Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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TL;DR: A new wavelet-based methodology for representing data on regular grids is introduced and studied in which the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme.
Abstract: A new wavelet-based methodology for representing data on regular grids is introduced and studied. The main attraction of this new "local compression-alignment-modified- prediction (L-CAMP)" methodology is in the way it scales with the spatial dimension, making it, thus, highly suitable for the representation of high dimensional data. The specific highlights of the L-CAMP methodology are three. First, it is computed and inverted by fast algorithms with linear complexity and very small constants; moreover, the constants in the complexity bound decay, rather than grow, with the spatial dimension. Second, the representation is accompanied by solid mathematical theory that reveals its performance in terms of the maximal level of smoothness that is accurately encoded by the representation. Third, the localness of the representation, measured as the sum of the volumes of the supports of the underlying mother wavelets, is extreme. An illustration of this last property is done by comparing the L-CAMP system that is marked in this paper as V with the widely used tensor-product biorthogonal 9/7. Both are essentially equivalent in terms of performance. However, the L-CAMP V has in 10D localness score 575 000 000 000.
11 citations
TL;DR: In this paper, the biorthogonal B-spline wavelet and cubic spline wavelets are used for areal surface analysis and the lifting algorithm is used as a simple tool for design of spline Wavelets.
Abstract: Research into techniques for areal surface analysis has recently included wavelet filtering. The requirements for filtering surface analysis include: linear phase shift; finite pulse response and perfect reconstruction — only biorthogonal wavelets satisfy these conditions. The biorthogonal spline wavelet is considered for areal surface characterisation due to the curvature of the surfaces considered as examples. In this paper, the models and algorithms of biorthogonal B-spline and cubic spline wavelets are introduced. The lifting algorithm is used as a simple tool for design of spline wavelets. Bioengineering surfaces are used to test and compare the two kinds of wavelet filters to obtain the rough surfaces.
10 citations
01 Sep 2019
TL;DR: The study is devoted to biorthogonal wavelets of the class of smoothness on a triangulable space, which is obtained from closed multidimensional cubes or their homeomorphic formations by gluing edges, and the use of wavelet coefficients allows obtaining fairly accurate approximations of the function.
Abstract: The article presents the results of the first part of the work on the development of CAD / CAM / CAE systems for the analysis and construction of fiber laying trajectories for 3D printing of structures made of composite materials. The study is devoted to biorthogonal wavelets of the class of smoothness on a triangulable space, which is obtained from closed multidimensional cubes or their homeomorphic formations by gluing edges. The use of wavelet coefficients allows obtaining fairly accurate approximations of the function. The proposed results can be applied for an approximate solution of the problems of the theory of elasticity.
10 citations
TL;DR: In this paper, it was shown that a 4-dimensional compact manifold with harmonic Weyl tensor must be either locally conformally flat or isometric to a complex projective space, provided that the biorthogonal curvature satisfies a suitable pinching condition.
Abstract: We prove that a 4-dimensional compact manifold \(M^4\) with harmonic Weyl tensor must be either locally conformally flat or isometric to a complex projective space \(\mathbb {CP}^2,\) provided that the biorthogonal (sectional) curvature satisfies a suitable pinching condition. In particular, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional compact manifolds.
10 citations
TL;DR: In this paper, the authors proposed a method to evaluate the weights of resonance structures embedded in a molecular orbital by utilizing singlet-coupling scheme of an electron pair, in which a biorthogonal operator related to Mulliken population was used together with the Boys−Foster (BF) localization scheme.
Abstract: We have recently proposed a method to evaluate the weights of resonance structures embedded in a molecular orbital by utilizing singlet-coupling scheme of an electron pair [J. Phys. Chem. A 2006, 110, 9028]. The method was formulated on the basis of the second quantization, in which a biorthogonal operator related to Mulliken population (MP) was used together with the Boys−Foster (BF) localization scheme. Our method is very easy to use; only a standard localization procedure is required to obtain the resonance weights. In addition, obtained results agreed well with our chemical intuition. In the present Article, the restrictions, namely MP and BF, were removed, and an operator related to Lowdin population (LP) and other various types of localization schemes were employed to examine the generality of the method. We found that computed resonance weights were virtually independent not only on the choice of these combinations but also on basis set. This new finding, the invariant nature in terms of resonance,...
10 citations