Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets

Bacchelli, B; Bozzini, M; Rabut, Christophe; Varas, M. Leonor

doi:10.1016/j.acha.2004.11.007

Cited by 28 publications

(38 citation statements)

References 21 publications

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“…Real-valued polyharmonic wavelets (i.e., ) have been proposed both for the quincunx subsampling matrix [28], which corresponds to a two-channel design in 2-D, and for the dyadic subsampling matrix [35], [36]. So far, complex polyharmonic spline wavelets have only been specified explicitly for the simpler quincunx case [29], which corresponds to a classical twochannel design with a single wavelet generator.…”

Section: Operator-like Waveletsmentioning

confidence: 99%

“…Our proposal is to consider appropriate linear combinations of functions within , which corresponds to a filter at scale 0. Here, we pick the Marr smoothing function to be the polyharmonic B-spline (36) This smoothing function closely resembles a Gaussian due to the properties of the polyharmonic B-spline. In fact, it quickly converges to a Gaussian with standard deviation as increases [28].…”

Section: ) Pyramid Structure and Dyadic Subsampling Schemementioning

confidence: 99%

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Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Ville

Unser

2008

IEEE Trans. on Image Process.

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Abstract-Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of , irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.Index Terms-Feature extraction, primal sketch, steerable filters, wavelet design. MULTISCALE transforms are powerful tools for signal and image processing, computer vision, and for modeling biological vision. A prominent example is the 1-D wavelet transform, which acts as a multiscale version of an th-order derivative operator, where is the number of vanishing moments of the wavelet [1]. Its extension to multiple dimensions and to 2-D, in particular, is typically achieved by forming tensorproduct basis functions. However, such separable wavelets are not well matched to the singularities occuring in images such as lines and edges which can be arbitrarily oriented and even curved. Consequently, there has been a considerable research effort in developing alternative multiscale transforms that are better tuned to the geometry of natural images. Notable examples of these "geometrical x-lets" include biologically-inspired 2-D Gabor transforms [2] [24]. The derivation of the filters is not based on wavelets, but rather obtained through a numerical optimization process. Special constraints are imposed to ensure that the frequency response of the filters is essentially polar-separable and that the decomposition is simple to invert numerically (approximate tight-frame property).In this paper, we present an alternative approach based on an explicit analytical and spline-based formulation of complex wavelet bases of . Special care is given to design basis functions that best match the properties of the visual system, in accordance with Marr's theory of early vision [25]. To motivate our construction, we specify a number of properties that are highly desirable and that are fulfilled, sometimes implicitly, by a number of classical image-processing algorithms.• Invariance. We aim at invariance with respect to elementary geometric operations such as translation, scaling, and rotation. Traditional wavelet transforms only satisfy these propert...

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Section: Operator-like Waveletsmentioning

confidence: 99%

Section: ) Pyramid Structure and Dyadic Subsampling Schemementioning

confidence: 99%

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Ville

Unser

2008

IEEE Trans. on Image Process.

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“…This combined with the fact that the polyharmonic splines for provide a valid multiresolution analysis of yields the theorem below, which is a slight extension of previously published results. Specifically, Bacchelli et al [35], [38] investigated the dyadic, nonfractional case (i.e., and ), while Van De Ville et al [34] (23) which are termed primal or dual depending on the type of reconstruction wavelets. The complementary wavelet functions are the unique duals of in the sense that they satisfy the biorthogonality property: with .…”

Section: Propositionmentioning

confidence: 99%

“…Following Bacchelli et al [35], we define the symmetric, Laplacian-like spline wavelet of (fractional) order (19) where is the polyharmonic spline interpolator of order [cf. (17)] and where is a given admissible 2-D dilation 4 A Riesz basis is a fundamental concept in functional analysis that was introduced by the Hungarian mathematician Frigyes Riesz; it has not much to do with the Riesz transform that is due to Marcel Riesz (Frigyes' younger brother), except perhaps that the complex version of the Riesz transform maps a Riesz basis into another one.…”

Section: Laplacian-like Waveletsmentioning

confidence: 99%

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Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

Unser

Sage

Ville

2009

IEEE Trans. on Image Process.

180

205

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Abstract-The monogenic signal is the natural 2-D counterpart of the 1-D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.

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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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Decomposition and reconstruction of multidimensional signals using polyharmonic pre-wavelets

Cited by 28 publications

References 21 publications

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

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