Uniqki: A Personal Wiki Builder
A Uniqki site

Introduction to frequency-domain filtering

Dr. Huidae Cho
Institute for Environmental and Spatial Analysis...University of North Georgia

1   Spatial domain

Do you remember these filters?

  • smoothing linear filters
  • order-statistics non-linear filters
  • Sobel filters
  • Laplacian filters?

These filters are called spatial filters because they are applied to images “spatially.”

The domain of these spatial filters is the spatial domain (i.e., $f(x,y)$ or the $x$-$y$ spatial coordinates).

2   Frequency domain

Now, a Fourier-transformed image is in the frequency domain (i.e., $F(u,v)$ or $u$-$v$ frequency coordinates).

2.1   Exercise: Plotting $F(u)e^{2i\pi ux/M}$

Do you want to see how a one-dimensional image is decomposed into multiple frequencies?

frequency_exercise.py

frequency-exercise-1-1.png

In exercise 1, observe what $F(u)e^{2i\pi ux/M}$ curves look like and think about why two different frequency curves ($F(1)$ and $F(3)$) overlap.

In exercise 2, you’ll see why.

frequency-exercise-1-2.png

2.2   Exercise: Reading the Fourier spectrum

See the effect of $K$ in frequency_exercise_2.py.

This is the two-dimensional version: frequency_exercise_3.py.

3   Frequency-domain filtering

The fundamental idea behind frequency-domain filtering is to enhance images in the spatial domain by changing their Fourier transform in the frequency domain.

Low-frequency components in the Fourier transform represent the overall gray-level appearance of an image while high-frequency components, details such as edges.

3.1   Exercise: Fourier transform components

Try $F(0)=0$ and $F(3)=0$, and discuss what happens and why.

frequency_exercise_4.py

4   Frequency-domain filters

4.1   Low-pass filters

Low-pass filters “pass” low frequencies while attenuating high frequencies. Effects?

Reduced details $\rightarrow$ Smoothing

4.2   High-pass filters

High-pass filters “pass” high frequencies while attenuating low frequencies. Effects?

Reduced gray-level variations in smooth areas $\rightarrow$ Emphasized transitions

4.3   How do they work?

Element-by-element multiplication with the Fourier transform: \begin{align*} G(u,v)&=H(u,v)F(u,v)\\ g(x,y)&=\mathfrak{F}^{-1}[G(u,v)] \end{align*} where $G(u,v)$, $H(u,v)$, and $F(u,v)$ are the output Fourier transform, frequency-domain filter, and Fourier transform of the input image, respectively, $g(x,y)$ is the transformed image, and $\mathfrak{F}^{-1}$ denotes the inverse Fourier transform.

4.4   Notch filter

Probably, one of the simplest frequency-domain filter. We’ve already tried it manually in frequency_exercise_4.py.

\[H(u,v)= \begin{cases} 0& \text{if }(u,v)=(0,0)\\ 1& \text{otherwise.} \end{cases}\]

Or the shifted version (Remember why we shift the Fourier transform?)

\[H(u,v)= \begin{cases} 0& \text{if }(u,v)=(M/2,N/2)\\ 1& \text{otherwise.} \end{cases}\]

What does it do?

Set $F(0,0)=0$. That’s it. Effects?