Fourier Filtering 2D

In the spatial frequency domain, the spectrum of the 2D spatial signal (the image) is represented by a 2D array $X[k,l]=X_r[k,l]+j X_i[k,l]$. An arbitrary location $(k,l)$ in the spectrum array represents a spatial sinusoid with

• frequency $\sqrt{k^2+l^2}$ and
• direction $tan^{-1}(k/l)$

And the complex coefficient $X[k,l]=X_r[k,l]+jX_i[k,j]$ specifies the

• magnitude $\sqrt{X_r[k,l]^2+X_i[k,l]^2}$, and
• phase angle $tan^{-1}(X_i[k,l]/X_r[k,l])$

of the sinusoid. All frequency components $X[k,l]$ can be modified according to specific filtering needs. We could enhance and/or reduce various frequency components by multiplying the spectrum by a 2D filter mask. Note that if the spectrum is centralized, the DC is in the middle $(N/2,N/2)$ of the spectrum. The farther a frequency component $(k,l)$ is from this point, the higher frequency it has.

Here are some typical 2D filters. We assume $u=k-N/2, v=l-N/2$ in the following.

• Ideal low-pass filter
  
  $$H_{ideal}[k,l]=\left\{ \begin{array}{ll} 1 & \sqrt{u^2+v^2} <D_0 \\ 0 & \mbox{otherwise} \end{array} \right.$$
  
  
  where $D_0$ is the ideal cut-off frequency.
• Gaussian low-pass filter
  
  $$H_{gauss}[k,l]=exp[-a(u^2+v^2)/D_0^2]$$
  
  
  where $D_0$ is the cut-off frequency at which ($u^2+v^2=D_0^2$) the magnitude is attenuated to $exp(-a)$, and $a$ and $c$ are two parameters. Compared with the ideal filter, the Gaussian filter is smooth and it no longer have the undesired ringing effect.

  

• Butterworth low-pass filter
  
  $$H_{butterworth}[k,l]=\frac{1}{1+((u^2+v^2)/D_0^2)^n} =\left... ...u^2+v^2=D_0^2 0 & u,v\rightarrow \infty \end{array} \right.$$
  
  
  Butterworth filter is also a smooth low-pass filter with a parameter $n$. Note that when $n\rightarrow \infty$, the Butterworth filter becomes an ideal filter.

  

  

• Hamming low-pass filter
  
  $$H_{hamming}=\left\{ \begin{array}{ll} 0.5(1+cos(\pi \sqrt{u^... ...t{u^2+v^2} < D_0 \\ 0 & \mbox{otherwise} \end{array} \right.$$
  
  

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

$$H_{high-pass}=1-H_{low-pass}$$

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

The filtering effects are shown in these demonstrations