## An Overview of Wavelet Based Multiresolution Analyses (1993)

Citations: | 117 - 2 self |

### BibTeX

@MISC{Jawerth93anoverview,

author = {Björn Jawerth and Wim Sweldens},

title = {An Overview of Wavelet Based Multiresolution Analyses},

year = {1993}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

2385 | A theory for multiresolution signal decomposition: The wavelet representation
- Mallat
- 1989
(Show Context)
Citation Context ...onal wavelet expansions. With the notion of multiresolution analysis, introduced by St'ephane Mallat and Yves Meyer, a systematic framework for understanding these orthogonal expansions was developed =-=[85, 86, 87]-=-. It also provided the connection with quadrature mirror filtering. Soon Ingrid Daubechies [37] gave a construction of wavelets, non-zero only on a finite interval and with arbitrarily high, but fixed... |

1587 | Biothogonal bases of compactly supported wavelets
- Cohen, Daubechies, et al.
- 1992
(Show Context)
Citation Context ... 1 + e \Gammai! 2 'N K(!); with K(0) = 1 and K(��) 6= 0. This factorization together with the (bi)orthogonality conditions is used as a starting point for construction of compactly supported wavel=-=ets [24, 37]. We-=- also have that i psOE (p) (2k��) = ffi k M p for 0sp ! N; (29) and, by the Poisson summation formula, that X l (x \Gamma l) p OE(x \Gamma l) = M p for 0sp ! N: By rearranging the last expression ... |

696 |
An introduction to wavelets
- Chui
- 1992
(Show Context)
Citation Context ...e field of fractal functions and the more applied areas are left out almost entirely. Although wavelets are a relatively recent phenomenon, there are already several books on the subject, for example =-=[14, 15, 33, 40, 60, 79, 88, 97]-=-. Partially supported by DARPA Grant AFOSR 89-0455 and ONR Grant N00014-90-J-1343. y Research Assistant of the National Fund of Scientific Research Belgium, partially supported by ONR Grant N0001490 -... |

553 | Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 - Donoho, Johnstone - 1994 |

514 | The wavelet transform, time-frequency localization and signal analysis - Daubechies - 1990 |

496 |
Characterization of signals from multiscale edges
- Mallat, Zhong
- 1992
(Show Context)
Citation Context ...orm of the image f . However, we are more interested in the case when the transform is the fast wavelet transform. There are in fact several ways to use the wavelet transform for compression purposes =-=[83, 84]. One way -=-is to consider compression to be an approximation problem [47, 48]. More specifically, let us fix an orthogonal wavelet /. Given an integer Ms1 we try to find the "best" approximation of f b... |

492 | Entropy-based algorithms for best basis selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ... The criterion is determined by the application, and which basis functions will end up in the basis depends on the data. Entropy based criteria for applications in image compression, were proposed in =-=[27]-=-. Applications in signal processing can be found in [30, 112]. This wavelet packets construction can also be combined with wavelets on closed sets and wavelets in higher dimensions. 13 Multidimensiona... |

449 |
Multiresolution approximations and wavelet orthonormal bases of L 2
- Mallat
(Show Context)
Citation Context ...onal wavelet expansions. With the notion of multiresolution analysis, introduced by St'ephane Mallat and Yves Meyer, a systematic framework for understanding these orthogonal expansions was developed =-=[85, 86, 87]-=-. It also provided the connection with quadrature mirror filtering. Soon Ingrid Daubechies [37] gave a construction of wavelets, non-zero only on a finite interval and with arbitrarily high, but fixed... |

400 |
Fast wavelet transforms and numerical algorithms
- Beylkin, Coifman, et al.
(Show Context)
Citation Context ...amma1 x p /(x)dx with ps0: Of course, these integrals only make sense if OE and / have sufficient decay. The scaling function has M 0 = 1. Recursion formulae to calculate these moments are derived in =-=[10, 105]-=-. The number of vanishing wavelet moments is denoted by ~ N where ~ N is at least 1: N p = 0 for 0sp ! ~ N and N ~ N 6= 0: This is equivalent withs/ (p) (0) = 0 for 0sp ! ~ N ; and, since OE(0) = M 0 ... |

