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Convergence Of Cascade Algorithms In Sobolev Spaces For Perturbed Refinement Masks
 J. Approx. Theory
, 2000
"... . In this paper the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in term ..."
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. In this paper the convergence of the cascade algorithm in a Sobolev space is considered if the refinement mask is perturbed. It is proved that the cascade algorithm corresponding to a slightly perturbed mask converges. Moreover, the perturbation of the resulting limit function is estimated in terms of that of the masks. x1. Introduction In this paper we are concerned with the following problem: Given a compactly supported multivariate refinable function OE, how does perturbation of its finite refinement mask affect the convergence of the cascade algorithm? Further, if the cascade algorithm for the perturbed mask also converges, how the resulting limit function is related with OE? We say that a compactly supported function OE is Mrefinable if it satisfies a refinement equation OE = X ff2ZZ s a(ff)OE(M \Delta \Gamma ff); (1:1) where the finitely supported sequence a = (a(ff)) ff2ZZ s is called the refinement mask. The s \Theta s matrix M is called a dilation matrix. We suppo...
Localization of stability and pframe in the Fourier domain
 In Proceeding of Special Session on Wavelets, Frames and Operator Theory
"... Abstract. In this paper, we introduce and study the localization of stability and pframe properties of a finitely generated shiftinvariant system in the Fourier domain, and then provide more information to that shift invariant system. Especially for a shiftinvariant system generated by finitely man ..."
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Abstract. In this paper, we introduce and study the localization of stability and pframe properties of a finitely generated shiftinvariant system in the Fourier domain, and then provide more information to that shift invariant system. Especially for a shiftinvariant system generated by finitely many compactly supported functions, we show that it has pframe property at almost all frequencies, and that it either has stability property at almost all frequencies or does not have stability property at all frequencies. 1.
Localization of Calderon Convolution in the Fourier Domain
"... Abstract. In this paper, we introduce and study the localization of Calderon convolution for a finitely generated shiftinvariant space in the Fourier domain. We say that a linear space V of functions on R d is shiftinvariant if f ∈ V implies that f( · − k) ∈ V for all k ∈ Z d. For instance, the ..."
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Abstract. In this paper, we introduce and study the localization of Calderon convolution for a finitely generated shiftinvariant space in the Fourier domain. We say that a linear space V of functions on R d is shiftinvariant if f ∈ V implies that f( · − k) ∈ V for all k ∈ Z d. For instance, the space of all polynomials of degree at most N, the space of all pintegrable functions, and the space of all bandlimited functions in L 2 are shiftinvariant spaces.
How Many Holes Can Locally Linearly Independent Refinable Function Vectors Have?
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A HARMONIC ANALYSIS VIEW ON NEUROSCIENCE IMAGING
"... ABSTRACT. After highlighting some of the current trends in neuroscience imaging, this work studies the approximation errors due to varying directional aliasing, arising when 2D or 3Dimages are subjected to the action of orthogonal transformations. Such errors are common in 3Dimages of neurons acqu ..."
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ABSTRACT. After highlighting some of the current trends in neuroscience imaging, this work studies the approximation errors due to varying directional aliasing, arising when 2D or 3Dimages are subjected to the action of orthogonal transformations. Such errors are common in 3Dimages of neurons acquired by confocal microscopes. We also present an algorithm for the construction of synthetic data (computational phantoms) for the validation of algorithms for the morphological reconstruction of neurons. Our approach delivers synthetic data that have a very high degree of fidelity with respect to their groundtruth specifications. 1. OVERTURE What is the substance of knowledge and memory? These fundamental questions have been at the center of philosophical debate for over three millenia, but only during the last fifty years our understanding of these essential human cognitive functions is finally becoming concrete. The quest for answers takes us back to the philosopher Plato (424/423 BC –348/347 BC) who, in the dialogue “Theaetetus”, written circa 360 B.C. when Athens’ glory was in decline amidst the Peloponnesian war, attempts to define knowledge from a philosophical viewpoint. In the dialogue, Euclid (not the famous geometer from Alexandria)