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I have this question on mathematics forum too, NotationsNotations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my notations in 3D centered fft. Consider the 1D centered fft of a 2D image of size $(N+1)\times (N+1)$, where $N$ is even, along X axis it can be computed using 1D FFT along each row such that, \begin{eqnarray} F^{}_x (r,c) = \sum\limits_{n=-N/2}^{N/2} f(r,n) e^{-i\frac{2\pi c n}{N+1}}, \nonumber \\ -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2 \end{eqnarray} Next we operate on each column such that the 2D centered FFT can be written as, \begin{eqnarray} F^{}_{xy}(r,c) = \sum\limits_{n=-N/2}^{N/2}F^{}_x (n,c) e^{-i\frac{2\pi r n}{N+1}},\nonumber \\ -N/2 \leq r \leq N/2 , -N/2 \leq c \leq N/2 \end{eqnarray}

The order of operation is not important. We can even begin with each column operation first then each row operation the calculation would not be affected.

I hope so far my notations are correct. Now comes the confusing 3D part (it has row, column and depth indexes), consider 1D fft operation along each row or X-axis, \begin{eqnarray} & F^{}_x (r,c,d) = \sum\limits_{n=-N/2}^{N/2} f(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Next taking 1D fft along each column or Y-axis we have the equation,

\begin{eqnarray} & F^{}_{xy} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_x(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Now how do I represent the final 1D operation along Z-axis or each depth index ? Can I write something like this ? \begin{eqnarray} &(i) \qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \qquad \qquad or\\ &(ii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \qquad \qquad or\\ &(iii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,c,n) e^{-j\frac{2\pi d n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray} Are the above equations representation for 1D fft along z-axis correct ? Can someone help to clarify my confusion ?

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my notations in 3D centered fft. Consider the 1D centered fft of a 2D image of size $(N+1)\times (N+1)$, where $N$ is even, along X axis it can be computed using 1D FFT along each row such that, \begin{eqnarray} F^{}_x (r,c) = \sum\limits_{n=-N/2}^{N/2} f(r,n) e^{-i\frac{2\pi c n}{N+1}}, \nonumber \\ -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2 \end{eqnarray} Next we operate on each column such that the 2D centered FFT can be written as, \begin{eqnarray} F^{}_{xy}(r,c) = \sum\limits_{n=-N/2}^{N/2}F^{}_x (n,c) e^{-i\frac{2\pi r n}{N+1}},\nonumber \\ -N/2 \leq r \leq N/2 , -N/2 \leq c \leq N/2 \end{eqnarray}

The order of operation is not important. We can even begin with each column operation first then each row operation the calculation would not be affected.

I hope so far my notations are correct. Now comes the confusing 3D part (it has row, column and depth indexes), consider 1D fft operation along each row or X-axis, \begin{eqnarray} & F^{}_x (r,c,d) = \sum\limits_{n=-N/2}^{N/2} f(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Next taking 1D fft along each column or Y-axis we have the equation,

\begin{eqnarray} & F^{}_{xy} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_x(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Now how do I represent the final 1D operation along Z-axis or each depth index ? Can I write something like this ? \begin{eqnarray} &(i) \qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \qquad \qquad or\\ &(ii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \qquad \qquad or\\ &(iii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,c,n) e^{-j\frac{2\pi d n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray} Are the above equations representation for 1D fft along z-axis correct ? Can someone help to clarify my confusion ?

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my notations in 3D centered fft. Consider the 1D centered fft of a 2D image of size $(N+1)\times (N+1)$, where $N$ is even, along X axis it can be computed using 1D FFT along each row such that, \begin{eqnarray} F^{}_x (r,c) = \sum\limits_{n=-N/2}^{N/2} f(r,n) e^{-i\frac{2\pi c n}{N+1}}, \nonumber \\ -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2 \end{eqnarray} Next we operate on each column such that the 2D centered FFT can be written as, \begin{eqnarray} F^{}_{xy}(r,c) = \sum\limits_{n=-N/2}^{N/2}F^{}_x (n,c) e^{-i\frac{2\pi r n}{N+1}},\nonumber \\ -N/2 \leq r \leq N/2 , -N/2 \leq c \leq N/2 \end{eqnarray}

The order of operation is not important. We can even begin with each column operation first then each row operation the calculation would not be affected.

I hope so far my notations are correct. Now comes the confusing 3D part (it has row, column and depth indexes), consider 1D fft operation along each row or X-axis, \begin{eqnarray} & F^{}_x (r,c,d) = \sum\limits_{n=-N/2}^{N/2} f(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Next taking 1D fft along each column or Y-axis we have the equation,

\begin{eqnarray} & F^{}_{xy} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_x(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Now how do I represent the final 1D operation along Z-axis or each depth index ? Can I write something like this ? \begin{eqnarray} &(i) \qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \qquad \qquad or\\ &(ii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \qquad \qquad or\\ &(iii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,c,n) e^{-j\frac{2\pi d n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray} Are the above equations representation for 1D fft along z-axis correct ? Can someone help to clarify my confusion ?

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Syed Alam Abbas
Syed Alam Abbas
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Help with notations from 2D to 3D FFT representations as 1D FFT

I have this question on mathematics forum too, Notations, I thought of posting here, which ever place I get an answer, I will try to close it in the other. I need some help and clarifications for my notations in 3D centered fft. Consider the 1D centered fft of a 2D image of size $(N+1)\times (N+1)$, where $N$ is even, along X axis it can be computed using 1D FFT along each row such that, \begin{eqnarray} F^{}_x (r,c) = \sum\limits_{n=-N/2}^{N/2} f(r,n) e^{-i\frac{2\pi c n}{N+1}}, \nonumber \\ -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2 \end{eqnarray} Next we operate on each column such that the 2D centered FFT can be written as, \begin{eqnarray} F^{}_{xy}(r,c) = \sum\limits_{n=-N/2}^{N/2}F^{}_x (n,c) e^{-i\frac{2\pi r n}{N+1}},\nonumber \\ -N/2 \leq r \leq N/2 , -N/2 \leq c \leq N/2 \end{eqnarray}

The order of operation is not important. We can even begin with each column operation first then each row operation the calculation would not be affected.

I hope so far my notations are correct. Now comes the confusing 3D part (it has row, column and depth indexes), consider 1D fft operation along each row or X-axis, \begin{eqnarray} & F^{}_x (r,c,d) = \sum\limits_{n=-N/2}^{N/2} f(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Next taking 1D fft along each column or Y-axis we have the equation,

\begin{eqnarray} & F^{}_{xy} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_x(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray}

Now how do I represent the final 1D operation along Z-axis or each depth index ? Can I write something like this ? \begin{eqnarray} &(i) \qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(n,c,d) e^{-j\frac{2\pi r n}{N+1}}, \qquad \qquad or\\ &(ii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,n,d) e^{-j\frac{2\pi c n}{N+1}}, \nonumber \qquad \qquad or\\ &(iii)\qquad F^{}_{xyz} (r,c,d) = \sum\limits_{n=-N/2}^{N/2} F^{}_{xy}(r,c,n) e^{-j\frac{2\pi d n}{N+1}}, \nonumber \\ & r,c,d = -\frac{N}{2},\cdots, \frac{N}{2} \nonumber \\ %& -N/2 \leq r \leq N/2, -N/2 \leq c \leq N/2, -N/2 \leq d \leq N/2 \nonumber \\ \end{eqnarray} Are the above equations representation for 1D fft along z-axis correct ? Can someone help to clarify my confusion ?