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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 377 (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...
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 207 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Interpolating Wavelet Transform
, 1992
"... We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function. ..."
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Cited by 127 (13 self)
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We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function.
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 127 (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
CHARMS: A Simple Framework for Adaptive Simulation
 ACM Transactions on Graphics
, 2002
"... Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, bui ..."
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Cited by 123 (9 self)
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Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piecewise linear, higher order Bsplines, Loop subdivision, etc.). The (un)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thinshell animations.
Nonlinear wavelet transforms for image coding via lifting
, 2003
"... We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We al ..."
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Cited by 91 (3 self)
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We investigate central issues such as invertibility, stability, synchronization, and frequency characteristics for nonlinear wavelet transforms built using the lifting framework. The nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework. We also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels. Preliminary compression results for model and realworld images demonstrate the promise of our techniques.
A chronology of interpolation: From ancient astronomy to modern signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into histo ..."
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Cited by 61 (0 self)
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This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolutionbased interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.
Evaluation and Design of Filters Using a Taylor Series Expansion
 IEEE Transactions on Visualization and Computer Graphics
, 1997
"... We describe a new method for analyzing, classifying, and evaluating filters that can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. Our analysis shows the need and derives the m ..."
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Cited by 60 (6 self)
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We describe a new method for analyzing, classifying, and evaluating filters that can be applied to interpolation filters as well as to arbitrary derivative filters of any order. Our analysis is based on the Taylor series expansion of the convolution sum. Our analysis shows the need and derives the method for the normalization of derivative filter weights. Under certain minimal restrictions of the underlying function, we are able to compute tight absolute error bounds of the reconstruction process. We demonstrate the utilization of our methods to the analysis of the class of cubic BCspline filters. As our technique is not restricted to interpolation filters, we are able to show that the CatmullRom spline filter and its derivative are the most accurate reconstruction and derivative filters, respectively, among the class of BCspline filters. We also present a new derivative filter which features better spatial accuracy than any derivative BCspline filter, and is optimal within our fra...
Multiresolution representations using the autocorrelation functions of compactly supported wavelets
 IEEE Trans. Signal Processing
, 1993
"... CT 06520 0 ..."
Matrix Refinement Equations: Existence and Uniqueness
 J. Fourier Anal. Appl
, 1996
"... . Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of sca ..."
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Cited by 51 (3 self)
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. Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar refinement equations (where the coefficients c k are scalars) are known. This paper considers analogous questions for matrix refinement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an infinite product of the matrix \Delta = 1 2 P c k with itself. Fundamental differences between solutions of matrix equations and scalar refinement equations are examined. In particular, it is shown that "constrained" solutions of the matrix refinement equation can exist even when the infinite product diverges. The existence of constrained solutions is related to the eigenvalue structure of \Delta; so...