The Central Slice Theorem: An Example

Fig. 12.12 from IPMB, showing how to do a projection.

The Projection

Figure 12.12 shows that the projection is an integral of the object along various lines in the direction θ, as a function of displacement perpendicular to each line, x’. The integral becomes

The One-Dimensional Fourier Transform

The next step is to evaluate the one-dimensional Fourier transform of the projection

Two-Dimensional Fourier Transform

To calculate the two-dimensional Fourier transform, we must evaluate the double integral

Select One Direction in Frequency Space

If we want to focus on one direction in frequency space, we must convert to polar coordinates: kx = k cos θ and ky = k sin θ. The result is



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Brad Roth

Brad Roth

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.