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342
Multiscale Representations of Markov Random Fields
 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL 41. NO 12. DECEMBER 1993
, 1993
"... Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this pap ..."
Abstract

Cited by 85 (27 self)
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Recently, a framework for multiscale stochastic modeling was introduced based on coarsetofine scalerecursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this paper, we show that this model class is also quite rich. In particular, we describe how 1D Markov processes and 2D Markov random fields (MRF’s) can be represented within this framework. The recursive structure of 1D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2D MRF’s are well known to be very difficult to analyze due to their noncausal structure, and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification. In contrast, our multiscale representations are based on scalerecursive models and thus lead naturally to scalerecursive algorithms, which can be substantially more efficient computationally than those associated with MRF models. In 1D, the multiscale representation is a generalization of the midpoint deflection construction of Brownian motion. The representation of 2D MRF’s is based on a further generalization to a “midline ” deflection construction. The exact representations of 2D MRF’s are used to motivate a class of multiscale approximate MRF models based on onedimensional wavelet transforms. We demonstrate the use of these latter models in the context of texture representation and, in particular, we show how they can be used as approximations for or alternatives to wellknown MRF texture models.
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 84 (46 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 80 (21 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Composite Wavelet Bases for Operator Equations
 MATH. COMP
, 1996
"... This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary ..."
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Cited by 77 (21 self)
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This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudodifferential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Diffusion Wavelets
, 2004
"... We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their ..."
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Cited by 72 (12 self)
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We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. In this case our construction can be viewed as a farreaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the nonstandard wavelet representation of CalderónZygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical LittlewoodPaley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.
Wavelet transforms versus Fourier transforms
 Department of Mathematics, MIT, Cambridge MA
, 213
"... Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transfo ..."
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Cited by 71 (2 self)
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Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higherorder wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including highdefinition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: W(x)
Simultaneous Noise Suppression and Signal Compression using a Library of Orthonormal Bases and the Minimum Description Length Criterion
 WAVELETS IN GEOPHYSICS
, 1994
"... We describe an algorithm to estimate a discrete signal from its noisy observation, using a library of orthonormal bases (consisting of various wavelets, wavelet packets, and local trigonometric bases) and the informationtheoretic criterion called minimum description length (MDL). The key to effecti ..."
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Cited by 69 (3 self)
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We describe an algorithm to estimate a discrete signal from its noisy observation, using a library of orthonormal bases (consisting of various wavelets, wavelet packets, and local trigonometric bases) and the informationtheoretic criterion called minimum description length (MDL). The key to effective random noise suppression is that the signal component in the data may be represented efficiently by one or more of the bases in the library, whereas the noise component cannot be represented efficiently by any basis in the library. The MDL criterion gives the best compromise between the fidelity of the estimation result to the data (noise suppression) and the efficiency of the representation of the estimated signal (signal compression): it selects the "best" basis and the "best" number of terms to be retained out of various bases in the library in an objective manner. Because of the use of the MDL criterion, our algorithm is free from any parameter setting or subjective judgments. This ...
The Wavelet Element Method Part I: Construction and Analysis
, 1997
"... . The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some IR n . The domain of interest is split into subdomains which are mapped to a simple re ..."
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Cited by 65 (17 self)
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. The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some IR n . The domain of interest is split into subdomains which are mapped to a simple reference domain, here ndimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way, that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimension n in order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and...
The Curvelet Representation of Wave Propagators is Optimally Sparse
, 2004
"... This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shape ..."
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Cited by 60 (13 self)
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This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ≈ length 2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. • It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e. faster than any negative polynomial), • and wellorganized in the sense that the very few nonnegligible entries occur near a few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows—hence the diagonal shifts in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
Multiresolution representations using the autocorrelation functions of compactly supported wavelets
 IEEE Trans. Signal Processing
, 1993
"... CT 06520 0 ..."