391 | Singularity detection and processing with wavelets
- Mallat, Hwang
- 1992
(Show Context)
Citation Context ...d as two two-dimensional images with color or grey value corresponding to the modulus and phase of W(a; b). The continuous wavelet transform is also used in singularity detection and characterization =-=[57, 82]-=-. A typical result in this direction is that if a function f is Holder (Lipschitz) continuous of order 0 ! ff ! 1, so that jf(x + h) \Gamma f(x)j = O(h ff ), then the continuous wavelet transform has ... |

302 |
Image compression through wavelet transform coding
- DeVore, Jawerth, et al.
- 1992
(Show Context)
Citation Context ...ansform is the fast wavelet transform. There are in fact several ways to use the wavelet transform for compression purposes [83, 84]. One way is to consider compression to be an approximation problem =-=[47, 48]. More spe-=-cifically, let us fix an orthogonal wavelet /. Given an integer Ms1 we try to find the "best" approximation of f by using a representation fM (x) = X kl b jk / jk (x) with M non-zero coeffic... |

252 |
Multifrequency channel decompositions of images and wavelet models
- Mallat
- 1989
(Show Context)
Citation Context ...onal wavelet expansions. With the notion of multiresolution analysis, introduced by St'ephane Mallat and Yves Meyer, a systematic framework for understanding these orthogonal expansions was developed =-=[85, 86, 87]-=-. It also provided the connection with quadrature mirror filtering. Soon Ingrid Daubechies [37] gave a construction of wavelets, non-zero only on a finite interval and with arbitrarily high, but fixed... |

248 |
Wavelet and signal processing
- Rioul, Vetterli
- 1991
(Show Context)
Citation Context ...ro between every two samples) followed by filtering and addition. One can show that the conditions (24) correspond to the exact reconstruction of a subband coding scheme. More details can be found in =-=[96, 108, 109, 110]-=-. An interesting problem is: given a function f , determine, with a certain accuracy and in a computationally favorable way, the coefficientssn;l of a function in the space Vn which are needed to star... |

233 | Wavelets and filter banks: Theory and design - Vetterli, Herley - 1992 |

213 |
Wavelets and dilation equations: A brief introduction
- Strang
- 1989
(Show Context)
Citation Context ...xpression for OE is available. However, there are fast algorithms that use the refinement equation to evaluate the scaling function OE at dyadic points (x = 2 \Gammaj k, j; k 2 ZZ) (see, for example, =-=[8, 12, 37, 42, 43, 102]-=-). In many applications, we never need the scaling function itself; instead we may often work directly with the h k . To be able to use the collection fOE(x \Gamma l) j l 2 ZZg to approximate even the... |

197 |
Zur theorie der orthogonalen Funktionsysteme
- Haar
- 1910
(Show Context)
Citation Context ...rst orthogonal wavelets were discovered by Stromberg [104]. A discrete version of the Calder'on formula had also been used for similar purposes in [74] and long before this there were results by Haar =-=[68]-=-, Franklin [56], Ciesielski [20], Peetre [93], and others. Independently from these developments in harmonic analysis, Alex Grossmann, Jean Morlet, and their coworkers studied the wavelet transform in... |

193 | Continuous and discrete wavelet transforms
- Heil, Walnut
- 1989
(Show Context)
Citation Context ...he body of the paper. 4 The continuous wavelet transform Since we are going to be brief, let us start by pointing out that more detailed treatments of the continuous wavelet transform can be found in =-=[14, 65, 64, 70]-=-. As mentioned above, a wavelet expansion consists of translations and dilations of one fixed function, the wavelet / 2 L 2 (IR). In the continuous wavelet transform the translation and dilation param... |

