## Wavelet Transforms that Map Integers to Integers

Citations: | 262 - 6 self |

### BibTeX

@MISC{Calderbank_wavelettransforms,

author = {Robert Calderbank and et al.},

title = {Wavelet Transforms that Map Integers to Integers},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1719 |
T'en lectures on wavelets
- Daubechies
- 1992
(Show Context)
Citation Context ...ments and minimal filter length, then the resulting filter coefficients are real numbers which can be computed with high precision, but for which we do not even have a closed form expression if N ? 3 =-=[8]-=-. When the input data consist of sequences of integers (as is the case for images), the resulting filtered outputs no longer consist of integers. Yet, for lossless coding it would be of interest to be... |

1569 | Orthonormal bases of compactly supported wavelets
- Daubechies
- 1988
(Show Context)
Citation Context ...nd which yields near optimal approximation in certain Besov spaces. Relaxing the constraint of orthonormality of the wavelets makes it possible to obtain filter coefficients that are dyadic rationals =-=[6, 14, 37, 31]-=-; up to scaling these filters can be viewed as mapping integers to integers. This rescaling amplifies the dynamic range of the 1991 Mathematics Subject Classification. 42C15, 94A29. Key words and phra... |

1246 |
Embedded image coding using zerotrees of wavelet coefficients
- Shapiro
- 1993
(Show Context)
Citation Context ...ssless image coder and compare the results to the literature. 1. Introduction Wavelets, wavelet packets, and local cosine transforms are used in a variety of applications, including image compression =-=[2, 29, 39, 10, 25]-=-. In most cases, the filters that are used have floating point coefficients. For instance, if one prefers to use orthonormal filters with an assigned number N (N ? 2) of vanishing moments and minimal ... |

849 |
Multirate Systems and Filter Banks
- Vaidyanathan
- 1993
(Show Context)
Citation Context ... 2(n\Gammak)+1 !/ s 0;2k s 0;2k\Gamma1 ! = N X k=0 / h 2k h 2k+1 g 2k g 2k+1 !/ s 0;2(n\Gammak) s 0;2(n\Gammak)\Gamma1 ! (2.4) (This corresponds to a polyphase decomposition of the filters; see e.g., =-=[30, 35, 38].) The sub-=-scripts 0, 1 correspond to the "level" of the filtered sequences; level 0 are the unprocessed data, while level 1 corresponds to one application of the H , G filters. More generally, s j;n a... |

472 |
Wavelets and subband coding
- Vetterli, Kovacevic
- 1995
(Show Context)
Citation Context ... 2(n\Gammak)+1 !/ s 0;2k s 0;2k\Gamma1 ! = N X k=0 / h 2k h 2k+1 g 2k g 2k+1 !/ s 0;2(n\Gammak) s 0;2(n\Gammak)\Gamma1 ! (2.4) (This corresponds to a polyphase decomposition of the filters; see e.g., =-=[30, 35, 38].) The sub-=-scripts 0, 1 correspond to the "level" of the filtered sequences; level 0 are the unprocessed data, while level 1 corresponds to one application of the H , G filters. More generally, s j;n a... |

434 | Factoring wavelet transforms into lifting steps
- Daubechies, Sweldens
- 1998
(Show Context)
Citation Context ...rimal and dual lifting steps and a scaling. This theorem follows from a factorization well known to both electrical engineers (in linear system theory) as well as algebraists (in algebraic K-theory); =-=[9]-=- gives a self-contained proof as well as many examples coming from wavelet filter banks.. The proof is constructive; for a given wavelet transform, the filters used in the lifting steps can be found u... |

432 |
The lifting scheme: A custom-design construction of biorthogonal wavelets
- Sweldens
- 1996
(Show Context)
Citation Context ...An example in [39] of a wavelet transform that maps integers to integers is seen to be an instance of a large family of similar transforms, obtained by combining the lifting constructions proposed in =-=[32, 33]-=- with rounding-off in a reversible way. In Section 3.1 we review the S transform, and in Section 3.2 two generalizations, the TS transform and the S+P transform. In Section 3.3 we briefly review lifti... |

