Кіріспе
Математиканың әр түрлі салаларында қолданылатын интегралды түрлендіру. Қосу түрлендіруі. Математикада Нильс Хенрик Абельдің есімімен аталған Абель түрлендіруі, көбінесе сфералық немесе осьтік симметриялық функцияларды талдау үшін қолданылатын интегралды түрлендіру. f(r) функциясының Абель түрлендіруі былай беріледі:
Integral transform used in various branches of mathematics
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
[формула]
Integral transform used in various branches of mathematics
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
f(r) функциясы 1/r-ден тез кемісе, кері Абель түрлендіруі былай беріледі:
Integral transform used in various branches of mathematics
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
[формула]
Integral transform used in various branches of mathematics
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
Бейне талдауында алдыңғы Абель түрлендіруі оптикалық тұрғысынан жұқа, осьтік симметриялық эмиссиялық функцияны жазықтыққа проекциялау үшін қолданылады, ал кері Абель түрлендіруі осы эмиссиялық функцияның проекциясы (яғни, сканерлеу немесе фотосурет) берілгенде эмиссиялық функцияны есептеу үшін қолданылады. Цилиндрлі жалынның немесе қалампаның жұтылу спектроскопиясында алдыңғы Абель түрлендіруі – жалын ортасынан ең жақын қашықтықтағы y сәуле бойымен интеграцияланған жұтылу, ал кері Абель түрлендіруі орталықтан r қашықтықтағы жергілікті жұтылу коэффициентін береді. Абель түрлендіруі осьтік симметриялық геометриялармен шектеледі. Көбірек жалпы асимметриялық жағдайларда алгебралық реконструкция техникасы (ART), максималды ықтималдық күтуді максимизациялау (MLEM), сүзгіленген кері проекция (FBP) сияқты жалпыға бағдарланған реконструкция алгоритмдерін қолдану керек. Соңғы жылдары кері Абель түрлендіруі (және оның варианттары) фотофрагменттік иондық бейнелеу және фотоэлектрондық бейнелеудегі деректерді талдаудың негізгі құралына айналды. Кері Абель түрлендіруінің соңғы ең маңызды кеңейтулерінің қатарында фотоэлектрондық және фотоиондық бейнелерді талдаудың "сарымсақ қабығын аршу" және "негіздік жиынтық кеңейту" (BASEX) әдістері бар.
Integral transform used in various branches of mathematics
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
summation transformation
In mathematics, the Abel transform, 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
Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by
In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse 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 transform (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 transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
Радон трансформациясына қатынасы
Абель трансформациясын изотроптық 2D функция f(r) Радон трансформациясы ретінде қарастыруға болады. f(r) изотропты болғандықтан, оның Радон трансформациясы көру осінің әртүрлі бұрыштарында бірдей болады. Осылайша, Абель трансформациясы тек көру осі бойындағы қашықтықтың ғана функциясы болып табылады.
Abel transform can be viewed as the Radon transform of an isotropic 2D function f(r). As f(r) is isotropic, its Radon transform is the same at different angles of the viewing axis. Thus, the Abel transform is a function of the distance along the viewing axis only.