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Das Zentralschnitt-Theorem (auch Fourier-S … Das Zentralschnitt-Theorem (auch Fourier-Schnitt-Theorem) ist ein Theorem aus dem Bereich der Signaltheorie. Es besagt, dass die Projektion einer Funktion in der Richtung die eindimensionale Fourier-Transformation des Schnitts durch in der Richtung ist, wobei bzw. die mit bzw. korrespondierenden Raumfrequenzen sind. Der Schnitt geht dabei stets durch den Ursprung im Fourier-Raum. Das Theorem geht auf den australischen Physiker Ronald Bracewell zurück, der es im Jahr 1956 zunächst im Bereich der Radioastronomie angewandt hatte.reich der Radioastronomie angewandt hatte.
, In mathematics, the projection-slice theor … In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
* Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection.
* Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if
* F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above,
* P1 is the projection operator (which projects a 2-D function onto a 1-D line),
* S1 is a slice operator (which extracts a 1-D central slice from a function), then This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medicalCT scans where a "projection" is an x-rayimage of an internal organ. The Fourier transforms of these images areseen to be slices through the Fourier transform of the 3-dimensionaldensity of the internal organ, and these slices can be interpolated to buildup a complete Fourier transform of that density. The inverse Fourier transformis then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem.ell in 1956 for a radio-astronomy problem.
, Em matemática, o teorema da projeção de fa … Em matemática, o teorema da projeção de fatia, teorema da fatia de Fourier ou ainda teorema da fatia central em duas dimensões estabelece que os resultados dos seguintes dois cálculos são iguais:
* Tomando-se uma função bidimensional f(x,y), circularmente simétrica, projetada sobre uma linha (monodimensional), e realizando uma transformada de Fourier desta projeção.
* Tomando-se a mesma função, mas realizando uma transformada de Fourier bidimensional primeiro, e então fatiando-a através de sua origem, a qual é paralela à linha de projeção. Em termos de operador, se
* F1 e F2 são operadores da transformada de Fourier mono- e bidimensionais mencionados acima,
* P1 é o operador de projeção (o qual projeta uma função 2-D em uma linha 1-D) e
* S1 é o operador "fatiador" (o qual extrai uma fatia 1-D de uma função 2-D), então: Como f(x,y) é circularmente simétrica, pode-se também escrever f(r), onde r é o de um ponto qualquer (x,y). Esta ideia pode ser estendida para dimensões mais altas.e ser estendida para dimensões mais altas.
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| rdfs:comment |
Das Zentralschnitt-Theorem (auch Fourier-S … Das Zentralschnitt-Theorem (auch Fourier-Schnitt-Theorem) ist ein Theorem aus dem Bereich der Signaltheorie. Es besagt, dass die Projektion einer Funktion in der Richtung die eindimensionale Fourier-Transformation des Schnitts durch in der Richtung ist, wobei bzw. die mit bzw. korrespondierenden Raumfrequenzen sind. Der Schnitt geht dabei stets durch den Ursprung im Fourier-Raum. Das Theorem geht auf den australischen Physiker Ronald Bracewell zurück, der es im Jahr 1956 zunächst im Bereich der Radioastronomie angewandt hatte.reich der Radioastronomie angewandt hatte.
, Em matemática, o teorema da projeção de fa … Em matemática, o teorema da projeção de fatia, teorema da fatia de Fourier ou ainda teorema da fatia central em duas dimensões estabelece que os resultados dos seguintes dois cálculos são iguais:
* Tomando-se uma função bidimensional f(x,y), circularmente simétrica, projetada sobre uma linha (monodimensional), e realizando uma transformada de Fourier desta projeção.
* Tomando-se a mesma função, mas realizando uma transformada de Fourier bidimensional primeiro, e então fatiando-a através de sua origem, a qual é paralela à linha de projeção. Em termos de operador, senha de projeção. Em termos de operador, se
, In mathematics, the projection-slice theor … In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:
* Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection.
* Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if then This idea can be extended to higher dimensions.idea can be extended to higher dimensions.
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Projection-slice theorem
, Zentralschnitt-Theorem
, Teorema da projeção de fatia
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