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In addition to generalizing to ''N'' dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis.<ref name="NgFourierSlicePhotography">{{cite journal |last = Ng |first = Ren |title = Fourier Slice Photography |journal = ACM Transactions on Graphics |year = 2005 |url = https://graphics.stanford.edu/papers/fourierphoto/fourierphoto-600dpi.pdf |volume = 24 |issue = 3 |pages = 735–744 |doi = 10.1145/1073204.1073256 }}</ref> For convenience of notation, we consider the change of basis to be represented as ''B'', an ''N''-by-''N'' invertible matrix operating on ''N''-dimensional column vectors. Then the '''generalized Fourier-slice theorem''' can be stated as
: <math>F_m P_m B = S_m \frac{B^{-T}}{|B^{-T}|} F_N
where <math>B^{-T}=(B^{-1})^T</math> is the transpose of the inverse of the change of basis transform.
== Proof in two dimensions ==
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