373 | The lifting scheme: A construction of second generation wavelets - Sweldens - 1998 |

299 |
Image compression through wavelet transform coding
- DeVore, Jawerth, et al.
- 1992
(Show Context)
Citation Context ...ssless image coder and compare the results to the literature. 1. Introduction Wavelets, wavelet packets, and local cosine transforms are used in a variety of applications, including image compression =-=[2, 29, 39, 10, 25]-=-. In most cases, the filters that are used have floating point coefficients. For instance, if one prefers to use orthonormal filters with an assigned number N (N ? 2) of vanishing moments and minimal ... |

232 |
Wavelets and filter banks: Theory and design
- Vetterli, Herley
- 1992
(Show Context)
Citation Context ...nd which yields near optimal approximation in certain Besov spaces. Relaxing the constraint of orthonormality of the wavelets makes it possible to obtain filter coefficients that are dyadic rationals =-=[6, 14, 37, 31]-=-; up to scaling these filters can be viewed as mapping integers to integers. This rescaling amplifies the dynamic range of the 1991 Mathematics Subject Classification. 42C15, 94A29. Key words and phra... |

229 | Spherical wavelets: efficiently representing functions on the sphere
- Schroder, Sweldens
- 1995
(Show Context)
Citation Context ...ique that has been used in several different settings, for an easy construction and implementation of "traditional" wavelets [32], and of "second generation" wavelets [33], such as=-= spherical wavelets [26]-=-. Lifting is also closely related to several other techniques [13, 22, 37, 34, 20, 4, 15, 7, 3, 19, 28]. Rather than giving the general structure of lifting at this point, we show how to rewrite the H... |

171 | An image multiresolution representation for lossless and lossy compression
- Said, Pearlman
- 1996
(Show Context)
Citation Context ...ssless image coder and compare the results to the literature. 1. Introduction Wavelets, wavelet packets, and local cosine transforms are used in a variety of applications, including image compression =-=[2, 29, 39, 10, 25]-=-. In most cases, the filters that are used have floating point coefficients. For instance, if one prefers to use orthonormal filters with an assigned number N (N ? 2) of vanishing moments and minimal ... |

122 | Interpolating wavelet transforms
- Donoho
- 1992
(Show Context)
Citation Context ... construction and implementation of "traditional" wavelets [32], and of "second generation" wavelets [33], such as spherical wavelets [26]. Lifting is also closely related to sever=-=al other techniques [13, 22, 37, 34, 20, 4, 15, 7, 3, 19, 28]-=-. Rather than giving the general structure of lifting at this point, we show how to rewrite the Haar and S transforms using lifting. We rewrite (3.1) in two steps which need to be executed sequentiall... |

64 |
Lattice Structures for optimal design and robust implementation of two channel perfect reconstruction QMF Banks
- Vaidyannthan, Hoang
- 1988
(Show Context)
Citation Context ...e matrices H 0 and H 1 (the only non-zero H k ) in this case as H 0 = U / 1 0 0 0 ! V = U e H 0 V; H 1 = U / 0 0 0 1 ! V = U e H 1 V : (This is in fact a different way of writing the factorization of =-=[36]-=- for this particular case.) It follows that H = U e HV, where U and V are matrices with the 2 \Theta 2 blocks U resp. V , on the diagonal, and zeros elsewhere, and where e H = 0 B B B B B @ e H 0 e H ... |

54 | Smooth wavelet decompositions with blocky coefficient kernels
- Donoho
- 1993
(Show Context)
Citation Context ...e new wavelet coefficient d 1;l is exactly zero. This assures that the dual wavelet has three vanishing moments. This prediction idea also connects to average-interpolation as introduced by Donoho in =-=[12]-=-. Substituting we see that the high pass filter is f1=8; 1=8; \Gamma1; 1; \Gamma1=8; \Gamma1=8g while the low pass filter is f1=2; 1=2g, and we thus indeed calculate the (3,1) transform from [6]. Here... |

