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## The Lifting Scheme: A Construction Of Second Generation Wavelets (1997)

Citations: | 537 - 15 self |

### Citations

3530 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...bechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat =-=[85, 84, 86]-=-, Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic operations in the Four... |

2200 | Orthonormal bases of compactly supported wavelets: II. variations on a theme
- Daubechies
- 1993
(Show Context)
Citation Context ...fer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau =-=[29]-=-, Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except fo... |

1035 |
An Introcution of Wavelets
- Chui
- 1992
(Show Context)
Citation Context ...eration wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang =-=[19, 25, 24, 23]-=-, Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77... |

676 |
Characterization of signals from multiscale edges
- MALLAT, ZHONG
- 1992
(Show Context)
Citation Context ...fined by those edges. Considerably higher compression ratios can be obtained this way. Wavelet probing thus provides an alternative for the zero crossing representation introduced by Mallat and Zhong =-=[82, 83]-=-. The advantage is that no iterative reconstruction is needed. Wavelet probing interacts easily with lifting and the adaptive wavelets mentioned above. 14.10. Wavelets adapted to irregular samples. As... |

623 |
Multiresolution approximations and wavelet orthonormal bases of L 2 (R
- Mallat
- 1989
(Show Context)
Citation Context ...bechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat =-=[85, 84, 86]-=-, Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic operations in the Four... |

588 |
The lifting scheme: a custom-design construction of biorthogonal wavelets
- Sweldens
(Show Context)
Citation Context ...nctions in a local neighborhood. We will 514 WIM SWELDENS show later how this can be seen as a special case of lifting. The lifting scheme can also be used to construct first generation wavelets; see =-=[105, 52]-=-. Although in this setting, the lifting will never come up with wavelets which could not have been found using the Cohen--Daubechies--Feauveau machinery in [29], it leads to two new insights: a custom... |

583 | Factoring Wavelet Transforms into Lifting Steps
- Daubechies, Sweldens
- 1998
(Show Context)
Citation Context ...nctions in a local neighborhood. We will 514 WIM SWELDENS show later how this can be seen as a special case of lifting. The lifting scheme can also be used to construct first generation wavelets; see =-=[105, 52]-=-. Although in this setting, the lifting will never come up with wavelets which could not have been found using the Cohen--Daubechies--Feauveau machinery in [29], it leads to two new insights: a custom... |

386 | Multiresolution analysis for surfaces of arbitrary topological type
- Lounsbery, TD, et al.
- 1997
(Show Context)
Citation Context ...erpolation and subdivision as construction tools rather than the Fourier transform. It thus can be generalized to interval constructions [59] or weighed wavelets [104]. Lounsbery, De Rose, and Warren =-=[79, 80]-=- construct wavelets for the approximation of polyhedral surfaces of arbitrary genus. The wavelets are constructed by orthogonalizing scaling functions in a local neighborhood. We will 514 WIM SWELDENS... |

357 | Wavelet transforms that map integers to integers
- Calderbank, Daubechies, et al.
- 1998
(Show Context)
Citation Context ...d closer to one on both the finer (j > n) and the coarser (jsn) levels. Future research also involves the study of these schemes in higher dimensions. 14.11. Integer to integer wavelet transforms. In =-=[16]-=- lifting is used to build reversible wavelets which map integers to integers for applications to lossless image coding. The idea is to introduce a nonlinear round-o# in each lifting step. This way the... |

344 | Multifrequency channel decompositions of images and wavelet models
- Mallat
- 1989
(Show Context)
Citation Context ...bechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat =-=[85, 84, 86]-=-, Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic operations in the Four... |

337 |
Wavelets and filter banks: Theory and design.
- Vetterli, Herley
- 1992
(Show Context)
Citation Context ..., 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli =-=[73, 110]-=-, Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translati... |

215 |
A discrete transform and decompositions of distribution spaces,”
- Frazier, Jawerth
- 1990
(Show Context)
Citation Context ... 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth =-=[65, 67, 66]-=-, Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction too... |

214 |
Symmetric iterative interpolation processes
- Deslauriers, Dubic
- 1989
(Show Context)
Citation Context ...ned, dual lifting provides interpolating scaling functions, and lifting yields wavelets with vanishing moments. The dual lifting can be seen as an instance of irregular Deslauriers--Dubuc subdivision =-=[54, 55]-=-. Current research involves the study of more advanced choices for the Lazy wavelet. One of the strategies is to choose the sample locations so that the ratio of the largest versus the smallest interv... |

