利萨曲线上多项式近似方法的吉布斯现象谱滤波及其在MPI中的应用

S. Marchi, W. Erb, F. Marchetti
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引用次数: 26

摘要

利用切比雪夫级数对Lissajous曲线上的采样点进行多项式插值和逼近是磁粒子成像中快速重建图像的有效方法。由于光谱方法的性质,如果底层函数具有不连续,则在重建图像中会出现吉布斯现象。这个问题的一个可能的解决方案是谱滤波方法作用于近似多项式的系数。在这项工作中,在描述了吉布斯现象和经典的一维和多维滤波技术之后,我们提出了一种自适应光谱滤波过程,用于解决这种现象,并改进了对底层函数或图像的近似。在这种自适应滤波技术中,光谱滤波取决于空间点到最近不连续点的距离。我们从理论上、数值模拟以及在磁粉成像中的应用上证明了这种滤波方法的有效性。
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Spectral filtering for the reduction of the Gibbs phenomenon for polynomial approximation methods on Lissajous curves with applications in MPI
Polynomial interpolation and approximation methods on sampling points along Lissajous curves using Chebyshev series is an effective way for a fast image reconstruction in Magnetic Particle Imaging. Due to the nature of spectral methods, a Gibbs phenomenon occurs in the reconstructed image if the underlying function has discontinuities. A possible solution for this problem are spectral filtering methods acting on the coefficients of the approximating polynomial. In this work, after a description of the Gibbs phenomenon and classical filtering techniques in one and several dimensions, we present an adaptive spectral filtering process for the resolution of this phenomenon and for an improved approximation of the underlying function or image. In this adaptive filtering technique, the spectral filter depends on the distance of a spatial point to the nearest discontinuity. We show the effectiveness of this filtering approach in theory, in numerical simulations as well as in the application in Magnetic Particle Imaging.
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CiteScore
1.70
自引率
7.70%
发文量
0
审稿时长
8 weeks
期刊介绍: Dolomites Research Notes on Approximation is an open access journal that publishes peer-reviewed papers. It also publishes lecture notes and slides of the tutorials presented at the annual Dolomites Research Weeks and Workshops, which have been organized regularly since 2006 by the Padova-Verona Research Group on Constructive Approximation and Applications (CAA) in Alba di Canazei (Trento, Italy). The journal publishes, on invitation, survey papers and summaries of Ph.D. theses on approximation theory, algorithms, and applications.
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