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Next: 2D Fourier Filtering Up: Fourier_Analysis Previous: A 2D DFT Example

Spectrum Centralization

From the previous example, we see that in the 2D spectrum array, the DC component is $X[0,0]$ at the upper-left corner, the highest frequency component is $X[M/2,N/2]$ in the middle, and the high frequency components are around the middle, while the low frequency components are around the four sides. Sometime it is preferable to centralize the spectrum by shifting the spectrum by M/2 vertically and N/2 horizontally, so that the DC component and the low frequency components are in the middle while the high frequency components are around the four sides.

Similar to 1D DFT, the 2D DFT of a M by N 2D array of spatial samples also has the space shift property:

\begin{displaymath}{\cal F}^{-1} X[m-M/2,n-N/2]= x[m,n]e^{j\pi(m+n)}=x[m,n](-1)^{m+n} \end{displaymath}

If we change the sign of all spatial sample points $x[m,n]$ if $m+n$ is an odd number, i.e.

\begin{displaymath}
\left[ \begin{array}{rrrr}
x[0,0] & -x[0,1] & x[0,2] & \cd...
...ots \\
\cdots & \cdots & \cdots & \cdots \end{array} \right] \end{displaymath}

then the resulting 2D Fourier spectrum will be centralized with DC component in the middle and high frequency components around the four edges.



Ruye Wang 2003-11-17