189 | Using the refinement equations for the construction of pre-wavelets II: Powers of two
- Jia, Micchelli
- 1991
(Show Context)
Citation Context ...s to severe axial directional dependencies. Instead one prefers to work with nonseparable wavelets which have more axial symmetry and which do not necessarily use a product lattice; see, for example, =-=[22, 28, 32, 76, 77, 89, 90, 92, 93]-=-. Here too lifting can help. Each lattice allows for the immediate definition of a Lazy wavelet or a Haar wavelet, either 2-band or M-band. Polynomial cancelation then leads to the filter coe#cients. ... |

176 | Unconditional bases are optimal bases for data compression and statistical estimation.
- Donoho
- 1993
(Show Context)
Citation Context ...is leads to the fast wavelet transform, which allows us to pass between the function f and its wavelet coe#cients # j,m in linear time. These properties result in the fact that, quoted from Donoho in =-=[58], "wa-=-velets are optimal bases for compressing, estimating, and recovering functions in F ." Roughly speaking, for a general class of functions, the essential information contained in a function is cap... |

165 | P.: Wavelet radiosity
- GORTLER, SCHRÖDER, et al.
- 1993
(Show Context)
Citation Context ...spherical images. Current research involves the generalization of the construction and the applications to arbitrary surfaces. 14.4. Adaptive wavelets. The idea of adaptive wavelets was introduced in =-=[69, 97, 98]-=- in the context of the numerical solution of integral equations for illumination computations. The idea is the following. Assume the solution can be approximated with su#cient accuracy in a linear spa... |

157 |
Decomposition of Besov spaces
- Frazier, Jawerth
- 1985
(Show Context)
Citation Context ... 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth =-=[65, 67, 66]-=-, Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction too... |

152 | Interpolating Wavelet Transforms.
- Donoho
- 1992
(Show Context)
Citation Context ...der) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho =-=[57, 56]-=-, Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier tra... |

150 | Building your own wavelets at home
- Sweldens, Schröder
- 1996
(Show Context)
Citation Context ...nterval. Lifting then only requires pulling in the right aunts (scaling functions on the coarser level) at the boundary of the interval. All calculations can be done in place. For details we refer to =-=[106]-=-. A software package, LIFTPACK, to calculate the wavelet transformation of images is currently available [64]. Its properties are in-place calculation, correct treatment of boundaries, arbitrary size ... |

143 |
Non-Separable Bidimensional Wavelet Bases
- Cohen, Daubechies
- 1991
(Show Context)
Citation Context ...tructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies =-=[28]-=-, Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86]... |

136 | On compactly supported spline wavelets and a duality principle”.
- Chui, Wang
- 1992
(Show Context)
Citation Context ...eration wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang =-=[19, 25, 24, 23]-=-, Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77... |

133 | editor, Wavelets: A Tutorial in Theory and Applications - Chui - 1992 |

131 |
Entropy based algorithms for best basis selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ...undant set. For a given function, one can choose the best basis with respect to a criterion such as the entropy of the basis coe#cients. A fast tree algorithm to find the best basis was introduced in =-=[36]-=-; see also [115]. This idea again carries over into the second generation setting and can be combined with lifting. The conditions for exact reconstruction have exactly the same algebraic structure as... |

113 |
Multiresolution analysis, wavelets and fast algorithms on an interval,
- Cohen, Daubechies, et al.
- 1992
(Show Context)
Citation Context ...ruction tool. A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval =-=[8, 10, 18, 30, 31, 88]-=-, wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other ins... |

109 | Multilevel preconditioning
- Dahmen, Kunoth
(Show Context)
Citation Context ...ase remains a topic for further study. 14.9. Wavelets on bounded domains and wavelet probing. One of the important application domains of wavelets is the solution of partial di#erential equations. In =-=[44]-=- it is shown that one can use wavelets to build multilevel preconditioners which result in sti#ness matrices with uniformly bounded condition numbers. This leads to linear solution algorithms. To solv... |

107 |
Waveletlike bases for the fast solution of second-kind integral equations,”
- Alpert, Beylkin, et al.
- 1993
(Show Context)
Citation Context ...f what they call tree wavelets. Tree wavelets have the property that each wavelet of level j is supported within the support of only one scaling function of level j. Haar wavelets and Alpert wavelets =-=[4, 5, 6]-=- have this property. The advantage is that subdividing the support of a scaling function on level j, and thus constructing the wavelets of level j associated with it, does not imply subdividing any ot... |

99 | Simple regularity criteria for subdivision schemes,”
- Rioul
- 1992
(Show Context)
Citation Context ...metrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil [37, 38], Daubechies and Lagarias [50, 51], Eirola [63], Rioul =-=[94]-=-, and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1: While first generation wavelets provided bases for functio... |

