A fast algorithm for the inversion of Abel’s transform

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• 作者：Enrico De Micheli ^{enrico.demicheli@cnr.it}
• 关键词：Abel inversion ; Legendre polynomials ; Radio occultation ; Plasma emission coefficients ; Inverse problems ; Stability estimates
• 刊名：Applied Mathematics and Computation
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：301
• 期：Complete
• 页码：12-24
• 全文大小：784 K
• 卷排序：301

文摘

We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.