161 | Using the refinement equations for the construction of pre-wavelets III: Elliptic splines
- Micchelli, Rabut, et al.
- 1991
(Show Context)
Citation Context ...d have exponential decay. The same wavelets, but in a different setting, were also derived by Akram Aldroubi, Murray Eden and Michael Unser in [107]. ffl Other semiorthogonal wavelets can be found in =-=[75, 90, 91, 94]. Some o-=-f these families and properties are summerized in table 1. Examples of interpolating scaling functions: ffl The Shannon sampling function OE Shannon = sin(��x) ��x ; is an interpolating scalin... |

152 |
A Fourier analysis of the finite element variational method. In: Constructive aspects of Functional Analysis. Edizione Cremonese
- Strang, Fix
- 1973
(Show Context)
Citation Context ...already saw that the number of vanishing wavelet moments is important for the characterization of singularities. It also defines the convergence rate of the wavelet approximation for smooth functions =-=[55, 102, 103]-=-, since if f 2 C N , then kP j f(x) \Gamma f(x)k = O(h N ) with h = 2 \Gamman : In fact, the conditions (29) are usually referred to as the Strang--Fix conditions, and these conditions were establishe... |

148 |
painless nonorthogonal Expansions
- Daubechies, Grossmann, et al.
(Show Context)
Citation Context ... Independently from these developments in harmonic analysis, Alex Grossmann, Jean Morlet, and their coworkers studied the wavelet transform in its continuous form [65, 66, 67]. The theory of "fra=-=mes" [41]-=- provided a suitable general framework for these investigations. In the early to mid 80's there were several groups, perhaps most notably the one associated with Yves Meyer and his collaborators, that... |

143 |
Litttlewood–Paley Theory and the Study of Function Spaces
- Frazier, Jawerth, et al.
(Show Context)
Citation Context ...e field of fractal functions and the more applied areas are left out almost entirely. Although wavelets are a relatively recent phenomenon, there are already several books on the subject, for example =-=[14, 15, 33, 40, 60, 79, 88, 97]-=-. Partially supported by DARPA Grant AFOSR 89-0455 and ONR Grant N00014-90-J-1343. y Research Assistant of the National Fund of Scientific Research Belgium, partially supported by ONR Grant N0001490 -... |

140 | Fractal functions and wavelet expansions based on several scaling functions - Geronimo, Hardin, et al. - 1994 |

134 |
Decomposition of Hardy functions into square integrable wavelets of constant shape
- Grossmand, Morlet
- 1984
(Show Context)
Citation Context ...ielski [20], Peetre [93], and others. Independently from these developments in harmonic analysis, Alex Grossmann, Jean Morlet, and their coworkers studied the wavelet transform in its continuous form =-=[65, 66, 67]. The theo-=-ry of "frames" [41] provided a suitable general framework for these investigations. In the early to mid 80's there were several groups, perhaps most notably the one associated with Yves Meye... |

124 |
A discrete transform and decompositions of distribution spaces
- Frazier, Jawerth
- 1990
(Show Context)
Citation Context ...ast potentially, could be effective substitutes for Fourier series in numerical applications. (The first named author of this paper came to this understanding through the joint work with Mike Frazier =-=[57, 58, 59]-=-.) As the emphasis shifted more towards the representations themselves, and the building blocks involved, the name also shifted: Yves Meyer and Jean Morlet suggested the word wavelet for the building ... |

123 | Interpolating wavelet transforms
- Donoho
- 1992
(Show Context)
Citation Context ...lds a family of interpolating functions which were studied by Gilles Deslauriers and Serge Dubuc in [45, 46]. These functions are smooth and compactly supported. More information can also be found in =-=[50, 98]-=-. 11 Wavelets on closed sets So far we have been discussing wavelet theory on the real line (and its higher dimensional analogs). For many applications the functions involved are only defined on a com... |

113 |
Nonseparable bidimensional wavelet bases
- Cohen, Daubechies
- 1993
(Show Context)
Citation Context ...sists of applying the one-dimensional fast wavelet transform to the rows and columns of a matrix. There are also several other extensions to higher dimensions. We mention nonseparable basis functions =-=[23, 44, 94, 101]-=-, other lattices corresponding to different symmetries [26], Clifford valued wavelets [3], etc. However we leave these topics for now. 14 Applications 14.1 Data compression One of the applications of ... |