97 |
Construction of orthogonal wavelets using fractal interpolation functions,”
- Donovan, Geronimo, et al.
- 1996
(Show Context)
Citation Context ...ose connections are pointed out in [105, 52]. Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions =-=[60, 61, 62, 70]-=-. They also introduced the concept of several generating functions (multiwavelets). As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second ... |

96 | Stability of multiscale transformations.
- Dahmen
- 1996
(Show Context)
Citation Context ...eration wavelets have been reported in the literature, e.g., the construction of scaling functions through subdivision [41], basis constructions [43], as well as the development of stability criteria =-=[41, 42]-=-. In this paper, we present the lifting scheme, a simple, general construction of second generation wavelets. The basic idea, which inspired the name, is to start with a very simple or trivial multire... |

93 |
Sobolev characterization of solutions of dilation equations,
- Eirola
- 1992
(Show Context)
Citation Context .... In the geometrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil [37, 38], Daubechies and Lagarias [50, 51], Eirola =-=[63]-=-, Rioul [94], and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1: While first generation wavelets provided bases... |

90 |
Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung gros̈er vollbesetzter Gleichungssysteme
- Schneider
- 1998
(Show Context)
Citation Context ... cascade out. Lifting thus opens the door to smooth adaptive wavelets. Current research involves the incorporation of these wavelets in illumination computations. A word of caution is needed here. In =-=[46, 96]-=- it is shown that adaptive wavelet algorithms require wavelets on manifolds satisfying specific conditions concerning stability, regularity, and norm equivalence. As pointed out earlier, lifting does ... |

81 |
On the asymptotic convergence of B-spline wavelets to Gabor functions.
- Unser, Aldroubi, et al.
- 1992
(Show Context)
Citation Context ...l and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser =-=[2, 3, 108, 107]-=-, Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 6... |

73 |
A family of polynomial spline wavelet transforms," Signal Process
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...l and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser =-=[2, 3, 108, 107]-=-, Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 6... |

71 | Wavelets on a closed subsets of the real line, in:
- Andersson, Hall, et al.
- 1994
(Show Context)
Citation Context ...ruction tool. A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval =-=[8, 10, 18, 30, 31, 88]-=-, wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other ins... |

71 |
Wavelets and Recursive Filter Banks,”
- Herley, Vetterli
- 1993
(Show Context)
Citation Context ..., 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli =-=[73, 110]-=-, Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translati... |

60 | Smooth wavelet decompositions with blocky coefficient kernels.
- Donoho
- 1993
(Show Context)
Citation Context ...der) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho =-=[57, 56]-=-, Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier tra... |

58 |
Wavelets and pre-wavelets in low dimensions
- Riemenschneider, Shen
- 1992
(Show Context)
Citation Context ...s to severe axial directional dependencies. Instead one prefers to work with nonseparable wavelets which have more axial symmetry and which do not necessarily use a product lattice; see, for example, =-=[22, 28, 32, 76, 77, 89, 90, 92, 93]-=-. Here too lifting can help. Each lattice allows for the immediate definition of a Lazy wavelet or a Haar wavelet, either 2-band or M-band. Polynomial cancelation then leads to the filter coe#cients. ... |

56 |
A block spin construction of ondelettes
- Battle
- 1987
(Show Context)
Citation Context ...eduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie =-=[13, 78]-=-, Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, ... |

56 |
A general framework for compactly supported splines and wavelets
- Chui, Wang
- 1992
(Show Context)
Citation Context ...eration wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie [13, 78], Chui and Wang =-=[19, 25, 24, 23]-=-, Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77... |

52 |
Wavelet Transform Maxima and Multiscale Edges
- Mallat, Zhong
- 1990
(Show Context)
Citation Context ...fined by those edges. Considerably higher compression ratios can be obtained this way. Wavelet probing thus provides an alternative for the zero crossing representation introduced by Mallat and Zhong =-=[82, 83]-=-. The advantage is that no iterative reconstruction is needed. Wavelet probing interacts easily with lifting and the adaptive wavelets mentioned above. 14.10. Wavelets adapted to irregular samples. As... |

51 |
Families of multiresolution and wavelet spaces with optimal properties.
- Aldroubi, Unser
- 1993
(Show Context)
Citation Context ...l and integral equations, and noise reduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser =-=[2, 3, 108, 107]-=-, Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 6... |

51 |
Multiresolution representation of data: a general framework
- Harten
- 1996
(Show Context)
Citation Context ...hor learned of two other very similar techniques developed independent of each other and of lifting. Harten and Abgrall developed a general multiresolution approximation framework based on prediction =-=[71, 1]-=-, while Dahmen and co-workers [17, 46] developed a mechanism to characterize all stable biorthogonal decomposition. We will come back to this toward the end of the paper. 3. Multiresolution analysis. ... |

