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The lifting scheme: A construction of second generation wavelets. (1988)

by W Sweldens
Venue:SIAM J. Math. Anal.
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Factoring wavelet transforms into lifting steps

by Ingrid Daubechies, Wim Sweldens - J. FOURIER ANAL. APPL , 1998
"... This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decompositio ..."
Abstract - Cited by 584 (8 self) - Add to MetaCart
This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.
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...varying update coefficients can be computed [46]. This thus immediately allows for a (2,2) type transform for irregular samples. These spatial lifting steps can also be used in higher dimensions (see =-=[45]-=-) and leads e.g., to wavelets on a sphere [40] or more complex manifolds. Note that the idea of using spatial wavelet constructions for building second generation wavelets has been proposed by several...

Wavelet Transforms that Map Integers to Integers

by Robert Calderbank, et al.
"... ..."
Abstract - Cited by 357 (6 self) - Add to MetaCart
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Spherical Wavelets: Efficiently Representing Functions on the Sphere

by Peter Schröder, Wim Sweldens , 1995
"... Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classic ..."
Abstract - Cited by 286 (14 self) - Add to MetaCart
Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

The JPEG2000 Still Image Coding System: an Overview

by Charilaos Christopoulos, Athanassios Skodras, Touradj Ebrahimi , 2000
"... With the increasing use of multimedia technologies, image compression requires higher performance as well as new features. To address this need in the specific area of still image encoding, a new standard is currently being developed, the JPEC2000. It is not only intended to provide rate-distortion ..."
Abstract - Cited by 259 (2 self) - Add to MetaCart
With the increasing use of multimedia technologies, image compression requires higher performance as well as new features. To address this need in the specific area of still image encoding, a new standard is currently being developed, the JPEC2000. It is not only intended to provide rate-distortion and subjective image quality performance superior to existing standards, but also to provide features and functionalities that current standards can either not address efficiently or in many cases cannot address at all. Lossless and lossy compression, embedded lossy to lossless coding, progressive transmission by pixel accuracy and by resolution, robustness to the presence of bit-errors and region-of-interest coding, are some representative features. It is interesting to note that JPEG2000 is being designed to address the requirements of a diversity of applications, e.g. Internet, color facsimile, printing, scanning, digital photography, remote sensing, mobile applications, medical imagery, digital library and E-commerce.

Computer Vision: Algorithms and Applications

by Richard Szeliski , 2010
"... ..."
Abstract - Cited by 252 (2 self) - Add to MetaCart
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Multiresolution signal processing for meshes

by Igor Guskov, Wim Sweldens, Peter Schröder
"... We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a non-uniform relaxation procedure whose weights depend on the geometry and we show its superiority over existing schemes wh ..."
Abstract - Cited by 245 (11 self) - Add to MetaCart
We generalize basic signal processing tools such as downsampling, upsampling, and filters to irregular connectivity triangle meshes. This is accomplished through the design of a non-uniform relaxation procedure whose weights depend on the geometry and we show its superiority over existing schemes whose weights depend only on connectivity. This is combined with known mesh simplification methods to build subdivision and pyramid algorithms. We demonstrate the power of these algorithms through a number of application examples including smoothing, enhancement, editing, and texture mapping.

Wavelet and Multiscale Methods for Operator Equations

by Wolfgang Dahmen - Acta Numerica , 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
Abstract - Cited by 220 (37 self) - Add to MetaCart
this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
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...1 , varying L j and K j produces a whole family of further stable completions and corresponding decompositions of the spaces S(\Phi j ). The special case K j = I covers the lifting scheme proposed in =-=[168, 169]-=-. In this case one has \Psi T j = \Phi T j+1 M j;1 = \Phi T j+1 M j;0 L j + \Phi T j+1 M j;1 = \Phi T j L+ \Psi j ; that is, in terms of individual functions, one has / j;k = X l2\Delta j (L j ) l;k O...

The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions

by Wim Sweldens - in Wavelet Applications in Signal and Image Processing III , 1995
"... In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in ..."
Abstract - Cited by 200 (0 self) - Add to MetaCart
In this paper we present the basic idea behind the lifting scheme, a new construction of biorthogonal wavelets which does not use the Fourier transform. In contrast with earlier papers we introduce lifting purely from a wavelet transform point of view and only consider the wavelet basis functions in a later stage. We show how lifting leads to a faster, fully in-place implementation of the wavelet transform. Moreover, it can be used in the construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one function. A typical example of the latter are wavelets on the sphere. Keywords: wavelet, biorthogonal, in-place calculation, lifting 1 Introduction At the present day it has become virtually impossible to give the definition of a "wavelet". The research field is growing so fast and novel contributions are made at such a rate that even if one manages to give a definition today, it might be obsolete tomorrow. One, very vague, way of thinking about...

The JPEG2000 still image compression standard

by Athanassios Skodras, Charilaos Christopoulos, Touradj Ebrahimi - IEEE Signal Proc. Mag , 2001
"... The development of standards (emerging and established) by the International Organization for Standardization (ISO), the International Telecommunications Union (ITU), and the International Electrotechnical Commission (IEC) for audio, image, and video, for both transmission and storage, has led to wo ..."
Abstract - Cited by 180 (11 self) - Add to MetaCart
The development of standards (emerging and established) by the International Organization for Standardization (ISO), the International Telecommunications Union (ITU), and the International Electrotechnical Commission (IEC) for audio, image, and video, for both transmission and storage, has led to worldwide activity in developing hardware and software systems and products applicable to a number of diverse disciplines [7], [22], [23], [55], [56], [73]. Although the standards implicitly address the basic encoding operations, there is freedom and flexibility in the actual design and development of devices. This is because only the syntax and semantics of the bit stream for decoding are specified by standards, their main objective being the compatibility and interoperability among the systems (hardware/software) manufactured by different companies. There is, thus, much room for innovation and ingenuity. Since the mid 1980s, members from both the ITU and the ISO have been working together to establish a joint international standard for the compression of grayscale and color still images. This effort has been known as JPEG, the Joint
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...tely odd sample values of the signal are updated with a weighted sum of even sample values, and even sample values are updated with a weighted sum of odd sample values (Fig. 8) [3], [45], [62], [64], =-=[65]-=-. For the reversible (lossless) case the results are rounded to integer values. The lifting-based filtering [2], [38] for the 5/3 analysis filter is achieved by means of ⎢ x ( 2n) + x ( 2n+ 2) ext ext...

Building Your Own Wavelets at Home

by Wim Sweldens, Peter Schröder
"... Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what mak ..."
Abstract - Cited by 150 (13 self) - Add to MetaCart
Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas. The main
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