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Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.


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TL;DR: A new pulse technique for counteracting RF inhomogeneity at high fields is reported, which makes use of the detailed knowledge of the voxels' B1 and B0 amplitude 2D histogram to generate gates where the flip angle is made uniform.
Abstract: A new pulse technique for counteracting RF inhomogeneity at high fields is reported. The pulses make use of the detailed knowledge of the voxels' B(1) and B(0) amplitude 2D histogram to generate, through an optimization procedure, gates where the flip angle is made uniform. Although most approaches to date require the use of parallel transmission, this method does not and therefore offers several advantages. The data necessary for the algorithm to determine an irradiation scheme requires only one transmit B(1) along with a B(0) inhomogeneity measurement. The use of a B(1) and B(0) amplitude 2D histogram instead of their spatial distribution also decreases substantially the complexity of the optimization problem, allowing the algorithm to find an RF solution in less than 30 s. Finally, the optimization procedure is based on an exact calculation and does not use any linear approximation. In this article, the theory behind the method in addition to spoiled gradient echo experimental data at 3T for 3D brain imaging are reported. The images obtained yield a reduction of the standard deviation of the sine of the flip angle by a factor of up to 15 around the desired value, compared to when a standard square pulse calibrated by the scanner is used.

25 citations

Journal ArticleDOI
TL;DR: In this paper, the authors proposed two new numerical schemes for a time discretization of the nonlinear equations describing a quasi-static electromagnetic field in a ferromagnetic material.

25 citations

Journal ArticleDOI
M. Davies1
TL;DR: In this paper, a simplex method of linear programming is proposed for the smoothing of discrete data by fitting functions which are linear in their parameters, such as polynomials and regression planes.
Abstract: SUMMARY A study is made of the fitting of discrete data by linear regression planes and by polynomials according to the criterion of least total absolute deviations. A form of solution is proposed which makes use of the simplex method of linear programming. An example is given to show the practical application of the method. THIS paper is concerned with the smoothing of discrete data by fitting functions which are linear in their parameters, such as polynomials and regression planes. The usual method adopted is the principle of least squares. As is well known, for observations having errors which are uncorrelated and which have constant variance, this method gives unbiased, minimum-variance estimates of the linear parameters, and accords with the maximum likelihood principle when the errors of observation are normally distributed. An alternative and sometimes preferable method would be to minimize the total sum of the absolute deviations of the observations from the approximating function. This method, which was first advocated by Edgeworth (see Bowley, 1928) has been largely neglected in the literature on account of the much greater labour of computation, but with the introduction of the high-speed digital computer this objection loses much of its force. It has been shown by Stiefel (1960) that there is a close connection between Chebyshev approximation (in which the maximum departure is minimized) and the simplex method of linear programming. The purpose of the present paper is to develop a similar correspondence for Edgeworth approximation. 2. COMPARISON WITH LEAST SQUARES

25 citations

Journal ArticleDOI
TL;DR: Theoretically, the proposed Delaunay-based surface reconstruction algorithm is justified by establishing a topological guarantee on the 3D shape-hull with the help of topological rules and the effectiveness of the approach is demonstrated with experimental results on models with sharp features and sparsely distributed point clouds.
Abstract: Given a finite set of points S ? R 2 , we define a proximity graph called as shape-hull graph ( SHG ( S ) ) that contains all Gabriel edges and a few non-Gabriel edges of Delaunay triangulation of S . For any S , SHG ( S ) is topologically regular with its boundary (referred to as shape-hull ( SH )) homeomorphic to a simple closed curve. We introduce the concept of divergent concavity for simple, closed, planar curves based on the alignment of curves in concave portions and discuss various measures to characterize curves having divergent concavity. Under sufficiently dense sampling, we prove that SH ( S ) , where S is sampled from a divergent concave curve Σ D , represents a piece-wise linear approximation of Σ D . We extend this result to provide a sculpting algorithm for closed surface reconstruction from a set of raw samples. The surface is constructed through a repeated elimination of Delaunay tetrahedra subjected to circumcenter and topological constraints. Theoretically, we justify our algorithm by establishing a topological guarantee on the 3D shape-hull with the help of topological rules. We demonstrate the effectiveness of our approach with experimental results on models with sharp features and sparsely distributed point clouds. Compared to existing sculpting approaches for surface reconstruction that require either a parameter tuning or several stages, our approach is simple, non-parametric, single stage and reconstructs topologically correct piece-wise linear approximation for divergent concave surfaces. Delaunay-based surface reconstruction algorithm has been proposed.It is a non-parametric and single stage approach.Theoretical guarantee has been discussed.

25 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202229
202197
2020134
2019124
2018147