48 |
Ondelettes, analyses multirésolutions et filtres miroirs en quadrature
- Cohen
- 1990
(Show Context)
Citation Context ...uld like to have a condition which relates convergence of the cascade algorithm and the Riesz basis property back to the filter coe#cients, similar to the Cohen criterion in the first generation case =-=[26]-=- or the Cohen--Daubechies--Feauveau theorem [29] or [48, Theorem 8.3.1]. This result is part of the analysis phase of the construction. As we mentioned earlier, this paper is mostly concerned with the... |

47 |
Energy moments in time and frequency for two-scale difference equation solutions and wavelets
- Villemoes
- 1992
(Show Context)
Citation Context ...checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil [37, 38], Daubechies and Lagarias [50, 51], Eirola [63], Rioul [94], and Villemoes =-=[113, 112]-=-. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1: While first generation wavelets provided bases for functions defined on R n , applic... |

46 | Wavelets on a bounded interval. In: Numerical Methods of Approximation Theory - CHUI, QUAK - 1992 |

46 |
A cardinal spline approach to wavelets
- Chui, Wang
- 1991
(Show Context)
Citation Context |

46 |
Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices,”
- Vaidyanathan, Nguyen, et al.
- 1989
(Show Context)
Citation Context ...ical literature, is to split a space V j+1 into M (as opposed to 2) subspaces: V j # W 1 j # �� �� �� # W M-1 j . For each subspace a di#erent filter is used. Several constructions were in=-=troduced in [72, 81, 102, 109]-=-. In some sense the second generation wavelet setting already incorporates this. Indeed, it even allows for di#erent filters for each individual wavelet. However, thinking of lifting combined with M-b... |

44 |
Compactly supported bidimensional wavelets bases with hexagonal symmetry
- Cohen, Schlenker
- 1993
(Show Context)
Citation Context ...s to severe axial directional dependencies. Instead one prefers to work with nonseparable wavelets which have more axial symmetry and which do not necessarily use a product lattice; see, for example, =-=[22, 28, 32, 76, 77, 89, 90, 92, 93]-=-. Here too lifting can help. Each lattice allows for the immediate definition of a Lazy wavelet or a Haar wavelet, either 2-band or M-band. Polynomial cancelation then leads to the filter coe#cients. ... |

43 |
Size Properties of Wavelet Packets
- Coifman, Meyer, et al.
- 1992
(Show Context)
Citation Context ..., the stability issue does not go away but rather manifests itself as a problem concerning ill-conditioning. 14.6. Wavelet packets. Wavelet packets were introduced by Coifman, Meyer, and Wickerhauser =-=[34, 35, 114, 115]-=-. The idea is to also further split the W j spaces with the help of the h and g filters. This way one obtains a better frequency localization. The splitting leads to a full binary tree of wavelet pack... |

40 | Multiscale decompositions on bounded domains, preprint
- Cohen, Dahmen, et al.
- 1995
(Show Context)
Citation Context ...results concerning the construction of wavelets adapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains =-=[27, 74]-=-, spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other instances of second generation wavelets h... |

39 |
A class of bases in l 2 for the sparse representation of integral operators
- Alpert
- 1993
(Show Context)
Citation Context ...f what they call tree wavelets. Tree wavelets have the property that each wavelet of level j is supported within the support of only one scaling function of level j. Haar wavelets and Alpert wavelets =-=[4, 5, 6]-=- have this property. The advantage is that subdividing the support of a scaling function on level j, and thus constructing the wavelets of level j associated with it, does not imply subdividing any ot... |

39 |
Signal processing and compression with wave packets
- Coifman, Meyer, et al.
- 1992
(Show Context)
Citation Context ..., the stability issue does not go away but rather manifests itself as a problem concerning ill-conditioning. 14.6. Wavelet packets. Wavelet packets were introduced by Coifman, Meyer, and Wickerhauser =-=[34, 35, 114, 115]-=-. The idea is to also further split the W j spaces with the help of the h and g filters. This way one obtains a better frequency localization. The splitting leads to a full binary tree of wavelet pack... |

38 | Wavelets and Other Bases for Fast Numerical Linear Algebra
- Alpert
- 1992
(Show Context)
Citation Context ...f what they call tree wavelets. Tree wavelets have the property that each wavelet of level j is supported within the support of only one scaling function of level j. Haar wavelets and Alpert wavelets =-=[4, 5, 6]-=- have this property. The advantage is that subdividing the support of a scaling function on level j, and thus constructing the wavelets of level j associated with it, does not imply subdividing any ot... |

