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.
Since the original introduction of wavelet theory for square integrable functions defined on the real line, numerous authors have introduced extensions and related transforms for signals on the plane and higher dimensional spaces. By taking separable products of one dimensional wavelets, one can construct orthogonal families of wavelets in any dimension @cite_6 . However, this yields wavelets with often undesirable bias for coordinate axis directions. A large family of alternative multiscale transforms has been developed and used extensively for image processing, including Laplacian pyramids @cite_13 , steerable wavelets @cite_40 , complex dual-tree wavelets @cite_43 , curvelets @cite_38 , and bandlets @cite_30 . Wavelet transforms have also been defined for certain non-Euclidean manifolds, most notably the sphere @cite_51 @cite_8 and other conic sections @cite_7 .
@cite_30 @cite_38 @cite_7 @cite_8 @cite_6 @cite_43 @cite_40 @cite_51 @cite_13
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We introduce a complete parameterization of the family of two-dimensional steerable wavelets that are polar-separable in the Fourier domain under the constraint of self-reversibility. These wavelets are constructed by multiorder generalized Riesz transformation of a primary isotropic bandpass pyramid. The backbone of the transform (pyramid) is characterized by a radial frequency profile function @math , while the directional wavelet components at each scale are encoded by an @math shaping matrix @math , where @math is the number of wavelet channels and @math the order of the Riesz transform. We provide general conditions on @math and @math for the underlying wavelet system to form a tight frame of @math (with a redundancy factor @math ). The proposed framework ensures that the wavelets are steerable and provides new degrees of freedom (shaping matrix @math ) that can be exploited for designing specific wavelet systems. It encompasses many known transforms as part... 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. In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are—unlike more traditional transforms like wavelets—able to efficiently handle data with features along edges. The main result in Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009) confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions ψ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution f with respect to the shearlet ψ can resolve the wavefront set of f. We demonstrate that the same result can be verified under much weaker assumptions on ψ, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for ( L^2( R ^2) ) from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure. A recently developed approach, called “empirical wavelet transform,” aims to build one-dimensional (1D) adaptive wavelet frames accordingly to the analyzed signal. In this paper, we present several extensions of this approach to two-dimensional (2D) signals (images). We revisit some well-known transforms (tensor wavelets, Littlewood--Paley wavelets, ridgelets, and curvelets) and show that it is possible to build their empirical counterparts. We prove that such constructions lead to different adaptive frames which show some promising properties for image analysis and processing.