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Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. 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 dec ..."
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Cited by 480 (7 self)
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ABSTRACT. 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 wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained 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, nonunitary 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 waveletlike transform that maps integers to integers. 1.
The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 427 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
ViewDependent Refinement of Progressive Meshes
"... Levelofdetail (LOD) representations are an important tool for realtime rendering of complex geometric environments. The previously introduced progressive mesh representation defines for an arbitrary triangle mesh a sequence of approximating meshes optimized for viewindependent LOD. In this paper, ..."
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Cited by 425 (5 self)
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Levelofdetail (LOD) representations are an important tool for realtime rendering of complex geometric environments. The previously introduced progressive mesh representation defines for an arbitrary triangle mesh a sequence of approximating meshes optimized for viewindependent LOD. In this paper, we introduce a framework for selectively refining an arbitrary progressive mesh according to changing view parameters. We define efficient refinement criteria based on the view frustum, surface orientation, and screenspace geometric error, and develop a realtime algorithm for incrementally refining and coarsening the mesh according to these criteria. The algorithm exploits view coherence, supports frame rate regulation, and is found to require less than 15 % of total frame time on a graphics workstation. Moreover, for continuous motions this work can be amortized over consecutive frames. In addition, smooth visual transitions (geomorphs) can be constructed between any two selectively refined meshes. A number of previous schemes create viewdependent LOD meshes for height fields (e.g. terrains) and parametric surfaces (e.g. NURBS). Our framework also performs well for these special cases. Notably, the absence of a rigid subdivision structure allows more accurate approximations than with existing schemes. We include results for these cases as well as for general meshes.
Spherical Wavelets: Efficiently Representing Functions on the Sphere
, 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 ..."
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Cited by 240 (14 self)
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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.
Interactive Multiresolution Mesh Editing
"... We describe a multiresolution representation for meshes based on subdivision. Subdivision is a natural extension of the existing patchbased surface representations. At the same time subdivision algorithms can be viewed as operating directly on polygonal meshes, which makes them a useful tool for me ..."
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Cited by 189 (20 self)
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We describe a multiresolution representation for meshes based on subdivision. Subdivision is a natural extension of the existing patchbased surface representations. At the same time subdivision algorithms can be viewed as operating directly on polygonal meshes, which makes them a useful tool for mesh manipulation. Combination of subdivision and smoothing algorithms of Taubin [26] allows us to construct a set of algorithms for interactive multiresolution editing of complex meshes of arbitrary topology. Simplicity of the essential algorithms for re nement and coarsi cation allows to make them local and adaptive, considerably improving their efficiency. We have built a scalable interactive multiresolution editing system based on such algorithms.
Multiresolution Curves
, 1994
"... We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present m ..."
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Cited by 157 (5 self)
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We describe a multiresolution curve representation, based on wavelets, that conveniently supports a variety of operations: smoothing a curve; editing the overall form of a curve while preserving its details; and approximating a curve within any given error tolerance for scan conversion. We present methods to support continuous levels of smoothing as well as direct manipulation of an arbitrary portion of the curve; the control points, as well as the discrete nature of the underlying hierarchical representation, can be hidden from the user. The multiresolution representation requires no extra storage beyond that of the original control points, and the algorithms using the representation are both simple and fast.
Building Your Own Wavelets at Home
"... 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 ..."
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Cited by 135 (13 self)
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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
Wavelets for Computer Graphics: A Primer  Part 2
 IEEE Computer Graphics and Applications
, 1995
"... this paper. Thanks also go to Ronen Barzel, Steven Gortler, Michael Shantzis, and the anonymous reviewers for their many helpful comments. This work was supported by NSF Presidential and National Young Investigator awards (CCR8957323 and CCR9357790), by NSF grant CDA9123308, by an NSF Graduate Res ..."
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Cited by 98 (1 self)
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this paper. Thanks also go to Ronen Barzel, Steven Gortler, Michael Shantzis, and the anonymous reviewers for their many helpful comments. This work was supported by NSF Presidential and National Young Investigator awards (CCR8957323 and CCR9357790), by NSF grant CDA9123308, by an NSF Graduate Research Fellowship, by the University of Washington Royalty Research Fund (659731), and by industrial gifts from Adobe, Aldus, Microsoft, and Xerox. References
Analysis and Application of Subdivision Surfaces
, 1996
"... Subdivision surfaces are a convenient representation for modeling objects of arbitrary topological type. In this dissertation, we investigate the analysis of a piecewise smooth subdivision scheme, and we apply the scheme to reconstruct objects from nonuniformly sampled data points. Defined as the ..."
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Cited by 65 (0 self)
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Subdivision surfaces are a convenient representation for modeling objects of arbitrary topological type. In this dissertation, we investigate the analysis of a piecewise smooth subdivision scheme, and we apply the scheme to reconstruct objects from nonuniformly sampled data points. Defined as the limit of repeated refinement of a mesh of 3D control points, subdivision surfaces require analysis to establish convergence to a welldefined, tangent plane smooth G1 surface. Recent research has focused on analyzing smooth surface schemes in which the rules are symmetrical about each vertex and edge. However, a scheme for creating surfaces with sharp features has rules that do not exhibit this symmetry. In this dissertation, we extend the use of eigenanalysis and characteristic maps to analyze a piecewise smoot...