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2D Fourier Filtering

The 2D DFT of an M by N 2D spatial signal $x[m,n]$ is also an M by N 2D array, with its (k,l)th component $X[k,l]=X_r[k,l]+X_i[k,l]$ representing a spatial sinusoid with

During filtering, The 2D spectrum $X[k,l]$ is multiplied by a 2D filter mask $H[k,l]$:

\begin{displaymath}Y[k,l]=H[k,l]\; X[k,l],\;\;\;\;\;(k=0,\cdots,M-1,\;\; l=1,\cdots,N-1) \end{displaymath}

so that the frequency components can be modified (enhanced or reduced). If the spectrum is centralized, the DC is in the center of the spectrum at $(k_0=M/2,\;l_0=N/2)$, and the farther a frequency component $X[k,l]$ is from the center, the higher spatial frequency it represents.

Fourier_filter.gif

Here are some typical 2D filters. (The filtering effects are demonstrated here.) We assume $M=N$ for simplicity. The cut-off frequency $D_0$ is in the range of $0<D_0<N/2$.

The high-pass filters corresponding to each of the low-pass filters above can be obtained by

\begin{displaymath}H_{high-pass}=1-H_{low-pass} \end{displaymath}

Alternatively, the same filters above can be used as high-pass filters if they are applied to the 2D spectrum without centralization.


next up previous
Next: Unitary and Orthogonal Transforms Up: Fourier_Analysis Previous: Spectrum Centralization
Ruye Wang 2003-11-17