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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Abel transform ： ウィキペディア英語版 Abel transform In mathematics, the Abel transform,〔N. H. Abel, Journal fur die reine und angewandte Mathematik, 1, pp. 153-157 (1826)〕 named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function ''f''(''r'') is given by: :$F(y)=2\backslash int\_y^\backslash infty\; \backslash frac\backslash int\_r^\backslash infty\backslash frac\backslash ,\backslash frac\{\backslash sqrt\{y^2-r^2\}\}.$ In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the reverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function. In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as Algebraic Reconstruction Technique (ART), Maximum Likelihood Expectation Maximization (MLEM), Filtered Back-Projection (FBP) algorithms should be employed. In recent years, the inverse Abel transformation (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transformation are the Onion Peeling and BAsis Set Expansion (BASEX) methods of photoelectron and photoion image analysis. == Geometrical interpretation == In two dimensions, the Abel transform ''F''(''y'') can be interpreted as the projection of a circularly symmetric function ''f''(''r'') along a set of parallel lines of sight which are a distance ''y'' from the origin. Referring to the figure on the right, the observer (I) will see :$F(y)=\backslash int\_^\backslash infty\; f\backslash !\backslash left(\backslash sqrt\backslash right)\backslash ,dx$ where ''f''(''r'') is the circularly symmetric function represented by the gray color in the figure. It is assumed that the observer is actually at ''x'' = ∞ so that the limits of integration are ±∞ and all lines of sight are parallel to the ''x''-axis. Realizing that the radius ''r'' is related to ''x'' and ''y'' via ''r''2 = ''x''2 + ''y''2, it follows that :$dx=\backslash frac^\backslash infty\; f(r)\backslash ,dx=2\backslash int\_^\backslash infty\; f(r)\backslash ,dx,\; \backslash ;\backslash ;\; r=\; \backslash sqrt.$ Substituting the expression for ''dx'' in terms of ''r'' and rewriting the integration limits accordingly yields the Abel transform. The Abel transform may be extended to higher dimensions. Of particular interest is the extension to three dimensions. If we have an axially symmetric function ''f''(ρ'',z'') where ρ2 = ''x''2 + ''y''2 is the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the ''z'' axis. Without loss of generality, we can take that plane to be the ''yz''-plane so that :$F(y,z)$ =\int_^\infty f(\rho,z)\,dx =2\int_y^\infty \frac^\infty f(r)\,dx =2\int_s^\infty \frac{\sqrt{r^2-s^2}} which is again, the Abel transform of ''f''(''r'') in ''r'' and ''s''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Abel transform」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.