1,721,038 research outputs found
Magnetic Tomography by Random Spatial Sampling
No full textThe Magnetic Tomography (MT) is an imaging technique that aims at reconstructing an unknown electric current distribution flowing within a volume conductor from the measurements of its magnetic field
in the outer space. Among the other imaging techniques, MT has the advantage to be noninvasive and to have a high temporal resolution. For these reasons MT has applications in several fields, from geophysics to archeology, from nondestructive analysis of structures to medical tomography.
MT devices do not give immediately an image of the electric current that flows in the conductor under study. Actually, to reconstruct the unknown current distribution from the magnetic data an highly ill-posed and ill-conditioned inverse problem has to be solved.
We propose to solve the MT inverse problem by an inversion method based on the random sampling of the source space. The main advantage of the method is the dimensionality reduction that makes the method fast and the storage requirements very low.
Moreover, the method can be easily applied to conductors of any shape.
Some numerical tests showing the performances of the method on both synthetic and real data will be shown
Bell-shaped nonstationary refinable ripplets
Full text linkWe study the approximation properties of the class of nonstationary refinable ripplets introduced in Gori and Pitolli (2008). These functions are the solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variationdiminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions, convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the C2 nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented
Ternary shape-preserving subdivision schemes
Full text linkWe analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order
Bases for shape preserving curves
Full text linkThe shape preserving properties of a curve in depend on the properties of the function basis we use in its representation. Both sign consistent and totally positive bases have shape preserving properties useful in Computer Aided Geometric Design. Some of the most useful properties are lightened and some examples of shape preserving bases are given
Subdivision schemes for shape preserving approximations
No full textWe use subdivision schemes with general dilation to efficiently evaluate shape preserving approximations. To fulfill our goal the refinement rules of the schemes are obtained by the refinement masks associated to refinable ripplets, i.e. refinable functions whose integer translates form a variation diminishing basis
Totally positive refinable functions with general dilation M
Full text linkWe construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given
On the numerical solution of fractional boundary value problems by a spline quasi-interpolant operator
Full text linkBoundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate
Optimal B-spline bases for the numerical solution of fractional differential problems
Full text linkEfficient numerical methods to solve fractional differential problems are particularly required in order to approximate accurately the nonlocal behavior of the fractional derivative. The aim of the paper is to show how optimal B-spline bases allow us to construct accurate numerical methods that have a low computational cost. First of all, we describe in detail how to construct optimal B-spline bases on bounded intervals and recall their main properties. Then, we give the analytical expression of their derivatives of fractional order and use these bases in the numerical solution of fractional differential problems. Some numerical tests showing the good performances of the bases in solving a time-fractional diffusion problem by a collocation-Galerkin method are also displayed
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