38 |
den Enden, “New networks for perfect inversion and perfect reconstruction
- Bruekers, van
- 1992
(Show Context)
Citation Context ... construction and implementation of "traditional" wavelets [32], and of "second generation" wavelets [33], such as spherical wavelets [26]. Lifting is also closely related to sever=-=al other techniques [13, 22, 37, 34, 20, 4, 15, 7, 3, 19, 28]-=-. Rather than giving the general structure of lifting at this point, we show how to rewrite the Haar and S transforms using lifting. We rewrite (3.1) in two steps which need to be executed sequentiall... |

34 |
Local decomposition of refinable spaces
- Carnicer
(Show Context)
Citation Context ... construction and implementation of "traditional" wavelets [32], and of "second generation" wavelets [33], such as spherical wavelets [26]. Lifting is also closely related to sever=-=al other techniques [13, 22, 37, 34, 20, 4, 15, 7, 3, 19, 28]-=-. Rather than giving the general structure of lifting at this point, we show how to rewrite the Haar and S transforms using lifting. We rewrite (3.1) in two steps which need to be executed sequentiall... |

32 |
Multiresolution representation of data: a general framework
- HARTEN
- 1996
(Show Context)
Citation Context |

31 |
Multiresolution surfaces of arbitrary topological type
- Lounsbery, DeRose, et al.
- 1997
(Show Context)
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29 |
Lossless integer wavelet transform
- Dewitte, Cornelis
- 1997
(Show Context)
Citation Context ...eton University in fall 1995. Note. After finishing this work, we learned that a construction similar to the one presented in Section 3 was obtained independently by Dewitte-Cornelis and Chao-Fischer =-=[11, 5]-=-. The Chao-Fischer approach has the extra feature that it uses modular arithmetic to eliminate all increase in dynamic range; thus if the original image uses 8 bits per pixel, all the wavelet coeffici... |

20 |
The FBI compression standard for digitized fingerprint images
- Brislawn, Bradley, et al.
- 1996
(Show Context)
Citation Context |

20 |
Banded matrices with banded inverses. II. Locally finite decomposition of spline spaces
- Dahmen, Micchelli
- 1993
(Show Context)
Citation Context |

19 |
A binary wavelet decomposition of binary images
- Tewfik, Swanson
- 1996
(Show Context)
Citation Context ...nd which yields near optimal approximation in certain Besov spaces. Relaxing the constraint of orthonormality of the wavelets makes it possible to obtain filter coefficients that are dyadic rationals =-=[6, 14, 37, 31]-=-; up to scaling these filters can be viewed as mapping integers to integers. This rescaling amplifies the dynamic range of the 1991 Mathematics Subject Classification. 42C15, 94A29. Key words and phra... |

17 |
A class of linear-phase regular biorthogonal wavelets, in
- Kim, Ansari, et al.
- 1992
(Show Context)
Citation Context |

17 | On ladder structures and linear phase conditions fo bi-orthogonal filter banks
- Shah, Kalker
- 1994
(Show Context)
Citation Context |

16 |
A Comparison of Reversible Methods for Data Compression
- Heer, Reinfelder
- 1990
(Show Context)
Citation Context ...th integers. In the case of the Fourier and cosine transforms, this problem has been solved in [17]. An integer version of the orthogonal Haar transform has been known for some time as the Stransform =-=[24, 16]-=-. In [10] a redundant integer version of the Haar transform is introduced which uses rounded integer arithmetic and which yields near optimal approximation in certain Besov spaces. Relaxing the constr... |

16 |
A Simple and Effective Precoding Scheme for Noise Whitening on Intersymbol Interference Channels
- Laroia, Tretter, et al.
- 1993
(Show Context)
Citation Context ...y's microprocessors floating point and integer computations are virtually equally fast. The first approach, in Section 2, is inspired by the precoding invented by Laroia, Tretter, and Farvardin (LTF) =-=[21]-=-, which is used for transmission over channels subject to intersymbol interference and Gaussian noise. When integer data undergo causal filtering with a leading filter coefficient equal to 1, this pre... |

