Image Denoising
Ideal low pass filter:
$$
\hat{h}_{p,q}= \left\{
\begin{array}{cc}
1 & \mbox{ if $D(p,q)\le D_0$} \\
0 & \mbox{ otherwise.}
\end{array}
\right.
$$
Butterworth low pass filter
$$
\hat{h}_{p,q}=\frac{1}{1+[D(p,q)/D_0]^{2n}} \, .
$$
Gaussian low pass filter
$$
\hat{h}_{p,q}=e^{-D(p,q)^2}{2\sigma^2} \, .
$$
Image Edge Detection
Ideal high pass filter:
$$
\hat{h}_{p,q}= \left\{
\begin{array}{cc}
0 & \mbox{ if $D(p,q)\le D_0$} \\
1 & \mbox{ otherwise.}
\end{array}
\right.
$$
Butterworth high pass filter
$$
\hat{h}_{p,q}=1-\frac{1}{1+[D(p,q)/D_0]^{2n}}=\frac{1}{1+[D_0/D(p,q)]^{2n}} \, .
$$
Gaussian low pass filter
$$
\hat{h}_{p,q}=1-e^{-D(p,q)^2/(2\sigma^2)} \, .
$$
Deconvolution
$g=h*f$, given both $g$ and $h$, want to determine $f$.If only $g$ is given, we have a blind deconvolution problem.
$$
f=\mathcal{F}^{-1}\left[ \frac{\mathcal{F}(g)}{\mathcal{F}(h)} \right] \, .
$$
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