36 | Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets
- Donovan, Geronimo, et al.
- 1996
(Show Context)
Citation Context ...ose connections are pointed out in [105, 52]. Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions =-=[60, 61, 62, 70]-=-. They also introduced the concept of several generating functions (multiwavelets). As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second ... |

35 | Some remarks on multiscale transformations, stability and biorthogonality
- DAHMEN
- 1994
(Show Context)
Citation Context ...tructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the construction of scaling functions through subdivision =-=[41]-=-, basis constructions [43], as well as the development of stability criteria [41, 42]. In this paper, we present the lifting scheme, a simple, general construction of second generation wavelets. The b... |

33 |
Interpolation dyadique. In: Fractals, Dimensions non entières et applications
- Deslauriers, Dubuc
- 1987
(Show Context)
Citation Context ...ned, dual lifting provides interpolating scaling functions, and lifting yields wavelets with vanishing moments. The dual lifting can be seen as an instance of irregular Deslauriers--Dubuc subdivision =-=[54, 55]-=-. Current research involves the study of more advanced choices for the Lazy wavelet. One of the strategies is to choose the sample locations so that the ratio of the largest versus the smallest interv... |

33 |
Rank M wavelets with N vanishing moments,
- Heller
- 1995
(Show Context)
Citation Context ...ical literature, is to split a space V j+1 into M (as opposed to 2) subspaces: V j # W 1 j # �� �� �� # W M-1 j . For each subspace a di#erent filter is used. Several constructions were in=-=troduced in [72, 81, 102, 109]-=-. In some sense the second generation wavelet setting already incorporates this. Indeed, it even allows for di#erent filters for each individual wavelet. However, thinking of lifting combined with M-b... |

32 |
Ondelettes et Operateurs; I: Ondelettes, II: Operateurs de Calder6n-Zygmund, III: (with R. Coifman) Operateurs multilineaires
- Meyer
- 1990
(Show Context)
Citation Context ... Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer =-=[87]-=-, and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic operations in the Fourier domain. ... |

31 |
Multiresolution surfaces of arbitrary topological type
- Lounsbery, DeRose, et al.
- 1997
(Show Context)
Citation Context ...erpolation and subdivision as construction tools rather than the Fourier transform. It thus can be generalized to interval constructions [59] or weighed wavelets [104]. Lounsbery, De Rose, and Warren =-=[79, 80]-=- construct wavelets for the approximation of polyhedral surfaces of arbitrary genus. The wavelets are constructed by orthogonalizing scaling functions in a local neighborhood. We will 514 WIM SWELDENS... |

28 | A new class of unbalanced Haar wavelets that form an unconditional basis for Lp on general measure spaces,
- Girardi, Sweldens
- 1997
(Show Context)
Citation Context ... scheme. The idea was first introduced by Coifman, Jones, and Semmes for dyadic cubes in [33], generalized for Cli#ord-valued measures in [9, 91], and later generalized for arbitrary partitionings in =-=[68]-=-. We first introduce the notion of a nested set of partitionings. Definition 11.1. A set of measurable subsets {X j,k | j, k} is a nested set of partitionings if it is a set of partitionings and if, f... |

27 |
The construction and application of wavelets in numerical analysis
- Sweldens
- 1994
(Show Context)
Citation Context ...G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets =-=[11, 12, 104]-=-. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the construction of scaling functions through ... |

26 |
Discrete spline filters for multiresolution and wavelets of l2,”
- Aldroubi, Eden, et al.
- 1994
(Show Context)
Citation Context |

26 | On Minimum Entropy Segmentation’
- DONOHO
- 1994
(Show Context)
Citation Context ...ion of first generation wavelets which relies on polynomial interpolation and subdivision as construction tools rather than the Fourier transform. It thus can be generalized to interval constructions =-=[59]-=- or weighed wavelets [104]. Lounsbery, De Rose, and Warren [79, 80] construct wavelets for the approximation of polyhedral surfaces of arbitrary genus. The wavelets are constructed by orthogonalizing ... |

25 |
Spline prewavelets for nonuniform knots, Numerische Mathematik 61
- Buhmann, Micchelli
- 1992
(Show Context)
Citation Context ...dapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, =-=[15, 7, 45]-=-, and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the const... |

23 |
J.Lagarias, Two-scale dierence equations. II. Local regularity, in products of matrices and fractals
- Daubechies
- 1992
(Show Context)
Citation Context ...ace (property P1). In the geometrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil [37, 38], Daubechies and Lagarias =-=[50, 51]-=-, Eirola [63], Rioul [94], and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1: While first generation wavelets p... |