14 | An approach to fast integer reversible wavelet transforms for image compression. [Online]. Available: http://www.compsci.com/wchao/Publication/Pub no. 1.ps.gz
- Chao, Fisher
(Show Context)
Citation Context ...eton University in fall 1995. Note. After finishing this work, we learned that a construction similar to the one presented in Section 3 was obtained independently by Dewitte-Cornelis and Chao-Fischer =-=[11, 5]-=-. The Chao-Fischer approach has the extra feature that it uses modular arithmetic to eliminate all increase in dynamic range; thus if the original image uses 8 bits per pixel, all the wavelet coeffici... |

13 | Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets
- Kalker, Shah
- 1992
(Show Context)
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9 | editors. Recent Advances in Wavelet Analysis - Schumaker, Webb - 1993 |

6 |
A comparison of prediction schemes proposed for a new lossless image compression standard’.Proc
- Memon, Sippy, et al.
- 1996
(Show Context)
Citation Context .... The decorrelating steps have always been performed in the spatial domains and they involve some form of non adaptive or adaptive prediction of a pixel values based on previously visited pixels (see =-=[23]-=- and the references therein for comparisons of state-of-the-art spatial-domain predication techniques). In fact, we can view wavelet transforms that use two lifting steps such as those built from Desl... |

6 | Recent Advances in Wavelet Analysis - SCHUMAKER, WEBB, et al. - 1993 |

6 | Crew Lossless/Lossy Medical Image Compression
- Zandi, Boliek, et al.
- 1995
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2 |
Discrete Fourier, Hartley and cosine transforms in signal processing
- Hong
- 1993
(Show Context)
Citation Context ...et, for lossless coding it would be of interest to be able to characterize the output completely again with integers. In the case of the Fourier and cosine transforms, this problem has been solved in =-=[17]-=-. An integer version of the orthogonal Haar transform has been known for some time as the Stransform [24, 16]. In [10] a redundant integer version of the Haar transform is introduced which uses rounde... |

1 |
1/SC29/WG1. Call for contributions - lossless compression of continuous-tone still pictures
- JTC
- 1994
(Show Context)
Citation Context .... (For Haar, the problem reduces to considering H 0 only in 2D; since H \Gamma1 0 = H t 0 , we can look at Figure 2a again, which shows the regions H 0 [0; 1) 2 + n; their mirror images are the H t 0 =-=[0; 1]-=- 2 + m. For ff = 1, they do intersect non-trivially.) If we do use an expansion factor ff, then we are dealing with the transform a \Gamma! ffHa + f instead, and the sufficient condition for invertibi... |

1 |
Binary friendly wavelets
- Gros
- 1995
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1 |
The lifting scheme and finite-precision-error-free filter banks
- Johnston
- 1996
(Show Context)
Citation Context ... however, so that this does not lead to reasonable lossless compression. In the special case of binary valued images it is possible to define binary valued wavelet transforms (and thus no bit growth) =-=[18, 31]-=-. In this paper, we present two approaches that lead to wavelet transforms that map integers to integers, and which can be used for lossless coding. The two approaches are independent of each other, a... |

1 | On constructing regular filter banks from domain bounded polynomials
- Tolhuizen, Shah, et al.
- 1994
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1 | Lossless/lossy medical image compression - CREW |

1 | Which one of these two do you want? W. Sweldens. The lifting scheme: A customdesign construction of biorthogonal wavelets - Sweldens - 1996 |

1 | this one correct? P.P. Vaidyanathan and P.-Q. Hoang. Lattice structures for optimal design and robust implementation of two-band perfect reconstruction QMF banks - Is - 1988 |

1 | Which one of these three do you want? P.P. Vaidyanathan. Theory and design of Mchannel maximally decimated quadrature mirror filters with arbitrary M, having perfect reconstruction property - Vaidyanathan, Nguyen, et al. - 1987 |