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Equations. More Organized Proof of The Central Slice Theorem. More Organized Proof of The Central Slice Theorem. More Organized Proof of The Central Slice Theorem. The Central Slice Theorem.

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Equations

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Equations l.jpg

Equations


More organized proof of the central slice theorem l.jpg

More Organized Proof of The Central Slice Theorem


More organized proof of the central slice theorem3 l.jpg

More Organized Proof of The Central Slice Theorem


More organized proof of the central slice theorem4 l.jpg

More Organized Proof of The Central Slice Theorem


The central slice theorem l.jpg

The Central Slice Theorem

Consider a 2-dimensional example of an emission imaging system. O(x,y) is the object function, describing the source distribution. The projection data, is the line integral along the projection direction.

The Central Slice Theorem can be seen as a consequence of the separability of a 2-D Fourier Transform.

The 1-D Fourier Transform of the projection is,


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The Central Slice Theorem

The one-dimensional Fourier transformation of a projection obtained at an angle J, is the same as the radical slice taken through the two-dimensional Fourier domain of the object at the same angle.

Radon

transform

2D

FT

1D

FT

´ü▒


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