111 |
New thoughts on Besov spaces
- Peetre
- 1976
(Show Context)
Citation Context ...romberg [104]. A discrete version of the Calder'on formula had also been used for similar purposes in [74] and long before this there were results by Haar [68], Franklin [56], Ciesielski [20], Peetre =-=[93]-=-, and others. Independently from these developments in harmonic analysis, Alex Grossmann, Jean Morlet, and their coworkers studied the wavelet transform in its continuous form [65, 66, 67]. The theory... |

108 | On compactly supported spline wavelets and a duality principle
- Chui, Wang
- 1992
(Show Context)
Citation Context ...s. A disadvantage is that for small filter lengths, the dual functions have very low regularity. ffl Examples of semiorthogonal wavelets are the ones constructed by Charles Chui and Jianzhong Wang in =-=[17, 18, 19]-=-. The scaling functions are cardinal B--splines of order m and the wavelet functions are splines with compact support [0; 2m \Gamma 1]. All primary and dual functions still have generalized linear pha... |

108 |
Lagarias, Two-scale difference equations: II. Local regularity, Infinite products of matrices and fractals
- Daubechies, C
(Show Context)
Citation Context ...xpression for OE is available. However, there are fast algorithms that use the refinement equation to evaluate the scaling function OE at dyadic points (x = 2 \Gammaj k, j; k 2 ZZ) (see, for example, =-=[8, 12, 37, 42, 43, 102]-=-). In many applications, we never need the scaling function itself; instead we may often work directly with the h k . To be able to use the collection fOE(x \Gamma l) j l 2 ZZg to approximate even the... |

103 | Oversampled filter banks
- Cvetkovic, Vetterli
- 1998
(Show Context)
Citation Context ...ro between every two samples) followed by filtering and addition. One can show that the conditions (24) correspond to the exact reconstruction of a subband coding scheme. More details can be found in =-=[96, 108, 109, 110]-=-. An interesting problem is: given a function f , determine, with a certain accuracy and in a computationally favorable way, the coefficientssn;l of a function in the space Vn which are needed to star... |

103 |
Entropy Based Algorithms for Best Basis Selection
- Coifman, Wickerhauser
- 1992
(Show Context)
Citation Context ... criterion is determined by the application, and which basis functions that will end up in the basis depends on the data. For applications in image processing, entropy-based criteria were proposed in =-=[40]-=-. The best basis selection in that case has a numerical complexity ofO(M). Applications in signal processing can be found in [36, 139]. This wavelet packets construction can also be combined with wave... |

100 |
Wavelets: "A tutorial of theory and applications
- Chui
- 121
(Show Context)
Citation Context ...e field of fractal functions and the more applied areas are left out almost entirely. Although wavelets are a relatively recent phenomenon, there are already several books on the subject, for example =-=[14, 15, 33, 40, 60, 79, 88, 97]-=-. Partially supported by DARPA Grant AFOSR 89-0455 and ONR Grant N00014-90-J-1343. y Research Assistant of the National Fund of Scientific Research Belgium, partially supported by ONR Grant N0001490 -... |

97 |
P.: Multiresolution analysis, wavelets, and fast algorithms on an interval
- Cohen, Daubechies, et al.
- 1993
(Show Context)
Citation Context ... V [0;1] j . Since this fact is directly linked to many of the approximation properties of wavelets, any construction of a multiresolution analysis on [0; 1] should preserve this. The construction in =-=[25]-=- uses this as a starting point and is slightly different than the one by Yves Meyer. Let us briefly describe this construction as well. Again we start with the scaling function OE from the Daubechies ... |

94 | Multilevel preconditioning
- DAHMEN, KUNOTH
- 1992
(Show Context)
Citation Context ...s where traditionally finite element methods are used, e.g. for solving boundary value problems [72]. There are interesting results showing that this might be fruitful; for example, it has been shown =-=[11, 36, 92, 113]-=-. that for many problems the condition number of the N \Theta N stiffness matrix remains bounded as the dimension N goes to infinity. This is in contrast with the situation for regular finite elements... |

