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Image Restoration and Reconstruction
Published in Scott E. Umbaugh, Digital Image Processing and Analysis, 2017
The Fourier-slice theorem, also called the Central-slice theorem and the Projection-slice theorem, establishes a link between the 1D Fourier transform of a projection and the 2D Fourier transform of the scene being imaged. The Fourier-slice theorem states that the 1D Fourier transform of a projection is equivalent to a slice through the 2D Fourier transform of the image at the same angle as the projection. This is illustrated in Figure 9.7-5. Here we see the image on the left and its Fourier spectrum on the right. A projection at an angle θ is shown as a red line. In the figure, the 1D Fourier transform of the projection is shown as a slice (line) through the 2D spectrum of the image. This is the essence of the Fourier-slice theorem.
Advanced Applications of Volume Visualization Methods in Medicine
Published in Alexander D. Poularikas, Stergios Stergiopoulos, Advanced Signal Processing, 2017
Georgios Sakas, Grigorios Karangelis, Andreas Pommert
While both surface- and volume-based rendering are operated in a 3D space, 3D images may also be created from other data representations. One such method is frequency domain rendering, which creates 3D images in Fourier space, based on the Fourier projection-slice theorem.112 This method is very fast, but the resulting images are somewhat similar to X-ray images, lacking real depth information.
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Published in Borko Furht, Darko Kirovski, Multimedia Security Handbook, 2004
Figure 11.8 shows the line integral procedure and an example the Radon transform of Lena. The projection slice theorem [14] states that the Fourier transform of the projection of an image onto a line is the 2-D Fourier transform of the image evaluated along a radial line. From the theorem, we can use 2-D Fourier transform instead of the Radon transform during implementation.
Generalised plane strain embedded in three-dimensional anisotropic elasticity
Published in Philosophical Magazine, 2021
Markus Lazar, Helmut O. K. Kirchner
In 3 dimensions, the projection-slice theorem states that the 2-dimensional Fourier transform of the projection of a 3-dimensional function onto a 2-dimensional linear submanifold is equal to a 2-dimensional slice of the 3-dimensional Fourier transform of that function consisting of a 2-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms, if and are the 2- and 3-dimensional Fourier transform operators, is the projection operator (which projects a 3D function onto a 2D plane), is a slice operator (which extracts a 2D central slice from a 3D function),