Real-space refinement of atomic models in macromolecular crystallography and cryo-electron microscopy fits a model to a map obtained with experimental data. To do so, the atomic model is converted into a map of limite...
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Real-space refinement of atomic models in macromolecular crystallography and cryo-electron microscopy fits a model to a map obtained with experimental data. To do so, the atomic model is converted into a map of limited resolution and then this map is compared quantitatively with the experimental one. For an appropriate comparison, the atomic contributions comprising the model map should reflect the resolution of the experimental map and the atomic displacement parameter (ADP) values. Such contributions are spherically symmetric oscillating functions, different for chemically different kinds of atoms, different ADPs and different resolution values, and their derivatives with respect to atomic parameters rule the model refinement. For given parameter values, every contribution may be calculated numerically using two Fourier transforms, which is highly time consuming and makes calculation of the respective derivatives problematic. Alternatively, for an atom of each required type its contribution can be expressed in an analytical form as a sum of specially designed terms. Each term is different from zero essentially inside a spherical shell, and changing the ADP value does not change its form but rather changes the value of one of its arguments. In general, these terms become a convenient tool for the decomposition of oscillating spherically symmetric functions. This work describes the algorithms and respective software, named dec3D, to carry out such a shell decomposition for density contributions of different kinds of atoms and ions.
The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals integral(2 pi)(0) e(iwx) phi(x)dx in the Sobolev space (H) over tilde (m)(2). Here, (H) o...
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The present work is devoted to the construction of optimal quadrature formulas for the approximate calculation of the integrals integral(2 pi)(0) e(iwx) phi(x)dx in the Sobolev space (H) over tilde (m)(2). Here, (H) over tilde (m)(2) is the Hilbert space of periodic and complex-valued functions whose m-th generalized derivatives are square-integrable. Here, firstly, in order to obtain an upper bound for the error of the quadrature formula, the norm of the error functional is calculated. For this, the extremal function of the considered quadrature formula is used. By minimizing the norm of the error functional with respect to the coefficients, an optimal quadrature formula is then obtained. Using the explicit form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula is calculated. The convergence of the constructed optimal quadrature formula is investigated, and it is shown that the rate of convergence of the optimal quadrature formula is O(h(m)) for vertical bar w vertical bar < N and O(vertical bar w vertical bar(-m)) for vertical bar w vertical bar >= N. Finally, we present numerical results of comparison for absolute errors of the optimal quadrature formula with the exp(iwx) weight in the case m = 2 and the Midpoint formula. There, one can see the advantage of the optimal quadrature formulas.
Numerical methods designed for the integration of oscillating functions are compared. The methods are applied to a quasi-three-dimensional electrodynamic problem.
Numerical methods designed for the integration of oscillating functions are compared. The methods are applied to a quasi-three-dimensional electrodynamic problem.
These last few years, image decomposition algorithms have been proposed to split an image into two parts: the structures and the textures. These algorithms are not adapted to the case of noisy images because the textu...
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These last few years, image decomposition algorithms have been proposed to split an image into two parts: the structures and the textures. These algorithms are not adapted to the case of noisy images because the textures are corrupted by noise. In this paper, we propose a new model which decomposes an image into three parts (structures, textures and noise) based on a local regularization scheme. We compare our results with the recent work of Aujol and Chambolle. We finish by giving another model which combines the advantages of the two previous ones.
This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S...
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This paper is devoted to the modeling of real textured images by functional minimization and partial differential equations. Following the ideas of Yves Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, we decompose a given (possible textured) image f into a sum of two functions u + v, where u is an element of BV is a function of bounded variation (a cartoon or sketchy approximation of f), while v is a function representing the texture or noise. To model v we use the space of oscillating functions introduced by Yves Meyer, which is in some sense the dual of the BV space. The new algorithm is very simple, making use of differential equations and is easily solved in practice. Finally, we implement the method by finite differences, and we present various numerical results on real textured images, showing the obtained decomposition u + v, but we also show how the method can be used for texture discrimination and texture segmentation.
Following a brief description of this After a brief examination of the method of analysis type of antenna and of the equation to be solved, the author presents some graphic representations of the oscillatory two-dimen...
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Following a brief description of this After a brief examination of the method of analysis type of antenna and of the equation to be solved, the author presents some graphic representations of the oscillatory two-dimensional integrands and compares various numerical integration methods. Résumé: Après une brève description de ce type d'antenne et des équations à résoudre nous donnons quelques visualisations des intègrants oscillants bidimensionnels ainsi que des comparaisons entre diverses méthodes classiques d'intègration numérique.
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