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| In [[mathematics]], the '''projection-slice theorem''' or '''Fourier slice theorem''' in two dimensions states that the results of the following two calculations are equal:
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| * Take a two-dimensional function ''f''('''r'''), [[Projection (mathematics)|project]] it onto a (one-dimensional) line, and do a [[Fourier transform]] of that projection.
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| * 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.
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| In operator terms, if
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| * ''F''<sub>1</sub> and ''F''<sub>2</sub> are the 1- and 2-dimensional Fourier transform operators mentioned above,
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| * ''P''<sub>1</sub> is the projection operator (which projects a 2-D function onto a 1-D line) and
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| * ''S''<sub>1</sub> is a slice operator (which extracts a 1-D central slice from a function),
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| then:
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| :<math>F_1 P_1=S_1 F_2\,</math>
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| This idea can be extended to higher dimensions.
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| This theorem is used, for example, in the analysis of medical
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| [[computed axial tomography|CT]] scans where a "projection" is an x-ray
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| image of an internal organ. The Fourier transforms of these images are
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| seen to be slices through the Fourier transform of the 3-dimensional
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| density of the internal organ, and these slices can be interpolated to build
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| up a complete Fourier transform of that density. The inverse Fourier transform
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| is then used to recover the 3-dimensional density of the object. This technique was first derived by Bracewell (1956) for a radio astronomy problem.
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| == The projection-slice theorem in ''N'' dimensions ==
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| In ''N'' dimensions, the '''projection-slice theorem''' states that the
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| [[Fourier transform]] of the '''projection''' of an ''N''-dimensional function
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| ''f''('''r''') onto an m-dimensional [[Euclidean space|linear submanifold]]
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| is equal to an m-dimensional '''slice''' of the ''N''-dimensional Fourier transform of that | |
| function consisting of an ''m''-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms:
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| :<math>F_mP_m=S_mF_N.\,</math>
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| == Proof in two dimensions ==
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| [[Image:ProjectionSlice.png|frame|center|512px|A graphical illustration of the projection slice theorem in two dimensions. ''f''('''r''') and ''F''('''k''') are 2-dimensional Fourier transform pairs. The projection of ''f''('''r''') onto the ''x''-axis is the integral of ''f''('''r''') along lines of sight parallel to the ''y''-axis and is labelled ''p''(''x''). The slice through ''F''('''k''') is on the ''k''<sub>''x''</sub> axis, which is parallel to the ''x'' axis and labelled ''s''(''k''<sub>''x''</sub>). The projection-slice theorem states that ''p''(''x'') and ''s''(''k''<sub>''x''</sub>) are 1-dimensional Fourier transform pairs.]]
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| The projection-slice theorem is easily proven for the case of two dimensions.
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| Without loss of generality, we can take the projection line to be the ''x''-axis.
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| There is no loss of generality because using a shifted and rotated line the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform. Rotated function is the Fourier pair of the rotated Fourier transform, this completes the explanation.
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| If ''f''(''x'', ''y'') is a two-dimensional function, then the projection of ''f''(''x'', ''y'') onto the ''x'' axis is ''p''(''x'') where
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| :<math>p(x)=\int_{-\infty}^\infty f(x,y)\,dy.</math>
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| The Fourier transform of <math>f(x,y)</math> is
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| :<math> | |
| F(k_x,k_y)=\int_{-\infty}^\infty \int_{-\infty}^\infty
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| f(x,y)\,e^{-2\pi i(xk_x+yk_y)}\,dxdy.
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| </math>
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| The slice is then <math>s(k_x)</math>
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| :<math>s(k_x)=F(k_x,0)
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| =\int_{-\infty}^\infty \int_{-\infty}^\infty f(x,y)\,e^{-2\pi ixk_x}\,dxdy
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| </math>
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| :::<math>=\int_{-\infty}^\infty | |
| \left[\int_{-\infty}^\infty f(x,y)\,dy\right]\,e^{-2\pi ixk_x} dx
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| </math>
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| :::<math>=\int_{-\infty}^\infty p(x)\,e^{-2\pi ixk_x} dx
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| </math>
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| which is just the Fourier transform of ''p''(''x''). The proof for higher dimensions is easily generalized from the above example. | |
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| == The FHA cycle ==
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| If the two-dimensional function ''f''('''r''') is circularly symmetric, it may be represented as ''f''(''r'') where ''r'' = |'''r'''|. In this case the projection onto any projection line
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| will be the [[Abel transform]] of ''f''(''r''). The two-dimensional [[Fourier transform]]
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| of ''f''('''r''') will be a circularly symmetric function given by the zeroth order [[Hankel transform]] of ''f''(''r''), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or
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| :<math>F_1A_1=H\,</math>
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| where ''A''<sub>1</sub> represents the Abel transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, ''F''<sub>1</sub> represents the 1-D Fourier transform
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| operator, and ''H'' represents the zeroth order Hankel transform operator.
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| == Extension to n-dimension signal ==
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| The n-dimensional projection-slice theorem was developed by Ng in 2005 for the application of digital refocusing of light field photographs.
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| == See also ==
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| *[[Radon transform#Relationship with the Fourier transform|Relationship with the Fourier Transform]]
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| == References ==
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| * {{cite journal | author=Bracewell, R.N. | title=Numerical Transforms | journal=Science | year=1990 | volume=248 | pages=697–704 | doi=10.1126/science.248.4956.697 | pmid=17812072 | issue=4956}}
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| * {{cite journal | author=Bracewell, R.N. | title= Strip Integration in Radio Astronomy| journal=Aust. J. Phys. | year=1956 | volume=9 | pages=198 | doi= 10.1071/PH560198| pmid= | issue=2}}
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| * {{cite book | author=Gaskill, Jack D. | title=Linear Systems, Fourier Transforms, and Optics | publisher=John Wiley & Sons, New York | year=1978 | isbn =0-471-29288-5 }}
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| * {{cite journal | author=Ng, R. | title = Fourier Slice Photography | journal=ACM Transactions on Graphics | year = 2005 | volume=24 | issue=3 | pages=735–744 | doi=10.1145/1073204.1073256}}
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| [[Category:Fourier analysis]]
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| [[Category:Integral transforms]]
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| [[Category:Image processing]]
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| [[Category:Theorems in analysis]]
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The child of your , knows patience and determination are key elements when it comes to a successful occupation- . His first album, Keep Me, generated the best reaches “All My Friends “Country and Say” Gentleman,” while his work, Doin’ Issue, discovered the artist-three right No. 8 single people: In addition Phoning Is actually a Great Point.”
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