
Banded matrices with banded inverses. II. Locally finite decomposition of spline spaces
- Dahmen, Micchelli
- 1993
(Show Context)
Citation Context ...ilar to second generation wavelets [7, 15]. Again independently, both of Dahmen and of lifting, Harten developed a general multiresolution approximation framework based on spatial prediction [23]. In =-=[14]-=-, Dahmen and Micchelli propose a construction of compactly supported wavelets that generates complementary spaces in a multiresolution analysis of univariate irregular knot splines. The construction o... |

20 | Wavelet multiresolution representation of curves and surfaces
- Reissell
- 1996
(Show Context)
Citation Context ... with all diagonal entries equal to one. It is a well known result in matrix algebra that any matrix This family was derived independently, but without the use of lifting, by several people: Reissell =-=[38]-=-, Tian and Wells [47], and Strang [43]. The derivation using lifting can be found in [44]. 4swith polynomial entries and determinant one can be factored into such elementary matrices. For those famili... |

17 | On ladder structures and linear phase conditions for bi-orthogonal filter banks
- Shah, Kalker
- 1994
(Show Context)
Citation Context ...k literature subband transform built using elementary matrices are known as ladder structures and were introduced in [5]. Later several constructions concerning factoring into ladder steps were given =-=[28, 41, 48, 32, 33]-=-. Vetterli and Herley [56] also use the Euclidean algorithm and the connection to diophantine equations to find all high pass filters that, together with a given low-pass filter, make a finite filter ... |

14 |
Running FIR and IIR filtering using multirate filter banks
- Vetterli
- 1988
(Show Context)
Citation Context ...e the standard algorithm. In the case of long filters, they suggest an FFT based scheme known as the Vetterli-algorithm [56]. In the case of short filters, they suggest a “fast running FIR” algorithm =-=[54]-=-. How these ideas combine with the idea of using lifting and which combination will be optimal for a certain wavelet goes beyond the scope of this paper and remains a topic of future research. 21s9. C... |

13 | Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets
- Kalker, Shah
- 1992
(Show Context)
Citation Context ...k literature subband transform built using elementary matrices are known as ladder structures and were introduced in [5]. Later several constructions concerning factoring into ladder steps were given =-=[28, 41, 48, 32, 33]-=-. Vetterli and Herley [56] also use the Euclidean algorithm and the connection to diophantine equations to find all high pass filters that, together with a given low-pass filter, make a finite filter ... |

11 | On the realizability of bi-orthogonal M-dimensional 2-band filter banks
- Tolhuizen, Hollimann, et al.
- 1995
(Show Context)
Citation Context ...k literature subband transform built using elementary matrices are known as ladder structures and were introduced in [5]. Later several constructions concerning factoring into ladder steps were given =-=[28, 41, 48, 32, 33]-=-. Vetterli and Herley [56] also use the Euclidean algorithm and the connection to diophantine equations to find all high pass filters that, together with a given low-pass filter, make a finite filter ... |

10 |
U-L block-triangular matrix and ladder realizations of subband coders
- Marshall
- 1993
(Show Context)
Citation Context |

6 | Vanishing moments and biorthogonal wavelet systems
- Tian, Wells
- 1996
(Show Context)
Citation Context ...tries equal to one. It is a well known result in matrix algebra that any matrix This family was derived independently, but without the use of lifting, by several people: Reissell [38], Tian and Wells =-=[47]-=-, and Strang [43]. The derivation using lifting can be found in [44]. 4swith polynomial entries and determinant one can be factored into such elementary matrices. For those familiar with the common no... |

5 |
editors. Wavelets: Time-Frequency Methods and Phase Space. Inverse problems and theoretical imaging
- Tchamitchian
- 1989
(Show Context)
Citation Context |

5 | Wavelet video coding with ladder structures and entropy-constrained quantization
- Dyck, Marshall, et al.
- 1996
(Show Context)
Citation Context ...tine equations to find all high pass filters that, together with a given low-pass filter, make a finite filter wavelet transform. Van Dyck et al. use ladder structures to design a wavelet video coder =-=[20]-=-. In this paper we give a self-contained constructive proof of the standard factorization result and apply it to several popular wavelets. We consider the Laurent polynomial setting as opposed to the ... |

4 |
A Fast Wavelet Transform Based Upon the Euclidean Algorithm
- Marshall
- 1993
(Show Context)
Citation Context |