We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian @math . Given a wavelet generating kernel @math and a scale parameter @math , we define the scaled wavelet operator @math . The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on @math , this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing @math . We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
Geller and Mayeli @cite_48 studied a construction for wavelets on compact differentiable manifolds that is formally similar to our approach on weighted graphs. In particular, they define scaling using a pseudodifferential operator @math , where @math is the manifold Laplace-Beltrami operator and @math is a scale parameter, and obtain wavelets by applying this to a delta impulse. They also study the localization of the resulting wavelets, however the methods and theoretical results in their paper are different as they are in the setting of smooth manifolds.
@cite_48
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We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains. Let M be a smooth compact oriented Riemannian manifold, and let ΔM be the Laplace–Beltrami operator on M. Say ( 0 f S ( R ^+) ) , and that f (0) = 0. For t > 0, let Kt(x, y) denote the kernel of f (t2 ΔM). We show that Kt is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t2Δ) on ( R ^n ) . We define continuous ( S )-wavelets on M, in such a manner that Kt(x, y) satisfies this definition, because of its localization near the diagonal. Continuous ( S )-wavelets on M are analogous to continuous wavelets on ( R ^n ) in ( S ) ( ( R ^n )). In particular, we are able to characterize the Holder continuous functions on M by the size of their continuous ( S )-wavelet transforms, for Holder exponents strictly between 0 and 1. If M is the torus ( T^2 ) or the sphere S2, and f (s) = se−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for Kt, one to be used when t is large, and one to be used when t is small. This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j 2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior. Abstract We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace–Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand, if the shrinking at the vertex parts of the manifold is sufficiently slower comparing to that of the edge parts, the limiting spectrum corresponds to decoupled edges with Dirichlet boundary conditions at the endpoints. At the borderline between the two regimes we have a third possibility when the limiting spectrum can be described by a nontrivial coupling at the vertices.