92 |
Two-scale difference equations I. Existence and global regularity of solutions
- Daubechies, Lagarias
(Show Context)
Citation Context ...he fact that the integral of OE does not vanish, we see that X k h k = 1: (5) The scaling function is, under very general conditions, uniquely defined by its refinement equation and the normalization =-=[42]-=-, Z +1 \Gamma1 OE(x) dx = 1: In many cases, no explicit expression for OE is available. However, there are fast algorithms that use the refinement equation to evaluate the scaling function OE at dyadi... |

91 |
Decomposition of Besov spaces
- Frazier, Jawerth
(Show Context)
Citation Context ...ast potentially, could be effective substitutes for Fourier series in numerical applications. (The first named author of this paper came to this understanding through the joint work with Mike Frazier =-=[57, 58, 59]-=-.) As the emphasis shifted more towards the representations themselves, and the building blocks involved, the name also shifted: Yves Meyer and Jean Morlet suggested the word wavelet for the building ... |

86 |
Sobolev characterization of solutions of dilation equations
- Eirola
- 1992
(Show Context)
Citation Context ...e in the regularity of OE. The regularity is N \Gamma 1 at most, but in many cases it is smaller due to the influence of K. The regularity of solutions of refinement equations is studied in detail in =-=[32, 42, 43, 54, 95]-=-. 9 The fast wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... |

78 | On the construction of multivariate (pre) wavelets
- Boor, DeVore, et al.
- 1993
(Show Context)
Citation Context ...onstructions of nonseparable wavelets that use this kind of splitting. One of the problems here is, given the scaling function, is there an easy way, cf. (19), to nd the wavelets? This was studied in =-=[54, 113, 121]-=-. Another idea is to still try to split into just two subspaces. This involves the use of di erent lattices [99]. In the bivariate case, Ingrid Daubechies and Albert Cohen constructed smooth, compactl... |

78 | Simple regularity criteria for subdivision schemes
- Rioul
- 1992
(Show Context)
Citation Context ...s a role in the regularity of '. The Holder regularity isN,1 at most, but in many cases it is lower due to the in uence ofK. The regularity of solutions of re nement equations is studied in detail in =-=[42, 41, 52, 53, 66, 114, 135, 136]-=-. Note that we never required the dual scaling function to satisfy a partition of unity property, nor the wavelet to have avanishing moment. In fact, it is possible to have awavelet with a non-vanishi... |

75 |
banks allowing perfect reconstruction
- Vetterli, “Filter
- 1986
(Show Context)
Citation Context ...ro between every two samples) followed by filtering and addition. One can show that the conditions (24) correspond to the exact reconstruction of a subband coding scheme. More details can be found in =-=[96, 108, 109, 110]-=-. An interesting problem is: given a function f , determine, with a certain accuracy and in a computationally favorable way, the coefficientssn;l of a function in the space Vn which are needed to star... |

74 |
Wavelet-like bases for the fast solution of second-kind integral equations
- Alpert, Beylkin, et al.
- 1993
(Show Context)
Citation Context ... written as H(0) = 1: Examples of scaling functions: ffl A well known family of scaling functions is the set of cardinal B--splines. The cardinal B--spline of order 1 is the box function N 1 (x) = �=-=� [0;1]-=- (x). For m ? 1 the cardinal B--spline Nm is defined recursively as a convolution, Nm = Nm\Gamma1sN 1 : These functions satisfy Nm (x) = 2 m\Gamma1 X k i m k j Nm (2x \Gamma k); and Nm (!) = ` 1 \Gamm... |

69 |
Transforms associated to square integrable group representations I
- Grossmann, Morlet, et al.
- 1985
(Show Context)
Citation Context ...ielski [20], Peetre [93], and others. Independently from these developments in harmonic analysis, Alex Grossmann, Jean Morlet, and their coworkers studied the wavelet transform in its continuous form =-=[65, 66, 67]. The theo-=-ry of "frames" [41] provided a suitable general framework for these investigations. In the early to mid 80's there were several groups, perhaps most notably the one associated with Yves Meye... |