22 |
Banded matrices with banded inverses. II. Locally finite decomposition of spline spaces
- Dahmen, Micchelli
- 1993
(Show Context)
Citation Context ...dapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, =-=[15, 7, 45]-=-, and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the const... |

21 | W.: ‘LIFTPACK: a software package for wavelet transforms using lifting’.
- Fernandez, Periaswamy, et al.
- 1996
(Show Context)
Citation Context ... boundary of the interval. All calculations can be done in place. For details we refer to [106]. A software package, LIFTPACK, to calculate the wavelet transformation of images is currently available =-=[64]-=-. Its properties are in-place calculation, correct treatment of boundaries, arbitrary size images (not only powers of two), and a faster implementation of existing biorthogonal wavelet filters (speedu... |

19 | A multiscale method for the double layer potential equation on a polyhedron
- Dahmen, Kleemann, et al.
- 1994
(Show Context)
Citation Context ...ard one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the construction of scaling functions through subdivision [41], basis constructions =-=[43]-=-, as well as the development of stability criteria [41, 42]. In this paper, we present the lifting scheme, a simple, general construction of second generation wavelets. The basic idea, which inspired ... |

18 |
Multiresolution representation in unstructured meshes
- Abgrall, Harten
- 1998
(Show Context)
Citation Context ...hor learned of two other very similar techniques developed independent of each other and of lifting. Harten and Abgrall developed a general multiresolution approximation framework based on prediction =-=[71, 1]-=-, while Dahmen and co-workers [17, 46] developed a mechanism to characterize all stable biorthogonal decomposition. We will come back to this toward the end of the paper. 3. Multiresolution analysis. ... |

18 |
Characterization of scaling functions: continuous solutions
- Collela, Heil
- 1994
(Show Context)
Citation Context ...a basis for the proper function space (property P1). In the geometrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil =-=[37, 38]-=-, Daubechies and Lagarias [50, 51], Eirola [63], Rioul [94], and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1:... |

18 |
Box splines, cardinal series, and wavelets
- Riemenschneider, Shen
- 1991
(Show Context)
Citation Context |

16 | Wavelet multiresolution analyses adapted for the fast solution of boundary value ordinary differential equations
- Jawerth, Sweldens
- 1993
(Show Context)
Citation Context ...ed wavelets will again be of the order of the number of vanishing moments. An example of this behavior is given in [106]. Weighted wavelets are also useful in the solution of boundary value ODEs; see =-=[75, 104]-=-. If the operator is of the form -DaD, then operator wavelets defined as the antiderivative of weighted wavelets with weight function w(x) = p a(x) diagonalize the operator. The solution algorithm is ... |

15 |
Two-scale dierence equations. I. Existence and global regularity of solutions
- Daubechies, Lagarias
- 1991
(Show Context)
Citation Context ...ace (property P1). In the geometrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil [37, 38], Daubechies and Lagarias =-=[50, 51]-=-, Eirola [63], Rioul [94], and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1: While first generation wavelets p... |

14 |
Compactly supported boxspline wavelets
- Chui, Stöckler, et al.
- 1992
(Show Context)
Citation Context |

13 |
Wavelets with boundary conditions on the interval. WaveletsA tutorial in Theory and Applications,
- Auscher
- 1992
(Show Context)
Citation Context ...ruction tool. A proper substitute is needed. Several results concerning the construction of wavelets adapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval =-=[8, 10, 18, 30, 31, 88]-=-, wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other ins... |

13 |
Two elementary proofs of the L 2 boundedness of Cauchy integrals on Lipschitz curves
- Coifman, Jones, et al.
- 1989
(Show Context)
Citation Context ...rthogonal Haar wavelets, which form a first example of an initial multiresolution analysis to start the lifting scheme. The idea was first introduced by Coifman, Jones, and Semmes for dyadic cubes in =-=[33]-=-, generalized for Cli#ord-valued measures in [9, 91], and later generalized for arbitrary partitionings in [68]. We first introduce the notion of a nested set of partitionings. Definition 11.1. A set ... |

13 |
Decomposition of refinable spaces and applications to operator equations
- Dahmen
- 1993
(Show Context)
Citation Context ...velets that generate complementary spaces in a multiresolution analysis of univariate irregular knot splines. Dahmen already made use of a technique related to lifting in the first generation setting =-=[40]-=- and later introduced a multiscale framework related to second generation wavelets [41]. Finally, after finishing this work, the author learned of two other very similar techniques developed independe... |