69 | Wavelets on a closed subsets of the real line
- Andersson, Hall, et al.
- 1994
(Show Context)
Citation Context ...nV [0;1] j . Since this fact is directly linked to many of the approximation properties of wavelets, any construction of a multiresolution analysis on [0; 1] should preserve this. The construction in =-=[5, 32, 33]-=- uses this as a starting point and is slightly di erent from the one by Yves Meyer. Let us brie y describe this construction as well. Again we start with an orthogonal Daubechies scaling function' wit... |

68 |
Wavelet methods for fast resolution of elliptic problems
- Jaffard
(Show Context)
Citation Context ... of the close similarities between the scaling function and nite elements, it seems natural to try wavelets where traditionally nite element methods are used, e.g. for solving boundary value problems =-=[84]-=-. There are interesting results showing that this might be fruitful; for example, it has been shown [17, 46, 111, 140] that for many problems the condition number of theNN sti ness matrix remains boun... |

66 |
Necessary and sufficient conditions for constructing orthonormal wavelet bases
- Lawton
- 1991
(Show Context)
Citation Context ... ZZ: The last two equations are equivalent but they provide only a necessary condition for the orthogonality of the scaling function and its translates. This relationship is investigated in detail in =-=[77]-=-. Now, an orthogonal wavelet is a function / such that the collection of functions f/(x \Gamma l) j l 2 ZZg is an orthonormal basis of W 0 . This is the case if h /; /(\Delta \Gamma l) i = ffi l Again... |

62 |
Y.: Ondelettes et bases hilbertiennes
- Lemarié, Meyer
- 1986
(Show Context)
Citation Context ...let suggested the word wavelet for the building blocks, and what earlier had been referred to as Littlewood-Paley theory now started to be called wavelet theory. Pierre-Gilles Lemari'e and Yves Meyer =-=[80]-=-, independently of Stromberg, constructed new orthogonal wavelet expansions. With the notion of multiresolution analysis, introduced by St'ephane Mallat and Yves Meyer, a systematic framework for unde... |

61 |
A family of polynomial spline wavelet transforms
- Unser, Aldroubi, et al.
- 1993
(Show Context)
Citation Context ...the dual functions do not have compact support but instead have exponential decay. The same wavelets, but in a different setting, were also derived by Akram Aldroubi, Murray Eden and Michael Unser in =-=[107]-=-. ffl Other semiorthogonal wavelets can be found in [75, 90, 91, 94]. Some of these families and properties are summerized in table 1. Examples of interpolating scaling functions: ffl The Shannon samp... |

54 |
Cardinal interpolation and spline functions
- Schoenberg
- 1969
(Show Context)
Citation Context ...minator does not vanish [111]. Even if OE is compactly supported, OE interpol is in general not compactly supported. The cardinal spline interpolation functions of even order are constructed this way =-=[99]-=-. ffl An interpolation scaling function can also be constructed from a pair of biorthogonal scaling functions as OE interpol (x) = Z +1 \Gamma1 OE(y + x) ~ OE(y) dy: The interpolation property immedia... |

53 | G.: Multiresolution representations using the autocorrelation functions of compactly supported wavelets
- Saito, Beylkin
(Show Context)
Citation Context ...lds a family of interpolating functions which were studied by Gilles Deslauriers and Serge Dubuc in [45, 46]. These functions are smooth and compactly supported. More information can also be found in =-=[50, 98]-=-. 11 Wavelets on closed sets So far we have been discussing wavelet theory on the real line (and its higher dimensional analogs). For many applications the functions involved are only defined on a com... |

53 |
A sampling theorem for wavelet subspaces
- Walter
- 1992
(Show Context)
Citation Context ...Vn which are needed to start the fast wavelet transform. A trivial solution could besn;l = f(l=2 n ): Other sampling procedures, such as (quasi-)interpolation and quadrature formulae were proposed in =-=[73, 100, 105, 111]-=- An implementation of a fast wavelet transform in pseudo code is given in the appendix. 10 Examples of wavelets Now that we have discussed the essentials of wavelet multiresolution analysis, we shall ... |