13 |
Multiscale methods for pseudodierential equations on smooth manifolds
- Dahmen, Prodorf, et al.
- 1994
(Show Context)
Citation Context ... techniques developed independent of each other and of lifting. Harten and Abgrall developed a general multiresolution approximation framework based on prediction [71, 1], while Dahmen and co-workers =-=[17, 46]-=- developed a mechanism to characterize all stable biorthogonal decomposition. We will come back to this toward the end of the paper. 3. Multiresolution analysis. In this section we present the second ... |

13 |
Theory of regular M-band wavelets
- Steffen, Heller, et al.
- 1993
(Show Context)
Citation Context ...ical literature, is to split a space V j+1 into M (as opposed to 2) subspaces: V j # W 1 j # �� �� �� # W M-1 j . For each subspace a di#erent filter is used. Several constructions were in=-=troduced in [72, 81, 102, 109]-=-. In some sense the second generation wavelet setting already incorporates this. Indeed, it even allows for di#erent filters for each individual wavelet. However, thinking of lifting combined with M-b... |

11 |
Construction of compact p-wavelets
- Welland, Lundberg
- 1993
(Show Context)
Citation Context |

10 |
Multiresolution analyses based on fractal functions
- Hardin, Kessler, et al.
- 1992
(Show Context)
Citation Context ...ose connections are pointed out in [105, 52]. Over the last few years Donovan, Hardin, Geronimo, and Massopust have developed techniques to construct wavelets based on fractal interpolation functions =-=[60, 61, 62, 70]-=-. They also introduced the concept of several generating functions (multiwavelets). As this technique does not rely on the Fourier transform either, it too potentially can be used to construct second ... |

9 |
Ondelettes sur l’intervalle, Rev
- Meyer
- 1992
(Show Context)
Citation Context |

7 | Wavelet Probing for Compression Based Segmentation
- Deng, Jawerth, et al.
- 1993
(Show Context)
Citation Context ...se adaptive grids to obtain the correct convergence order. There is another important application of wavelets on domains. It is a technique called wavelet probing introduced independently in [59] and =-=[8, 53]-=-. Let us discuss the idea first on the real line. Consider a function which is smooth except for jump discontinuities at isolated points. We know that the decay of the wavelet coe#cients is fast away ... |

6 | A Class of Orthogonal Multiresolution Analysis in 2D
- Donovan, Geronimo, et al.
- 1995
(Show Context)
Citation Context |

6 |
The -transform and applications to distribution spaces
- Frazier, Jawerth
- 1986
(Show Context)
Citation Context ... 108, 107], Battle and Lemarie [13, 78], Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth =-=[65, 67, 66]-=-, Herley and Vetterli [73, 110], Kovacevic and Vetterli [77, 111], Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction too... |

5 |
Bases d’ondelettes sur les courbes corde-arc, noyau de Cauchy et espaces de Hardy associés
- Tchamitchian
- 1989
(Show Context)
Citation Context ...G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets =-=[11, 12, 104]-=-. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the construction of scaling functions through ... |

4 |
Ondelettes localisation exponentielles
- Lemari
- 1988
(Show Context)
Citation Context ...eduction. Many first generation wavelet families have been constructed over the last ten years. We refer to the work of (in alphabetical order) Aldroubi and Unser [2, 3, 108, 107], Battle and Lemarie =-=[13, 78]-=-, Chui and Wang [19, 25, 24, 23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, ... |

4 | W.: Spherical wavelets: Texture processing - oder, Sweldens - 1995 |

3 |
Ondelettes et conjecture de
- Tchamitchian
- 1991
(Show Context)
Citation Context ...G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, [15, 7, 45], and weighted wavelets =-=[11, 12, 104]-=-. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the construction of scaling functions through ... |

3 |
The characterization of continuous, four-coe#cient scaling functions and wavelets
- Colella, Heil
- 1992
(Show Context)
Citation Context ...a basis for the proper function space (property P1). In the geometrical stage, one checks the smoothness of the basis functions (property P3). In this context, we mention the work of Collela and Heil =-=[37, 38]-=-, Daubechies and Lagarias [50, 51], Eirola [63], Rioul [94], and Villemoes [113, 112]. Let us next consider applications which illustrate the need for generalizations of first generation wavelets. G1:... |

3 |
Spherical wavelets: E#ciently representing functions on the sphere
- oder, Sweldens
- 1995
(Show Context)
Citation Context ...floating point calculations will be bounded. As we mentioned before, lifting does not guarantee stability and bounded condition numbers. However, in a practical situation involving spherical wavelets =-=[99]-=- we numerically estimated the condition number and found it to vary little with the number of levels. For a spherical wavelet transform involving roughly 650, 000 coe#cients we found the condition num... |

3 |
Str omberg, A modified Franklin system and higher order spline systems on R n as unconditional bases for Hardy spaces
- O
- 1981
(Show Context)
Citation Context ...arly 1980s, several years before the above developments, Stromberg discovered the first orthogonal wavelets with a technique based on spline interpolation which does not rely on the Fourier transform =-=[103]-=-. The construction as initiated by Daubechies and coworkers essentially consists of three stages. The algebraic stage involves constructing the filters that are used in the fast wavelet transform; mor... |

2 |
The Cauchy singular integral operator and Cli#ord wavelets, in [14
- Andersson, Jawerth, et al.
(Show Context)
Citation Context ...le of an initial multiresolution analysis to start the lifting scheme. The idea was first introduced by Coifman, Jones, and Semmes for dyadic cubes in [33], generalized for Cli#ord-valued measures in =-=[9, 91]-=-, and later generalized for arbitrary partitionings in [68]. We first introduce the notion of a nested set of partitionings. Definition 11.1. A set of measurable subsets {X j,k | j, k} is a nested set... |

2 |
Pe na, Local decompositions of refinable spaces
- Carnicer, Dahmen, et al.
- 1996
(Show Context)
Citation Context ... techniques developed independent of each other and of lifting. Harten and Abgrall developed a general multiresolution approximation framework based on prediction [71, 1], while Dahmen and co-workers =-=[17, 46]-=- developed a mechanism to characterize all stable biorthogonal decomposition. We will come back to this toward the end of the paper. 3. Multiresolution analysis. In this section we present the second ... |

2 |
cevi c and M. Vetterli, Nonseparable multidimensional perfect reconstruction filterbanks
- Kova
- 1992
(Show Context)
Citation Context ...23], Cohen and Daubechies [28], Cohen, Daubechies, and Feauveau [29], Daubechies [47, 49, 48], Donoho [57, 56], Frazier and Jawerth [65, 67, 66], Herley and Vetterli [73, 110], Kovacevic and Vetterli =-=[77, 111]-=-, Mallat [85, 84, 86], Meyer [87], and many more. Except for Donoho, they all rely on the Fourier transform as a basic construction tool. The reason is that translation and dilation become algebraic o... |

2 |
P.: Wavelet Algorithms for Illumination Computations
- ODER
- 1994
(Show Context)
Citation Context ...spherical images. Current research involves the generalization of the construction and the applications to arbitrary surfaces. 14.4. Adaptive wavelets. The idea of adaptive wavelets was introduced in =-=[69, 97, 98]-=- in the context of the numerical solution of integral equations for illumination computations. The idea is the following. Assume the solution can be approximated with su#cient accuracy in a linear spa... |

2 |
Wavelet projections for radiosity, Computer Graphics Forum
- oder, Gortler, et al.
- 1994
(Show Context)
Citation Context ...spherical images. Current research involves the generalization of the construction and the applications to arbitrary surfaces. 14.4. Adaptive wavelets. The idea of adaptive wavelets was introduced in =-=[69, 97, 98]-=- in the context of the numerical solution of integral equations for illumination computations. The idea is the following. Assume the solution can be approximated with su#cient accuracy in a linear spa... |

1 |
Lemari e-Rieusset, Ondelettes splines sur grilles irregulieres, manuscript
- Amellaoui, G
(Show Context)
Citation Context ...dapted to some of the cases in G1--G3 already exist. For example, we have wavelets on an interval [8, 10, 18, 30, 31, 88], wavelets on bounded domains [27, 74], spline wavelets for irregular samples, =-=[15, 7, 45]-=-, and weighted wavelets [11, 12, 104]. These constructions are tailored toward one specific setting. Other instances of second generation wavelets have been reported in the literature, e.g., the const... |

1 |
Elliptic Boundary Value Problems and Besov Spaces
- Dahlke, DeVore
- 1996
(Show Context)
Citation Context ...imple case of the Laplace equation and a nonsmooth domain it was recently shown that one cannot obtain an O(M -2 ) accuracy (where M is the number of elements) unless one uses nonlinear approximation =-=[39]-=-. The underlying reason is that the solution does not belong to the second-order Sobolev space but rather to a second-order Besov space. In other words, one has to use adaptive grids to obtain the cor... |

1 |
Singular integrals, Hardy spaces and Cli#ord wavelets
- Mitrea
- 1994
(Show Context)
Citation Context ...le of an initial multiresolution analysis to start the lifting scheme. The idea was first introduced by Coifman, Jones, and Semmes for dyadic cubes in [33], generalized for Cli#ord-valued measures in =-=[9, 91]-=-, and later generalized for arbitrary partitionings in [68]. We first introduce the notion of a nested set of partitionings. Definition 11.1. A set of measurable subsets {X j,k | j, k} is a nested set... |